8342 lines
704 KiB
Plaintext
8342 lines
704 KiB
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HÿÈH‰E „¼ I‹E …Àx
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HÿÈI‰E „· H‹|$PH…ÿtH‹…Àx HÿÈH‰„ª H‹\$ H‹…Àx HÿÈH‰„¢ H‹„$ dH+%( …¼Š H�Ä( L‰à[]A\A]A^A_Ãf D H‰Ïÿ— éüþÿÿf�H‰÷ÿ‡ é ÿÿÿf�H‰×ÿw é ÿÿÿf�H‰ßÿg é,ÿÿÿf�H‰ïÿW é6ÿÿÿf�L‰ïÿG é;ÿÿÿf�ÿ: éKÿÿÿ D H‰ßÿ' éPÿÿÿf�H‰Ïÿ éNýÿÿf�H‰÷ÿ é[ýÿÿf�H‰×ÿ÷ éhýÿÿf�H‰ßÿç éuýÿÿf�ÿÚ é…ýÿÿ D H‰ÏÿÇ é�ýÿÿf�H‰÷ÿ· éœýÿÿf�H‰×ÿ§ é¬ýÿÿf�H‰ßÿ— é¼ýÿÿf�ÿŠ éÏýÿÿ D ‰”$À L‰ßL‰”$¸ ÿh ‹”$À L‹”$¸ éDüÿÿ @ ‰”$¸ L‰×ÿ@ ‹”$¸ é<üÿÿ @ L‰œ$Ð L‰÷H‰Œ$È ‰”$À L‰”$¸ ÿ L‹œ$Ð H‹Œ$È ‹”$À L‹”$¸ érûÿÿ @ L‰œ$Ð L‰çH‰Œ$È ‰”$À L‰”$¸ ÿ¸ L‹œ$Ð H‹Œ$È ‹”$À L‹”$¸ é<ûÿÿ @ L‰œ$Ð L‰ÿH‰Œ$È ‰”$À L‰”$¸ ÿh L‹œ$Ð H‹Œ$È ‹”$À L‹”$¸ éûÿÿ @ L‰œ$È H‰Ï‰”$À L‰”$¸ ÿ L‹œ$È ‹”$À L‹”$¸ éÜúÿÿ @ L‰ïÿ÷ éÃéÿÿf�L‰ÿÿç é’éÿÿf�L‰÷ÿ× écéÿÿf�L‰çÿÇ é@éÿÿf�ÿš H…À…Z' L‰çè)ýÿH‰$H‰„$è H…À…èÿÿHÇ„$¨ 1É1íE1ÒHÇ„$� E1Û»2 º© HÇ„$ˆ HÇ„$€ HÇ„$˜ HÇ„$° HÇ„$ HÇD$` HÇD$p HÇD$@ HÇD$X HÇD$0 HÇD$ é±ùÿÿ�A‹E ÿÀtA‰E M‹}A‹ÿÀtA‰L‰ïÿE H‰D$é]ëÿÿ HÇ„$¨ 1ÉL‹´$è E1ÿHÇ„$� E1Ò1íE1ÛHÇ„$ˆ »2 º© HÇ„$€ HÇ„$˜ HÇ„$° HÇ„$ HÇD$` HÇD$p HÇD$@ HÇD$X HÇD$0 HÇD$
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ÿ; …À…³q 1ÒH‰ßL‰îAÿÔH‰ÃH‰D$ÿŠ H…Û„‚q H‹D$H‰„$ø H‹\$H‰ßè�ýÿH‹…Àx HÿÈH‰„ V I‰í1É»€3 º· HÇ„$ø HÇ„$¨ HÇ„$� E1Ò1íE1ÛHÇ„$ˆ HÇ„$€ HÇ„$˜ HÇ„$° HÇ„$ HÇD$` HÇD$p HÇD$@ HÇD$X HÇD$0 HÇD$ éZóÿÿ1ÒL‰æH‰ßÿÔ H‰$H‰„$ð H…À…ÄãÿÿE1ÿE1öI‰í1ÉL‰´$¨ E1ÒE1Û1íL‰¼$� »(3 º³ L‰¼$ˆ L‰¼$€ L‰¼$˜ L‰¼$° L‰¼$ L‰|$`L‰|$pL‰|$@L‰|$XL‰|$0L‰|$é¿òÿÿ € D¶àéaçÿÿ € HÇD$ L‹-H H‹D$PH; …V A‹E ÿÀtA‰E HÇ„$ð Ç„$Ð L‰l$0H‹
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¯ ‹H‰L$X‰D$@ÿÀt‰H‹— H‰D$XL‹%k+ A‹$ÿÀt A‰$L‹%X+ H‹D$PH‹ …ÀxH‹L$PHÿÈH‰„g; L‰d$PH‹t$`L9î”ÀH;5Ä ”ÂÂ…O H;5R „B H‰÷ÿS …ÀˆiI …À„! H‹|$ H;=' „ú8 H‹GH‹5î H‹€� H…À„êc ÿÐH‰D$@Hƒ|$@ „ìc H‹ H‹D$@H‰”$À H9P…�S H‹XH‰œ$ð H…Û„ˆS L‹P‹ÿÀt‰A‹ÿÀtA‰H‹T$@H‹…Àx HÿÈH‰„žS L‰×º H�´$ L‰T$@H‰œ$ HÇ„$ è ýÿL‹T$@H‰D$pH‰„$è H‹…Àx HÿÈH‰„ÎV Hƒ|$p HÇ„$ð „?i I‹…Àx HÿÈI‰„¬R H‹|$p1ɺ HÇÆÿÿÿÿè$ùüÿH‰„$è I‰ÆH…À„òg H‹|$pH‰Æÿº H‰D$@H…À„‚n I‹…Àx HÿÈI‰„,W H‹T$pHÇ„$è H‹…Àx HÿÈH‰„W H‹¼$¸ H‹5ž% H‹GH‹€� H…À„Õm ÿÐH‰D$`H‹D$`H‰„$è H…À„Tm H‹D$`H‹Œ$À H9H….Y H‹XfHnÃH‰œ$ð D$XH…Û„Y L‹`‹ÿÀt‰A‹$ÿÀtA‰$H‹t$`L‰¤$è H‹…Àx HÿÈH‰„Y H�´$ º L‰ç)„$ è‚ýÿH‰D$pH‹…Àx HÿÈH‰„ä` L‰d$`Hƒ|$p HÇ„$ð „_s H‹\$`H‹…Àx HÿÈH‰„_X H‹|$@H‹5I% HÇ„$è H‹GH‹€� H…À„s ÿÐI‰ÂM…Ò„šr L‰T$`¿ ÿ£ L‹T$`H…ÀH‰„$è I‰Æ„r H‹L$p‹‰D$`ÿÀt‰H‹D$pL‰T$`I‰Fÿ5 L‹T$`H…ÀH‰„$ð I‰Ç„bq L‰T$`H‹x$ H‰ÇH‹5% ÿ˜ L‹T$`…Àˆ^g I‹BH‹˜€ H…Û„˜} L‰T$`H�=› ÿí L‹T$`…À…G� L‰”$€ L‰×L‰úL‰öÿÓH‰ÃH‰D$`ÿ/ H…ÛL‹”$€ „ü� H‹D$`H‰„$ø I‹…Àx HÿÈI‰„¦p I‹…Àx HÿÈI‰„wp HÇ„$è I‹…Àx HÿÈI‰„fp H‹ ]! H‹=¶ HÇ„$ð H‹SH‰Þÿ�
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HÿÈI‰$„È� Hc„$Ð H‹œ$ð H‹Œ$ø ¾ �PH)ÆHcÒfHnÃH‰ÏH�´ô H‰Œ$¨ ö! )„$ èéýÿH‰ßI‰ÄH‰„$è è¦ëüÿE1ÛM…äH‹Œ$¨ L‰œ$ð „l¬ H‹…Àx HÿÈH‰„B� H‹t$`E1ÒL‰”$ø H‹…Àx HÿÈH‰„€‡ L‰d$`H‹EH‹5Œ H‰ïH‹€� H…À„du ÿÐI‰ÆL‰´$è M…ö„u H‹5D! L9ö„a I‹FH; …� A‹^ƒãI‹…Àx HÿÈI‰„�m HÇ„$è …Û…Œa ‹¼$Ð …ÿ…°e H‹5þ H‹|$`èÜéüÿH‰„$è I‰ÆH…À„‹® H‹5Á 1ÒH‰Çè÷òüÿ‰Ã…ÀˆOª I‹…Àx HÿÈI‰„/– 1öH‰´$è …Û„Ke H‹=1 èüûüÿH‰„$è I‰ÆH‹„$à H‰„$¨ M…ö„ H‹5á L‰÷èIéüÿH‰„$ø H‰ÁH…À„Ь I‹…Àx HÿÈI‰„F} H‰Œ$¸ ÿœ H‹Œ$¸ H…ÀH‰„$è I‰Æ„q¬ H‹5y H‰ïH‰Œ$¸ èáèüÿH‹Œ$¸ H…ÀH‰„$ð I‰Ç„÷¬ H‹5F H‰ÂL‰÷ÿÒ H‹Œ$¸ …Àˆ·¬ I‹…Àx HÿÈI‰„�| H‹5� H‰ÏL‰òH‰Œ$¸ è•çüÿH‹Œ$¸ H…ÀH‰„$ð I‰Ä„C¬ H‹…Àx HÿÈH‰„M¬ I‹1ÉH‰Œ$ø …Àx HÿÈI‰„n H‹t$`1ÒH‰ïH‰”$è è2ýÿH‰„$ð I‰ÇH…À„& H‹5 H‰ÂL‰çÿ …Àˆî¬ I‹…Àx HÿÈI‰„# A‹$ÿÀ„| I‰íA‰$L‰åéååÿÿH‹€ H9„áûÿÿH…ÀuëH;Q …ýÿÿéÊûÿÿf D ÿr H…Àu H‹®ÿ H�5ßû H‹8ÿÞ f D HÇ„$¨ 1É1í» 2 HÇ„$� E1Òº© E1ÛHÇ„$ˆ HÇ„$€ HÇ„$˜ HÇ„$° HÇ„$ HÇD$` HÇD$p HÇD$@ HÇD$X HÇD$0 HÇD$ HÇ$ é;äÿÿ D ÿr H‰Áé¦Óÿÿf�I‰íHÇ$ E1Ò1íHÇ„$¨ E1Û»/2 ºª HÇ„$� HÇ„$ˆ HÇ„$€ HÇ„$˜ HÇ„$° HÇ„$ HÇD$` HÇD$p HÇD$@ HÇD$X HÇD$0 HÇD$ éÄãÿÿ @ H‰ßÿ÷þ é Öÿÿf�H;Iþ „R H‰L$H‰Ïº ÿøþ H‹L$H…ÀH‰$H‰Æ„¿D H;$ A”ÅH;A ”ÀDè…)
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º ëà º ëÖ HƒìHƒÄà an integer is required __pyx_capi__ _rand_uint64 _rand_uint32 _rand_uint16 _rand_uint8 _rand_bool _rand_int64 _rand_int32 _rand_int16 _rand_int8 numpy.random._common check_constraint check_array_constraint double (double *, npy_intp) kahan_sum double_fill validate_output_shape cont disc cont_broadcast_3 discrete_broadcast_iii name __loader__ loader __file__ origin __package__ parent __path__ submodule_
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RandomState(seed=None)
|
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Container for the slow Mersenne Twister pseudo-random number generator.
|
|||
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|
Consider using a different BitGenerator with the Generator container
|
|||
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instead.
|
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`RandomState` and `Generator` expose a number of methods for generating
|
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|
random numbers drawn from a variety of probability distributions. In
|
|||
|
|
addition to the distribution-specific arguments, each method takes a
|
|||
|
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keyword argument `size` that defaults to ``None``. If `size` is ``None``,
|
|||
|
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then a single value is generated and returned. If `size` is an integer,
|
|||
|
|
then a 1-D array filled with generated values is returned. If `size` is a
|
|||
|
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tuple, then an array with that shape is filled and returned.
|
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**Compatibility Guarantee**
|
|||
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|
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|
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A fixed bit generator using a fixed seed and a fixed series of calls to
|
|||
|
|
'RandomState' methods using the same parameters will always produce the
|
|||
|
|
same results up to roundoff error except when the values were incorrect.
|
|||
|
|
`RandomState` is effectively frozen and will only receive updates that
|
|||
|
|
are required by changes in the internals of Numpy. More substantial
|
|||
|
|
changes, including algorithmic improvements, are reserved for
|
|||
|
|
`Generator`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
seed : {None, int, array_like, BitGenerator}, optional
|
|||
|
|
Random seed used to initialize the pseudo-random number generator or
|
|||
|
|
an instantized BitGenerator. If an integer or array, used as a seed for
|
|||
|
|
the MT19937 BitGenerator. Values can be any integer between 0 and
|
|||
|
|
2**32 - 1 inclusive, an array (or other sequence) of such integers,
|
|||
|
|
or ``None`` (the default). If `seed` is ``None``, then the `MT19937`
|
|||
|
|
BitGenerator is initialized by reading data from ``/dev/urandom``
|
|||
|
|
(or the Windows analogue) if available or seed from the clock
|
|||
|
|
otherwise.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The Python stdlib module "random" also contains a Mersenne Twister
|
|||
|
|
pseudo-random number generator with a number of methods that are similar
|
|||
|
|
to the ones available in `RandomState`. `RandomState`, besides being
|
|||
|
|
NumPy-aware, has the advantage that it provides a much larger number
|
|||
|
|
of probability distributions to choose from.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
Generator
|
|||
|
|
MT19937
|
|||
|
|
numpy.random.BitGenerator
|
|||
|
|
|
|||
|
|
à¿üÿ`ÂüÿÀÁüÿØÁüÿ Âüÿ’ýÿ\�ýÿ¼œýÿÔœýÿ �ýÿ·ýÿØÀýÿ8ÀýÿPÀýÿ˜Àýÿ$þÿ”þÿÔþÿþÿDþÿ¸.þÿ0þÿ 0þÿh0þÿH/þÿÀ0þÿà�ÿÿð‘ÿÿˆ‘ÿÿ¸‘ÿÿ0’ÿÿ
|
|||
|
|
This is an alias of `random_sample`. See `random_sample` for the complete
|
|||
|
|
documentation.
|
|||
|
|
|
|||
|
|
This is an alias of `random_sample`. See `random_sample` for the complete
|
|||
|
|
documentation.
|
|||
|
|
|
|||
|
|
Sets the singleton RandomState's bit generator
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
bitgen
|
|||
|
|
A bit generator instance
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The singleton RandomState provides the random variate generators in the
|
|||
|
|
``numpy.random``namespace. This function, and its counterpart get method,
|
|||
|
|
provides a path to hot-swap the default MT19937 bit generator with a
|
|||
|
|
user provided alternative. These function are intended to provide
|
|||
|
|
a continuous path where a single underlying bit generator can be
|
|||
|
|
used both with an instance of ``Generator`` and with the singleton
|
|||
|
|
instance of RandomState.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
get_bit_generator
|
|||
|
|
numpy.random.Generator
|
|||
|
|
|
|||
|
|
Returns the singleton RandomState's bit generator
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
BitGenerator
|
|||
|
|
The bit generator that underlies the singleton RandomState instance
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The singleton RandomState provides the random variate generators in the
|
|||
|
|
``numpy.random`` namespace. This function, and its counterpart set method,
|
|||
|
|
provides a path to hot-swap the default MT19937 bit generator with a
|
|||
|
|
user provided alternative. These function are intended to provide
|
|||
|
|
a continuous path where a single underlying bit generator can be
|
|||
|
|
used both with an instance of ``Generator`` and with the singleton
|
|||
|
|
instance of RandomState.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
set_bit_generator
|
|||
|
|
numpy.random.Generator
|
|||
|
|
|
|||
|
|
seed(seed=None)
|
|||
|
|
|
|||
|
|
Reseed the singleton RandomState instance.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
This is a convenience, legacy function that exists to support
|
|||
|
|
older code that uses the singleton RandomState. Best practice
|
|||
|
|
is to use a dedicated ``Generator`` instance rather than
|
|||
|
|
the random variate generation methods exposed directly in
|
|||
|
|
the random module.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
numpy.random.Generator
|
|||
|
|
|
|||
|
|
permutation(x)
|
|||
|
|
|
|||
|
|
Randomly permute a sequence, or return a permuted range.
|
|||
|
|
|
|||
|
|
If `x` is a multi-dimensional array, it is only shuffled along its
|
|||
|
|
first index.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the
|
|||
|
|
`~numpy.random.Generator.permutation`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
x : int or array_like
|
|||
|
|
If `x` is an integer, randomly permute ``np.arange(x)``.
|
|||
|
|
If `x` is an array, make a copy and shuffle the elements
|
|||
|
|
randomly.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray
|
|||
|
|
Permuted sequence or array range.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.permutation: which should be used for new code.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> np.random.permutation(10)
|
|||
|
|
array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6]) # random
|
|||
|
|
|
|||
|
|
>>> np.random.permutation([1, 4, 9, 12, 15])
|
|||
|
|
array([15, 1, 9, 4, 12]) # random
|
|||
|
|
|
|||
|
|
>>> arr = np.arange(9).reshape((3, 3))
|
|||
|
|
>>> np.random.permutation(arr)
|
|||
|
|
array([[6, 7, 8], # random
|
|||
|
|
[0, 1, 2],
|
|||
|
|
[3, 4, 5]])
|
|||
|
|
|
|||
|
|
|
|||
|
|
shuffle(x)
|
|||
|
|
|
|||
|
|
Modify a sequence in-place by shuffling its contents.
|
|||
|
|
|
|||
|
|
This function only shuffles the array along the first axis of a
|
|||
|
|
multi-dimensional array. The order of sub-arrays is changed but
|
|||
|
|
their contents remains the same.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.shuffle`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
x : ndarray or MutableSequence
|
|||
|
|
The array, list or mutable sequence to be shuffled.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
None
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.shuffle: which should be used for new code.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> arr = np.arange(10)
|
|||
|
|
>>> np.random.shuffle(arr)
|
|||
|
|
>>> arr
|
|||
|
|
[1 7 5 2 9 4 3 6 0 8] # random
|
|||
|
|
|
|||
|
|
Multi-dimensional arrays are only shuffled along the first axis:
|
|||
|
|
|
|||
|
|
>>> arr = np.arange(9).reshape((3, 3))
|
|||
|
|
>>> np.random.shuffle(arr)
|
|||
|
|
>>> arr
|
|||
|
|
array([[3, 4, 5], # random
|
|||
|
|
[6, 7, 8],
|
|||
|
|
[0, 1, 2]])
|
|||
|
|
|
|||
|
|
|
|||
|
|
dirichlet(alpha, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from the Dirichlet distribution.
|
|||
|
|
|
|||
|
|
Draw `size` samples of dimension k from a Dirichlet distribution. A
|
|||
|
|
Dirichlet-distributed random variable can be seen as a multivariate
|
|||
|
|
generalization of a Beta distribution. The Dirichlet distribution
|
|||
|
|
is a conjugate prior of a multinomial distribution in Bayesian
|
|||
|
|
inference.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.dirichlet`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
alpha : sequence of floats, length k
|
|||
|
|
Parameter of the distribution (length ``k`` for sample of
|
|||
|
|
length ``k``).
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n)``, then
|
|||
|
|
``m * n * k`` samples are drawn. Default is None, in which case a
|
|||
|
|
vector of length ``k`` is returned.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
samples : ndarray,
|
|||
|
|
The drawn samples, of shape ``(size, k)``.
|
|||
|
|
|
|||
|
|
Raises
|
|||
|
|
------
|
|||
|
|
ValueError
|
|||
|
|
If any value in ``alpha`` is less than or equal to zero
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.dirichlet: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The Dirichlet distribution is a distribution over vectors
|
|||
|
|
:math:`x` that fulfil the conditions :math:`x_i>0` and
|
|||
|
|
:math:`\sum_{i=1}^k x_i = 1`.
|
|||
|
|
|
|||
|
|
The probability density function :math:`p` of a
|
|||
|
|
Dirichlet-distributed random vector :math:`X` is
|
|||
|
|
proportional to
|
|||
|
|
|
|||
|
|
.. math:: p(x) \propto \prod_{i=1}^{k}{x^{\alpha_i-1}_i},
|
|||
|
|
|
|||
|
|
where :math:`\alpha` is a vector containing the positive
|
|||
|
|
concentration parameters.
|
|||
|
|
|
|||
|
|
The method uses the following property for computation: let :math:`Y`
|
|||
|
|
be a random vector which has components that follow a standard gamma
|
|||
|
|
distribution, then :math:`X = \frac{1}{\sum_{i=1}^k{Y_i}} Y`
|
|||
|
|
is Dirichlet-distributed
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] David McKay, "Information Theory, Inference and Learning
|
|||
|
|
Algorithms," chapter 23,
|
|||
|
|
http://www.inference.org.uk/mackay/itila/
|
|||
|
|
.. [2] Wikipedia, "Dirichlet distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Dirichlet_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Taking an example cited in Wikipedia, this distribution can be used if
|
|||
|
|
one wanted to cut strings (each of initial length 1.0) into K pieces
|
|||
|
|
with different lengths, where each piece had, on average, a designated
|
|||
|
|
average length, but allowing some variation in the relative sizes of
|
|||
|
|
the pieces.
|
|||
|
|
|
|||
|
|
>>> s = np.random.dirichlet((10, 5, 3), 20).transpose()
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> plt.barh(range(20), s[0])
|
|||
|
|
>>> plt.barh(range(20), s[1], left=s[0], color='g')
|
|||
|
|
>>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r')
|
|||
|
|
>>> plt.title("Lengths of Strings")
|
|||
|
|
|
|||
|
|
|
|||
|
|
multinomial(n, pvals, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a multinomial distribution.
|
|||
|
|
|
|||
|
|
The multinomial distribution is a multivariate generalization of the
|
|||
|
|
binomial distribution. Take an experiment with one of ``p``
|
|||
|
|
possible outcomes. An example of such an experiment is throwing a dice,
|
|||
|
|
where the outcome can be 1 through 6. Each sample drawn from the
|
|||
|
|
distribution represents `n` such experiments. Its values,
|
|||
|
|
``X_i = [X_0, X_1, ..., X_p]``, represent the number of times the
|
|||
|
|
outcome was ``i``.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.multinomial`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
n : int
|
|||
|
|
Number of experiments.
|
|||
|
|
pvals : sequence of floats, length p
|
|||
|
|
Probabilities of each of the ``p`` different outcomes. These
|
|||
|
|
must sum to 1 (however, the last element is always assumed to
|
|||
|
|
account for the remaining probability, as long as
|
|||
|
|
``sum(pvals[:-1]) <= 1)``.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. Default is None, in which case a
|
|||
|
|
single value is returned.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray
|
|||
|
|
The drawn samples, of shape *size*, if that was provided. If not,
|
|||
|
|
the shape is ``(N,)``.
|
|||
|
|
|
|||
|
|
In other words, each entry ``out[i,j,...,:]`` is an N-dimensional
|
|||
|
|
value drawn from the distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.multinomial: which should be used for new code.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Throw a dice 20 times:
|
|||
|
|
|
|||
|
|
>>> np.random.multinomial(20, [1/6.]*6, size=1)
|
|||
|
|
array([[4, 1, 7, 5, 2, 1]]) # random
|
|||
|
|
|
|||
|
|
It landed 4 times on 1, once on 2, etc.
|
|||
|
|
|
|||
|
|
Now, throw the dice 20 times, and 20 times again:
|
|||
|
|
|
|||
|
|
>>> np.random.multinomial(20, [1/6.]*6, size=2)
|
|||
|
|
array([[3, 4, 3, 3, 4, 3], # random
|
|||
|
|
[2, 4, 3, 4, 0, 7]])
|
|||
|
|
|
|||
|
|
For the first run, we threw 3 times 1, 4 times 2, etc. For the second,
|
|||
|
|
we threw 2 times 1, 4 times 2, etc.
|
|||
|
|
|
|||
|
|
A loaded die is more likely to land on number 6:
|
|||
|
|
|
|||
|
|
>>> np.random.multinomial(100, [1/7.]*5 + [2/7.])
|
|||
|
|
array([11, 16, 14, 17, 16, 26]) # random
|
|||
|
|
|
|||
|
|
The probability inputs should be normalized. As an implementation
|
|||
|
|
detail, the value of the last entry is ignored and assumed to take
|
|||
|
|
up any leftover probability mass, but this should not be relied on.
|
|||
|
|
A biased coin which has twice as much weight on one side as on the
|
|||
|
|
other should be sampled like so:
|
|||
|
|
|
|||
|
|
>>> np.random.multinomial(100, [1.0 / 3, 2.0 / 3]) # RIGHT
|
|||
|
|
array([38, 62]) # random
|
|||
|
|
|
|||
|
|
not like:
|
|||
|
|
|
|||
|
|
>>> np.random.multinomial(100, [1.0, 2.0]) # WRONG
|
|||
|
|
Traceback (most recent call last):
|
|||
|
|
ValueError: pvals < 0, pvals > 1 or pvals contains NaNs
|
|||
|
|
|
|||
|
|
|
|||
|
|
multivariate_normal(mean, cov, size=None, check_valid='warn', tol=1e-8)
|
|||
|
|
|
|||
|
|
Draw random samples from a multivariate normal distribution.
|
|||
|
|
|
|||
|
|
The multivariate normal, multinormal or Gaussian distribution is a
|
|||
|
|
generalization of the one-dimensional normal distribution to higher
|
|||
|
|
dimensions. Such a distribution is specified by its mean and
|
|||
|
|
covariance matrix. These parameters are analogous to the mean
|
|||
|
|
(average or "center") and variance (standard deviation, or "width,"
|
|||
|
|
squared) of the one-dimensional normal distribution.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the
|
|||
|
|
`~numpy.random.Generator.multivariate_normal`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
mean : 1-D array_like, of length N
|
|||
|
|
Mean of the N-dimensional distribution.
|
|||
|
|
cov : 2-D array_like, of shape (N, N)
|
|||
|
|
Covariance matrix of the distribution. It must be symmetric and
|
|||
|
|
positive-semidefinite for proper sampling.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Given a shape of, for example, ``(m,n,k)``, ``m*n*k`` samples are
|
|||
|
|
generated, and packed in an `m`-by-`n`-by-`k` arrangement. Because
|
|||
|
|
each sample is `N`-dimensional, the output shape is ``(m,n,k,N)``.
|
|||
|
|
If no shape is specified, a single (`N`-D) sample is returned.
|
|||
|
|
check_valid : { 'warn', 'raise', 'ignore' }, optional
|
|||
|
|
Behavior when the covariance matrix is not positive semidefinite.
|
|||
|
|
tol : float, optional
|
|||
|
|
Tolerance when checking the singular values in covariance matrix.
|
|||
|
|
cov is cast to double before the check.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray
|
|||
|
|
The drawn samples, of shape *size*, if that was provided. If not,
|
|||
|
|
the shape is ``(N,)``.
|
|||
|
|
|
|||
|
|
In other words, each entry ``out[i,j,...,:]`` is an N-dimensional
|
|||
|
|
value drawn from the distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.multivariate_normal: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The mean is a coordinate in N-dimensional space, which represents the
|
|||
|
|
location where samples are most likely to be generated. This is
|
|||
|
|
analogous to the peak of the bell curve for the one-dimensional or
|
|||
|
|
univariate normal distribution.
|
|||
|
|
|
|||
|
|
Covariance indicates the level to which two variables vary together.
|
|||
|
|
From the multivariate normal distribution, we draw N-dimensional
|
|||
|
|
samples, :math:`X = [x_1, x_2, ... x_N]`. The covariance matrix
|
|||
|
|
element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`.
|
|||
|
|
The element :math:`C_{ii}` is the variance of :math:`x_i` (i.e. its
|
|||
|
|
"spread").
|
|||
|
|
|
|||
|
|
Instead of specifying the full covariance matrix, popular
|
|||
|
|
approximations include:
|
|||
|
|
|
|||
|
|
- Spherical covariance (`cov` is a multiple of the identity matrix)
|
|||
|
|
- Diagonal covariance (`cov` has non-negative elements, and only on
|
|||
|
|
the diagonal)
|
|||
|
|
|
|||
|
|
This geometrical property can be seen in two dimensions by plotting
|
|||
|
|
generated data-points:
|
|||
|
|
|
|||
|
|
>>> mean = [0, 0]
|
|||
|
|
>>> cov = [[1, 0], [0, 100]] # diagonal covariance
|
|||
|
|
|
|||
|
|
Diagonal covariance means that points are oriented along x or y-axis:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> x, y = np.random.multivariate_normal(mean, cov, 5000).T
|
|||
|
|
>>> plt.plot(x, y, 'x')
|
|||
|
|
>>> plt.axis('equal')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
Note that the covariance matrix must be positive semidefinite (a.k.a.
|
|||
|
|
nonnegative-definite). Otherwise, the behavior of this method is
|
|||
|
|
undefined and backwards compatibility is not guaranteed.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Papoulis, A., "Probability, Random Variables, and Stochastic
|
|||
|
|
Processes," 3rd ed., New York: McGraw-Hill, 1991.
|
|||
|
|
.. [2] Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern
|
|||
|
|
Classification," 2nd ed., New York: Wiley, 2001.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> mean = (1, 2)
|
|||
|
|
>>> cov = [[1, 0], [0, 1]]
|
|||
|
|
>>> x = np.random.multivariate_normal(mean, cov, (3, 3))
|
|||
|
|
>>> x.shape
|
|||
|
|
(3, 3, 2)
|
|||
|
|
|
|||
|
|
Here we generate 800 samples from the bivariate normal distribution
|
|||
|
|
with mean [0, 0] and covariance matrix [[6, -3], [-3, 3.5]]. The
|
|||
|
|
expected variances of the first and second components of the sample
|
|||
|
|
are 6 and 3.5, respectively, and the expected correlation
|
|||
|
|
coefficient is -3/sqrt(6*3.5) ≈ -0.65465.
|
|||
|
|
|
|||
|
|
>>> cov = np.array([[6, -3], [-3, 3.5]])
|
|||
|
|
>>> pts = np.random.multivariate_normal([0, 0], cov, size=800)
|
|||
|
|
|
|||
|
|
Check that the mean, covariance, and correlation coefficient of the
|
|||
|
|
sample are close to the expected values:
|
|||
|
|
|
|||
|
|
>>> pts.mean(axis=0)
|
|||
|
|
array([ 0.0326911 , -0.01280782]) # may vary
|
|||
|
|
>>> np.cov(pts.T)
|
|||
|
|
array([[ 5.96202397, -2.85602287],
|
|||
|
|
[-2.85602287, 3.47613949]]) # may vary
|
|||
|
|
>>> np.corrcoef(pts.T)[0, 1]
|
|||
|
|
-0.6273591314603949 # may vary
|
|||
|
|
|
|||
|
|
We can visualize this data with a scatter plot. The orientation
|
|||
|
|
of the point cloud illustrates the negative correlation of the
|
|||
|
|
components of this sample.
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> plt.plot(pts[:, 0], pts[:, 1], '.', alpha=0.5)
|
|||
|
|
>>> plt.axis('equal')
|
|||
|
|
>>> plt.grid()
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
logseries(p, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a logarithmic series distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from a log series distribution with specified
|
|||
|
|
shape parameter, 0 <= ``p`` < 1.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.logseries`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
p : float or array_like of floats
|
|||
|
|
Shape parameter for the distribution. Must be in the range [0, 1).
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``p`` is a scalar. Otherwise,
|
|||
|
|
``np.array(p).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized logarithmic series distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.logser : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.logseries: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the Log Series distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(k) = \frac{-p^k}{k \ln(1-p)},
|
|||
|
|
|
|||
|
|
where p = probability.
|
|||
|
|
|
|||
|
|
The log series distribution is frequently used to represent species
|
|||
|
|
richness and occurrence, first proposed by Fisher, Corbet, and
|
|||
|
|
Williams in 1943 [2]. It may also be used to model the numbers of
|
|||
|
|
occupants seen in cars [3].
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Buzas, Martin A.; Culver, Stephen J., Understanding regional
|
|||
|
|
species diversity through the log series distribution of
|
|||
|
|
occurrences: BIODIVERSITY RESEARCH Diversity & Distributions,
|
|||
|
|
Volume 5, Number 5, September 1999 , pp. 187-195(9).
|
|||
|
|
.. [2] Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The
|
|||
|
|
relation between the number of species and the number of
|
|||
|
|
individuals in a random sample of an animal population.
|
|||
|
|
Journal of Animal Ecology, 12:42-58.
|
|||
|
|
.. [3] D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small
|
|||
|
|
Data Sets, CRC Press, 1994.
|
|||
|
|
.. [4] Wikipedia, "Logarithmic distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Logarithmic_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> a = .6
|
|||
|
|
>>> s = np.random.logseries(a, 10000)
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(s)
|
|||
|
|
|
|||
|
|
# plot against distribution
|
|||
|
|
|
|||
|
|
>>> def logseries(k, p):
|
|||
|
|
... return -p**k/(k*np.log(1-p))
|
|||
|
|
>>> plt.plot(bins, logseries(bins, a)*count.max()/
|
|||
|
|
... logseries(bins, a).max(), 'r')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
hypergeometric(ngood, nbad, nsample, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a Hypergeometric distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from a hypergeometric distribution with specified
|
|||
|
|
parameters, `ngood` (ways to make a good selection), `nbad` (ways to make
|
|||
|
|
a bad selection), and `nsample` (number of items sampled, which is less
|
|||
|
|
than or equal to the sum ``ngood + nbad``).
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the
|
|||
|
|
`~numpy.random.Generator.hypergeometric`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
ngood : int or array_like of ints
|
|||
|
|
Number of ways to make a good selection. Must be nonnegative.
|
|||
|
|
nbad : int or array_like of ints
|
|||
|
|
Number of ways to make a bad selection. Must be nonnegative.
|
|||
|
|
nsample : int or array_like of ints
|
|||
|
|
Number of items sampled. Must be at least 1 and at most
|
|||
|
|
``ngood + nbad``.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if `ngood`, `nbad`, and `nsample`
|
|||
|
|
are all scalars. Otherwise, ``np.broadcast(ngood, nbad, nsample).size``
|
|||
|
|
samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized hypergeometric distribution. Each
|
|||
|
|
sample is the number of good items within a randomly selected subset of
|
|||
|
|
size `nsample` taken from a set of `ngood` good items and `nbad` bad items.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.hypergeom : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.hypergeometric: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the Hypergeometric distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(x) = \frac{\binom{g}{x}\binom{b}{n-x}}{\binom{g+b}{n}},
|
|||
|
|
|
|||
|
|
where :math:`0 \le x \le n` and :math:`n-b \le x \le g`
|
|||
|
|
|
|||
|
|
for P(x) the probability of ``x`` good results in the drawn sample,
|
|||
|
|
g = `ngood`, b = `nbad`, and n = `nsample`.
|
|||
|
|
|
|||
|
|
Consider an urn with black and white marbles in it, `ngood` of them
|
|||
|
|
are black and `nbad` are white. If you draw `nsample` balls without
|
|||
|
|
replacement, then the hypergeometric distribution describes the
|
|||
|
|
distribution of black balls in the drawn sample.
|
|||
|
|
|
|||
|
|
Note that this distribution is very similar to the binomial
|
|||
|
|
distribution, except that in this case, samples are drawn without
|
|||
|
|
replacement, whereas in the Binomial case samples are drawn with
|
|||
|
|
replacement (or the sample space is infinite). As the sample space
|
|||
|
|
becomes large, this distribution approaches the binomial.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Lentner, Marvin, "Elementary Applied Statistics", Bogden
|
|||
|
|
and Quigley, 1972.
|
|||
|
|
.. [2] Weisstein, Eric W. "Hypergeometric Distribution." From
|
|||
|
|
MathWorld--A Wolfram Web Resource.
|
|||
|
|
http://mathworld.wolfram.com/HypergeometricDistribution.html
|
|||
|
|
.. [3] Wikipedia, "Hypergeometric distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Hypergeometric_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> ngood, nbad, nsamp = 100, 2, 10
|
|||
|
|
# number of good, number of bad, and number of samples
|
|||
|
|
>>> s = np.random.hypergeometric(ngood, nbad, nsamp, 1000)
|
|||
|
|
>>> from matplotlib.pyplot import hist
|
|||
|
|
>>> hist(s)
|
|||
|
|
# note that it is very unlikely to grab both bad items
|
|||
|
|
|
|||
|
|
Suppose you have an urn with 15 white and 15 black marbles.
|
|||
|
|
If you pull 15 marbles at random, how likely is it that
|
|||
|
|
12 or more of them are one color?
|
|||
|
|
|
|||
|
|
>>> s = np.random.hypergeometric(15, 15, 15, 100000)
|
|||
|
|
>>> sum(s>=12)/100000. + sum(s<=3)/100000.
|
|||
|
|
# answer = 0.003 ... pretty unlikely!
|
|||
|
|
|
|||
|
|
|
|||
|
|
geometric(p, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from the geometric distribution.
|
|||
|
|
|
|||
|
|
Bernoulli trials are experiments with one of two outcomes:
|
|||
|
|
success or failure (an example of such an experiment is flipping
|
|||
|
|
a coin). The geometric distribution models the number of trials
|
|||
|
|
that must be run in order to achieve success. It is therefore
|
|||
|
|
supported on the positive integers, ``k = 1, 2, ...``.
|
|||
|
|
|
|||
|
|
The probability mass function of the geometric distribution is
|
|||
|
|
|
|||
|
|
.. math:: f(k) = (1 - p)^{k - 1} p
|
|||
|
|
|
|||
|
|
where `p` is the probability of success of an individual trial.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.geometric`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
p : float or array_like of floats
|
|||
|
|
The probability of success of an individual trial.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``p`` is a scalar. Otherwise,
|
|||
|
|
``np.array(p).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized geometric distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.geometric: which should be used for new code.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw ten thousand values from the geometric distribution,
|
|||
|
|
with the probability of an individual success equal to 0.35:
|
|||
|
|
|
|||
|
|
>>> z = np.random.geometric(p=0.35, size=10000)
|
|||
|
|
|
|||
|
|
How many trials succeeded after a single run?
|
|||
|
|
|
|||
|
|
>>> (z == 1).sum() / 10000.
|
|||
|
|
0.34889999999999999 #random
|
|||
|
|
|
|||
|
|
|
|||
|
|
zipf(a, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a Zipf distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from a Zipf distribution with specified parameter
|
|||
|
|
`a` > 1.
|
|||
|
|
|
|||
|
|
The Zipf distribution (also known as the zeta distribution) is a
|
|||
|
|
discrete probability distribution that satisfies Zipf's law: the
|
|||
|
|
frequency of an item is inversely proportional to its rank in a
|
|||
|
|
frequency table.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.zipf`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
a : float or array_like of floats
|
|||
|
|
Distribution parameter. Must be greater than 1.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``a`` is a scalar. Otherwise,
|
|||
|
|
``np.array(a).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized Zipf distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.zipf : probability density function, distribution, or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.zipf: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the Zipf distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(k) = \frac{k^{-a}}{\zeta(a)},
|
|||
|
|
|
|||
|
|
for integers :math:`k \geq 1`, where :math:`\zeta` is the Riemann Zeta
|
|||
|
|
function.
|
|||
|
|
|
|||
|
|
It is named for the American linguist George Kingsley Zipf, who noted
|
|||
|
|
that the frequency of any word in a sample of a language is inversely
|
|||
|
|
proportional to its rank in the frequency table.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Zipf, G. K., "Selected Studies of the Principle of Relative
|
|||
|
|
Frequency in Language," Cambridge, MA: Harvard Univ. Press,
|
|||
|
|
1932.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> a = 4.0
|
|||
|
|
>>> n = 20000
|
|||
|
|
>>> s = np.random.zipf(a, n)
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the expected histogram based on the probability
|
|||
|
|
density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> from scipy.special import zeta # doctest: +SKIP
|
|||
|
|
|
|||
|
|
`bincount` provides a fast histogram for small integers.
|
|||
|
|
|
|||
|
|
>>> count = np.bincount(s)
|
|||
|
|
>>> k = np.arange(1, s.max() + 1)
|
|||
|
|
|
|||
|
|
>>> plt.bar(k, count[1:], alpha=0.5, label='sample count')
|
|||
|
|
>>> plt.plot(k, n*(k**-a)/zeta(a), 'k.-', alpha=0.5,
|
|||
|
|
... label='expected count') # doctest: +SKIP
|
|||
|
|
>>> plt.semilogy()
|
|||
|
|
>>> plt.grid(alpha=0.4)
|
|||
|
|
>>> plt.legend()
|
|||
|
|
>>> plt.title(f'Zipf sample, a={a}, size={n}')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
poisson(lam=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a Poisson distribution.
|
|||
|
|
|
|||
|
|
The Poisson distribution is the limit of the binomial distribution
|
|||
|
|
for large N.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.poisson`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
lam : float or array_like of floats
|
|||
|
|
Expected number of events occurring in a fixed-time interval,
|
|||
|
|
must be >= 0. A sequence must be broadcastable over the requested
|
|||
|
|
size.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``lam`` is a scalar. Otherwise,
|
|||
|
|
``np.array(lam).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized Poisson distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.poisson: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The Poisson distribution
|
|||
|
|
|
|||
|
|
.. math:: f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!}
|
|||
|
|
|
|||
|
|
For events with an expected separation :math:`\lambda` the Poisson
|
|||
|
|
distribution :math:`f(k; \lambda)` describes the probability of
|
|||
|
|
:math:`k` events occurring within the observed
|
|||
|
|
interval :math:`\lambda`.
|
|||
|
|
|
|||
|
|
Because the output is limited to the range of the C int64 type, a
|
|||
|
|
ValueError is raised when `lam` is within 10 sigma of the maximum
|
|||
|
|
representable value.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Weisstein, Eric W. "Poisson Distribution."
|
|||
|
|
From MathWorld--A Wolfram Web Resource.
|
|||
|
|
http://mathworld.wolfram.com/PoissonDistribution.html
|
|||
|
|
.. [2] Wikipedia, "Poisson distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Poisson_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> import numpy as np
|
|||
|
|
>>> s = np.random.poisson(5, 10000)
|
|||
|
|
|
|||
|
|
Display histogram of the sample:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(s, 14, density=True)
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
Draw each 100 values for lambda 100 and 500:
|
|||
|
|
|
|||
|
|
>>> s = np.random.poisson(lam=(100., 500.), size=(100, 2))
|
|||
|
|
|
|||
|
|
|
|||
|
|
negative_binomial(n, p, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a negative binomial distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from a negative binomial distribution with specified
|
|||
|
|
parameters, `n` successes and `p` probability of success where `n`
|
|||
|
|
is > 0 and `p` is in the interval [0, 1].
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the
|
|||
|
|
`~numpy.random.Generator.negative_binomial`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
n : float or array_like of floats
|
|||
|
|
Parameter of the distribution, > 0.
|
|||
|
|
p : float or array_like of floats
|
|||
|
|
Parameter of the distribution, >= 0 and <=1.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``n`` and ``p`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(n, p).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized negative binomial distribution,
|
|||
|
|
where each sample is equal to N, the number of failures that
|
|||
|
|
occurred before a total of n successes was reached.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.negative_binomial: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability mass function of the negative binomial distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(N;n,p) = \frac{\Gamma(N+n)}{N!\Gamma(n)}p^{n}(1-p)^{N},
|
|||
|
|
|
|||
|
|
where :math:`n` is the number of successes, :math:`p` is the
|
|||
|
|
probability of success, :math:`N+n` is the number of trials, and
|
|||
|
|
:math:`\Gamma` is the gamma function. When :math:`n` is an integer,
|
|||
|
|
:math:`\frac{\Gamma(N+n)}{N!\Gamma(n)} = \binom{N+n-1}{N}`, which is
|
|||
|
|
the more common form of this term in the pmf. The negative
|
|||
|
|
binomial distribution gives the probability of N failures given n
|
|||
|
|
successes, with a success on the last trial.
|
|||
|
|
|
|||
|
|
If one throws a die repeatedly until the third time a "1" appears,
|
|||
|
|
then the probability distribution of the number of non-"1"s that
|
|||
|
|
appear before the third "1" is a negative binomial distribution.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Weisstein, Eric W. "Negative Binomial Distribution." From
|
|||
|
|
MathWorld--A Wolfram Web Resource.
|
|||
|
|
http://mathworld.wolfram.com/NegativeBinomialDistribution.html
|
|||
|
|
.. [2] Wikipedia, "Negative binomial distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Negative_binomial_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
A real world example. A company drills wild-cat oil
|
|||
|
|
exploration wells, each with an estimated probability of
|
|||
|
|
success of 0.1. What is the probability of having one success
|
|||
|
|
for each successive well, that is what is the probability of a
|
|||
|
|
single success after drilling 5 wells, after 6 wells, etc.?
|
|||
|
|
|
|||
|
|
>>> s = np.random.negative_binomial(1, 0.1, 100000)
|
|||
|
|
>>> for i in range(1, 11): # doctest: +SKIP
|
|||
|
|
... probability = sum(s<i) / 100000.
|
|||
|
|
... print(i, "wells drilled, probability of one success =", probability)
|
|||
|
|
|
|||
|
|
|
|||
|
|
binomial(n, p, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a binomial distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from a binomial distribution with specified
|
|||
|
|
parameters, n trials and p probability of success where
|
|||
|
|
n an integer >= 0 and p is in the interval [0,1]. (n may be
|
|||
|
|
input as a float, but it is truncated to an integer in use)
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.binomial`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
n : int or array_like of ints
|
|||
|
|
Parameter of the distribution, >= 0. Floats are also accepted,
|
|||
|
|
but they will be truncated to integers.
|
|||
|
|
p : float or array_like of floats
|
|||
|
|
Parameter of the distribution, >= 0 and <=1.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``n`` and ``p`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(n, p).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized binomial distribution, where
|
|||
|
|
each sample is equal to the number of successes over the n trials.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.binom : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.binomial: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the binomial distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(N) = \binom{n}{N}p^N(1-p)^{n-N},
|
|||
|
|
|
|||
|
|
where :math:`n` is the number of trials, :math:`p` is the probability
|
|||
|
|
of success, and :math:`N` is the number of successes.
|
|||
|
|
|
|||
|
|
When estimating the standard error of a proportion in a population by
|
|||
|
|
using a random sample, the normal distribution works well unless the
|
|||
|
|
product p*n <=5, where p = population proportion estimate, and n =
|
|||
|
|
number of samples, in which case the binomial distribution is used
|
|||
|
|
instead. For example, a sample of 15 people shows 4 who are left
|
|||
|
|
handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4,
|
|||
|
|
so the binomial distribution should be used in this case.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Dalgaard, Peter, "Introductory Statistics with R",
|
|||
|
|
Springer-Verlag, 2002.
|
|||
|
|
.. [2] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill,
|
|||
|
|
Fifth Edition, 2002.
|
|||
|
|
.. [3] Lentner, Marvin, "Elementary Applied Statistics", Bogden
|
|||
|
|
and Quigley, 1972.
|
|||
|
|
.. [4] Weisstein, Eric W. "Binomial Distribution." From MathWorld--A
|
|||
|
|
Wolfram Web Resource.
|
|||
|
|
http://mathworld.wolfram.com/BinomialDistribution.html
|
|||
|
|
.. [5] Wikipedia, "Binomial distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Binomial_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> n, p = 10, .5 # number of trials, probability of each trial
|
|||
|
|
>>> s = np.random.binomial(n, p, 1000)
|
|||
|
|
# result of flipping a coin 10 times, tested 1000 times.
|
|||
|
|
|
|||
|
|
A real world example. A company drills 9 wild-cat oil exploration
|
|||
|
|
wells, each with an estimated probability of success of 0.1. All nine
|
|||
|
|
wells fail. What is the probability of that happening?
|
|||
|
|
|
|||
|
|
Let's do 20,000 trials of the model, and count the number that
|
|||
|
|
generate zero positive results.
|
|||
|
|
|
|||
|
|
>>> sum(np.random.binomial(9, 0.1, 20000) == 0)/20000.
|
|||
|
|
# answer = 0.38885, or 38%.
|
|||
|
|
|
|||
|
|
|
|||
|
|
triangular(left, mode, right, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from the triangular distribution over the
|
|||
|
|
interval ``[left, right]``.
|
|||
|
|
|
|||
|
|
The triangular distribution is a continuous probability
|
|||
|
|
distribution with lower limit left, peak at mode, and upper
|
|||
|
|
limit right. Unlike the other distributions, these parameters
|
|||
|
|
directly define the shape of the pdf.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.triangular`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
left : float or array_like of floats
|
|||
|
|
Lower limit.
|
|||
|
|
mode : float or array_like of floats
|
|||
|
|
The value where the peak of the distribution occurs.
|
|||
|
|
The value must fulfill the condition ``left <= mode <= right``.
|
|||
|
|
right : float or array_like of floats
|
|||
|
|
Upper limit, must be larger than `left`.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``left``, ``mode``, and ``right``
|
|||
|
|
are all scalars. Otherwise, ``np.broadcast(left, mode, right).size``
|
|||
|
|
samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized triangular distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.triangular: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density function for the triangular distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(x;l, m, r) = \begin{cases}
|
|||
|
|
\frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\
|
|||
|
|
\frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\
|
|||
|
|
0& \text{otherwise}.
|
|||
|
|
\end{cases}
|
|||
|
|
|
|||
|
|
The triangular distribution is often used in ill-defined
|
|||
|
|
problems where the underlying distribution is not known, but
|
|||
|
|
some knowledge of the limits and mode exists. Often it is used
|
|||
|
|
in simulations.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Wikipedia, "Triangular distribution"
|
|||
|
|
https://en.wikipedia.org/wiki/Triangular_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw values from the distribution and plot the histogram:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> h = plt.hist(np.random.triangular(-3, 0, 8, 100000), bins=200,
|
|||
|
|
... density=True)
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
wald(mean, scale, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a Wald, or inverse Gaussian, distribution.
|
|||
|
|
|
|||
|
|
As the scale approaches infinity, the distribution becomes more like a
|
|||
|
|
Gaussian. Some references claim that the Wald is an inverse Gaussian
|
|||
|
|
with mean equal to 1, but this is by no means universal.
|
|||
|
|
|
|||
|
|
The inverse Gaussian distribution was first studied in relationship to
|
|||
|
|
Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian
|
|||
|
|
because there is an inverse relationship between the time to cover a
|
|||
|
|
unit distance and distance covered in unit time.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.wald`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
mean : float or array_like of floats
|
|||
|
|
Distribution mean, must be > 0.
|
|||
|
|
scale : float or array_like of floats
|
|||
|
|
Scale parameter, must be > 0.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``mean`` and ``scale`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(mean, scale).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized Wald distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.wald: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density function for the Wald distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^
|
|||
|
|
\frac{-scale(x-mean)^2}{2\cdotp mean^2x}
|
|||
|
|
|
|||
|
|
As noted above the inverse Gaussian distribution first arise
|
|||
|
|
from attempts to model Brownian motion. It is also a
|
|||
|
|
competitor to the Weibull for use in reliability modeling and
|
|||
|
|
modeling stock returns and interest rate processes.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Brighton Webs Ltd., Wald Distribution,
|
|||
|
|
https://web.archive.org/web/20090423014010/http://www.brighton-webs.co.uk:80/distributions/wald.asp
|
|||
|
|
.. [2] Chhikara, Raj S., and Folks, J. Leroy, "The Inverse Gaussian
|
|||
|
|
Distribution: Theory : Methodology, and Applications", CRC Press,
|
|||
|
|
1988.
|
|||
|
|
.. [3] Wikipedia, "Inverse Gaussian distribution"
|
|||
|
|
https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw values from the distribution and plot the histogram:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> h = plt.hist(np.random.wald(3, 2, 100000), bins=200, density=True)
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
rayleigh(scale=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a Rayleigh distribution.
|
|||
|
|
|
|||
|
|
The :math:`\chi` and Weibull distributions are generalizations of the
|
|||
|
|
Rayleigh.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.rayleigh`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
scale : float or array_like of floats, optional
|
|||
|
|
Scale, also equals the mode. Must be non-negative. Default is 1.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``scale`` is a scalar. Otherwise,
|
|||
|
|
``np.array(scale).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized Rayleigh distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.rayleigh: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density function for the Rayleigh distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}}
|
|||
|
|
|
|||
|
|
The Rayleigh distribution would arise, for example, if the East
|
|||
|
|
and North components of the wind velocity had identical zero-mean
|
|||
|
|
Gaussian distributions. Then the wind speed would have a Rayleigh
|
|||
|
|
distribution.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Brighton Webs Ltd., "Rayleigh Distribution,"
|
|||
|
|
https://web.archive.org/web/20090514091424/http://brighton-webs.co.uk:80/distributions/rayleigh.asp
|
|||
|
|
.. [2] Wikipedia, "Rayleigh distribution"
|
|||
|
|
https://en.wikipedia.org/wiki/Rayleigh_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw values from the distribution and plot the histogram
|
|||
|
|
|
|||
|
|
>>> from matplotlib.pyplot import hist
|
|||
|
|
>>> values = hist(np.random.rayleigh(3, 100000), bins=200, density=True)
|
|||
|
|
|
|||
|
|
Wave heights tend to follow a Rayleigh distribution. If the mean wave
|
|||
|
|
height is 1 meter, what fraction of waves are likely to be larger than 3
|
|||
|
|
meters?
|
|||
|
|
|
|||
|
|
>>> meanvalue = 1
|
|||
|
|
>>> modevalue = np.sqrt(2 / np.pi) * meanvalue
|
|||
|
|
>>> s = np.random.rayleigh(modevalue, 1000000)
|
|||
|
|
|
|||
|
|
The percentage of waves larger than 3 meters is:
|
|||
|
|
|
|||
|
|
>>> 100.*sum(s>3)/1000000.
|
|||
|
|
0.087300000000000003 # random
|
|||
|
|
|
|||
|
|
|
|||
|
|
lognormal(mean=0.0, sigma=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a log-normal distribution.
|
|||
|
|
|
|||
|
|
Draw samples from a log-normal distribution with specified mean,
|
|||
|
|
standard deviation, and array shape. Note that the mean and standard
|
|||
|
|
deviation are not the values for the distribution itself, but of the
|
|||
|
|
underlying normal distribution it is derived from.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.lognormal`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
mean : float or array_like of floats, optional
|
|||
|
|
Mean value of the underlying normal distribution. Default is 0.
|
|||
|
|
sigma : float or array_like of floats, optional
|
|||
|
|
Standard deviation of the underlying normal distribution. Must be
|
|||
|
|
non-negative. Default is 1.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``mean`` and ``sigma`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(mean, sigma).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized log-normal distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.lognorm : probability density function, distribution,
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.lognormal: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
A variable `x` has a log-normal distribution if `log(x)` is normally
|
|||
|
|
distributed. The probability density function for the log-normal
|
|||
|
|
distribution is:
|
|||
|
|
|
|||
|
|
.. math:: p(x) = \frac{1}{\sigma x \sqrt{2\pi}}
|
|||
|
|
e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}
|
|||
|
|
|
|||
|
|
where :math:`\mu` is the mean and :math:`\sigma` is the standard
|
|||
|
|
deviation of the normally distributed logarithm of the variable.
|
|||
|
|
A log-normal distribution results if a random variable is the *product*
|
|||
|
|
of a large number of independent, identically-distributed variables in
|
|||
|
|
the same way that a normal distribution results if the variable is the
|
|||
|
|
*sum* of a large number of independent, identically-distributed
|
|||
|
|
variables.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Limpert, E., Stahel, W. A., and Abbt, M., "Log-normal
|
|||
|
|
Distributions across the Sciences: Keys and Clues,"
|
|||
|
|
BioScience, Vol. 51, No. 5, May, 2001.
|
|||
|
|
https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf
|
|||
|
|
.. [2] Reiss, R.D. and Thomas, M., "Statistical Analysis of Extreme
|
|||
|
|
Values," Basel: Birkhauser Verlag, 2001, pp. 31-32.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> mu, sigma = 3., 1. # mean and standard deviation
|
|||
|
|
>>> s = np.random.lognormal(mu, sigma, 1000)
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(s, 100, density=True, align='mid')
|
|||
|
|
|
|||
|
|
>>> x = np.linspace(min(bins), max(bins), 10000)
|
|||
|
|
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
|
|||
|
|
... / (x * sigma * np.sqrt(2 * np.pi)))
|
|||
|
|
|
|||
|
|
>>> plt.plot(x, pdf, linewidth=2, color='r')
|
|||
|
|
>>> plt.axis('tight')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
Demonstrate that taking the products of random samples from a uniform
|
|||
|
|
distribution can be fit well by a log-normal probability density
|
|||
|
|
function.
|
|||
|
|
|
|||
|
|
>>> # Generate a thousand samples: each is the product of 100 random
|
|||
|
|
>>> # values, drawn from a normal distribution.
|
|||
|
|
>>> b = []
|
|||
|
|
>>> for i in range(1000):
|
|||
|
|
... a = 10. + np.random.standard_normal(100)
|
|||
|
|
... b.append(np.prod(a))
|
|||
|
|
|
|||
|
|
>>> b = np.array(b) / np.min(b) # scale values to be positive
|
|||
|
|
>>> count, bins, ignored = plt.hist(b, 100, density=True, align='mid')
|
|||
|
|
>>> sigma = np.std(np.log(b))
|
|||
|
|
>>> mu = np.mean(np.log(b))
|
|||
|
|
|
|||
|
|
>>> x = np.linspace(min(bins), max(bins), 10000)
|
|||
|
|
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
|
|||
|
|
... / (x * sigma * np.sqrt(2 * np.pi)))
|
|||
|
|
|
|||
|
|
>>> plt.plot(x, pdf, color='r', linewidth=2)
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
logistic(loc=0.0, scale=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a logistic distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from a logistic distribution with specified
|
|||
|
|
parameters, loc (location or mean, also median), and scale (>0).
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.logistic`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
loc : float or array_like of floats, optional
|
|||
|
|
Parameter of the distribution. Default is 0.
|
|||
|
|
scale : float or array_like of floats, optional
|
|||
|
|
Parameter of the distribution. Must be non-negative.
|
|||
|
|
Default is 1.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``loc`` and ``scale`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized logistic distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.logistic : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.logistic: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the Logistic distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(x) = P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2},
|
|||
|
|
|
|||
|
|
where :math:`\mu` = location and :math:`s` = scale.
|
|||
|
|
|
|||
|
|
The Logistic distribution is used in Extreme Value problems where it
|
|||
|
|
can act as a mixture of Gumbel distributions, in Epidemiology, and by
|
|||
|
|
the World Chess Federation (FIDE) where it is used in the Elo ranking
|
|||
|
|
system, assuming the performance of each player is a logistically
|
|||
|
|
distributed random variable.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Reiss, R.-D. and Thomas M. (2001), "Statistical Analysis of
|
|||
|
|
Extreme Values, from Insurance, Finance, Hydrology and Other
|
|||
|
|
Fields," Birkhauser Verlag, Basel, pp 132-133.
|
|||
|
|
.. [2] Weisstein, Eric W. "Logistic Distribution." From
|
|||
|
|
MathWorld--A Wolfram Web Resource.
|
|||
|
|
http://mathworld.wolfram.com/LogisticDistribution.html
|
|||
|
|
.. [3] Wikipedia, "Logistic-distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Logistic_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> loc, scale = 10, 1
|
|||
|
|
>>> s = np.random.logistic(loc, scale, 10000)
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(s, bins=50)
|
|||
|
|
|
|||
|
|
# plot against distribution
|
|||
|
|
|
|||
|
|
>>> def logist(x, loc, scale):
|
|||
|
|
... return np.exp((loc-x)/scale)/(scale*(1+np.exp((loc-x)/scale))**2)
|
|||
|
|
>>> lgst_val = logist(bins, loc, scale)
|
|||
|
|
>>> plt.plot(bins, lgst_val * count.max() / lgst_val.max())
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
gumbel(loc=0.0, scale=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a Gumbel distribution.
|
|||
|
|
|
|||
|
|
Draw samples from a Gumbel distribution with specified location and
|
|||
|
|
scale. For more information on the Gumbel distribution, see
|
|||
|
|
Notes and References below.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.gumbel`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
loc : float or array_like of floats, optional
|
|||
|
|
The location of the mode of the distribution. Default is 0.
|
|||
|
|
scale : float or array_like of floats, optional
|
|||
|
|
The scale parameter of the distribution. Default is 1. Must be non-
|
|||
|
|
negative.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``loc`` and ``scale`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized Gumbel distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.gumbel_l
|
|||
|
|
scipy.stats.gumbel_r
|
|||
|
|
scipy.stats.genextreme
|
|||
|
|
weibull
|
|||
|
|
random.Generator.gumbel: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme
|
|||
|
|
Value Type I) distribution is one of a class of Generalized Extreme
|
|||
|
|
Value (GEV) distributions used in modeling extreme value problems.
|
|||
|
|
The Gumbel is a special case of the Extreme Value Type I distribution
|
|||
|
|
for maximums from distributions with "exponential-like" tails.
|
|||
|
|
|
|||
|
|
The probability density for the Gumbel distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/
|
|||
|
|
\beta}},
|
|||
|
|
|
|||
|
|
where :math:`\mu` is the mode, a location parameter, and
|
|||
|
|
:math:`\beta` is the scale parameter.
|
|||
|
|
|
|||
|
|
The Gumbel (named for German mathematician Emil Julius Gumbel) was used
|
|||
|
|
very early in the hydrology literature, for modeling the occurrence of
|
|||
|
|
flood events. It is also used for modeling maximum wind speed and
|
|||
|
|
rainfall rates. It is a "fat-tailed" distribution - the probability of
|
|||
|
|
an event in the tail of the distribution is larger than if one used a
|
|||
|
|
Gaussian, hence the surprisingly frequent occurrence of 100-year
|
|||
|
|
floods. Floods were initially modeled as a Gaussian process, which
|
|||
|
|
underestimated the frequency of extreme events.
|
|||
|
|
|
|||
|
|
It is one of a class of extreme value distributions, the Generalized
|
|||
|
|
Extreme Value (GEV) distributions, which also includes the Weibull and
|
|||
|
|
Frechet.
|
|||
|
|
|
|||
|
|
The function has a mean of :math:`\mu + 0.57721\beta` and a variance
|
|||
|
|
of :math:`\frac{\pi^2}{6}\beta^2`.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Gumbel, E. J., "Statistics of Extremes,"
|
|||
|
|
New York: Columbia University Press, 1958.
|
|||
|
|
.. [2] Reiss, R.-D. and Thomas, M., "Statistical Analysis of Extreme
|
|||
|
|
Values from Insurance, Finance, Hydrology and Other Fields,"
|
|||
|
|
Basel: Birkhauser Verlag, 2001.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> mu, beta = 0, 0.1 # location and scale
|
|||
|
|
>>> s = np.random.gumbel(mu, beta, 1000)
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(s, 30, density=True)
|
|||
|
|
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
|
|||
|
|
... * np.exp( -np.exp( -(bins - mu) /beta) ),
|
|||
|
|
... linewidth=2, color='r')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
Show how an extreme value distribution can arise from a Gaussian process
|
|||
|
|
and compare to a Gaussian:
|
|||
|
|
|
|||
|
|
>>> means = []
|
|||
|
|
>>> maxima = []
|
|||
|
|
>>> for i in range(0,1000) :
|
|||
|
|
... a = np.random.normal(mu, beta, 1000)
|
|||
|
|
... means.append(a.mean())
|
|||
|
|
... maxima.append(a.max())
|
|||
|
|
>>> count, bins, ignored = plt.hist(maxima, 30, density=True)
|
|||
|
|
>>> beta = np.std(maxima) * np.sqrt(6) / np.pi
|
|||
|
|
>>> mu = np.mean(maxima) - 0.57721*beta
|
|||
|
|
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
|
|||
|
|
... * np.exp(-np.exp(-(bins - mu)/beta)),
|
|||
|
|
... linewidth=2, color='r')
|
|||
|
|
>>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi))
|
|||
|
|
... * np.exp(-(bins - mu)**2 / (2 * beta**2)),
|
|||
|
|
... linewidth=2, color='g')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
laplace(loc=0.0, scale=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from the Laplace or double exponential distribution with
|
|||
|
|
specified location (or mean) and scale (decay).
|
|||
|
|
|
|||
|
|
The Laplace distribution is similar to the Gaussian/normal distribution,
|
|||
|
|
but is sharper at the peak and has fatter tails. It represents the
|
|||
|
|
difference between two independent, identically distributed exponential
|
|||
|
|
random variables.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.laplace`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
loc : float or array_like of floats, optional
|
|||
|
|
The position, :math:`\mu`, of the distribution peak. Default is 0.
|
|||
|
|
scale : float or array_like of floats, optional
|
|||
|
|
:math:`\lambda`, the exponential decay. Default is 1. Must be non-
|
|||
|
|
negative.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``loc`` and ``scale`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized Laplace distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.laplace: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
It has the probability density function
|
|||
|
|
|
|||
|
|
.. math:: f(x; \mu, \lambda) = \frac{1}{2\lambda}
|
|||
|
|
\exp\left(-\frac{|x - \mu|}{\lambda}\right).
|
|||
|
|
|
|||
|
|
The first law of Laplace, from 1774, states that the frequency
|
|||
|
|
of an error can be expressed as an exponential function of the
|
|||
|
|
absolute magnitude of the error, which leads to the Laplace
|
|||
|
|
distribution. For many problems in economics and health
|
|||
|
|
sciences, this distribution seems to model the data better
|
|||
|
|
than the standard Gaussian distribution.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of
|
|||
|
|
Mathematical Functions with Formulas, Graphs, and Mathematical
|
|||
|
|
Tables, 9th printing," New York: Dover, 1972.
|
|||
|
|
.. [2] Kotz, Samuel, et. al. "The Laplace Distribution and
|
|||
|
|
Generalizations, " Birkhauser, 2001.
|
|||
|
|
.. [3] Weisstein, Eric W. "Laplace Distribution."
|
|||
|
|
From MathWorld--A Wolfram Web Resource.
|
|||
|
|
http://mathworld.wolfram.com/LaplaceDistribution.html
|
|||
|
|
.. [4] Wikipedia, "Laplace distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Laplace_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution
|
|||
|
|
|
|||
|
|
>>> loc, scale = 0., 1.
|
|||
|
|
>>> s = np.random.laplace(loc, scale, 1000)
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(s, 30, density=True)
|
|||
|
|
>>> x = np.arange(-8., 8., .01)
|
|||
|
|
>>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale)
|
|||
|
|
>>> plt.plot(x, pdf)
|
|||
|
|
|
|||
|
|
Plot Gaussian for comparison:
|
|||
|
|
|
|||
|
|
>>> g = (1/(scale * np.sqrt(2 * np.pi)) *
|
|||
|
|
... np.exp(-(x - loc)**2 / (2 * scale**2)))
|
|||
|
|
>>> plt.plot(x,g)
|
|||
|
|
|
|||
|
|
|
|||
|
|
power(a, size=None)
|
|||
|
|
|
|||
|
|
Draws samples in [0, 1] from a power distribution with positive
|
|||
|
|
exponent a - 1.
|
|||
|
|
|
|||
|
|
Also known as the power function distribution.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.power`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
a : float or array_like of floats
|
|||
|
|
Parameter of the distribution. Must be non-negative.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``a`` is a scalar. Otherwise,
|
|||
|
|
``np.array(a).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized power distribution.
|
|||
|
|
|
|||
|
|
Raises
|
|||
|
|
------
|
|||
|
|
ValueError
|
|||
|
|
If a <= 0.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.power: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density function is
|
|||
|
|
|
|||
|
|
.. math:: P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0.
|
|||
|
|
|
|||
|
|
The power function distribution is just the inverse of the Pareto
|
|||
|
|
distribution. It may also be seen as a special case of the Beta
|
|||
|
|
distribution.
|
|||
|
|
|
|||
|
|
It is used, for example, in modeling the over-reporting of insurance
|
|||
|
|
claims.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Christian Kleiber, Samuel Kotz, "Statistical size distributions
|
|||
|
|
in economics and actuarial sciences", Wiley, 2003.
|
|||
|
|
.. [2] Heckert, N. A. and Filliben, James J. "NIST Handbook 148:
|
|||
|
|
Dataplot Reference Manual, Volume 2: Let Subcommands and Library
|
|||
|
|
Functions", National Institute of Standards and Technology
|
|||
|
|
Handbook Series, June 2003.
|
|||
|
|
https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> a = 5. # shape
|
|||
|
|
>>> samples = 1000
|
|||
|
|
>>> s = np.random.power(a, samples)
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(s, bins=30)
|
|||
|
|
>>> x = np.linspace(0, 1, 100)
|
|||
|
|
>>> y = a*x**(a-1.)
|
|||
|
|
>>> normed_y = samples*np.diff(bins)[0]*y
|
|||
|
|
>>> plt.plot(x, normed_y)
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
Compare the power function distribution to the inverse of the Pareto.
|
|||
|
|
|
|||
|
|
>>> from scipy import stats # doctest: +SKIP
|
|||
|
|
>>> rvs = np.random.power(5, 1000000)
|
|||
|
|
>>> rvsp = np.random.pareto(5, 1000000)
|
|||
|
|
>>> xx = np.linspace(0,1,100)
|
|||
|
|
>>> powpdf = stats.powerlaw.pdf(xx,5) # doctest: +SKIP
|
|||
|
|
|
|||
|
|
>>> plt.figure()
|
|||
|
|
>>> plt.hist(rvs, bins=50, density=True)
|
|||
|
|
>>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP
|
|||
|
|
>>> plt.title('np.random.power(5)')
|
|||
|
|
|
|||
|
|
>>> plt.figure()
|
|||
|
|
>>> plt.hist(1./(1.+rvsp), bins=50, density=True)
|
|||
|
|
>>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP
|
|||
|
|
>>> plt.title('inverse of 1 + np.random.pareto(5)')
|
|||
|
|
|
|||
|
|
>>> plt.figure()
|
|||
|
|
>>> plt.hist(1./(1.+rvsp), bins=50, density=True)
|
|||
|
|
>>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP
|
|||
|
|
>>> plt.title('inverse of stats.pareto(5)')
|
|||
|
|
|
|||
|
|
|
|||
|
|
weibull(a, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a Weibull distribution.
|
|||
|
|
|
|||
|
|
Draw samples from a 1-parameter Weibull distribution with the given
|
|||
|
|
shape parameter `a`.
|
|||
|
|
|
|||
|
|
.. math:: X = (-ln(U))^{1/a}
|
|||
|
|
|
|||
|
|
Here, U is drawn from the uniform distribution over (0,1].
|
|||
|
|
|
|||
|
|
The more common 2-parameter Weibull, including a scale parameter
|
|||
|
|
:math:`\lambda` is just :math:`X = \lambda(-ln(U))^{1/a}`.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.weibull`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
a : float or array_like of floats
|
|||
|
|
Shape parameter of the distribution. Must be nonnegative.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``a`` is a scalar. Otherwise,
|
|||
|
|
``np.array(a).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized Weibull distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.weibull_max
|
|||
|
|
scipy.stats.weibull_min
|
|||
|
|
scipy.stats.genextreme
|
|||
|
|
gumbel
|
|||
|
|
random.Generator.weibull: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The Weibull (or Type III asymptotic extreme value distribution
|
|||
|
|
for smallest values, SEV Type III, or Rosin-Rammler
|
|||
|
|
distribution) is one of a class of Generalized Extreme Value
|
|||
|
|
(GEV) distributions used in modeling extreme value problems.
|
|||
|
|
This class includes the Gumbel and Frechet distributions.
|
|||
|
|
|
|||
|
|
The probability density for the Weibull distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(x) = \frac{a}
|
|||
|
|
{\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},
|
|||
|
|
|
|||
|
|
where :math:`a` is the shape and :math:`\lambda` the scale.
|
|||
|
|
|
|||
|
|
The function has its peak (the mode) at
|
|||
|
|
:math:`\lambda(\frac{a-1}{a})^{1/a}`.
|
|||
|
|
|
|||
|
|
When ``a = 1``, the Weibull distribution reduces to the exponential
|
|||
|
|
distribution.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Waloddi Weibull, Royal Technical University, Stockholm,
|
|||
|
|
1939 "A Statistical Theory Of The Strength Of Materials",
|
|||
|
|
Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939,
|
|||
|
|
Generalstabens Litografiska Anstalts Forlag, Stockholm.
|
|||
|
|
.. [2] Waloddi Weibull, "A Statistical Distribution Function of
|
|||
|
|
Wide Applicability", Journal Of Applied Mechanics ASME Paper
|
|||
|
|
1951.
|
|||
|
|
.. [3] Wikipedia, "Weibull distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Weibull_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> a = 5. # shape
|
|||
|
|
>>> s = np.random.weibull(a, 1000)
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> x = np.arange(1,100.)/50.
|
|||
|
|
>>> def weib(x,n,a):
|
|||
|
|
... return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)
|
|||
|
|
|
|||
|
|
>>> count, bins, ignored = plt.hist(np.random.weibull(5.,1000))
|
|||
|
|
>>> x = np.arange(1,100.)/50.
|
|||
|
|
>>> scale = count.max()/weib(x, 1., 5.).max()
|
|||
|
|
>>> plt.plot(x, weib(x, 1., 5.)*scale)
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
pareto(a, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a Pareto II or Lomax distribution with
|
|||
|
|
specified shape.
|
|||
|
|
|
|||
|
|
The Lomax or Pareto II distribution is a shifted Pareto
|
|||
|
|
distribution. The classical Pareto distribution can be
|
|||
|
|
obtained from the Lomax distribution by adding 1 and
|
|||
|
|
multiplying by the scale parameter ``m`` (see Notes). The
|
|||
|
|
smallest value of the Lomax distribution is zero while for the
|
|||
|
|
classical Pareto distribution it is ``mu``, where the standard
|
|||
|
|
Pareto distribution has location ``mu = 1``. Lomax can also
|
|||
|
|
be considered as a simplified version of the Generalized
|
|||
|
|
Pareto distribution (available in SciPy), with the scale set
|
|||
|
|
to one and the location set to zero.
|
|||
|
|
|
|||
|
|
The Pareto distribution must be greater than zero, and is
|
|||
|
|
unbounded above. It is also known as the "80-20 rule". In
|
|||
|
|
this distribution, 80 percent of the weights are in the lowest
|
|||
|
|
20 percent of the range, while the other 20 percent fill the
|
|||
|
|
remaining 80 percent of the range.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.pareto`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
a : float or array_like of floats
|
|||
|
|
Shape of the distribution. Must be positive.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``a`` is a scalar. Otherwise,
|
|||
|
|
``np.array(a).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized Pareto distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.lomax : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
scipy.stats.genpareto : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.pareto: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the Pareto distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(x) = \frac{am^a}{x^{a+1}}
|
|||
|
|
|
|||
|
|
where :math:`a` is the shape and :math:`m` the scale.
|
|||
|
|
|
|||
|
|
The Pareto distribution, named after the Italian economist
|
|||
|
|
Vilfredo Pareto, is a power law probability distribution
|
|||
|
|
useful in many real world problems. Outside the field of
|
|||
|
|
economics it is generally referred to as the Bradford
|
|||
|
|
distribution. Pareto developed the distribution to describe
|
|||
|
|
the distribution of wealth in an economy. It has also found
|
|||
|
|
use in insurance, web page access statistics, oil field sizes,
|
|||
|
|
and many other problems, including the download frequency for
|
|||
|
|
projects in Sourceforge [1]_. It is one of the so-called
|
|||
|
|
"fat-tailed" distributions.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Francis Hunt and Paul Johnson, On the Pareto Distribution of
|
|||
|
|
Sourceforge projects.
|
|||
|
|
.. [2] Pareto, V. (1896). Course of Political Economy. Lausanne.
|
|||
|
|
.. [3] Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme
|
|||
|
|
Values, Birkhauser Verlag, Basel, pp 23-30.
|
|||
|
|
.. [4] Wikipedia, "Pareto distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Pareto_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> a, m = 3., 2. # shape and mode
|
|||
|
|
>>> s = (np.random.pareto(a, 1000) + 1) * m
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with the probability
|
|||
|
|
density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, _ = plt.hist(s, 100, density=True)
|
|||
|
|
>>> fit = a*m**a / bins**(a+1)
|
|||
|
|
>>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color='r')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
vonmises(mu, kappa, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a von Mises distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from a von Mises distribution with specified mode
|
|||
|
|
(mu) and dispersion (kappa), on the interval [-pi, pi].
|
|||
|
|
|
|||
|
|
The von Mises distribution (also known as the circular normal
|
|||
|
|
distribution) is a continuous probability distribution on the unit
|
|||
|
|
circle. It may be thought of as the circular analogue of the normal
|
|||
|
|
distribution.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.vonmises`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
mu : float or array_like of floats
|
|||
|
|
Mode ("center") of the distribution.
|
|||
|
|
kappa : float or array_like of floats
|
|||
|
|
Dispersion of the distribution, has to be >=0.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``mu`` and ``kappa`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(mu, kappa).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized von Mises distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.vonmises : probability density function, distribution, or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.vonmises: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the von Mises distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)},
|
|||
|
|
|
|||
|
|
where :math:`\mu` is the mode and :math:`\kappa` the dispersion,
|
|||
|
|
and :math:`I_0(\kappa)` is the modified Bessel function of order 0.
|
|||
|
|
|
|||
|
|
The von Mises is named for Richard Edler von Mises, who was born in
|
|||
|
|
Austria-Hungary, in what is now the Ukraine. He fled to the United
|
|||
|
|
States in 1939 and became a professor at Harvard. He worked in
|
|||
|
|
probability theory, aerodynamics, fluid mechanics, and philosophy of
|
|||
|
|
science.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of
|
|||
|
|
Mathematical Functions with Formulas, Graphs, and Mathematical
|
|||
|
|
Tables, 9th printing," New York: Dover, 1972.
|
|||
|
|
.. [2] von Mises, R., "Mathematical Theory of Probability
|
|||
|
|
and Statistics", New York: Academic Press, 1964.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> mu, kappa = 0.0, 4.0 # mean and dispersion
|
|||
|
|
>>> s = np.random.vonmises(mu, kappa, 1000)
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> from scipy.special import i0 # doctest: +SKIP
|
|||
|
|
>>> plt.hist(s, 50, density=True)
|
|||
|
|
>>> x = np.linspace(-np.pi, np.pi, num=51)
|
|||
|
|
>>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa)) # doctest: +SKIP
|
|||
|
|
>>> plt.plot(x, y, linewidth=2, color='r') # doctest: +SKIP
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
standard_t(df, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a standard Student's t distribution with `df` degrees
|
|||
|
|
of freedom.
|
|||
|
|
|
|||
|
|
A special case of the hyperbolic distribution. As `df` gets
|
|||
|
|
large, the result resembles that of the standard normal
|
|||
|
|
distribution (`standard_normal`).
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.standard_t`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
df : float or array_like of floats
|
|||
|
|
Degrees of freedom, must be > 0.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``df`` is a scalar. Otherwise,
|
|||
|
|
``np.array(df).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized standard Student's t distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.standard_t: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density function for the t distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df}
|
|||
|
|
\Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2}
|
|||
|
|
|
|||
|
|
The t test is based on an assumption that the data come from a
|
|||
|
|
Normal distribution. The t test provides a way to test whether
|
|||
|
|
the sample mean (that is the mean calculated from the data) is
|
|||
|
|
a good estimate of the true mean.
|
|||
|
|
|
|||
|
|
The derivation of the t-distribution was first published in
|
|||
|
|
1908 by William Gosset while working for the Guinness Brewery
|
|||
|
|
in Dublin. Due to proprietary issues, he had to publish under
|
|||
|
|
a pseudonym, and so he used the name Student.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Dalgaard, Peter, "Introductory Statistics With R",
|
|||
|
|
Springer, 2002.
|
|||
|
|
.. [2] Wikipedia, "Student's t-distribution"
|
|||
|
|
https://en.wikipedia.org/wiki/Student's_t-distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
From Dalgaard page 83 [1]_, suppose the daily energy intake for 11
|
|||
|
|
women in kilojoules (kJ) is:
|
|||
|
|
|
|||
|
|
>>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \
|
|||
|
|
... 7515, 8230, 8770])
|
|||
|
|
|
|||
|
|
Does their energy intake deviate systematically from the recommended
|
|||
|
|
value of 7725 kJ? Our null hypothesis will be the absence of deviation,
|
|||
|
|
and the alternate hypothesis will be the presence of an effect that could be
|
|||
|
|
either positive or negative, hence making our test 2-tailed.
|
|||
|
|
|
|||
|
|
Because we are estimating the mean and we have N=11 values in our sample,
|
|||
|
|
we have N-1=10 degrees of freedom. We set our significance level to 95% and
|
|||
|
|
compute the t statistic using the empirical mean and empirical standard
|
|||
|
|
deviation of our intake. We use a ddof of 1 to base the computation of our
|
|||
|
|
empirical standard deviation on an unbiased estimate of the variance (note:
|
|||
|
|
the final estimate is not unbiased due to the concave nature of the square
|
|||
|
|
root).
|
|||
|
|
|
|||
|
|
>>> np.mean(intake)
|
|||
|
|
6753.636363636364
|
|||
|
|
>>> intake.std(ddof=1)
|
|||
|
|
1142.1232221373727
|
|||
|
|
>>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake)))
|
|||
|
|
>>> t
|
|||
|
|
-2.8207540608310198
|
|||
|
|
|
|||
|
|
We draw 1000000 samples from Student's t distribution with the adequate
|
|||
|
|
degrees of freedom.
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> s = np.random.standard_t(10, size=1000000)
|
|||
|
|
>>> h = plt.hist(s, bins=100, density=True)
|
|||
|
|
|
|||
|
|
Does our t statistic land in one of the two critical regions found at
|
|||
|
|
both tails of the distribution?
|
|||
|
|
|
|||
|
|
>>> np.sum(np.abs(t) < np.abs(s)) / float(len(s))
|
|||
|
|
0.018318 #random < 0.05, statistic is in critical region
|
|||
|
|
|
|||
|
|
The probability value for this 2-tailed test is about 1.83%, which is
|
|||
|
|
lower than the 5% pre-determined significance threshold.
|
|||
|
|
|
|||
|
|
Therefore, the probability of observing values as extreme as our intake
|
|||
|
|
conditionally on the null hypothesis being true is too low, and we reject
|
|||
|
|
the null hypothesis of no deviation.
|
|||
|
|
|
|||
|
|
|
|||
|
|
standard_cauchy(size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a standard Cauchy distribution with mode = 0.
|
|||
|
|
|
|||
|
|
Also known as the Lorentz distribution.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the
|
|||
|
|
`~numpy.random.Generator.standard_cauchy`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. Default is None, in which case a
|
|||
|
|
single value is returned.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
samples : ndarray or scalar
|
|||
|
|
The drawn samples.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.standard_cauchy: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density function for the full Cauchy distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(x; x_0, \gamma) = \frac{1}{\pi \gamma \bigl[ 1+
|
|||
|
|
(\frac{x-x_0}{\gamma})^2 \bigr] }
|
|||
|
|
|
|||
|
|
and the Standard Cauchy distribution just sets :math:`x_0=0` and
|
|||
|
|
:math:`\gamma=1`
|
|||
|
|
|
|||
|
|
The Cauchy distribution arises in the solution to the driven harmonic
|
|||
|
|
oscillator problem, and also describes spectral line broadening. It
|
|||
|
|
also describes the distribution of values at which a line tilted at
|
|||
|
|
a random angle will cut the x axis.
|
|||
|
|
|
|||
|
|
When studying hypothesis tests that assume normality, seeing how the
|
|||
|
|
tests perform on data from a Cauchy distribution is a good indicator of
|
|||
|
|
their sensitivity to a heavy-tailed distribution, since the Cauchy looks
|
|||
|
|
very much like a Gaussian distribution, but with heavier tails.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] NIST/SEMATECH e-Handbook of Statistical Methods, "Cauchy
|
|||
|
|
Distribution",
|
|||
|
|
https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm
|
|||
|
|
.. [2] Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A
|
|||
|
|
Wolfram Web Resource.
|
|||
|
|
http://mathworld.wolfram.com/CauchyDistribution.html
|
|||
|
|
.. [3] Wikipedia, "Cauchy distribution"
|
|||
|
|
https://en.wikipedia.org/wiki/Cauchy_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples and plot the distribution:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> s = np.random.standard_cauchy(1000000)
|
|||
|
|
>>> s = s[(s>-25) & (s<25)] # truncate distribution so it plots well
|
|||
|
|
>>> plt.hist(s, bins=100)
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
noncentral_chisquare(df, nonc, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a noncentral chi-square distribution.
|
|||
|
|
|
|||
|
|
The noncentral :math:`\chi^2` distribution is a generalization of
|
|||
|
|
the :math:`\chi^2` distribution.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the
|
|||
|
|
`~numpy.random.Generator.noncentral_chisquare`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
df : float or array_like of floats
|
|||
|
|
Degrees of freedom, must be > 0.
|
|||
|
|
|
|||
|
|
.. versionchanged:: 1.10.0
|
|||
|
|
Earlier NumPy versions required dfnum > 1.
|
|||
|
|
nonc : float or array_like of floats
|
|||
|
|
Non-centrality, must be non-negative.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``df`` and ``nonc`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(df, nonc).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized noncentral chi-square distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.noncentral_chisquare: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density function for the noncentral Chi-square
|
|||
|
|
distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(x;df,nonc) = \sum^{\infty}_{i=0}
|
|||
|
|
\frac{e^{-nonc/2}(nonc/2)^{i}}{i!}
|
|||
|
|
P_{Y_{df+2i}}(x),
|
|||
|
|
|
|||
|
|
where :math:`Y_{q}` is the Chi-square with q degrees of freedom.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Wikipedia, "Noncentral chi-squared distribution"
|
|||
|
|
https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw values from the distribution and plot the histogram
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
|
|||
|
|
... bins=200, density=True)
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
Draw values from a noncentral chisquare with very small noncentrality,
|
|||
|
|
and compare to a chisquare.
|
|||
|
|
|
|||
|
|
>>> plt.figure()
|
|||
|
|
>>> values = plt.hist(np.random.noncentral_chisquare(3, .0000001, 100000),
|
|||
|
|
... bins=np.arange(0., 25, .1), density=True)
|
|||
|
|
>>> values2 = plt.hist(np.random.chisquare(3, 100000),
|
|||
|
|
... bins=np.arange(0., 25, .1), density=True)
|
|||
|
|
>>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
Demonstrate how large values of non-centrality lead to a more symmetric
|
|||
|
|
distribution.
|
|||
|
|
|
|||
|
|
>>> plt.figure()
|
|||
|
|
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
|
|||
|
|
... bins=200, density=True)
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
chisquare(df, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a chi-square distribution.
|
|||
|
|
|
|||
|
|
When `df` independent random variables, each with standard normal
|
|||
|
|
distributions (mean 0, variance 1), are squared and summed, the
|
|||
|
|
resulting distribution is chi-square (see Notes). This distribution
|
|||
|
|
is often used in hypothesis testing.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.chisquare`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
df : float or array_like of floats
|
|||
|
|
Number of degrees of freedom, must be > 0.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``df`` is a scalar. Otherwise,
|
|||
|
|
``np.array(df).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized chi-square distribution.
|
|||
|
|
|
|||
|
|
Raises
|
|||
|
|
------
|
|||
|
|
ValueError
|
|||
|
|
When `df` <= 0 or when an inappropriate `size` (e.g. ``size=-1``)
|
|||
|
|
is given.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.chisquare: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The variable obtained by summing the squares of `df` independent,
|
|||
|
|
standard normally distributed random variables:
|
|||
|
|
|
|||
|
|
.. math:: Q = \sum_{i=0}^{\mathtt{df}} X^2_i
|
|||
|
|
|
|||
|
|
is chi-square distributed, denoted
|
|||
|
|
|
|||
|
|
.. math:: Q \sim \chi^2_k.
|
|||
|
|
|
|||
|
|
The probability density function of the chi-squared distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)}
|
|||
|
|
x^{k/2 - 1} e^{-x/2},
|
|||
|
|
|
|||
|
|
where :math:`\Gamma` is the gamma function,
|
|||
|
|
|
|||
|
|
.. math:: \Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] NIST "Engineering Statistics Handbook"
|
|||
|
|
https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> np.random.chisquare(2,4)
|
|||
|
|
array([ 1.89920014, 9.00867716, 3.13710533, 5.62318272]) # random
|
|||
|
|
|
|||
|
|
noncentral_f(dfnum, dfden, nonc, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from the noncentral F distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from an F distribution with specified parameters,
|
|||
|
|
`dfnum` (degrees of freedom in numerator) and `dfden` (degrees of
|
|||
|
|
freedom in denominator), where both parameters > 1.
|
|||
|
|
`nonc` is the non-centrality parameter.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the
|
|||
|
|
`~numpy.random.Generator.noncentral_f`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
dfnum : float or array_like of floats
|
|||
|
|
Numerator degrees of freedom, must be > 0.
|
|||
|
|
|
|||
|
|
.. versionchanged:: 1.14.0
|
|||
|
|
Earlier NumPy versions required dfnum > 1.
|
|||
|
|
dfden : float or array_like of floats
|
|||
|
|
Denominator degrees of freedom, must be > 0.
|
|||
|
|
nonc : float or array_like of floats
|
|||
|
|
Non-centrality parameter, the sum of the squares of the numerator
|
|||
|
|
means, must be >= 0.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``dfnum``, ``dfden``, and ``nonc``
|
|||
|
|
are all scalars. Otherwise, ``np.broadcast(dfnum, dfden, nonc).size``
|
|||
|
|
samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized noncentral Fisher distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.noncentral_f: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
When calculating the power of an experiment (power = probability of
|
|||
|
|
rejecting the null hypothesis when a specific alternative is true) the
|
|||
|
|
non-central F statistic becomes important. When the null hypothesis is
|
|||
|
|
true, the F statistic follows a central F distribution. When the null
|
|||
|
|
hypothesis is not true, then it follows a non-central F statistic.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Weisstein, Eric W. "Noncentral F-Distribution."
|
|||
|
|
From MathWorld--A Wolfram Web Resource.
|
|||
|
|
http://mathworld.wolfram.com/NoncentralF-Distribution.html
|
|||
|
|
.. [2] Wikipedia, "Noncentral F-distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Noncentral_F-distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
In a study, testing for a specific alternative to the null hypothesis
|
|||
|
|
requires use of the Noncentral F distribution. We need to calculate the
|
|||
|
|
area in the tail of the distribution that exceeds the value of the F
|
|||
|
|
distribution for the null hypothesis. We'll plot the two probability
|
|||
|
|
distributions for comparison.
|
|||
|
|
|
|||
|
|
>>> dfnum = 3 # between group deg of freedom
|
|||
|
|
>>> dfden = 20 # within groups degrees of freedom
|
|||
|
|
>>> nonc = 3.0
|
|||
|
|
>>> nc_vals = np.random.noncentral_f(dfnum, dfden, nonc, 1000000)
|
|||
|
|
>>> NF = np.histogram(nc_vals, bins=50, density=True)
|
|||
|
|
>>> c_vals = np.random.f(dfnum, dfden, 1000000)
|
|||
|
|
>>> F = np.histogram(c_vals, bins=50, density=True)
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> plt.plot(F[1][1:], F[0])
|
|||
|
|
>>> plt.plot(NF[1][1:], NF[0])
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
f(dfnum, dfden, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from an F distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from an F distribution with specified parameters,
|
|||
|
|
`dfnum` (degrees of freedom in numerator) and `dfden` (degrees of
|
|||
|
|
freedom in denominator), where both parameters must be greater than
|
|||
|
|
zero.
|
|||
|
|
|
|||
|
|
The random variate of the F distribution (also known as the
|
|||
|
|
Fisher distribution) is a continuous probability distribution
|
|||
|
|
that arises in ANOVA tests, and is the ratio of two chi-square
|
|||
|
|
variates.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.f`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
dfnum : float or array_like of floats
|
|||
|
|
Degrees of freedom in numerator, must be > 0.
|
|||
|
|
dfden : float or array_like of float
|
|||
|
|
Degrees of freedom in denominator, must be > 0.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``dfnum`` and ``dfden`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(dfnum, dfden).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized Fisher distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.f : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.f: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The F statistic is used to compare in-group variances to between-group
|
|||
|
|
variances. Calculating the distribution depends on the sampling, and
|
|||
|
|
so it is a function of the respective degrees of freedom in the
|
|||
|
|
problem. The variable `dfnum` is the number of samples minus one, the
|
|||
|
|
between-groups degrees of freedom, while `dfden` is the within-groups
|
|||
|
|
degrees of freedom, the sum of the number of samples in each group
|
|||
|
|
minus the number of groups.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill,
|
|||
|
|
Fifth Edition, 2002.
|
|||
|
|
.. [2] Wikipedia, "F-distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/F-distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
An example from Glantz[1], pp 47-40:
|
|||
|
|
|
|||
|
|
Two groups, children of diabetics (25 people) and children from people
|
|||
|
|
without diabetes (25 controls). Fasting blood glucose was measured,
|
|||
|
|
case group had a mean value of 86.1, controls had a mean value of
|
|||
|
|
82.2. Standard deviations were 2.09 and 2.49 respectively. Are these
|
|||
|
|
data consistent with the null hypothesis that the parents diabetic
|
|||
|
|
status does not affect their children's blood glucose levels?
|
|||
|
|
Calculating the F statistic from the data gives a value of 36.01.
|
|||
|
|
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> dfnum = 1. # between group degrees of freedom
|
|||
|
|
>>> dfden = 48. # within groups degrees of freedom
|
|||
|
|
>>> s = np.random.f(dfnum, dfden, 1000)
|
|||
|
|
|
|||
|
|
The lower bound for the top 1% of the samples is :
|
|||
|
|
|
|||
|
|
>>> np.sort(s)[-10]
|
|||
|
|
7.61988120985 # random
|
|||
|
|
|
|||
|
|
So there is about a 1% chance that the F statistic will exceed 7.62,
|
|||
|
|
the measured value is 36, so the null hypothesis is rejected at the 1%
|
|||
|
|
level.
|
|||
|
|
|
|||
|
|
|
|||
|
|
gamma(shape, scale=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a Gamma distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from a Gamma distribution with specified parameters,
|
|||
|
|
`shape` (sometimes designated "k") and `scale` (sometimes designated
|
|||
|
|
"theta"), where both parameters are > 0.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.gamma`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
shape : float or array_like of floats
|
|||
|
|
The shape of the gamma distribution. Must be non-negative.
|
|||
|
|
scale : float or array_like of floats, optional
|
|||
|
|
The scale of the gamma distribution. Must be non-negative.
|
|||
|
|
Default is equal to 1.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``shape`` and ``scale`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(shape, scale).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized gamma distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.gamma : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.gamma: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the Gamma distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},
|
|||
|
|
|
|||
|
|
where :math:`k` is the shape and :math:`\theta` the scale,
|
|||
|
|
and :math:`\Gamma` is the Gamma function.
|
|||
|
|
|
|||
|
|
The Gamma distribution is often used to model the times to failure of
|
|||
|
|
electronic components, and arises naturally in processes for which the
|
|||
|
|
waiting times between Poisson distributed events are relevant.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A
|
|||
|
|
Wolfram Web Resource.
|
|||
|
|
http://mathworld.wolfram.com/GammaDistribution.html
|
|||
|
|
.. [2] Wikipedia, "Gamma distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Gamma_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> shape, scale = 2., 2. # mean=4, std=2*sqrt(2)
|
|||
|
|
>>> s = np.random.gamma(shape, scale, 1000)
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> import scipy.special as sps # doctest: +SKIP
|
|||
|
|
>>> count, bins, ignored = plt.hist(s, 50, density=True)
|
|||
|
|
>>> y = bins**(shape-1)*(np.exp(-bins/scale) / # doctest: +SKIP
|
|||
|
|
... (sps.gamma(shape)*scale**shape))
|
|||
|
|
>>> plt.plot(bins, y, linewidth=2, color='r') # doctest: +SKIP
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
standard_gamma(shape, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a standard Gamma distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from a Gamma distribution with specified parameters,
|
|||
|
|
shape (sometimes designated "k") and scale=1.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the
|
|||
|
|
`~numpy.random.Generator.standard_gamma`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
shape : float or array_like of floats
|
|||
|
|
Parameter, must be non-negative.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``shape`` is a scalar. Otherwise,
|
|||
|
|
``np.array(shape).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized standard gamma distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.gamma : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.standard_gamma: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the Gamma distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},
|
|||
|
|
|
|||
|
|
where :math:`k` is the shape and :math:`\theta` the scale,
|
|||
|
|
and :math:`\Gamma` is the Gamma function.
|
|||
|
|
|
|||
|
|
The Gamma distribution is often used to model the times to failure of
|
|||
|
|
electronic components, and arises naturally in processes for which the
|
|||
|
|
waiting times between Poisson distributed events are relevant.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A
|
|||
|
|
Wolfram Web Resource.
|
|||
|
|
http://mathworld.wolfram.com/GammaDistribution.html
|
|||
|
|
.. [2] Wikipedia, "Gamma distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Gamma_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> shape, scale = 2., 1. # mean and width
|
|||
|
|
>>> s = np.random.standard_gamma(shape, 1000000)
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> import scipy.special as sps # doctest: +SKIP
|
|||
|
|
>>> count, bins, ignored = plt.hist(s, 50, density=True)
|
|||
|
|
>>> y = bins**(shape-1) * ((np.exp(-bins/scale))/ # doctest: +SKIP
|
|||
|
|
... (sps.gamma(shape) * scale**shape))
|
|||
|
|
>>> plt.plot(bins, y, linewidth=2, color='r') # doctest: +SKIP
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
normal(loc=0.0, scale=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw random samples from a normal (Gaussian) distribution.
|
|||
|
|
|
|||
|
|
The probability density function of the normal distribution, first
|
|||
|
|
derived by De Moivre and 200 years later by both Gauss and Laplace
|
|||
|
|
independently [2]_, is often called the bell curve because of
|
|||
|
|
its characteristic shape (see the example below).
|
|||
|
|
|
|||
|
|
The normal distributions occurs often in nature. For example, it
|
|||
|
|
describes the commonly occurring distribution of samples influenced
|
|||
|
|
by a large number of tiny, random disturbances, each with its own
|
|||
|
|
unique distribution [2]_.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.normal`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
loc : float or array_like of floats
|
|||
|
|
Mean ("centre") of the distribution.
|
|||
|
|
scale : float or array_like of floats
|
|||
|
|
Standard deviation (spread or "width") of the distribution. Must be
|
|||
|
|
non-negative.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``loc`` and ``scale`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized normal distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.norm : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.normal: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the Gaussian distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }}
|
|||
|
|
e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },
|
|||
|
|
|
|||
|
|
where :math:`\mu` is the mean and :math:`\sigma` the standard
|
|||
|
|
deviation. The square of the standard deviation, :math:`\sigma^2`,
|
|||
|
|
is called the variance.
|
|||
|
|
|
|||
|
|
The function has its peak at the mean, and its "spread" increases with
|
|||
|
|
the standard deviation (the function reaches 0.607 times its maximum at
|
|||
|
|
:math:`x + \sigma` and :math:`x - \sigma` [2]_). This implies that
|
|||
|
|
normal is more likely to return samples lying close to the mean, rather
|
|||
|
|
than those far away.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Wikipedia, "Normal distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Normal_distribution
|
|||
|
|
.. [2] P. R. Peebles Jr., "Central Limit Theorem" in "Probability,
|
|||
|
|
Random Variables and Random Signal Principles", 4th ed., 2001,
|
|||
|
|
pp. 51, 51, 125.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> mu, sigma = 0, 0.1 # mean and standard deviation
|
|||
|
|
>>> s = np.random.normal(mu, sigma, 1000)
|
|||
|
|
|
|||
|
|
Verify the mean and the variance:
|
|||
|
|
|
|||
|
|
>>> abs(mu - np.mean(s))
|
|||
|
|
0.0 # may vary
|
|||
|
|
|
|||
|
|
>>> abs(sigma - np.std(s, ddof=1))
|
|||
|
|
0.1 # may vary
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(s, 30, density=True)
|
|||
|
|
>>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
|
|||
|
|
... np.exp( - (bins - mu)**2 / (2 * sigma**2) ),
|
|||
|
|
... linewidth=2, color='r')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
Two-by-four array of samples from the normal distribution with
|
|||
|
|
mean 3 and standard deviation 2.5:
|
|||
|
|
|
|||
|
|
>>> np.random.normal(3, 2.5, size=(2, 4))
|
|||
|
|
array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random
|
|||
|
|
[ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random
|
|||
|
|
|
|||
|
|
|
|||
|
|
standard_normal(size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a standard Normal distribution (mean=0, stdev=1).
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the
|
|||
|
|
`~numpy.random.Generator.standard_normal`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. Default is None, in which case a
|
|||
|
|
single value is returned.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : float or ndarray
|
|||
|
|
A floating-point array of shape ``size`` of drawn samples, or a
|
|||
|
|
single sample if ``size`` was not specified.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
normal :
|
|||
|
|
Equivalent function with additional ``loc`` and ``scale`` arguments
|
|||
|
|
for setting the mean and standard deviation.
|
|||
|
|
random.Generator.standard_normal: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
For random samples from the normal distribution with mean ``mu`` and
|
|||
|
|
standard deviation ``sigma``, use one of::
|
|||
|
|
|
|||
|
|
mu + sigma * np.random.standard_normal(size=...)
|
|||
|
|
np.random.normal(mu, sigma, size=...)
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> np.random.standard_normal()
|
|||
|
|
2.1923875335537315 #random
|
|||
|
|
|
|||
|
|
>>> s = np.random.standard_normal(8000)
|
|||
|
|
>>> s
|
|||
|
|
array([ 0.6888893 , 0.78096262, -0.89086505, ..., 0.49876311, # random
|
|||
|
|
-0.38672696, -0.4685006 ]) # random
|
|||
|
|
>>> s.shape
|
|||
|
|
(8000,)
|
|||
|
|
>>> s = np.random.standard_normal(size=(3, 4, 2))
|
|||
|
|
>>> s.shape
|
|||
|
|
(3, 4, 2)
|
|||
|
|
|
|||
|
|
Two-by-four array of samples from the normal distribution with
|
|||
|
|
mean 3 and standard deviation 2.5:
|
|||
|
|
|
|||
|
|
>>> 3 + 2.5 * np.random.standard_normal(size=(2, 4))
|
|||
|
|
array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random
|
|||
|
|
[ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random
|
|||
|
|
|
|||
|
|
|
|||
|
|
random_integers(low, high=None, size=None)
|
|||
|
|
|
|||
|
|
Random integers of type `np.int_` between `low` and `high`, inclusive.
|
|||
|
|
|
|||
|
|
Return random integers of type `np.int_` from the "discrete uniform"
|
|||
|
|
distribution in the closed interval [`low`, `high`]. If `high` is
|
|||
|
|
None (the default), then results are from [1, `low`]. The `np.int_`
|
|||
|
|
type translates to the C long integer type and its precision
|
|||
|
|
is platform dependent.
|
|||
|
|
|
|||
|
|
This function has been deprecated. Use randint instead.
|
|||
|
|
|
|||
|
|
.. deprecated:: 1.11.0
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
low : int
|
|||
|
|
Lowest (signed) integer to be drawn from the distribution (unless
|
|||
|
|
``high=None``, in which case this parameter is the *highest* such
|
|||
|
|
integer).
|
|||
|
|
high : int, optional
|
|||
|
|
If provided, the largest (signed) integer to be drawn from the
|
|||
|
|
distribution (see above for behavior if ``high=None``).
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. Default is None, in which case a
|
|||
|
|
single value is returned.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : int or ndarray of ints
|
|||
|
|
`size`-shaped array of random integers from the appropriate
|
|||
|
|
distribution, or a single such random int if `size` not provided.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
randint : Similar to `random_integers`, only for the half-open
|
|||
|
|
interval [`low`, `high`), and 0 is the lowest value if `high` is
|
|||
|
|
omitted.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
To sample from N evenly spaced floating-point numbers between a and b,
|
|||
|
|
use::
|
|||
|
|
|
|||
|
|
a + (b - a) * (np.random.random_integers(N) - 1) / (N - 1.)
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> np.random.random_integers(5)
|
|||
|
|
4 # random
|
|||
|
|
>>> type(np.random.random_integers(5))
|
|||
|
|
<class 'numpy.int64'>
|
|||
|
|
>>> np.random.random_integers(5, size=(3,2))
|
|||
|
|
array([[5, 4], # random
|
|||
|
|
[3, 3],
|
|||
|
|
[4, 5]])
|
|||
|
|
|
|||
|
|
Choose five random numbers from the set of five evenly-spaced
|
|||
|
|
numbers between 0 and 2.5, inclusive (*i.e.*, from the set
|
|||
|
|
:math:`{0, 5/8, 10/8, 15/8, 20/8}`):
|
|||
|
|
|
|||
|
|
>>> 2.5 * (np.random.random_integers(5, size=(5,)) - 1) / 4.
|
|||
|
|
array([ 0.625, 1.25 , 0.625, 0.625, 2.5 ]) # random
|
|||
|
|
|
|||
|
|
Roll two six sided dice 1000 times and sum the results:
|
|||
|
|
|
|||
|
|
>>> d1 = np.random.random_integers(1, 6, 1000)
|
|||
|
|
>>> d2 = np.random.random_integers(1, 6, 1000)
|
|||
|
|
>>> dsums = d1 + d2
|
|||
|
|
|
|||
|
|
Display results as a histogram:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(dsums, 11, density=True)
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
randn(d0, d1, ..., dn)
|
|||
|
|
|
|||
|
|
Return a sample (or samples) from the "standard normal" distribution.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
This is a convenience function for users porting code from Matlab,
|
|||
|
|
and wraps `standard_normal`. That function takes a
|
|||
|
|
tuple to specify the size of the output, which is consistent with
|
|||
|
|
other NumPy functions like `numpy.zeros` and `numpy.ones`.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the
|
|||
|
|
`~numpy.random.Generator.standard_normal`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
If positive int_like arguments are provided, `randn` generates an array
|
|||
|
|
of shape ``(d0, d1, ..., dn)``, filled
|
|||
|
|
with random floats sampled from a univariate "normal" (Gaussian)
|
|||
|
|
distribution of mean 0 and variance 1. A single float randomly sampled
|
|||
|
|
from the distribution is returned if no argument is provided.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
d0, d1, ..., dn : int, optional
|
|||
|
|
The dimensions of the returned array, must be non-negative.
|
|||
|
|
If no argument is given a single Python float is returned.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
Z : ndarray or float
|
|||
|
|
A ``(d0, d1, ..., dn)``-shaped array of floating-point samples from
|
|||
|
|
the standard normal distribution, or a single such float if
|
|||
|
|
no parameters were supplied.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
standard_normal : Similar, but takes a tuple as its argument.
|
|||
|
|
normal : Also accepts mu and sigma arguments.
|
|||
|
|
random.Generator.standard_normal: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
For random samples from the normal distribution with mean ``mu`` and
|
|||
|
|
standard deviation ``sigma``, use::
|
|||
|
|
|
|||
|
|
sigma * np.random.randn(...) + mu
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> np.random.randn()
|
|||
|
|
2.1923875335537315 # random
|
|||
|
|
|
|||
|
|
Two-by-four array of samples from the normal distribution with
|
|||
|
|
mean 3 and standard deviation 2.5:
|
|||
|
|
|
|||
|
|
>>> 3 + 2.5 * np.random.randn(2, 4)
|
|||
|
|
array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random
|
|||
|
|
[ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random
|
|||
|
|
|
|||
|
|
|
|||
|
|
rand(d0, d1, ..., dn)
|
|||
|
|
|
|||
|
|
Random values in a given shape.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
This is a convenience function for users porting code from Matlab,
|
|||
|
|
and wraps `random_sample`. That function takes a
|
|||
|
|
tuple to specify the size of the output, which is consistent with
|
|||
|
|
other NumPy functions like `numpy.zeros` and `numpy.ones`.
|
|||
|
|
|
|||
|
|
Create an array of the given shape and populate it with
|
|||
|
|
random samples from a uniform distribution
|
|||
|
|
over ``[0, 1)``.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
d0, d1, ..., dn : int, optional
|
|||
|
|
The dimensions of the returned array, must be non-negative.
|
|||
|
|
If no argument is given a single Python float is returned.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray, shape ``(d0, d1, ..., dn)``
|
|||
|
|
Random values.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> np.random.rand(3,2)
|
|||
|
|
array([[ 0.14022471, 0.96360618], #random
|
|||
|
|
[ 0.37601032, 0.25528411], #random
|
|||
|
|
[ 0.49313049, 0.94909878]]) #random
|
|||
|
|
|
|||
|
|
|
|||
|
|
uniform(low=0.0, high=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a uniform distribution.
|
|||
|
|
|
|||
|
|
Samples are uniformly distributed over the half-open interval
|
|||
|
|
``[low, high)`` (includes low, but excludes high). In other words,
|
|||
|
|
any value within the given interval is equally likely to be drawn
|
|||
|
|
by `uniform`.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.uniform`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
low : float or array_like of floats, optional
|
|||
|
|
Lower boundary of the output interval. All values generated will be
|
|||
|
|
greater than or equal to low. The default value is 0.
|
|||
|
|
high : float or array_like of floats
|
|||
|
|
Upper boundary of the output interval. All values generated will be
|
|||
|
|
less than or equal to high. The high limit may be included in the
|
|||
|
|
returned array of floats due to floating-point rounding in the
|
|||
|
|
equation ``low + (high-low) * random_sample()``. The default value
|
|||
|
|
is 1.0.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``low`` and ``high`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(low, high).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized uniform distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
randint : Discrete uniform distribution, yielding integers.
|
|||
|
|
random_integers : Discrete uniform distribution over the closed
|
|||
|
|
interval ``[low, high]``.
|
|||
|
|
random_sample : Floats uniformly distributed over ``[0, 1)``.
|
|||
|
|
random : Alias for `random_sample`.
|
|||
|
|
rand : Convenience function that accepts dimensions as input, e.g.,
|
|||
|
|
``rand(2,2)`` would generate a 2-by-2 array of floats,
|
|||
|
|
uniformly distributed over ``[0, 1)``.
|
|||
|
|
random.Generator.uniform: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density function of the uniform distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(x) = \frac{1}{b - a}
|
|||
|
|
|
|||
|
|
anywhere within the interval ``[a, b)``, and zero elsewhere.
|
|||
|
|
|
|||
|
|
When ``high`` == ``low``, values of ``low`` will be returned.
|
|||
|
|
If ``high`` < ``low``, the results are officially undefined
|
|||
|
|
and may eventually raise an error, i.e. do not rely on this
|
|||
|
|
function to behave when passed arguments satisfying that
|
|||
|
|
inequality condition. The ``high`` limit may be included in the
|
|||
|
|
returned array of floats due to floating-point rounding in the
|
|||
|
|
equation ``low + (high-low) * random_sample()``. For example:
|
|||
|
|
|
|||
|
|
>>> x = np.float32(5*0.99999999)
|
|||
|
|
>>> x
|
|||
|
|
5.0
|
|||
|
|
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> s = np.random.uniform(-1,0,1000)
|
|||
|
|
|
|||
|
|
All values are within the given interval:
|
|||
|
|
|
|||
|
|
>>> np.all(s >= -1)
|
|||
|
|
True
|
|||
|
|
>>> np.all(s < 0)
|
|||
|
|
True
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with the
|
|||
|
|
probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(s, 15, density=True)
|
|||
|
|
>>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
choice(a, size=None, replace=True, p=None)
|
|||
|
|
|
|||
|
|
Generates a random sample from a given 1-D array
|
|||
|
|
|
|||
|
|
.. versionadded:: 1.7.0
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.choice`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
a : 1-D array-like or int
|
|||
|
|
If an ndarray, a random sample is generated from its elements.
|
|||
|
|
If an int, the random sample is generated as if it were ``np.arange(a)``
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. Default is None, in which case a
|
|||
|
|
single value is returned.
|
|||
|
|
replace : boolean, optional
|
|||
|
|
Whether the sample is with or without replacement. Default is True,
|
|||
|
|
meaning that a value of ``a`` can be selected multiple times.
|
|||
|
|
p : 1-D array-like, optional
|
|||
|
|
The probabilities associated with each entry in a.
|
|||
|
|
If not given, the sample assumes a uniform distribution over all
|
|||
|
|
entries in ``a``.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
samples : single item or ndarray
|
|||
|
|
The generated random samples
|
|||
|
|
|
|||
|
|
Raises
|
|||
|
|
------
|
|||
|
|
ValueError
|
|||
|
|
If a is an int and less than zero, if a or p are not 1-dimensional,
|
|||
|
|
if a is an array-like of size 0, if p is not a vector of
|
|||
|
|
probabilities, if a and p have different lengths, or if
|
|||
|
|
replace=False and the sample size is greater than the population
|
|||
|
|
size
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
randint, shuffle, permutation
|
|||
|
|
random.Generator.choice: which should be used in new code
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
Setting user-specified probabilities through ``p`` uses a more general but less
|
|||
|
|
efficient sampler than the default. The general sampler produces a different sample
|
|||
|
|
than the optimized sampler even if each element of ``p`` is 1 / len(a).
|
|||
|
|
|
|||
|
|
Sampling random rows from a 2-D array is not possible with this function,
|
|||
|
|
but is possible with `Generator.choice` through its ``axis`` keyword.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Generate a uniform random sample from np.arange(5) of size 3:
|
|||
|
|
|
|||
|
|
>>> np.random.choice(5, 3)
|
|||
|
|
array([0, 3, 4]) # random
|
|||
|
|
>>> #This is equivalent to np.random.randint(0,5,3)
|
|||
|
|
|
|||
|
|
Generate a non-uniform random sample from np.arange(5) of size 3:
|
|||
|
|
|
|||
|
|
>>> np.random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0])
|
|||
|
|
array([3, 3, 0]) # random
|
|||
|
|
|
|||
|
|
Generate a uniform random sample from np.arange(5) of size 3 without
|
|||
|
|
replacement:
|
|||
|
|
|
|||
|
|
>>> np.random.choice(5, 3, replace=False)
|
|||
|
|
array([3,1,0]) # random
|
|||
|
|
>>> #This is equivalent to np.random.permutation(np.arange(5))[:3]
|
|||
|
|
|
|||
|
|
Generate a non-uniform random sample from np.arange(5) of size
|
|||
|
|
3 without replacement:
|
|||
|
|
|
|||
|
|
>>> np.random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0])
|
|||
|
|
array([2, 3, 0]) # random
|
|||
|
|
|
|||
|
|
Any of the above can be repeated with an arbitrary array-like
|
|||
|
|
instead of just integers. For instance:
|
|||
|
|
|
|||
|
|
>>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher']
|
|||
|
|
>>> np.random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3])
|
|||
|
|
array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], # random
|
|||
|
|
dtype='<U11')
|
|||
|
|
|
|||
|
|
|
|||
|
|
bytes(length)
|
|||
|
|
|
|||
|
|
Return random bytes.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.bytes`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
length : int
|
|||
|
|
Number of random bytes.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : bytes
|
|||
|
|
String of length `length`.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.bytes: which should be used for new code.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> np.random.bytes(10)
|
|||
|
|
b' eh\x85\x022SZ\xbf\xa4' #random
|
|||
|
|
|
|||
|
|
randint(low, high=None, size=None, dtype=int)
|
|||
|
|
|
|||
|
|
Return random integers from `low` (inclusive) to `high` (exclusive).
|
|||
|
|
|
|||
|
|
Return random integers from the "discrete uniform" distribution of
|
|||
|
|
the specified dtype in the "half-open" interval [`low`, `high`). If
|
|||
|
|
`high` is None (the default), then results are from [0, `low`).
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.integers`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
low : int or array-like of ints
|
|||
|
|
Lowest (signed) integers to be drawn from the distribution (unless
|
|||
|
|
``high=None``, in which case this parameter is one above the
|
|||
|
|
*highest* such integer).
|
|||
|
|
high : int or array-like of ints, optional
|
|||
|
|
If provided, one above the largest (signed) integer to be drawn
|
|||
|
|
from the distribution (see above for behavior if ``high=None``).
|
|||
|
|
If array-like, must contain integer values
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. Default is None, in which case a
|
|||
|
|
single value is returned.
|
|||
|
|
dtype : dtype, optional
|
|||
|
|
Desired dtype of the result. Byteorder must be native.
|
|||
|
|
The default value is int.
|
|||
|
|
|
|||
|
|
.. versionadded:: 1.11.0
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : int or ndarray of ints
|
|||
|
|
`size`-shaped array of random integers from the appropriate
|
|||
|
|
distribution, or a single such random int if `size` not provided.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random_integers : similar to `randint`, only for the closed
|
|||
|
|
interval [`low`, `high`], and 1 is the lowest value if `high` is
|
|||
|
|
omitted.
|
|||
|
|
random.Generator.integers: which should be used for new code.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> np.random.randint(2, size=10)
|
|||
|
|
array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) # random
|
|||
|
|
>>> np.random.randint(1, size=10)
|
|||
|
|
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
|
|||
|
|
|
|||
|
|
Generate a 2 x 4 array of ints between 0 and 4, inclusive:
|
|||
|
|
|
|||
|
|
>>> np.random.randint(5, size=(2, 4))
|
|||
|
|
array([[4, 0, 2, 1], # random
|
|||
|
|
[3, 2, 2, 0]])
|
|||
|
|
|
|||
|
|
Generate a 1 x 3 array with 3 different upper bounds
|
|||
|
|
|
|||
|
|
>>> np.random.randint(1, [3, 5, 10])
|
|||
|
|
array([2, 2, 9]) # random
|
|||
|
|
|
|||
|
|
Generate a 1 by 3 array with 3 different lower bounds
|
|||
|
|
|
|||
|
|
>>> np.random.randint([1, 5, 7], 10)
|
|||
|
|
array([9, 8, 7]) # random
|
|||
|
|
|
|||
|
|
Generate a 2 by 4 array using broadcasting with dtype of uint8
|
|||
|
|
|
|||
|
|
>>> np.random.randint([1, 3, 5, 7], [[10], [20]], dtype=np.uint8)
|
|||
|
|
array([[ 8, 6, 9, 7], # random
|
|||
|
|
[ 1, 16, 9, 12]], dtype=uint8)
|
|||
|
|
|
|||
|
|
tomaxint(size=None)
|
|||
|
|
|
|||
|
|
Return a sample of uniformly distributed random integers in the interval
|
|||
|
|
[0, ``np.iinfo(np.int_).max``]. The `np.int_` type translates to the C long
|
|||
|
|
integer type and its precision is platform dependent.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. Default is None, in which case a
|
|||
|
|
single value is returned.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray
|
|||
|
|
Drawn samples, with shape `size`.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
randint : Uniform sampling over a given half-open interval of integers.
|
|||
|
|
random_integers : Uniform sampling over a given closed interval of
|
|||
|
|
integers.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> rs = np.random.RandomState() # need a RandomState object
|
|||
|
|
>>> rs.tomaxint((2,2,2))
|
|||
|
|
array([[[1170048599, 1600360186], # random
|
|||
|
|
[ 739731006, 1947757578]],
|
|||
|
|
[[1871712945, 752307660],
|
|||
|
|
[1601631370, 1479324245]]])
|
|||
|
|
>>> rs.tomaxint((2,2,2)) < np.iinfo(np.int_).max
|
|||
|
|
array([[[ True, True],
|
|||
|
|
[ True, True]],
|
|||
|
|
[[ True, True],
|
|||
|
|
[ True, True]]])
|
|||
|
|
|
|||
|
|
|
|||
|
|
standard_exponential(size=None)
|
|||
|
|
|
|||
|
|
Draw samples from the standard exponential distribution.
|
|||
|
|
|
|||
|
|
`standard_exponential` is identical to the exponential distribution
|
|||
|
|
with a scale parameter of 1.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the
|
|||
|
|
`~numpy.random.Generator.standard_exponential`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. Default is None, in which case a
|
|||
|
|
single value is returned.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : float or ndarray
|
|||
|
|
Drawn samples.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.standard_exponential: which should be used for new code.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Output a 3x8000 array:
|
|||
|
|
|
|||
|
|
>>> n = np.random.standard_exponential((3, 8000))
|
|||
|
|
|
|||
|
|
|
|||
|
|
exponential(scale=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from an exponential distribution.
|
|||
|
|
|
|||
|
|
Its probability density function is
|
|||
|
|
|
|||
|
|
.. math:: f(x; \frac{1}{\beta}) = \frac{1}{\beta} \exp(-\frac{x}{\beta}),
|
|||
|
|
|
|||
|
|
for ``x > 0`` and 0 elsewhere. :math:`\beta` is the scale parameter,
|
|||
|
|
which is the inverse of the rate parameter :math:`\lambda = 1/\beta`.
|
|||
|
|
The rate parameter is an alternative, widely used parameterization
|
|||
|
|
of the exponential distribution [3]_.
|
|||
|
|
|
|||
|
|
The exponential distribution is a continuous analogue of the
|
|||
|
|
geometric distribution. It describes many common situations, such as
|
|||
|
|
the size of raindrops measured over many rainstorms [1]_, or the time
|
|||
|
|
between page requests to Wikipedia [2]_.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.exponential`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
scale : float or array_like of floats
|
|||
|
|
The scale parameter, :math:`\beta = 1/\lambda`. Must be
|
|||
|
|
non-negative.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``scale`` is a scalar. Otherwise,
|
|||
|
|
``np.array(scale).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized exponential distribution.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
A real world example: Assume a company has 10000 customer support
|
|||
|
|
agents and the average time between customer calls is 4 minutes.
|
|||
|
|
|
|||
|
|
>>> n = 10000
|
|||
|
|
>>> time_between_calls = np.random.default_rng().exponential(scale=4, size=n)
|
|||
|
|
|
|||
|
|
What is the probability that a customer will call in the next
|
|||
|
|
4 to 5 minutes?
|
|||
|
|
|
|||
|
|
>>> x = ((time_between_calls < 5).sum())/n
|
|||
|
|
>>> y = ((time_between_calls < 4).sum())/n
|
|||
|
|
>>> x-y
|
|||
|
|
0.08 # may vary
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.exponential: which should be used for new code.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Peyton Z. Peebles Jr., "Probability, Random Variables and
|
|||
|
|
Random Signal Principles", 4th ed, 2001, p. 57.
|
|||
|
|
.. [2] Wikipedia, "Poisson process",
|
|||
|
|
https://en.wikipedia.org/wiki/Poisson_process
|
|||
|
|
.. [3] Wikipedia, "Exponential distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Exponential_distribution
|
|||
|
|
|
|||
|
|
|
|||
|
|
beta(a, b, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a Beta distribution.
|
|||
|
|
|
|||
|
|
The Beta distribution is a special case of the Dirichlet distribution,
|
|||
|
|
and is related to the Gamma distribution. It has the probability
|
|||
|
|
distribution function
|
|||
|
|
|
|||
|
|
.. math:: f(x; a,b) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1}
|
|||
|
|
(1 - x)^{\beta - 1},
|
|||
|
|
|
|||
|
|
where the normalization, B, is the beta function,
|
|||
|
|
|
|||
|
|
.. math:: B(\alpha, \beta) = \int_0^1 t^{\alpha - 1}
|
|||
|
|
(1 - t)^{\beta - 1} dt.
|
|||
|
|
|
|||
|
|
It is often seen in Bayesian inference and order statistics.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.beta`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
a : float or array_like of floats
|
|||
|
|
Alpha, positive (>0).
|
|||
|
|
b : float or array_like of floats
|
|||
|
|
Beta, positive (>0).
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``a`` and ``b`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(a, b).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized beta distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.beta: which should be used for new code.
|
|||
|
|
|
|||
|
|
random(size=None)
|
|||
|
|
|
|||
|
|
Return random floats in the half-open interval [0.0, 1.0). Alias for
|
|||
|
|
`random_sample` to ease forward-porting to the new random API.
|
|||
|
|
|
|||
|
|
random_sample(size=None)
|
|||
|
|
|
|||
|
|
Return random floats in the half-open interval [0.0, 1.0).
|
|||
|
|
|
|||
|
|
Results are from the "continuous uniform" distribution over the
|
|||
|
|
stated interval. To sample :math:`Unif[a, b), b > a` multiply
|
|||
|
|
the output of `random_sample` by `(b-a)` and add `a`::
|
|||
|
|
|
|||
|
|
(b - a) * random_sample() + a
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.random`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. Default is None, in which case a
|
|||
|
|
single value is returned.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : float or ndarray of floats
|
|||
|
|
Array of random floats of shape `size` (unless ``size=None``, in which
|
|||
|
|
case a single float is returned).
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.random: which should be used for new code.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> np.random.random_sample()
|
|||
|
|
0.47108547995356098 # random
|
|||
|
|
>>> type(np.random.random_sample())
|
|||
|
|
<class 'float'>
|
|||
|
|
>>> np.random.random_sample((5,))
|
|||
|
|
array([ 0.30220482, 0.86820401, 0.1654503 , 0.11659149, 0.54323428]) # random
|
|||
|
|
|
|||
|
|
Three-by-two array of random numbers from [-5, 0):
|
|||
|
|
|
|||
|
|
>>> 5 * np.random.random_sample((3, 2)) - 5
|
|||
|
|
array([[-3.99149989, -0.52338984], # random
|
|||
|
|
[-2.99091858, -0.79479508],
|
|||
|
|
[-1.23204345, -1.75224494]])
|
|||
|
|
|
|||
|
|
|
|||
|
|
set_state(state)
|
|||
|
|
|
|||
|
|
Set the internal state of the generator from a tuple.
|
|||
|
|
|
|||
|
|
For use if one has reason to manually (re-)set the internal state of
|
|||
|
|
the bit generator used by the RandomState instance. By default,
|
|||
|
|
RandomState uses the "Mersenne Twister"[1]_ pseudo-random number
|
|||
|
|
generating algorithm.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
state : {tuple(str, ndarray of 624 uints, int, int, float), dict}
|
|||
|
|
The `state` tuple has the following items:
|
|||
|
|
|
|||
|
|
1. the string 'MT19937', specifying the Mersenne Twister algorithm.
|
|||
|
|
2. a 1-D array of 624 unsigned integers ``keys``.
|
|||
|
|
3. an integer ``pos``.
|
|||
|
|
4. an integer ``has_gauss``.
|
|||
|
|
5. a float ``cached_gaussian``.
|
|||
|
|
|
|||
|
|
If state is a dictionary, it is directly set using the BitGenerators
|
|||
|
|
`state` property.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : None
|
|||
|
|
Returns 'None' on success.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
get_state
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
`set_state` and `get_state` are not needed to work with any of the
|
|||
|
|
random distributions in NumPy. If the internal state is manually altered,
|
|||
|
|
the user should know exactly what he/she is doing.
|
|||
|
|
|
|||
|
|
For backwards compatibility, the form (str, array of 624 uints, int) is
|
|||
|
|
also accepted although it is missing some information about the cached
|
|||
|
|
Gaussian value: ``state = ('MT19937', keys, pos)``.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] M. Matsumoto and T. Nishimura, "Mersenne Twister: A
|
|||
|
|
623-dimensionally equidistributed uniform pseudorandom number
|
|||
|
|
generator," *ACM Trans. on Modeling and Computer Simulation*,
|
|||
|
|
Vol. 8, No. 1, pp. 3-30, Jan. 1998.
|
|||
|
|
|
|||
|
|
|
|||
|
|
get_state(legacy=True)
|
|||
|
|
|
|||
|
|
Return a tuple representing the internal state of the generator.
|
|||
|
|
|
|||
|
|
For more details, see `set_state`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
legacy : bool, optional
|
|||
|
|
Flag indicating to return a legacy tuple state when the BitGenerator
|
|||
|
|
is MT19937, instead of a dict. Raises ValueError if the underlying
|
|||
|
|
bit generator is not an instance of MT19937.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : {tuple(str, ndarray of 624 uints, int, int, float), dict}
|
|||
|
|
If legacy is True, the returned tuple has the following items:
|
|||
|
|
|
|||
|
|
1. the string 'MT19937'.
|
|||
|
|
2. a 1-D array of 624 unsigned integer keys.
|
|||
|
|
3. an integer ``pos``.
|
|||
|
|
4. an integer ``has_gauss``.
|
|||
|
|
5. a float ``cached_gaussian``.
|
|||
|
|
|
|||
|
|
If `legacy` is False, or the BitGenerator is not MT19937, then
|
|||
|
|
state is returned as a dictionary.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
set_state
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
`set_state` and `get_state` are not needed to work with any of the
|
|||
|
|
random distributions in NumPy. If the internal state is manually altered,
|
|||
|
|
the user should know exactly what he/she is doing.
|
|||
|
|
|
|||
|
|
|
|||
|
|
seed(seed=None)
|
|||
|
|
|
|||
|
|
Reseed a legacy MT19937 BitGenerator
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
This is a convenience, legacy function.
|
|||
|
|
|
|||
|
|
The best practice is to **not** reseed a BitGenerator, rather to
|
|||
|
|
recreate a new one. This method is here for legacy reasons.
|
|||
|
|
This example demonstrates best practice.
|
|||
|
|
|
|||
|
|
>>> from numpy.random import MT19937
|
|||
|
|
>>> from numpy.random import RandomState, SeedSequence
|
|||
|
|
>>> rs = RandomState(MT19937(SeedSequence(123456789)))
|
|||
|
|
# Later, you want to restart the stream
|
|||
|
|
>>> rs = RandomState(MT19937(SeedSequence(987654321)))
|
|||
|
|
This function is deprecated. Please call randint({low}, {high} + 1) instead x must be an integer or at least 1-dimensional probabilities are not non-negative numpy.core.umath failed to import mean and cov must have same length legacy can only be True when the underlyign bitgenerator is an instance of MT19937. get_state and legacy can only be used with the MT19937 BitGenerator. To silence this warning, set `legacy` to False. covariance is not symmetric positive-semidefinite. cov must be 2 dimensional and square check_valid must equal 'warn', 'raise', or 'ignore' can only re-seed a MT19937 BitGenerator a must be 1-dimensional or an integer This function is deprecated. Please call randint(1, {low} + 1) instead Shuffling a one dimensional array subclass containing objects gives incorrect results for most array subclasses. Please use the new random number API instead: https://numpy.org/doc/stable/reference/random/index.html
|
|||
|
|
The new API fixes this issue. This version will not be fixed due to stability guarantees of the API. RandomState.triangular (line 3244) RandomState.standard_t (line 2150) RandomState.standard_normal (line 1385) RandomState.standard_exponential (line 577) RandomState.standard_cauchy (line 2075) RandomState.random_sample (line 385) RandomState.random_integers (line 1289) RandomState.permutation (line 4668) RandomState.noncentral_f (line 1823) RandomState.noncentral_chisquare (line 1986) RandomState.negative_binomial (line 3505) RandomState.multinomial (line 4257) RandomState.exponential (line 500) Providing a dtype with a non-native byteorder is not supported. If you require platform-independent byteorder, call byteswap when required.
|
|||
|
|
In future version, providing byteorder will raise a ValueError Negative dimensions are not allowed Invalid bit generator. The bit generator must be instantized. Fewer non-zero entries in p than size Cannot take a larger sample than population when 'replace=False'
|
|||
|
|
zipf(a, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a Zipf distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from a Zipf distribution with specified parameter
|
|||
|
|
`a` > 1.
|
|||
|
|
|
|||
|
|
The Zipf distribution (also known as the zeta distribution) is a
|
|||
|
|
discrete probability distribution that satisfies Zipf's law: the
|
|||
|
|
frequency of an item is inversely proportional to its rank in a
|
|||
|
|
frequency table.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.zipf`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
a : float or array_like of floats
|
|||
|
|
Distribution parameter. Must be greater than 1.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``a`` is a scalar. Otherwise,
|
|||
|
|
``np.array(a).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized Zipf distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.zipf : probability density function, distribution, or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.zipf: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the Zipf distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(k) = \frac{k^{-a}}{\zeta(a)},
|
|||
|
|
|
|||
|
|
for integers :math:`k \geq 1`, where :math:`\zeta` is the Riemann Zeta
|
|||
|
|
function.
|
|||
|
|
|
|||
|
|
It is named for the American linguist George Kingsley Zipf, who noted
|
|||
|
|
that the frequency of any word in a sample of a language is inversely
|
|||
|
|
proportional to its rank in the frequency table.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Zipf, G. K., "Selected Studies of the Principle of Relative
|
|||
|
|
Frequency in Language," Cambridge, MA: Harvard Univ. Press,
|
|||
|
|
1932.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> a = 4.0
|
|||
|
|
>>> n = 20000
|
|||
|
|
>>> s = np.random.zipf(a, n)
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the expected histogram based on the probability
|
|||
|
|
density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> from scipy.special import zeta # doctest: +SKIP
|
|||
|
|
|
|||
|
|
`bincount` provides a fast histogram for small integers.
|
|||
|
|
|
|||
|
|
>>> count = np.bincount(s)
|
|||
|
|
>>> k = np.arange(1, s.max() + 1)
|
|||
|
|
|
|||
|
|
>>> plt.bar(k, count[1:], alpha=0.5, label='sample count')
|
|||
|
|
>>> plt.plot(k, n*(k**-a)/zeta(a), 'k.-', alpha=0.5,
|
|||
|
|
... label='expected count') # doctest: +SKIP
|
|||
|
|
>>> plt.semilogy()
|
|||
|
|
>>> plt.grid(alpha=0.4)
|
|||
|
|
>>> plt.legend()
|
|||
|
|
>>> plt.title(f'Zipf sample, a={a}, size={n}')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
weibull(a, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a Weibull distribution.
|
|||
|
|
|
|||
|
|
Draw samples from a 1-parameter Weibull distribution with the given
|
|||
|
|
shape parameter `a`.
|
|||
|
|
|
|||
|
|
.. math:: X = (-ln(U))^{1/a}
|
|||
|
|
|
|||
|
|
Here, U is drawn from the uniform distribution over (0,1].
|
|||
|
|
|
|||
|
|
The more common 2-parameter Weibull, including a scale parameter
|
|||
|
|
:math:`\lambda` is just :math:`X = \lambda(-ln(U))^{1/a}`.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.weibull`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
a : float or array_like of floats
|
|||
|
|
Shape parameter of the distribution. Must be nonnegative.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``a`` is a scalar. Otherwise,
|
|||
|
|
``np.array(a).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized Weibull distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.weibull_max
|
|||
|
|
scipy.stats.weibull_min
|
|||
|
|
scipy.stats.genextreme
|
|||
|
|
gumbel
|
|||
|
|
random.Generator.weibull: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The Weibull (or Type III asymptotic extreme value distribution
|
|||
|
|
for smallest values, SEV Type III, or Rosin-Rammler
|
|||
|
|
distribution) is one of a class of Generalized Extreme Value
|
|||
|
|
(GEV) distributions used in modeling extreme value problems.
|
|||
|
|
This class includes the Gumbel and Frechet distributions.
|
|||
|
|
|
|||
|
|
The probability density for the Weibull distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(x) = \frac{a}
|
|||
|
|
{\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},
|
|||
|
|
|
|||
|
|
where :math:`a` is the shape and :math:`\lambda` the scale.
|
|||
|
|
|
|||
|
|
The function has its peak (the mode) at
|
|||
|
|
:math:`\lambda(\frac{a-1}{a})^{1/a}`.
|
|||
|
|
|
|||
|
|
When ``a = 1``, the Weibull distribution reduces to the exponential
|
|||
|
|
distribution.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Waloddi Weibull, Royal Technical University, Stockholm,
|
|||
|
|
1939 "A Statistical Theory Of The Strength Of Materials",
|
|||
|
|
Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939,
|
|||
|
|
Generalstabens Litografiska Anstalts Forlag, Stockholm.
|
|||
|
|
.. [2] Waloddi Weibull, "A Statistical Distribution Function of
|
|||
|
|
Wide Applicability", Journal Of Applied Mechanics ASME Paper
|
|||
|
|
1951.
|
|||
|
|
.. [3] Wikipedia, "Weibull distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Weibull_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> a = 5. # shape
|
|||
|
|
>>> s = np.random.weibull(a, 1000)
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> x = np.arange(1,100.)/50.
|
|||
|
|
>>> def weib(x,n,a):
|
|||
|
|
... return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)
|
|||
|
|
|
|||
|
|
>>> count, bins, ignored = plt.hist(np.random.weibull(5.,1000))
|
|||
|
|
>>> x = np.arange(1,100.)/50.
|
|||
|
|
>>> scale = count.max()/weib(x, 1., 5.).max()
|
|||
|
|
>>> plt.plot(x, weib(x, 1., 5.)*scale)
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
vonmises(mu, kappa, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a von Mises distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from a von Mises distribution with specified mode
|
|||
|
|
(mu) and dispersion (kappa), on the interval [-pi, pi].
|
|||
|
|
|
|||
|
|
The von Mises distribution (also known as the circular normal
|
|||
|
|
distribution) is a continuous probability distribution on the unit
|
|||
|
|
circle. It may be thought of as the circular analogue of the normal
|
|||
|
|
distribution.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.vonmises`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
mu : float or array_like of floats
|
|||
|
|
Mode ("center") of the distribution.
|
|||
|
|
kappa : float or array_like of floats
|
|||
|
|
Dispersion of the distribution, has to be >=0.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``mu`` and ``kappa`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(mu, kappa).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized von Mises distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.vonmises : probability density function, distribution, or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.vonmises: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the von Mises distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)},
|
|||
|
|
|
|||
|
|
where :math:`\mu` is the mode and :math:`\kappa` the dispersion,
|
|||
|
|
and :math:`I_0(\kappa)` is the modified Bessel function of order 0.
|
|||
|
|
|
|||
|
|
The von Mises is named for Richard Edler von Mises, who was born in
|
|||
|
|
Austria-Hungary, in what is now the Ukraine. He fled to the United
|
|||
|
|
States in 1939 and became a professor at Harvard. He worked in
|
|||
|
|
probability theory, aerodynamics, fluid mechanics, and philosophy of
|
|||
|
|
science.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of
|
|||
|
|
Mathematical Functions with Formulas, Graphs, and Mathematical
|
|||
|
|
Tables, 9th printing," New York: Dover, 1972.
|
|||
|
|
.. [2] von Mises, R., "Mathematical Theory of Probability
|
|||
|
|
and Statistics", New York: Academic Press, 1964.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> mu, kappa = 0.0, 4.0 # mean and dispersion
|
|||
|
|
>>> s = np.random.vonmises(mu, kappa, 1000)
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> from scipy.special import i0 # doctest: +SKIP
|
|||
|
|
>>> plt.hist(s, 50, density=True)
|
|||
|
|
>>> x = np.linspace(-np.pi, np.pi, num=51)
|
|||
|
|
>>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa)) # doctest: +SKIP
|
|||
|
|
>>> plt.plot(x, y, linewidth=2, color='r') # doctest: +SKIP
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
uniform(low=0.0, high=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a uniform distribution.
|
|||
|
|
|
|||
|
|
Samples are uniformly distributed over the half-open interval
|
|||
|
|
``[low, high)`` (includes low, but excludes high). In other words,
|
|||
|
|
any value within the given interval is equally likely to be drawn
|
|||
|
|
by `uniform`.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.uniform`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
low : float or array_like of floats, optional
|
|||
|
|
Lower boundary of the output interval. All values generated will be
|
|||
|
|
greater than or equal to low. The default value is 0.
|
|||
|
|
high : float or array_like of floats
|
|||
|
|
Upper boundary of the output interval. All values generated will be
|
|||
|
|
less than or equal to high. The high limit may be included in the
|
|||
|
|
returned array of floats due to floating-point rounding in the
|
|||
|
|
equation ``low + (high-low) * random_sample()``. The default value
|
|||
|
|
is 1.0.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``low`` and ``high`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(low, high).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized uniform distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
randint : Discrete uniform distribution, yielding integers.
|
|||
|
|
random_integers : Discrete uniform distribution over the closed
|
|||
|
|
interval ``[low, high]``.
|
|||
|
|
random_sample : Floats uniformly distributed over ``[0, 1)``.
|
|||
|
|
random : Alias for `random_sample`.
|
|||
|
|
rand : Convenience function that accepts dimensions as input, e.g.,
|
|||
|
|
``rand(2,2)`` would generate a 2-by-2 array of floats,
|
|||
|
|
uniformly distributed over ``[0, 1)``.
|
|||
|
|
random.Generator.uniform: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density function of the uniform distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(x) = \frac{1}{b - a}
|
|||
|
|
|
|||
|
|
anywhere within the interval ``[a, b)``, and zero elsewhere.
|
|||
|
|
|
|||
|
|
When ``high`` == ``low``, values of ``low`` will be returned.
|
|||
|
|
If ``high`` < ``low``, the results are officially undefined
|
|||
|
|
and may eventually raise an error, i.e. do not rely on this
|
|||
|
|
function to behave when passed arguments satisfying that
|
|||
|
|
inequality condition. The ``high`` limit may be included in the
|
|||
|
|
returned array of floats due to floating-point rounding in the
|
|||
|
|
equation ``low + (high-low) * random_sample()``. For example:
|
|||
|
|
|
|||
|
|
>>> x = np.float32(5*0.99999999)
|
|||
|
|
>>> x
|
|||
|
|
5.0
|
|||
|
|
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> s = np.random.uniform(-1,0,1000)
|
|||
|
|
|
|||
|
|
All values are within the given interval:
|
|||
|
|
|
|||
|
|
>>> np.all(s >= -1)
|
|||
|
|
True
|
|||
|
|
>>> np.all(s < 0)
|
|||
|
|
True
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with the
|
|||
|
|
probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(s, 15, density=True)
|
|||
|
|
>>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
triangular(left, mode, right, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from the triangular distribution over the
|
|||
|
|
interval ``[left, right]``.
|
|||
|
|
|
|||
|
|
The triangular distribution is a continuous probability
|
|||
|
|
distribution with lower limit left, peak at mode, and upper
|
|||
|
|
limit right. Unlike the other distributions, these parameters
|
|||
|
|
directly define the shape of the pdf.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.triangular`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
left : float or array_like of floats
|
|||
|
|
Lower limit.
|
|||
|
|
mode : float or array_like of floats
|
|||
|
|
The value where the peak of the distribution occurs.
|
|||
|
|
The value must fulfill the condition ``left <= mode <= right``.
|
|||
|
|
right : float or array_like of floats
|
|||
|
|
Upper limit, must be larger than `left`.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``left``, ``mode``, and ``right``
|
|||
|
|
are all scalars. Otherwise, ``np.broadcast(left, mode, right).size``
|
|||
|
|
samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized triangular distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.triangular: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density function for the triangular distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(x;l, m, r) = \begin{cases}
|
|||
|
|
\frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\
|
|||
|
|
\frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\
|
|||
|
|
0& \text{otherwise}.
|
|||
|
|
\end{cases}
|
|||
|
|
|
|||
|
|
The triangular distribution is often used in ill-defined
|
|||
|
|
problems where the underlying distribution is not known, but
|
|||
|
|
some knowledge of the limits and mode exists. Often it is used
|
|||
|
|
in simulations.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Wikipedia, "Triangular distribution"
|
|||
|
|
https://en.wikipedia.org/wiki/Triangular_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw values from the distribution and plot the histogram:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> h = plt.hist(np.random.triangular(-3, 0, 8, 100000), bins=200,
|
|||
|
|
... density=True)
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
tomaxint(size=None)
|
|||
|
|
|
|||
|
|
Return a sample of uniformly distributed random integers in the interval
|
|||
|
|
[0, ``np.iinfo(np.int_).max``]. The `np.int_` type translates to the C long
|
|||
|
|
integer type and its precision is platform dependent.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. Default is None, in which case a
|
|||
|
|
single value is returned.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray
|
|||
|
|
Drawn samples, with shape `size`.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
randint : Uniform sampling over a given half-open interval of integers.
|
|||
|
|
random_integers : Uniform sampling over a given closed interval of
|
|||
|
|
integers.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> rs = np.random.RandomState() # need a RandomState object
|
|||
|
|
>>> rs.tomaxint((2,2,2))
|
|||
|
|
array([[[1170048599, 1600360186], # random
|
|||
|
|
[ 739731006, 1947757578]],
|
|||
|
|
[[1871712945, 752307660],
|
|||
|
|
[1601631370, 1479324245]]])
|
|||
|
|
>>> rs.tomaxint((2,2,2)) < np.iinfo(np.int_).max
|
|||
|
|
array([[[ True, True],
|
|||
|
|
[ True, True]],
|
|||
|
|
[[ True, True],
|
|||
|
|
[ True, True]]])
|
|||
|
|
|
|||
|
|
sum(pvals[:-1].astype(np.float64)) > 1.0. The pvals array is cast to 64-bit floating point prior to checking the sum. Precision changes when casting may cause problems even if the sum of the original pvals is valid. state must be a dict or a tuple.
|
|||
|
|
standard_t(df, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a standard Student's t distribution with `df` degrees
|
|||
|
|
of freedom.
|
|||
|
|
|
|||
|
|
A special case of the hyperbolic distribution. As `df` gets
|
|||
|
|
large, the result resembles that of the standard normal
|
|||
|
|
distribution (`standard_normal`).
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.standard_t`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
df : float or array_like of floats
|
|||
|
|
Degrees of freedom, must be > 0.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``df`` is a scalar. Otherwise,
|
|||
|
|
``np.array(df).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized standard Student's t distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.standard_t: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density function for the t distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df}
|
|||
|
|
\Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2}
|
|||
|
|
|
|||
|
|
The t test is based on an assumption that the data come from a
|
|||
|
|
Normal distribution. The t test provides a way to test whether
|
|||
|
|
the sample mean (that is the mean calculated from the data) is
|
|||
|
|
a good estimate of the true mean.
|
|||
|
|
|
|||
|
|
The derivation of the t-distribution was first published in
|
|||
|
|
1908 by William Gosset while working for the Guinness Brewery
|
|||
|
|
in Dublin. Due to proprietary issues, he had to publish under
|
|||
|
|
a pseudonym, and so he used the name Student.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Dalgaard, Peter, "Introductory Statistics With R",
|
|||
|
|
Springer, 2002.
|
|||
|
|
.. [2] Wikipedia, "Student's t-distribution"
|
|||
|
|
https://en.wikipedia.org/wiki/Student's_t-distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
From Dalgaard page 83 [1]_, suppose the daily energy intake for 11
|
|||
|
|
women in kilojoules (kJ) is:
|
|||
|
|
|
|||
|
|
>>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \
|
|||
|
|
... 7515, 8230, 8770])
|
|||
|
|
|
|||
|
|
Does their energy intake deviate systematically from the recommended
|
|||
|
|
value of 7725 kJ? Our null hypothesis will be the absence of deviation,
|
|||
|
|
and the alternate hypothesis will be the presence of an effect that could be
|
|||
|
|
either positive or negative, hence making our test 2-tailed.
|
|||
|
|
|
|||
|
|
Because we are estimating the mean and we have N=11 values in our sample,
|
|||
|
|
we have N-1=10 degrees of freedom. We set our significance level to 95% and
|
|||
|
|
compute the t statistic using the empirical mean and empirical standard
|
|||
|
|
deviation of our intake. We use a ddof of 1 to base the computation of our
|
|||
|
|
empirical standard deviation on an unbiased estimate of the variance (note:
|
|||
|
|
the final estimate is not unbiased due to the concave nature of the square
|
|||
|
|
root).
|
|||
|
|
|
|||
|
|
>>> np.mean(intake)
|
|||
|
|
6753.636363636364
|
|||
|
|
>>> intake.std(ddof=1)
|
|||
|
|
1142.1232221373727
|
|||
|
|
>>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake)))
|
|||
|
|
>>> t
|
|||
|
|
-2.8207540608310198
|
|||
|
|
|
|||
|
|
We draw 1000000 samples from Student's t distribution with the adequate
|
|||
|
|
degrees of freedom.
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> s = np.random.standard_t(10, size=1000000)
|
|||
|
|
>>> h = plt.hist(s, bins=100, density=True)
|
|||
|
|
|
|||
|
|
Does our t statistic land in one of the two critical regions found at
|
|||
|
|
both tails of the distribution?
|
|||
|
|
|
|||
|
|
>>> np.sum(np.abs(t) < np.abs(s)) / float(len(s))
|
|||
|
|
0.018318 #random < 0.05, statistic is in critical region
|
|||
|
|
|
|||
|
|
The probability value for this 2-tailed test is about 1.83%, which is
|
|||
|
|
lower than the 5% pre-determined significance threshold.
|
|||
|
|
|
|||
|
|
Therefore, the probability of observing values as extreme as our intake
|
|||
|
|
conditionally on the null hypothesis being true is too low, and we reject
|
|||
|
|
the null hypothesis of no deviation.
|
|||
|
|
|
|||
|
|
set_state can only be used with legacy MT19937 state instances.
|
|||
|
|
rayleigh(scale=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a Rayleigh distribution.
|
|||
|
|
|
|||
|
|
The :math:`\chi` and Weibull distributions are generalizations of the
|
|||
|
|
Rayleigh.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.rayleigh`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
scale : float or array_like of floats, optional
|
|||
|
|
Scale, also equals the mode. Must be non-negative. Default is 1.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``scale`` is a scalar. Otherwise,
|
|||
|
|
``np.array(scale).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized Rayleigh distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.rayleigh: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density function for the Rayleigh distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}}
|
|||
|
|
|
|||
|
|
The Rayleigh distribution would arise, for example, if the East
|
|||
|
|
and North components of the wind velocity had identical zero-mean
|
|||
|
|
Gaussian distributions. Then the wind speed would have a Rayleigh
|
|||
|
|
distribution.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Brighton Webs Ltd., "Rayleigh Distribution,"
|
|||
|
|
https://web.archive.org/web/20090514091424/http://brighton-webs.co.uk:80/distributions/rayleigh.asp
|
|||
|
|
.. [2] Wikipedia, "Rayleigh distribution"
|
|||
|
|
https://en.wikipedia.org/wiki/Rayleigh_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw values from the distribution and plot the histogram
|
|||
|
|
|
|||
|
|
>>> from matplotlib.pyplot import hist
|
|||
|
|
>>> values = hist(np.random.rayleigh(3, 100000), bins=200, density=True)
|
|||
|
|
|
|||
|
|
Wave heights tend to follow a Rayleigh distribution. If the mean wave
|
|||
|
|
height is 1 meter, what fraction of waves are likely to be larger than 3
|
|||
|
|
meters?
|
|||
|
|
|
|||
|
|
>>> meanvalue = 1
|
|||
|
|
>>> modevalue = np.sqrt(2 / np.pi) * meanvalue
|
|||
|
|
>>> s = np.random.rayleigh(modevalue, 1000000)
|
|||
|
|
|
|||
|
|
The percentage of waves larger than 3 meters is:
|
|||
|
|
|
|||
|
|
>>> 100.*sum(s>3)/1000000.
|
|||
|
|
0.087300000000000003 # random
|
|||
|
|
|
|||
|
|
|
|||
|
|
random_integers(low, high=None, size=None)
|
|||
|
|
|
|||
|
|
Random integers of type `np.int_` between `low` and `high`, inclusive.
|
|||
|
|
|
|||
|
|
Return random integers of type `np.int_` from the "discrete uniform"
|
|||
|
|
distribution in the closed interval [`low`, `high`]. If `high` is
|
|||
|
|
None (the default), then results are from [1, `low`]. The `np.int_`
|
|||
|
|
type translates to the C long integer type and its precision
|
|||
|
|
is platform dependent.
|
|||
|
|
|
|||
|
|
This function has been deprecated. Use randint instead.
|
|||
|
|
|
|||
|
|
.. deprecated:: 1.11.0
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
low : int
|
|||
|
|
Lowest (signed) integer to be drawn from the distribution (unless
|
|||
|
|
``high=None``, in which case this parameter is the *highest* such
|
|||
|
|
integer).
|
|||
|
|
high : int, optional
|
|||
|
|
If provided, the largest (signed) integer to be drawn from the
|
|||
|
|
distribution (see above for behavior if ``high=None``).
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. Default is None, in which case a
|
|||
|
|
single value is returned.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : int or ndarray of ints
|
|||
|
|
`size`-shaped array of random integers from the appropriate
|
|||
|
|
distribution, or a single such random int if `size` not provided.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
randint : Similar to `random_integers`, only for the half-open
|
|||
|
|
interval [`low`, `high`), and 0 is the lowest value if `high` is
|
|||
|
|
omitted.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
To sample from N evenly spaced floating-point numbers between a and b,
|
|||
|
|
use::
|
|||
|
|
|
|||
|
|
a + (b - a) * (np.random.random_integers(N) - 1) / (N - 1.)
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> np.random.random_integers(5)
|
|||
|
|
4 # random
|
|||
|
|
>>> type(np.random.random_integers(5))
|
|||
|
|
<class 'numpy.int64'>
|
|||
|
|
>>> np.random.random_integers(5, size=(3,2))
|
|||
|
|
array([[5, 4], # random
|
|||
|
|
[3, 3],
|
|||
|
|
[4, 5]])
|
|||
|
|
|
|||
|
|
Choose five random numbers from the set of five evenly-spaced
|
|||
|
|
numbers between 0 and 2.5, inclusive (*i.e.*, from the set
|
|||
|
|
:math:`{0, 5/8, 10/8, 15/8, 20/8}`):
|
|||
|
|
|
|||
|
|
>>> 2.5 * (np.random.random_integers(5, size=(5,)) - 1) / 4.
|
|||
|
|
array([ 0.625, 1.25 , 0.625, 0.625, 2.5 ]) # random
|
|||
|
|
|
|||
|
|
Roll two six sided dice 1000 times and sum the results:
|
|||
|
|
|
|||
|
|
>>> d1 = np.random.random_integers(1, 6, 1000)
|
|||
|
|
>>> d2 = np.random.random_integers(1, 6, 1000)
|
|||
|
|
>>> dsums = d1 + d2
|
|||
|
|
|
|||
|
|
Display results as a histogram:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(dsums, 11, density=True)
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
randint(low, high=None, size=None, dtype=int)
|
|||
|
|
|
|||
|
|
Return random integers from `low` (inclusive) to `high` (exclusive).
|
|||
|
|
|
|||
|
|
Return random integers from the "discrete uniform" distribution of
|
|||
|
|
the specified dtype in the "half-open" interval [`low`, `high`). If
|
|||
|
|
`high` is None (the default), then results are from [0, `low`).
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.integers`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
low : int or array-like of ints
|
|||
|
|
Lowest (signed) integers to be drawn from the distribution (unless
|
|||
|
|
``high=None``, in which case this parameter is one above the
|
|||
|
|
*highest* such integer).
|
|||
|
|
high : int or array-like of ints, optional
|
|||
|
|
If provided, one above the largest (signed) integer to be drawn
|
|||
|
|
from the distribution (see above for behavior if ``high=None``).
|
|||
|
|
If array-like, must contain integer values
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. Default is None, in which case a
|
|||
|
|
single value is returned.
|
|||
|
|
dtype : dtype, optional
|
|||
|
|
Desired dtype of the result. Byteorder must be native.
|
|||
|
|
The default value is int.
|
|||
|
|
|
|||
|
|
.. versionadded:: 1.11.0
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : int or ndarray of ints
|
|||
|
|
`size`-shaped array of random integers from the appropriate
|
|||
|
|
distribution, or a single such random int if `size` not provided.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random_integers : similar to `randint`, only for the closed
|
|||
|
|
interval [`low`, `high`], and 1 is the lowest value if `high` is
|
|||
|
|
omitted.
|
|||
|
|
random.Generator.integers: which should be used for new code.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> np.random.randint(2, size=10)
|
|||
|
|
array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) # random
|
|||
|
|
>>> np.random.randint(1, size=10)
|
|||
|
|
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
|
|||
|
|
|
|||
|
|
Generate a 2 x 4 array of ints between 0 and 4, inclusive:
|
|||
|
|
|
|||
|
|
>>> np.random.randint(5, size=(2, 4))
|
|||
|
|
array([[4, 0, 2, 1], # random
|
|||
|
|
[3, 2, 2, 0]])
|
|||
|
|
|
|||
|
|
Generate a 1 x 3 array with 3 different upper bounds
|
|||
|
|
|
|||
|
|
>>> np.random.randint(1, [3, 5, 10])
|
|||
|
|
array([2, 2, 9]) # random
|
|||
|
|
|
|||
|
|
Generate a 1 by 3 array with 3 different lower bounds
|
|||
|
|
|
|||
|
|
>>> np.random.randint([1, 5, 7], 10)
|
|||
|
|
array([9, 8, 7]) # random
|
|||
|
|
|
|||
|
|
Generate a 2 by 4 array using broadcasting with dtype of uint8
|
|||
|
|
|
|||
|
|
>>> np.random.randint([1, 3, 5, 7], [[10], [20]], dtype=np.uint8)
|
|||
|
|
array([[ 8, 6, 9, 7], # random
|
|||
|
|
[ 1, 16, 9, 12]], dtype=uint8)
|
|||
|
|
|
|||
|
|
power(a, size=None)
|
|||
|
|
|
|||
|
|
Draws samples in [0, 1] from a power distribution with positive
|
|||
|
|
exponent a - 1.
|
|||
|
|
|
|||
|
|
Also known as the power function distribution.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.power`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
a : float or array_like of floats
|
|||
|
|
Parameter of the distribution. Must be non-negative.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``a`` is a scalar. Otherwise,
|
|||
|
|
``np.array(a).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized power distribution.
|
|||
|
|
|
|||
|
|
Raises
|
|||
|
|
------
|
|||
|
|
ValueError
|
|||
|
|
If a <= 0.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.power: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density function is
|
|||
|
|
|
|||
|
|
.. math:: P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0.
|
|||
|
|
|
|||
|
|
The power function distribution is just the inverse of the Pareto
|
|||
|
|
distribution. It may also be seen as a special case of the Beta
|
|||
|
|
distribution.
|
|||
|
|
|
|||
|
|
It is used, for example, in modeling the over-reporting of insurance
|
|||
|
|
claims.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Christian Kleiber, Samuel Kotz, "Statistical size distributions
|
|||
|
|
in economics and actuarial sciences", Wiley, 2003.
|
|||
|
|
.. [2] Heckert, N. A. and Filliben, James J. "NIST Handbook 148:
|
|||
|
|
Dataplot Reference Manual, Volume 2: Let Subcommands and Library
|
|||
|
|
Functions", National Institute of Standards and Technology
|
|||
|
|
Handbook Series, June 2003.
|
|||
|
|
https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> a = 5. # shape
|
|||
|
|
>>> samples = 1000
|
|||
|
|
>>> s = np.random.power(a, samples)
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(s, bins=30)
|
|||
|
|
>>> x = np.linspace(0, 1, 100)
|
|||
|
|
>>> y = a*x**(a-1.)
|
|||
|
|
>>> normed_y = samples*np.diff(bins)[0]*y
|
|||
|
|
>>> plt.plot(x, normed_y)
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
Compare the power function distribution to the inverse of the Pareto.
|
|||
|
|
|
|||
|
|
>>> from scipy import stats # doctest: +SKIP
|
|||
|
|
>>> rvs = np.random.power(5, 1000000)
|
|||
|
|
>>> rvsp = np.random.pareto(5, 1000000)
|
|||
|
|
>>> xx = np.linspace(0,1,100)
|
|||
|
|
>>> powpdf = stats.powerlaw.pdf(xx,5) # doctest: +SKIP
|
|||
|
|
|
|||
|
|
>>> plt.figure()
|
|||
|
|
>>> plt.hist(rvs, bins=50, density=True)
|
|||
|
|
>>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP
|
|||
|
|
>>> plt.title('np.random.power(5)')
|
|||
|
|
|
|||
|
|
>>> plt.figure()
|
|||
|
|
>>> plt.hist(1./(1.+rvsp), bins=50, density=True)
|
|||
|
|
>>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP
|
|||
|
|
>>> plt.title('inverse of 1 + np.random.pareto(5)')
|
|||
|
|
|
|||
|
|
>>> plt.figure()
|
|||
|
|
>>> plt.hist(1./(1.+rvsp), bins=50, density=True)
|
|||
|
|
>>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP
|
|||
|
|
>>> plt.title('inverse of stats.pareto(5)')
|
|||
|
|
|
|||
|
|
|
|||
|
|
pareto(a, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a Pareto II or Lomax distribution with
|
|||
|
|
specified shape.
|
|||
|
|
|
|||
|
|
The Lomax or Pareto II distribution is a shifted Pareto
|
|||
|
|
distribution. The classical Pareto distribution can be
|
|||
|
|
obtained from the Lomax distribution by adding 1 and
|
|||
|
|
multiplying by the scale parameter ``m`` (see Notes). The
|
|||
|
|
smallest value of the Lomax distribution is zero while for the
|
|||
|
|
classical Pareto distribution it is ``mu``, where the standard
|
|||
|
|
Pareto distribution has location ``mu = 1``. Lomax can also
|
|||
|
|
be considered as a simplified version of the Generalized
|
|||
|
|
Pareto distribution (available in SciPy), with the scale set
|
|||
|
|
to one and the location set to zero.
|
|||
|
|
|
|||
|
|
The Pareto distribution must be greater than zero, and is
|
|||
|
|
unbounded above. It is also known as the "80-20 rule". In
|
|||
|
|
this distribution, 80 percent of the weights are in the lowest
|
|||
|
|
20 percent of the range, while the other 20 percent fill the
|
|||
|
|
remaining 80 percent of the range.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.pareto`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
a : float or array_like of floats
|
|||
|
|
Shape of the distribution. Must be positive.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``a`` is a scalar. Otherwise,
|
|||
|
|
``np.array(a).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized Pareto distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.lomax : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
scipy.stats.genpareto : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.pareto: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the Pareto distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(x) = \frac{am^a}{x^{a+1}}
|
|||
|
|
|
|||
|
|
where :math:`a` is the shape and :math:`m` the scale.
|
|||
|
|
|
|||
|
|
The Pareto distribution, named after the Italian economist
|
|||
|
|
Vilfredo Pareto, is a power law probability distribution
|
|||
|
|
useful in many real world problems. Outside the field of
|
|||
|
|
economics it is generally referred to as the Bradford
|
|||
|
|
distribution. Pareto developed the distribution to describe
|
|||
|
|
the distribution of wealth in an economy. It has also found
|
|||
|
|
use in insurance, web page access statistics, oil field sizes,
|
|||
|
|
and many other problems, including the download frequency for
|
|||
|
|
projects in Sourceforge [1]_. It is one of the so-called
|
|||
|
|
"fat-tailed" distributions.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Francis Hunt and Paul Johnson, On the Pareto Distribution of
|
|||
|
|
Sourceforge projects.
|
|||
|
|
.. [2] Pareto, V. (1896). Course of Political Economy. Lausanne.
|
|||
|
|
.. [3] Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme
|
|||
|
|
Values, Birkhauser Verlag, Basel, pp 23-30.
|
|||
|
|
.. [4] Wikipedia, "Pareto distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Pareto_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> a, m = 3., 2. # shape and mode
|
|||
|
|
>>> s = (np.random.pareto(a, 1000) + 1) * m
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with the probability
|
|||
|
|
density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, _ = plt.hist(s, 100, density=True)
|
|||
|
|
>>> fit = a*m**a / bins**(a+1)
|
|||
|
|
>>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color='r')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
numpy.core.multiarray failed to import
|
|||
|
|
normal(loc=0.0, scale=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw random samples from a normal (Gaussian) distribution.
|
|||
|
|
|
|||
|
|
The probability density function of the normal distribution, first
|
|||
|
|
derived by De Moivre and 200 years later by both Gauss and Laplace
|
|||
|
|
independently [2]_, is often called the bell curve because of
|
|||
|
|
its characteristic shape (see the example below).
|
|||
|
|
|
|||
|
|
The normal distributions occurs often in nature. For example, it
|
|||
|
|
describes the commonly occurring distribution of samples influenced
|
|||
|
|
by a large number of tiny, random disturbances, each with its own
|
|||
|
|
unique distribution [2]_.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.normal`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
loc : float or array_like of floats
|
|||
|
|
Mean ("centre") of the distribution.
|
|||
|
|
scale : float or array_like of floats
|
|||
|
|
Standard deviation (spread or "width") of the distribution. Must be
|
|||
|
|
non-negative.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``loc`` and ``scale`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized normal distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.norm : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.normal: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the Gaussian distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }}
|
|||
|
|
e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },
|
|||
|
|
|
|||
|
|
where :math:`\mu` is the mean and :math:`\sigma` the standard
|
|||
|
|
deviation. The square of the standard deviation, :math:`\sigma^2`,
|
|||
|
|
is called the variance.
|
|||
|
|
|
|||
|
|
The function has its peak at the mean, and its "spread" increases with
|
|||
|
|
the standard deviation (the function reaches 0.607 times its maximum at
|
|||
|
|
:math:`x + \sigma` and :math:`x - \sigma` [2]_). This implies that
|
|||
|
|
normal is more likely to return samples lying close to the mean, rather
|
|||
|
|
than those far away.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Wikipedia, "Normal distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Normal_distribution
|
|||
|
|
.. [2] P. R. Peebles Jr., "Central Limit Theorem" in "Probability,
|
|||
|
|
Random Variables and Random Signal Principles", 4th ed., 2001,
|
|||
|
|
pp. 51, 51, 125.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> mu, sigma = 0, 0.1 # mean and standard deviation
|
|||
|
|
>>> s = np.random.normal(mu, sigma, 1000)
|
|||
|
|
|
|||
|
|
Verify the mean and the variance:
|
|||
|
|
|
|||
|
|
>>> abs(mu - np.mean(s))
|
|||
|
|
0.0 # may vary
|
|||
|
|
|
|||
|
|
>>> abs(sigma - np.std(s, ddof=1))
|
|||
|
|
0.1 # may vary
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(s, 30, density=True)
|
|||
|
|
>>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
|
|||
|
|
... np.exp( - (bins - mu)**2 / (2 * sigma**2) ),
|
|||
|
|
... linewidth=2, color='r')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
Two-by-four array of samples from the normal distribution with
|
|||
|
|
mean 3 and standard deviation 2.5:
|
|||
|
|
|
|||
|
|
>>> np.random.normal(3, 2.5, size=(2, 4))
|
|||
|
|
array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random
|
|||
|
|
[ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random
|
|||
|
|
|
|||
|
|
|
|||
|
|
noncentral_f(dfnum, dfden, nonc, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from the noncentral F distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from an F distribution with specified parameters,
|
|||
|
|
`dfnum` (degrees of freedom in numerator) and `dfden` (degrees of
|
|||
|
|
freedom in denominator), where both parameters > 1.
|
|||
|
|
`nonc` is the non-centrality parameter.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the
|
|||
|
|
`~numpy.random.Generator.noncentral_f`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
dfnum : float or array_like of floats
|
|||
|
|
Numerator degrees of freedom, must be > 0.
|
|||
|
|
|
|||
|
|
.. versionchanged:: 1.14.0
|
|||
|
|
Earlier NumPy versions required dfnum > 1.
|
|||
|
|
dfden : float or array_like of floats
|
|||
|
|
Denominator degrees of freedom, must be > 0.
|
|||
|
|
nonc : float or array_like of floats
|
|||
|
|
Non-centrality parameter, the sum of the squares of the numerator
|
|||
|
|
means, must be >= 0.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``dfnum``, ``dfden``, and ``nonc``
|
|||
|
|
are all scalars. Otherwise, ``np.broadcast(dfnum, dfden, nonc).size``
|
|||
|
|
samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized noncentral Fisher distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.noncentral_f: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
When calculating the power of an experiment (power = probability of
|
|||
|
|
rejecting the null hypothesis when a specific alternative is true) the
|
|||
|
|
non-central F statistic becomes important. When the null hypothesis is
|
|||
|
|
true, the F statistic follows a central F distribution. When the null
|
|||
|
|
hypothesis is not true, then it follows a non-central F statistic.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Weisstein, Eric W. "Noncentral F-Distribution."
|
|||
|
|
From MathWorld--A Wolfram Web Resource.
|
|||
|
|
http://mathworld.wolfram.com/NoncentralF-Distribution.html
|
|||
|
|
.. [2] Wikipedia, "Noncentral F-distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Noncentral_F-distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
In a study, testing for a specific alternative to the null hypothesis
|
|||
|
|
requires use of the Noncentral F distribution. We need to calculate the
|
|||
|
|
area in the tail of the distribution that exceeds the value of the F
|
|||
|
|
distribution for the null hypothesis. We'll plot the two probability
|
|||
|
|
distributions for comparison.
|
|||
|
|
|
|||
|
|
>>> dfnum = 3 # between group deg of freedom
|
|||
|
|
>>> dfden = 20 # within groups degrees of freedom
|
|||
|
|
>>> nonc = 3.0
|
|||
|
|
>>> nc_vals = np.random.noncentral_f(dfnum, dfden, nonc, 1000000)
|
|||
|
|
>>> NF = np.histogram(nc_vals, bins=50, density=True)
|
|||
|
|
>>> c_vals = np.random.f(dfnum, dfden, 1000000)
|
|||
|
|
>>> F = np.histogram(c_vals, bins=50, density=True)
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> plt.plot(F[1][1:], F[0])
|
|||
|
|
>>> plt.plot(NF[1][1:], NF[0])
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
noncentral_chisquare(df, nonc, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a noncentral chi-square distribution.
|
|||
|
|
|
|||
|
|
The noncentral :math:`\chi^2` distribution is a generalization of
|
|||
|
|
the :math:`\chi^2` distribution.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the
|
|||
|
|
`~numpy.random.Generator.noncentral_chisquare`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
df : float or array_like of floats
|
|||
|
|
Degrees of freedom, must be > 0.
|
|||
|
|
|
|||
|
|
.. versionchanged:: 1.10.0
|
|||
|
|
Earlier NumPy versions required dfnum > 1.
|
|||
|
|
nonc : float or array_like of floats
|
|||
|
|
Non-centrality, must be non-negative.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``df`` and ``nonc`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(df, nonc).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized noncentral chi-square distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.noncentral_chisquare: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density function for the noncentral Chi-square
|
|||
|
|
distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(x;df,nonc) = \sum^{\infty}_{i=0}
|
|||
|
|
\frac{e^{-nonc/2}(nonc/2)^{i}}{i!}
|
|||
|
|
P_{Y_{df+2i}}(x),
|
|||
|
|
|
|||
|
|
where :math:`Y_{q}` is the Chi-square with q degrees of freedom.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Wikipedia, "Noncentral chi-squared distribution"
|
|||
|
|
https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw values from the distribution and plot the histogram
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
|
|||
|
|
... bins=200, density=True)
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
Draw values from a noncentral chisquare with very small noncentrality,
|
|||
|
|
and compare to a chisquare.
|
|||
|
|
|
|||
|
|
>>> plt.figure()
|
|||
|
|
>>> values = plt.hist(np.random.noncentral_chisquare(3, .0000001, 100000),
|
|||
|
|
... bins=np.arange(0., 25, .1), density=True)
|
|||
|
|
>>> values2 = plt.hist(np.random.chisquare(3, 100000),
|
|||
|
|
... bins=np.arange(0., 25, .1), density=True)
|
|||
|
|
>>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
Demonstrate how large values of non-centrality lead to a more symmetric
|
|||
|
|
distribution.
|
|||
|
|
|
|||
|
|
>>> plt.figure()
|
|||
|
|
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
|
|||
|
|
... bins=200, density=True)
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
negative_binomial(n, p, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a negative binomial distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from a negative binomial distribution with specified
|
|||
|
|
parameters, `n` successes and `p` probability of success where `n`
|
|||
|
|
is > 0 and `p` is in the interval [0, 1].
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the
|
|||
|
|
`~numpy.random.Generator.negative_binomial`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
n : float or array_like of floats
|
|||
|
|
Parameter of the distribution, > 0.
|
|||
|
|
p : float or array_like of floats
|
|||
|
|
Parameter of the distribution, >= 0 and <=1.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``n`` and ``p`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(n, p).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized negative binomial distribution,
|
|||
|
|
where each sample is equal to N, the number of failures that
|
|||
|
|
occurred before a total of n successes was reached.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.negative_binomial: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability mass function of the negative binomial distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(N;n,p) = \frac{\Gamma(N+n)}{N!\Gamma(n)}p^{n}(1-p)^{N},
|
|||
|
|
|
|||
|
|
where :math:`n` is the number of successes, :math:`p` is the
|
|||
|
|
probability of success, :math:`N+n` is the number of trials, and
|
|||
|
|
:math:`\Gamma` is the gamma function. When :math:`n` is an integer,
|
|||
|
|
:math:`\frac{\Gamma(N+n)}{N!\Gamma(n)} = \binom{N+n-1}{N}`, which is
|
|||
|
|
the more common form of this term in the pmf. The negative
|
|||
|
|
binomial distribution gives the probability of N failures given n
|
|||
|
|
successes, with a success on the last trial.
|
|||
|
|
|
|||
|
|
If one throws a die repeatedly until the third time a "1" appears,
|
|||
|
|
then the probability distribution of the number of non-"1"s that
|
|||
|
|
appear before the third "1" is a negative binomial distribution.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Weisstein, Eric W. "Negative Binomial Distribution." From
|
|||
|
|
MathWorld--A Wolfram Web Resource.
|
|||
|
|
http://mathworld.wolfram.com/NegativeBinomialDistribution.html
|
|||
|
|
.. [2] Wikipedia, "Negative binomial distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Negative_binomial_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
A real world example. A company drills wild-cat oil
|
|||
|
|
exploration wells, each with an estimated probability of
|
|||
|
|
success of 0.1. What is the probability of having one success
|
|||
|
|
for each successive well, that is what is the probability of a
|
|||
|
|
single success after drilling 5 wells, after 6 wells, etc.?
|
|||
|
|
|
|||
|
|
>>> s = np.random.negative_binomial(1, 0.1, 100000)
|
|||
|
|
>>> for i in range(1, 11): # doctest: +SKIP
|
|||
|
|
... probability = sum(s<i) / 100000.
|
|||
|
|
... print(i, "wells drilled, probability of one success =", probability)
|
|||
|
|
|
|||
|
|
|
|||
|
|
multivariate_normal(mean, cov, size=None, check_valid='warn', tol=1e-8)
|
|||
|
|
|
|||
|
|
Draw random samples from a multivariate normal distribution.
|
|||
|
|
|
|||
|
|
The multivariate normal, multinormal or Gaussian distribution is a
|
|||
|
|
generalization of the one-dimensional normal distribution to higher
|
|||
|
|
dimensions. Such a distribution is specified by its mean and
|
|||
|
|
covariance matrix. These parameters are analogous to the mean
|
|||
|
|
(average or "center") and variance (standard deviation, or "width,"
|
|||
|
|
squared) of the one-dimensional normal distribution.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the
|
|||
|
|
`~numpy.random.Generator.multivariate_normal`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
mean : 1-D array_like, of length N
|
|||
|
|
Mean of the N-dimensional distribution.
|
|||
|
|
cov : 2-D array_like, of shape (N, N)
|
|||
|
|
Covariance matrix of the distribution. It must be symmetric and
|
|||
|
|
positive-semidefinite for proper sampling.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Given a shape of, for example, ``(m,n,k)``, ``m*n*k`` samples are
|
|||
|
|
generated, and packed in an `m`-by-`n`-by-`k` arrangement. Because
|
|||
|
|
each sample is `N`-dimensional, the output shape is ``(m,n,k,N)``.
|
|||
|
|
If no shape is specified, a single (`N`-D) sample is returned.
|
|||
|
|
check_valid : { 'warn', 'raise', 'ignore' }, optional
|
|||
|
|
Behavior when the covariance matrix is not positive semidefinite.
|
|||
|
|
tol : float, optional
|
|||
|
|
Tolerance when checking the singular values in covariance matrix.
|
|||
|
|
cov is cast to double before the check.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray
|
|||
|
|
The drawn samples, of shape *size*, if that was provided. If not,
|
|||
|
|
the shape is ``(N,)``.
|
|||
|
|
|
|||
|
|
In other words, each entry ``out[i,j,...,:]`` is an N-dimensional
|
|||
|
|
value drawn from the distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.multivariate_normal: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The mean is a coordinate in N-dimensional space, which represents the
|
|||
|
|
location where samples are most likely to be generated. This is
|
|||
|
|
analogous to the peak of the bell curve for the one-dimensional or
|
|||
|
|
univariate normal distribution.
|
|||
|
|
|
|||
|
|
Covariance indicates the level to which two variables vary together.
|
|||
|
|
From the multivariate normal distribution, we draw N-dimensional
|
|||
|
|
samples, :math:`X = [x_1, x_2, ... x_N]`. The covariance matrix
|
|||
|
|
element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`.
|
|||
|
|
The element :math:`C_{ii}` is the variance of :math:`x_i` (i.e. its
|
|||
|
|
"spread").
|
|||
|
|
|
|||
|
|
Instead of specifying the full covariance matrix, popular
|
|||
|
|
approximations include:
|
|||
|
|
|
|||
|
|
- Spherical covariance (`cov` is a multiple of the identity matrix)
|
|||
|
|
- Diagonal covariance (`cov` has non-negative elements, and only on
|
|||
|
|
the diagonal)
|
|||
|
|
|
|||
|
|
This geometrical property can be seen in two dimensions by plotting
|
|||
|
|
generated data-points:
|
|||
|
|
|
|||
|
|
>>> mean = [0, 0]
|
|||
|
|
>>> cov = [[1, 0], [0, 100]] # diagonal covariance
|
|||
|
|
|
|||
|
|
Diagonal covariance means that points are oriented along x or y-axis:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> x, y = np.random.multivariate_normal(mean, cov, 5000).T
|
|||
|
|
>>> plt.plot(x, y, 'x')
|
|||
|
|
>>> plt.axis('equal')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
Note that the covariance matrix must be positive semidefinite (a.k.a.
|
|||
|
|
nonnegative-definite). Otherwise, the behavior of this method is
|
|||
|
|
undefined and backwards compatibility is not guaranteed.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Papoulis, A., "Probability, Random Variables, and Stochastic
|
|||
|
|
Processes," 3rd ed., New York: McGraw-Hill, 1991.
|
|||
|
|
.. [2] Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern
|
|||
|
|
Classification," 2nd ed., New York: Wiley, 2001.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> mean = (1, 2)
|
|||
|
|
>>> cov = [[1, 0], [0, 1]]
|
|||
|
|
>>> x = np.random.multivariate_normal(mean, cov, (3, 3))
|
|||
|
|
>>> x.shape
|
|||
|
|
(3, 3, 2)
|
|||
|
|
|
|||
|
|
Here we generate 800 samples from the bivariate normal distribution
|
|||
|
|
with mean [0, 0] and covariance matrix [[6, -3], [-3, 3.5]]. The
|
|||
|
|
expected variances of the first and second components of the sample
|
|||
|
|
are 6 and 3.5, respectively, and the expected correlation
|
|||
|
|
coefficient is -3/sqrt(6*3.5) ≈ -0.65465.
|
|||
|
|
|
|||
|
|
>>> cov = np.array([[6, -3], [-3, 3.5]])
|
|||
|
|
>>> pts = np.random.multivariate_normal([0, 0], cov, size=800)
|
|||
|
|
|
|||
|
|
Check that the mean, covariance, and correlation coefficient of the
|
|||
|
|
sample are close to the expected values:
|
|||
|
|
|
|||
|
|
>>> pts.mean(axis=0)
|
|||
|
|
array([ 0.0326911 , -0.01280782]) # may vary
|
|||
|
|
>>> np.cov(pts.T)
|
|||
|
|
array([[ 5.96202397, -2.85602287],
|
|||
|
|
[-2.85602287, 3.47613949]]) # may vary
|
|||
|
|
>>> np.corrcoef(pts.T)[0, 1]
|
|||
|
|
-0.6273591314603949 # may vary
|
|||
|
|
|
|||
|
|
We can visualize this data with a scatter plot. The orientation
|
|||
|
|
of the point cloud illustrates the negative correlation of the
|
|||
|
|
components of this sample.
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> plt.plot(pts[:, 0], pts[:, 1], '.', alpha=0.5)
|
|||
|
|
>>> plt.axis('equal')
|
|||
|
|
>>> plt.grid()
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
multinomial(n, pvals, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a multinomial distribution.
|
|||
|
|
|
|||
|
|
The multinomial distribution is a multivariate generalization of the
|
|||
|
|
binomial distribution. Take an experiment with one of ``p``
|
|||
|
|
possible outcomes. An example of such an experiment is throwing a dice,
|
|||
|
|
where the outcome can be 1 through 6. Each sample drawn from the
|
|||
|
|
distribution represents `n` such experiments. Its values,
|
|||
|
|
``X_i = [X_0, X_1, ..., X_p]``, represent the number of times the
|
|||
|
|
outcome was ``i``.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.multinomial`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
n : int
|
|||
|
|
Number of experiments.
|
|||
|
|
pvals : sequence of floats, length p
|
|||
|
|
Probabilities of each of the ``p`` different outcomes. These
|
|||
|
|
must sum to 1 (however, the last element is always assumed to
|
|||
|
|
account for the remaining probability, as long as
|
|||
|
|
``sum(pvals[:-1]) <= 1)``.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. Default is None, in which case a
|
|||
|
|
single value is returned.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray
|
|||
|
|
The drawn samples, of shape *size*, if that was provided. If not,
|
|||
|
|
the shape is ``(N,)``.
|
|||
|
|
|
|||
|
|
In other words, each entry ``out[i,j,...,:]`` is an N-dimensional
|
|||
|
|
value drawn from the distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.multinomial: which should be used for new code.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Throw a dice 20 times:
|
|||
|
|
|
|||
|
|
>>> np.random.multinomial(20, [1/6.]*6, size=1)
|
|||
|
|
array([[4, 1, 7, 5, 2, 1]]) # random
|
|||
|
|
|
|||
|
|
It landed 4 times on 1, once on 2, etc.
|
|||
|
|
|
|||
|
|
Now, throw the dice 20 times, and 20 times again:
|
|||
|
|
|
|||
|
|
>>> np.random.multinomial(20, [1/6.]*6, size=2)
|
|||
|
|
array([[3, 4, 3, 3, 4, 3], # random
|
|||
|
|
[2, 4, 3, 4, 0, 7]])
|
|||
|
|
|
|||
|
|
For the first run, we threw 3 times 1, 4 times 2, etc. For the second,
|
|||
|
|
we threw 2 times 1, 4 times 2, etc.
|
|||
|
|
|
|||
|
|
A loaded die is more likely to land on number 6:
|
|||
|
|
|
|||
|
|
>>> np.random.multinomial(100, [1/7.]*5 + [2/7.])
|
|||
|
|
array([11, 16, 14, 17, 16, 26]) # random
|
|||
|
|
|
|||
|
|
The probability inputs should be normalized. As an implementation
|
|||
|
|
detail, the value of the last entry is ignored and assumed to take
|
|||
|
|
up any leftover probability mass, but this should not be relied on.
|
|||
|
|
A biased coin which has twice as much weight on one side as on the
|
|||
|
|
other should be sampled like so:
|
|||
|
|
|
|||
|
|
>>> np.random.multinomial(100, [1.0 / 3, 2.0 / 3]) # RIGHT
|
|||
|
|
array([38, 62]) # random
|
|||
|
|
|
|||
|
|
not like:
|
|||
|
|
|
|||
|
|
>>> np.random.multinomial(100, [1.0, 2.0]) # WRONG
|
|||
|
|
Traceback (most recent call last):
|
|||
|
|
ValueError: pvals < 0, pvals > 1 or pvals contains NaNs
|
|||
|
|
|
|||
|
|
|
|||
|
|
logseries(p, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a logarithmic series distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from a log series distribution with specified
|
|||
|
|
shape parameter, 0 <= ``p`` < 1.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.logseries`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
p : float or array_like of floats
|
|||
|
|
Shape parameter for the distribution. Must be in the range [0, 1).
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``p`` is a scalar. Otherwise,
|
|||
|
|
``np.array(p).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized logarithmic series distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.logser : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.logseries: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the Log Series distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(k) = \frac{-p^k}{k \ln(1-p)},
|
|||
|
|
|
|||
|
|
where p = probability.
|
|||
|
|
|
|||
|
|
The log series distribution is frequently used to represent species
|
|||
|
|
richness and occurrence, first proposed by Fisher, Corbet, and
|
|||
|
|
Williams in 1943 [2]. It may also be used to model the numbers of
|
|||
|
|
occupants seen in cars [3].
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Buzas, Martin A.; Culver, Stephen J., Understanding regional
|
|||
|
|
species diversity through the log series distribution of
|
|||
|
|
occurrences: BIODIVERSITY RESEARCH Diversity & Distributions,
|
|||
|
|
Volume 5, Number 5, September 1999 , pp. 187-195(9).
|
|||
|
|
.. [2] Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The
|
|||
|
|
relation between the number of species and the number of
|
|||
|
|
individuals in a random sample of an animal population.
|
|||
|
|
Journal of Animal Ecology, 12:42-58.
|
|||
|
|
.. [3] D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small
|
|||
|
|
Data Sets, CRC Press, 1994.
|
|||
|
|
.. [4] Wikipedia, "Logarithmic distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Logarithmic_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> a = .6
|
|||
|
|
>>> s = np.random.logseries(a, 10000)
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(s)
|
|||
|
|
|
|||
|
|
# plot against distribution
|
|||
|
|
|
|||
|
|
>>> def logseries(k, p):
|
|||
|
|
... return -p**k/(k*np.log(1-p))
|
|||
|
|
>>> plt.plot(bins, logseries(bins, a)*count.max()/
|
|||
|
|
... logseries(bins, a).max(), 'r')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
lognormal(mean=0.0, sigma=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a log-normal distribution.
|
|||
|
|
|
|||
|
|
Draw samples from a log-normal distribution with specified mean,
|
|||
|
|
standard deviation, and array shape. Note that the mean and standard
|
|||
|
|
deviation are not the values for the distribution itself, but of the
|
|||
|
|
underlying normal distribution it is derived from.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.lognormal`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
mean : float or array_like of floats, optional
|
|||
|
|
Mean value of the underlying normal distribution. Default is 0.
|
|||
|
|
sigma : float or array_like of floats, optional
|
|||
|
|
Standard deviation of the underlying normal distribution. Must be
|
|||
|
|
non-negative. Default is 1.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``mean`` and ``sigma`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(mean, sigma).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized log-normal distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.lognorm : probability density function, distribution,
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.lognormal: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
A variable `x` has a log-normal distribution if `log(x)` is normally
|
|||
|
|
distributed. The probability density function for the log-normal
|
|||
|
|
distribution is:
|
|||
|
|
|
|||
|
|
.. math:: p(x) = \frac{1}{\sigma x \sqrt{2\pi}}
|
|||
|
|
e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}
|
|||
|
|
|
|||
|
|
where :math:`\mu` is the mean and :math:`\sigma` is the standard
|
|||
|
|
deviation of the normally distributed logarithm of the variable.
|
|||
|
|
A log-normal distribution results if a random variable is the *product*
|
|||
|
|
of a large number of independent, identically-distributed variables in
|
|||
|
|
the same way that a normal distribution results if the variable is the
|
|||
|
|
*sum* of a large number of independent, identically-distributed
|
|||
|
|
variables.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Limpert, E., Stahel, W. A., and Abbt, M., "Log-normal
|
|||
|
|
Distributions across the Sciences: Keys and Clues,"
|
|||
|
|
BioScience, Vol. 51, No. 5, May, 2001.
|
|||
|
|
https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf
|
|||
|
|
.. [2] Reiss, R.D. and Thomas, M., "Statistical Analysis of Extreme
|
|||
|
|
Values," Basel: Birkhauser Verlag, 2001, pp. 31-32.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> mu, sigma = 3., 1. # mean and standard deviation
|
|||
|
|
>>> s = np.random.lognormal(mu, sigma, 1000)
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(s, 100, density=True, align='mid')
|
|||
|
|
|
|||
|
|
>>> x = np.linspace(min(bins), max(bins), 10000)
|
|||
|
|
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
|
|||
|
|
... / (x * sigma * np.sqrt(2 * np.pi)))
|
|||
|
|
|
|||
|
|
>>> plt.plot(x, pdf, linewidth=2, color='r')
|
|||
|
|
>>> plt.axis('tight')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
Demonstrate that taking the products of random samples from a uniform
|
|||
|
|
distribution can be fit well by a log-normal probability density
|
|||
|
|
function.
|
|||
|
|
|
|||
|
|
>>> # Generate a thousand samples: each is the product of 100 random
|
|||
|
|
>>> # values, drawn from a normal distribution.
|
|||
|
|
>>> b = []
|
|||
|
|
>>> for i in range(1000):
|
|||
|
|
... a = 10. + np.random.standard_normal(100)
|
|||
|
|
... b.append(np.prod(a))
|
|||
|
|
|
|||
|
|
>>> b = np.array(b) / np.min(b) # scale values to be positive
|
|||
|
|
>>> count, bins, ignored = plt.hist(b, 100, density=True, align='mid')
|
|||
|
|
>>> sigma = np.std(np.log(b))
|
|||
|
|
>>> mu = np.mean(np.log(b))
|
|||
|
|
|
|||
|
|
>>> x = np.linspace(min(bins), max(bins), 10000)
|
|||
|
|
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
|
|||
|
|
... / (x * sigma * np.sqrt(2 * np.pi)))
|
|||
|
|
|
|||
|
|
>>> plt.plot(x, pdf, color='r', linewidth=2)
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
logistic(loc=0.0, scale=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a logistic distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from a logistic distribution with specified
|
|||
|
|
parameters, loc (location or mean, also median), and scale (>0).
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.logistic`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
loc : float or array_like of floats, optional
|
|||
|
|
Parameter of the distribution. Default is 0.
|
|||
|
|
scale : float or array_like of floats, optional
|
|||
|
|
Parameter of the distribution. Must be non-negative.
|
|||
|
|
Default is 1.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``loc`` and ``scale`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized logistic distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.logistic : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.logistic: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the Logistic distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(x) = P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2},
|
|||
|
|
|
|||
|
|
where :math:`\mu` = location and :math:`s` = scale.
|
|||
|
|
|
|||
|
|
The Logistic distribution is used in Extreme Value problems where it
|
|||
|
|
can act as a mixture of Gumbel distributions, in Epidemiology, and by
|
|||
|
|
the World Chess Federation (FIDE) where it is used in the Elo ranking
|
|||
|
|
system, assuming the performance of each player is a logistically
|
|||
|
|
distributed random variable.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Reiss, R.-D. and Thomas M. (2001), "Statistical Analysis of
|
|||
|
|
Extreme Values, from Insurance, Finance, Hydrology and Other
|
|||
|
|
Fields," Birkhauser Verlag, Basel, pp 132-133.
|
|||
|
|
.. [2] Weisstein, Eric W. "Logistic Distribution." From
|
|||
|
|
MathWorld--A Wolfram Web Resource.
|
|||
|
|
http://mathworld.wolfram.com/LogisticDistribution.html
|
|||
|
|
.. [3] Wikipedia, "Logistic-distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Logistic_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> loc, scale = 10, 1
|
|||
|
|
>>> s = np.random.logistic(loc, scale, 10000)
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(s, bins=50)
|
|||
|
|
|
|||
|
|
# plot against distribution
|
|||
|
|
|
|||
|
|
>>> def logist(x, loc, scale):
|
|||
|
|
... return np.exp((loc-x)/scale)/(scale*(1+np.exp((loc-x)/scale))**2)
|
|||
|
|
>>> lgst_val = logist(bins, loc, scale)
|
|||
|
|
>>> plt.plot(bins, lgst_val * count.max() / lgst_val.max())
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
hypergeometric(ngood, nbad, nsample, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a Hypergeometric distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from a hypergeometric distribution with specified
|
|||
|
|
parameters, `ngood` (ways to make a good selection), `nbad` (ways to make
|
|||
|
|
a bad selection), and `nsample` (number of items sampled, which is less
|
|||
|
|
than or equal to the sum ``ngood + nbad``).
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the
|
|||
|
|
`~numpy.random.Generator.hypergeometric`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
ngood : int or array_like of ints
|
|||
|
|
Number of ways to make a good selection. Must be nonnegative.
|
|||
|
|
nbad : int or array_like of ints
|
|||
|
|
Number of ways to make a bad selection. Must be nonnegative.
|
|||
|
|
nsample : int or array_like of ints
|
|||
|
|
Number of items sampled. Must be at least 1 and at most
|
|||
|
|
``ngood + nbad``.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if `ngood`, `nbad`, and `nsample`
|
|||
|
|
are all scalars. Otherwise, ``np.broadcast(ngood, nbad, nsample).size``
|
|||
|
|
samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized hypergeometric distribution. Each
|
|||
|
|
sample is the number of good items within a randomly selected subset of
|
|||
|
|
size `nsample` taken from a set of `ngood` good items and `nbad` bad items.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.hypergeom : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.hypergeometric: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the Hypergeometric distribution is
|
|||
|
|
|
|||
|
|
.. math:: P(x) = \frac{\binom{g}{x}\binom{b}{n-x}}{\binom{g+b}{n}},
|
|||
|
|
|
|||
|
|
where :math:`0 \le x \le n` and :math:`n-b \le x \le g`
|
|||
|
|
|
|||
|
|
for P(x) the probability of ``x`` good results in the drawn sample,
|
|||
|
|
g = `ngood`, b = `nbad`, and n = `nsample`.
|
|||
|
|
|
|||
|
|
Consider an urn with black and white marbles in it, `ngood` of them
|
|||
|
|
are black and `nbad` are white. If you draw `nsample` balls without
|
|||
|
|
replacement, then the hypergeometric distribution describes the
|
|||
|
|
distribution of black balls in the drawn sample.
|
|||
|
|
|
|||
|
|
Note that this distribution is very similar to the binomial
|
|||
|
|
distribution, except that in this case, samples are drawn without
|
|||
|
|
replacement, whereas in the Binomial case samples are drawn with
|
|||
|
|
replacement (or the sample space is infinite). As the sample space
|
|||
|
|
becomes large, this distribution approaches the binomial.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Lentner, Marvin, "Elementary Applied Statistics", Bogden
|
|||
|
|
and Quigley, 1972.
|
|||
|
|
.. [2] Weisstein, Eric W. "Hypergeometric Distribution." From
|
|||
|
|
MathWorld--A Wolfram Web Resource.
|
|||
|
|
http://mathworld.wolfram.com/HypergeometricDistribution.html
|
|||
|
|
.. [3] Wikipedia, "Hypergeometric distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Hypergeometric_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> ngood, nbad, nsamp = 100, 2, 10
|
|||
|
|
# number of good, number of bad, and number of samples
|
|||
|
|
>>> s = np.random.hypergeometric(ngood, nbad, nsamp, 1000)
|
|||
|
|
>>> from matplotlib.pyplot import hist
|
|||
|
|
>>> hist(s)
|
|||
|
|
# note that it is very unlikely to grab both bad items
|
|||
|
|
|
|||
|
|
Suppose you have an urn with 15 white and 15 black marbles.
|
|||
|
|
If you pull 15 marbles at random, how likely is it that
|
|||
|
|
12 or more of them are one color?
|
|||
|
|
|
|||
|
|
>>> s = np.random.hypergeometric(15, 15, 15, 100000)
|
|||
|
|
>>> sum(s>=12)/100000. + sum(s<=3)/100000.
|
|||
|
|
# answer = 0.003 ... pretty unlikely!
|
|||
|
|
|
|||
|
|
|
|||
|
|
gumbel(loc=0.0, scale=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a Gumbel distribution.
|
|||
|
|
|
|||
|
|
Draw samples from a Gumbel distribution with specified location and
|
|||
|
|
scale. For more information on the Gumbel distribution, see
|
|||
|
|
Notes and References below.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.gumbel`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
loc : float or array_like of floats, optional
|
|||
|
|
The location of the mode of the distribution. Default is 0.
|
|||
|
|
scale : float or array_like of floats, optional
|
|||
|
|
The scale parameter of the distribution. Default is 1. Must be non-
|
|||
|
|
negative.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``loc`` and ``scale`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized Gumbel distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.gumbel_l
|
|||
|
|
scipy.stats.gumbel_r
|
|||
|
|
scipy.stats.genextreme
|
|||
|
|
weibull
|
|||
|
|
random.Generator.gumbel: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme
|
|||
|
|
Value Type I) distribution is one of a class of Generalized Extreme
|
|||
|
|
Value (GEV) distributions used in modeling extreme value problems.
|
|||
|
|
The Gumbel is a special case of the Extreme Value Type I distribution
|
|||
|
|
for maximums from distributions with "exponential-like" tails.
|
|||
|
|
|
|||
|
|
The probability density for the Gumbel distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/
|
|||
|
|
\beta}},
|
|||
|
|
|
|||
|
|
where :math:`\mu` is the mode, a location parameter, and
|
|||
|
|
:math:`\beta` is the scale parameter.
|
|||
|
|
|
|||
|
|
The Gumbel (named for German mathematician Emil Julius Gumbel) was used
|
|||
|
|
very early in the hydrology literature, for modeling the occurrence of
|
|||
|
|
flood events. It is also used for modeling maximum wind speed and
|
|||
|
|
rainfall rates. It is a "fat-tailed" distribution - the probability of
|
|||
|
|
an event in the tail of the distribution is larger than if one used a
|
|||
|
|
Gaussian, hence the surprisingly frequent occurrence of 100-year
|
|||
|
|
floods. Floods were initially modeled as a Gaussian process, which
|
|||
|
|
underestimated the frequency of extreme events.
|
|||
|
|
|
|||
|
|
It is one of a class of extreme value distributions, the Generalized
|
|||
|
|
Extreme Value (GEV) distributions, which also includes the Weibull and
|
|||
|
|
Frechet.
|
|||
|
|
|
|||
|
|
The function has a mean of :math:`\mu + 0.57721\beta` and a variance
|
|||
|
|
of :math:`\frac{\pi^2}{6}\beta^2`.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Gumbel, E. J., "Statistics of Extremes,"
|
|||
|
|
New York: Columbia University Press, 1958.
|
|||
|
|
.. [2] Reiss, R.-D. and Thomas, M., "Statistical Analysis of Extreme
|
|||
|
|
Values from Insurance, Finance, Hydrology and Other Fields,"
|
|||
|
|
Basel: Birkhauser Verlag, 2001.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> mu, beta = 0, 0.1 # location and scale
|
|||
|
|
>>> s = np.random.gumbel(mu, beta, 1000)
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> count, bins, ignored = plt.hist(s, 30, density=True)
|
|||
|
|
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
|
|||
|
|
... * np.exp( -np.exp( -(bins - mu) /beta) ),
|
|||
|
|
... linewidth=2, color='r')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
Show how an extreme value distribution can arise from a Gaussian process
|
|||
|
|
and compare to a Gaussian:
|
|||
|
|
|
|||
|
|
>>> means = []
|
|||
|
|
>>> maxima = []
|
|||
|
|
>>> for i in range(0,1000) :
|
|||
|
|
... a = np.random.normal(mu, beta, 1000)
|
|||
|
|
... means.append(a.mean())
|
|||
|
|
... maxima.append(a.max())
|
|||
|
|
>>> count, bins, ignored = plt.hist(maxima, 30, density=True)
|
|||
|
|
>>> beta = np.std(maxima) * np.sqrt(6) / np.pi
|
|||
|
|
>>> mu = np.mean(maxima) - 0.57721*beta
|
|||
|
|
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
|
|||
|
|
... * np.exp(-np.exp(-(bins - mu)/beta)),
|
|||
|
|
... linewidth=2, color='r')
|
|||
|
|
>>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi))
|
|||
|
|
... * np.exp(-(bins - mu)**2 / (2 * beta**2)),
|
|||
|
|
... linewidth=2, color='g')
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
geometric(p, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from the geometric distribution.
|
|||
|
|
|
|||
|
|
Bernoulli trials are experiments with one of two outcomes:
|
|||
|
|
success or failure (an example of such an experiment is flipping
|
|||
|
|
a coin). The geometric distribution models the number of trials
|
|||
|
|
that must be run in order to achieve success. It is therefore
|
|||
|
|
supported on the positive integers, ``k = 1, 2, ...``.
|
|||
|
|
|
|||
|
|
The probability mass function of the geometric distribution is
|
|||
|
|
|
|||
|
|
.. math:: f(k) = (1 - p)^{k - 1} p
|
|||
|
|
|
|||
|
|
where `p` is the probability of success of an individual trial.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.geometric`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
p : float or array_like of floats
|
|||
|
|
The probability of success of an individual trial.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``p`` is a scalar. Otherwise,
|
|||
|
|
``np.array(p).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized geometric distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.geometric: which should be used for new code.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw ten thousand values from the geometric distribution,
|
|||
|
|
with the probability of an individual success equal to 0.35:
|
|||
|
|
|
|||
|
|
>>> z = np.random.geometric(p=0.35, size=10000)
|
|||
|
|
|
|||
|
|
How many trials succeeded after a single run?
|
|||
|
|
|
|||
|
|
>>> (z == 1).sum() / 10000.
|
|||
|
|
0.34889999999999999 #random
|
|||
|
|
|
|||
|
|
|
|||
|
|
gamma(shape, scale=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a Gamma distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from a Gamma distribution with specified parameters,
|
|||
|
|
`shape` (sometimes designated "k") and `scale` (sometimes designated
|
|||
|
|
"theta"), where both parameters are > 0.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.gamma`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
shape : float or array_like of floats
|
|||
|
|
The shape of the gamma distribution. Must be non-negative.
|
|||
|
|
scale : float or array_like of floats, optional
|
|||
|
|
The scale of the gamma distribution. Must be non-negative.
|
|||
|
|
Default is equal to 1.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``shape`` and ``scale`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(shape, scale).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized gamma distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.gamma : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.gamma: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The probability density for the Gamma distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},
|
|||
|
|
|
|||
|
|
where :math:`k` is the shape and :math:`\theta` the scale,
|
|||
|
|
and :math:`\Gamma` is the Gamma function.
|
|||
|
|
|
|||
|
|
The Gamma distribution is often used to model the times to failure of
|
|||
|
|
electronic components, and arises naturally in processes for which the
|
|||
|
|
waiting times between Poisson distributed events are relevant.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A
|
|||
|
|
Wolfram Web Resource.
|
|||
|
|
http://mathworld.wolfram.com/GammaDistribution.html
|
|||
|
|
.. [2] Wikipedia, "Gamma distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Gamma_distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> shape, scale = 2., 2. # mean=4, std=2*sqrt(2)
|
|||
|
|
>>> s = np.random.gamma(shape, scale, 1000)
|
|||
|
|
|
|||
|
|
Display the histogram of the samples, along with
|
|||
|
|
the probability density function:
|
|||
|
|
|
|||
|
|
>>> import matplotlib.pyplot as plt
|
|||
|
|
>>> import scipy.special as sps # doctest: +SKIP
|
|||
|
|
>>> count, bins, ignored = plt.hist(s, 50, density=True)
|
|||
|
|
>>> y = bins**(shape-1)*(np.exp(-bins/scale) / # doctest: +SKIP
|
|||
|
|
... (sps.gamma(shape)*scale**shape))
|
|||
|
|
>>> plt.plot(bins, y, linewidth=2, color='r') # doctest: +SKIP
|
|||
|
|
>>> plt.show()
|
|||
|
|
|
|||
|
|
|
|||
|
|
f(dfnum, dfden, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from an F distribution.
|
|||
|
|
|
|||
|
|
Samples are drawn from an F distribution with specified parameters,
|
|||
|
|
`dfnum` (degrees of freedom in numerator) and `dfden` (degrees of
|
|||
|
|
freedom in denominator), where both parameters must be greater than
|
|||
|
|
zero.
|
|||
|
|
|
|||
|
|
The random variate of the F distribution (also known as the
|
|||
|
|
Fisher distribution) is a continuous probability distribution
|
|||
|
|
that arises in ANOVA tests, and is the ratio of two chi-square
|
|||
|
|
variates.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.f`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
dfnum : float or array_like of floats
|
|||
|
|
Degrees of freedom in numerator, must be > 0.
|
|||
|
|
dfden : float or array_like of float
|
|||
|
|
Degrees of freedom in denominator, must be > 0.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``dfnum`` and ``dfden`` are both scalars.
|
|||
|
|
Otherwise, ``np.broadcast(dfnum, dfden).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized Fisher distribution.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
scipy.stats.f : probability density function, distribution or
|
|||
|
|
cumulative density function, etc.
|
|||
|
|
random.Generator.f: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The F statistic is used to compare in-group variances to between-group
|
|||
|
|
variances. Calculating the distribution depends on the sampling, and
|
|||
|
|
so it is a function of the respective degrees of freedom in the
|
|||
|
|
problem. The variable `dfnum` is the number of samples minus one, the
|
|||
|
|
between-groups degrees of freedom, while `dfden` is the within-groups
|
|||
|
|
degrees of freedom, the sum of the number of samples in each group
|
|||
|
|
minus the number of groups.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill,
|
|||
|
|
Fifth Edition, 2002.
|
|||
|
|
.. [2] Wikipedia, "F-distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/F-distribution
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
An example from Glantz[1], pp 47-40:
|
|||
|
|
|
|||
|
|
Two groups, children of diabetics (25 people) and children from people
|
|||
|
|
without diabetes (25 controls). Fasting blood glucose was measured,
|
|||
|
|
case group had a mean value of 86.1, controls had a mean value of
|
|||
|
|
82.2. Standard deviations were 2.09 and 2.49 respectively. Are these
|
|||
|
|
data consistent with the null hypothesis that the parents diabetic
|
|||
|
|
status does not affect their children's blood glucose levels?
|
|||
|
|
Calculating the F statistic from the data gives a value of 36.01.
|
|||
|
|
|
|||
|
|
Draw samples from the distribution:
|
|||
|
|
|
|||
|
|
>>> dfnum = 1. # between group degrees of freedom
|
|||
|
|
>>> dfden = 48. # within groups degrees of freedom
|
|||
|
|
>>> s = np.random.f(dfnum, dfden, 1000)
|
|||
|
|
|
|||
|
|
The lower bound for the top 1% of the samples is :
|
|||
|
|
|
|||
|
|
>>> np.sort(s)[-10]
|
|||
|
|
7.61988120985 # random
|
|||
|
|
|
|||
|
|
So there is about a 1% chance that the F statistic will exceed 7.62,
|
|||
|
|
the measured value is 36, so the null hypothesis is rejected at the 1%
|
|||
|
|
level.
|
|||
|
|
|
|||
|
|
|
|||
|
|
exponential(scale=1.0, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from an exponential distribution.
|
|||
|
|
|
|||
|
|
Its probability density function is
|
|||
|
|
|
|||
|
|
.. math:: f(x; \frac{1}{\beta}) = \frac{1}{\beta} \exp(-\frac{x}{\beta}),
|
|||
|
|
|
|||
|
|
for ``x > 0`` and 0 elsewhere. :math:`\beta` is the scale parameter,
|
|||
|
|
which is the inverse of the rate parameter :math:`\lambda = 1/\beta`.
|
|||
|
|
The rate parameter is an alternative, widely used parameterization
|
|||
|
|
of the exponential distribution [3]_.
|
|||
|
|
|
|||
|
|
The exponential distribution is a continuous analogue of the
|
|||
|
|
geometric distribution. It describes many common situations, such as
|
|||
|
|
the size of raindrops measured over many rainstorms [1]_, or the time
|
|||
|
|
between page requests to Wikipedia [2]_.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.exponential`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
scale : float or array_like of floats
|
|||
|
|
The scale parameter, :math:`\beta = 1/\lambda`. Must be
|
|||
|
|
non-negative.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``scale`` is a scalar. Otherwise,
|
|||
|
|
``np.array(scale).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized exponential distribution.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
A real world example: Assume a company has 10000 customer support
|
|||
|
|
agents and the average time between customer calls is 4 minutes.
|
|||
|
|
|
|||
|
|
>>> n = 10000
|
|||
|
|
>>> time_between_calls = np.random.default_rng().exponential(scale=4, size=n)
|
|||
|
|
|
|||
|
|
What is the probability that a customer will call in the next
|
|||
|
|
4 to 5 minutes?
|
|||
|
|
|
|||
|
|
>>> x = ((time_between_calls < 5).sum())/n
|
|||
|
|
>>> y = ((time_between_calls < 4).sum())/n
|
|||
|
|
>>> x-y
|
|||
|
|
0.08 # may vary
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.exponential: which should be used for new code.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] Peyton Z. Peebles Jr., "Probability, Random Variables and
|
|||
|
|
Random Signal Principles", 4th ed, 2001, p. 57.
|
|||
|
|
.. [2] Wikipedia, "Poisson process",
|
|||
|
|
https://en.wikipedia.org/wiki/Poisson_process
|
|||
|
|
.. [3] Wikipedia, "Exponential distribution",
|
|||
|
|
https://en.wikipedia.org/wiki/Exponential_distribution
|
|||
|
|
|
|||
|
|
|
|||
|
|
choice(a, size=None, replace=True, p=None)
|
|||
|
|
|
|||
|
|
Generates a random sample from a given 1-D array
|
|||
|
|
|
|||
|
|
.. versionadded:: 1.7.0
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.choice`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
a : 1-D array-like or int
|
|||
|
|
If an ndarray, a random sample is generated from its elements.
|
|||
|
|
If an int, the random sample is generated as if it were ``np.arange(a)``
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. Default is None, in which case a
|
|||
|
|
single value is returned.
|
|||
|
|
replace : boolean, optional
|
|||
|
|
Whether the sample is with or without replacement. Default is True,
|
|||
|
|
meaning that a value of ``a`` can be selected multiple times.
|
|||
|
|
p : 1-D array-like, optional
|
|||
|
|
The probabilities associated with each entry in a.
|
|||
|
|
If not given, the sample assumes a uniform distribution over all
|
|||
|
|
entries in ``a``.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
samples : single item or ndarray
|
|||
|
|
The generated random samples
|
|||
|
|
|
|||
|
|
Raises
|
|||
|
|
------
|
|||
|
|
ValueError
|
|||
|
|
If a is an int and less than zero, if a or p are not 1-dimensional,
|
|||
|
|
if a is an array-like of size 0, if p is not a vector of
|
|||
|
|
probabilities, if a and p have different lengths, or if
|
|||
|
|
replace=False and the sample size is greater than the population
|
|||
|
|
size
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
randint, shuffle, permutation
|
|||
|
|
random.Generator.choice: which should be used in new code
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
Setting user-specified probabilities through ``p`` uses a more general but less
|
|||
|
|
efficient sampler than the default. The general sampler produces a different sample
|
|||
|
|
than the optimized sampler even if each element of ``p`` is 1 / len(a).
|
|||
|
|
|
|||
|
|
Sampling random rows from a 2-D array is not possible with this function,
|
|||
|
|
but is possible with `Generator.choice` through its ``axis`` keyword.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
Generate a uniform random sample from np.arange(5) of size 3:
|
|||
|
|
|
|||
|
|
>>> np.random.choice(5, 3)
|
|||
|
|
array([0, 3, 4]) # random
|
|||
|
|
>>> #This is equivalent to np.random.randint(0,5,3)
|
|||
|
|
|
|||
|
|
Generate a non-uniform random sample from np.arange(5) of size 3:
|
|||
|
|
|
|||
|
|
>>> np.random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0])
|
|||
|
|
array([3, 3, 0]) # random
|
|||
|
|
|
|||
|
|
Generate a uniform random sample from np.arange(5) of size 3 without
|
|||
|
|
replacement:
|
|||
|
|
|
|||
|
|
>>> np.random.choice(5, 3, replace=False)
|
|||
|
|
array([3,1,0]) # random
|
|||
|
|
>>> #This is equivalent to np.random.permutation(np.arange(5))[:3]
|
|||
|
|
|
|||
|
|
Generate a non-uniform random sample from np.arange(5) of size
|
|||
|
|
3 without replacement:
|
|||
|
|
|
|||
|
|
>>> np.random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0])
|
|||
|
|
array([2, 3, 0]) # random
|
|||
|
|
|
|||
|
|
Any of the above can be repeated with an arbitrary array-like
|
|||
|
|
instead of just integers. For instance:
|
|||
|
|
|
|||
|
|
>>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher']
|
|||
|
|
>>> np.random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3])
|
|||
|
|
array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], # random
|
|||
|
|
dtype='<U11')
|
|||
|
|
|
|||
|
|
|
|||
|
|
chisquare(df, size=None)
|
|||
|
|
|
|||
|
|
Draw samples from a chi-square distribution.
|
|||
|
|
|
|||
|
|
When `df` independent random variables, each with standard normal
|
|||
|
|
distributions (mean 0, variance 1), are squared and summed, the
|
|||
|
|
resulting distribution is chi-square (see Notes). This distribution
|
|||
|
|
is often used in hypothesis testing.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.chisquare`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
df : float or array_like of floats
|
|||
|
|
Number of degrees of freedom, must be > 0.
|
|||
|
|
size : int or tuple of ints, optional
|
|||
|
|
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
|
|||
|
|
``m * n * k`` samples are drawn. If size is ``None`` (default),
|
|||
|
|
a single value is returned if ``df`` is a scalar. Otherwise,
|
|||
|
|
``np.array(df).size`` samples are drawn.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : ndarray or scalar
|
|||
|
|
Drawn samples from the parameterized chi-square distribution.
|
|||
|
|
|
|||
|
|
Raises
|
|||
|
|
------
|
|||
|
|
ValueError
|
|||
|
|
When `df` <= 0 or when an inappropriate `size` (e.g. ``size=-1``)
|
|||
|
|
is given.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.chisquare: which should be used for new code.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
The variable obtained by summing the squares of `df` independent,
|
|||
|
|
standard normally distributed random variables:
|
|||
|
|
|
|||
|
|
.. math:: Q = \sum_{i=0}^{\mathtt{df}} X^2_i
|
|||
|
|
|
|||
|
|
is chi-square distributed, denoted
|
|||
|
|
|
|||
|
|
.. math:: Q \sim \chi^2_k.
|
|||
|
|
|
|||
|
|
The probability density function of the chi-squared distribution is
|
|||
|
|
|
|||
|
|
.. math:: p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)}
|
|||
|
|
x^{k/2 - 1} e^{-x/2},
|
|||
|
|
|
|||
|
|
where :math:`\Gamma` is the gamma function,
|
|||
|
|
|
|||
|
|
.. math:: \Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt.
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
.. [1] NIST "Engineering Statistics Handbook"
|
|||
|
|
https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> np.random.chisquare(2,4)
|
|||
|
|
array([ 1.89920014, 9.00867716, 3.13710533, 5.62318272]) # random
|
|||
|
|
|
|||
|
|
bytes(length)
|
|||
|
|
|
|||
|
|
Return random bytes.
|
|||
|
|
|
|||
|
|
.. note::
|
|||
|
|
New code should use the `~numpy.random.Generator.bytes`
|
|||
|
|
method of a `~numpy.random.Generator` instance instead;
|
|||
|
|
please see the :ref:`random-quick-start`.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
length : int
|
|||
|
|
Number of random bytes.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
out : bytes
|
|||
|
|
String of length `length`.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
random.Generator.bytes: which should be used for new code.
|
|||
|
|
|
|||
|
|
Examples
|
|||
|
|
--------
|
|||
|
|
>>> np.random.bytes(10)
|
|||
|
|
b' eh\x85\x022SZ\xbf\xa4' #random
|