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# Copyright 2002 Gary Strangman. All rights reserved 

# Copyright 2002-2016 The SciPy Developers 

# 

# The original code from Gary Strangman was heavily adapted for 

# use in SciPy by Travis Oliphant. The original code came with the 

# following disclaimer: 

# 

# This software is provided "as-is". There are no expressed or implied 

# warranties of any kind, including, but not limited to, the warranties 

# of merchantability and fitness for a given application. In no event 

# shall Gary Strangman be liable for any direct, indirect, incidental, 

# special, exemplary or consequential damages (including, but not limited 

# to, loss of use, data or profits, or business interruption) however 

# caused and on any theory of liability, whether in contract, strict 

# liability or tort (including negligence or otherwise) arising in any way 

# out of the use of this software, even if advised of the possibility of 

# such damage. 

 

""" 

A collection of basic statistical functions for Python. The function 

names appear below. 

 

Some scalar functions defined here are also available in the scipy.special 

package where they work on arbitrary sized arrays. 

 

Disclaimers: The function list is obviously incomplete and, worse, the 

functions are not optimized. All functions have been tested (some more 

so than others), but they are far from bulletproof. Thus, as with any 

free software, no warranty or guarantee is expressed or implied. :-) A 

few extra functions that don't appear in the list below can be found by 

interested treasure-hunters. These functions don't necessarily have 

both list and array versions but were deemed useful. 

 

Central Tendency 

---------------- 

.. autosummary:: 

:toctree: generated/ 

 

gmean 

hmean 

mode 

 

Moments 

------- 

.. autosummary:: 

:toctree: generated/ 

 

moment 

variation 

skew 

kurtosis 

normaltest 

 

Altered Versions 

---------------- 

.. autosummary:: 

:toctree: generated/ 

 

tmean 

tvar 

tstd 

tsem 

describe 

 

Frequency Stats 

--------------- 

.. autosummary:: 

:toctree: generated/ 

 

itemfreq 

scoreatpercentile 

percentileofscore 

cumfreq 

relfreq 

 

Variability 

----------- 

.. autosummary:: 

:toctree: generated/ 

 

obrientransform 

sem 

zmap 

zscore 

iqr 

 

Trimming Functions 

------------------ 

.. autosummary:: 

:toctree: generated/ 

 

trimboth 

trim1 

 

Correlation Functions 

--------------------- 

.. autosummary:: 

:toctree: generated/ 

 

pearsonr 

fisher_exact 

spearmanr 

pointbiserialr 

kendalltau 

weightedtau 

linregress 

theilslopes 

 

Inferential Stats 

----------------- 

.. autosummary:: 

:toctree: generated/ 

 

ttest_1samp 

ttest_ind 

ttest_ind_from_stats 

ttest_rel 

chisquare 

power_divergence 

ks_2samp 

mannwhitneyu 

ranksums 

wilcoxon 

kruskal 

friedmanchisquare 

combine_pvalues 

 

Statistical Distances 

--------------------- 

.. autosummary:: 

:toctree: generated/ 

 

wasserstein_distance 

energy_distance 

 

ANOVA Functions 

--------------- 

.. autosummary:: 

:toctree: generated/ 

 

f_oneway 

 

Support Functions 

----------------- 

.. autosummary:: 

:toctree: generated/ 

 

rankdata 

 

References 

---------- 

.. [CRCProbStat2000] Zwillinger, D. and Kokoska, S. (2000). CRC Standard 

Probability and Statistics Tables and Formulae. Chapman & Hall: New 

York. 2000. 

 

""" 

 

from __future__ import division, print_function, absolute_import 

 

import warnings 

import math 

from collections import namedtuple 

 

import numpy as np 

from numpy import array, asarray, ma, zeros 

 

from scipy._lib.six import callable, string_types 

from scipy._lib._version import NumpyVersion 

import scipy.special as special 

import scipy.linalg as linalg 

from . import distributions 

from . import mstats_basic 

from ._distn_infrastructure import _lazywhere 

from ._stats_mstats_common import _find_repeats, linregress, theilslopes 

from ._stats import _kendall_dis, _toint64, _weightedrankedtau 

 

 

__all__ = ['find_repeats', 'gmean', 'hmean', 'mode', 'tmean', 'tvar', 

'tmin', 'tmax', 'tstd', 'tsem', 'moment', 'variation', 

'skew', 'kurtosis', 'describe', 'skewtest', 'kurtosistest', 

'normaltest', 'jarque_bera', 'itemfreq', 

'scoreatpercentile', 'percentileofscore', 

'cumfreq', 'relfreq', 'obrientransform', 

'sem', 'zmap', 'zscore', 'iqr', 

'sigmaclip', 'trimboth', 'trim1', 'trim_mean', 'f_oneway', 

'pearsonr', 'fisher_exact', 'spearmanr', 'pointbiserialr', 

'kendalltau', 'weightedtau', 

'linregress', 'theilslopes', 'ttest_1samp', 

'ttest_ind', 'ttest_ind_from_stats', 'ttest_rel', 'kstest', 

'chisquare', 'power_divergence', 'ks_2samp', 'mannwhitneyu', 

'tiecorrect', 'ranksums', 'kruskal', 'friedmanchisquare', 

'rankdata', 

'combine_pvalues', 'wasserstein_distance', 'energy_distance'] 

 

 

def _chk_asarray(a, axis): 

if axis is None: 

a = np.ravel(a) 

outaxis = 0 

else: 

a = np.asarray(a) 

outaxis = axis 

 

if a.ndim == 0: 

a = np.atleast_1d(a) 

 

return a, outaxis 

 

 

def _chk2_asarray(a, b, axis): 

if axis is None: 

a = np.ravel(a) 

b = np.ravel(b) 

outaxis = 0 

else: 

a = np.asarray(a) 

b = np.asarray(b) 

outaxis = axis 

 

if a.ndim == 0: 

a = np.atleast_1d(a) 

if b.ndim == 0: 

b = np.atleast_1d(b) 

 

return a, b, outaxis 

 

 

def _contains_nan(a, nan_policy='propagate'): 

policies = ['propagate', 'raise', 'omit'] 

if nan_policy not in policies: 

raise ValueError("nan_policy must be one of {%s}" % 

', '.join("'%s'" % s for s in policies)) 

try: 

# Calling np.sum to avoid creating a huge array into memory 

# e.g. np.isnan(a).any() 

with np.errstate(invalid='ignore'): 

contains_nan = np.isnan(np.sum(a)) 

except TypeError: 

# If the check cannot be properly performed we fallback to omitting 

# nan values and raising a warning. This can happen when attempting to 

# sum things that are not numbers (e.g. as in the function `mode`). 

contains_nan = False 

nan_policy = 'omit' 

warnings.warn("The input array could not be properly checked for nan " 

"values. nan values will be ignored.", RuntimeWarning) 

 

if contains_nan and nan_policy == 'raise': 

raise ValueError("The input contains nan values") 

 

return (contains_nan, nan_policy) 

 

 

def gmean(a, axis=0, dtype=None): 

""" 

Compute the geometric mean along the specified axis. 

 

Return the geometric average of the array elements. 

That is: n-th root of (x1 * x2 * ... * xn) 

 

Parameters 

---------- 

a : array_like 

Input array or object that can be converted to an array. 

axis : int or None, optional 

Axis along which the geometric mean is computed. Default is 0. 

If None, compute over the whole array `a`. 

dtype : dtype, optional 

Type of the returned array and of the accumulator in which the 

elements are summed. If dtype is not specified, it defaults to the 

dtype of a, unless a has an integer dtype with a precision less than 

that of the default platform integer. In that case, the default 

platform integer is used. 

 

Returns 

------- 

gmean : ndarray 

see dtype parameter above 

 

See Also 

-------- 

numpy.mean : Arithmetic average 

numpy.average : Weighted average 

hmean : Harmonic mean 

 

Notes 

----- 

The geometric average is computed over a single dimension of the input 

array, axis=0 by default, or all values in the array if axis=None. 

float64 intermediate and return values are used for integer inputs. 

 

Use masked arrays to ignore any non-finite values in the input or that 

arise in the calculations such as Not a Number and infinity because masked 

arrays automatically mask any non-finite values. 

 

Examples 

-------- 

>>> from scipy.stats import gmean 

>>> gmean([1, 4]) 

2.0 

>>> gmean([1, 2, 3, 4, 5, 6, 7]) 

3.3800151591412964 

""" 

if not isinstance(a, np.ndarray): 

# if not an ndarray object attempt to convert it 

log_a = np.log(np.array(a, dtype=dtype)) 

elif dtype: 

# Must change the default dtype allowing array type 

if isinstance(a, np.ma.MaskedArray): 

log_a = np.log(np.ma.asarray(a, dtype=dtype)) 

else: 

log_a = np.log(np.asarray(a, dtype=dtype)) 

else: 

log_a = np.log(a) 

return np.exp(log_a.mean(axis=axis)) 

 

 

def hmean(a, axis=0, dtype=None): 

""" 

Calculate the harmonic mean along the specified axis. 

 

That is: n / (1/x1 + 1/x2 + ... + 1/xn) 

 

Parameters 

---------- 

a : array_like 

Input array, masked array or object that can be converted to an array. 

axis : int or None, optional 

Axis along which the harmonic mean is computed. Default is 0. 

If None, compute over the whole array `a`. 

dtype : dtype, optional 

Type of the returned array and of the accumulator in which the 

elements are summed. If `dtype` is not specified, it defaults to the 

dtype of `a`, unless `a` has an integer `dtype` with a precision less 

than that of the default platform integer. In that case, the default 

platform integer is used. 

 

Returns 

------- 

hmean : ndarray 

see `dtype` parameter above 

 

See Also 

-------- 

numpy.mean : Arithmetic average 

numpy.average : Weighted average 

gmean : Geometric mean 

 

Notes 

----- 

The harmonic mean is computed over a single dimension of the input 

array, axis=0 by default, or all values in the array if axis=None. 

float64 intermediate and return values are used for integer inputs. 

 

Use masked arrays to ignore any non-finite values in the input or that 

arise in the calculations such as Not a Number and infinity. 

 

Examples 

-------- 

>>> from scipy.stats import hmean 

>>> hmean([1, 4]) 

1.6000000000000001 

>>> hmean([1, 2, 3, 4, 5, 6, 7]) 

2.6997245179063363 

""" 

if not isinstance(a, np.ndarray): 

a = np.array(a, dtype=dtype) 

if np.all(a > 0): 

# Harmonic mean only defined if greater than zero 

if isinstance(a, np.ma.MaskedArray): 

size = a.count(axis) 

else: 

if axis is None: 

a = a.ravel() 

size = a.shape[0] 

else: 

size = a.shape[axis] 

return size / np.sum(1.0 / a, axis=axis, dtype=dtype) 

else: 

raise ValueError("Harmonic mean only defined if all elements greater " 

"than zero") 

 

 

ModeResult = namedtuple('ModeResult', ('mode', 'count')) 

 

 

def mode(a, axis=0, nan_policy='propagate'): 

""" 

Return an array of the modal (most common) value in the passed array. 

 

If there is more than one such value, only the smallest is returned. 

The bin-count for the modal bins is also returned. 

 

Parameters 

---------- 

a : array_like 

n-dimensional array of which to find mode(s). 

axis : int or None, optional 

Axis along which to operate. Default is 0. If None, compute over 

the whole array `a`. 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. 

 

Returns 

------- 

mode : ndarray 

Array of modal values. 

count : ndarray 

Array of counts for each mode. 

 

Examples 

-------- 

>>> a = np.array([[6, 8, 3, 0], 

... [3, 2, 1, 7], 

... [8, 1, 8, 4], 

... [5, 3, 0, 5], 

... [4, 7, 5, 9]]) 

>>> from scipy import stats 

>>> stats.mode(a) 

(array([[3, 1, 0, 0]]), array([[1, 1, 1, 1]])) 

 

To get mode of whole array, specify ``axis=None``: 

 

>>> stats.mode(a, axis=None) 

(array([3]), array([3])) 

 

""" 

a, axis = _chk_asarray(a, axis) 

if a.size == 0: 

return ModeResult(np.array([]), np.array([])) 

 

contains_nan, nan_policy = _contains_nan(a, nan_policy) 

 

if contains_nan and nan_policy == 'omit': 

a = ma.masked_invalid(a) 

return mstats_basic.mode(a, axis) 

 

scores = np.unique(np.ravel(a)) # get ALL unique values 

testshape = list(a.shape) 

testshape[axis] = 1 

oldmostfreq = np.zeros(testshape, dtype=a.dtype) 

oldcounts = np.zeros(testshape, dtype=int) 

for score in scores: 

template = (a == score) 

counts = np.expand_dims(np.sum(template, axis), axis) 

mostfrequent = np.where(counts > oldcounts, score, oldmostfreq) 

oldcounts = np.maximum(counts, oldcounts) 

oldmostfreq = mostfrequent 

 

return ModeResult(mostfrequent, oldcounts) 

 

 

def _mask_to_limits(a, limits, inclusive): 

"""Mask an array for values outside of given limits. 

 

This is primarily a utility function. 

 

Parameters 

---------- 

a : array 

limits : (float or None, float or None) 

A tuple consisting of the (lower limit, upper limit). Values in the 

input array less than the lower limit or greater than the upper limit 

will be masked out. None implies no limit. 

inclusive : (bool, bool) 

A tuple consisting of the (lower flag, upper flag). These flags 

determine whether values exactly equal to lower or upper are allowed. 

 

Returns 

------- 

A MaskedArray. 

 

Raises 

------ 

A ValueError if there are no values within the given limits. 

""" 

lower_limit, upper_limit = limits 

lower_include, upper_include = inclusive 

am = ma.MaskedArray(a) 

if lower_limit is not None: 

if lower_include: 

am = ma.masked_less(am, lower_limit) 

else: 

am = ma.masked_less_equal(am, lower_limit) 

 

if upper_limit is not None: 

if upper_include: 

am = ma.masked_greater(am, upper_limit) 

else: 

am = ma.masked_greater_equal(am, upper_limit) 

 

if am.count() == 0: 

raise ValueError("No array values within given limits") 

 

return am 

 

 

def tmean(a, limits=None, inclusive=(True, True), axis=None): 

""" 

Compute the trimmed mean. 

 

This function finds the arithmetic mean of given values, ignoring values 

outside the given `limits`. 

 

Parameters 

---------- 

a : array_like 

Array of values. 

limits : None or (lower limit, upper limit), optional 

Values in the input array less than the lower limit or greater than the 

upper limit will be ignored. When limits is None (default), then all 

values are used. Either of the limit values in the tuple can also be 

None representing a half-open interval. 

inclusive : (bool, bool), optional 

A tuple consisting of the (lower flag, upper flag). These flags 

determine whether values exactly equal to the lower or upper limits 

are included. The default value is (True, True). 

axis : int or None, optional 

Axis along which to compute test. Default is None. 

 

Returns 

------- 

tmean : float 

 

See also 

-------- 

trim_mean : returns mean after trimming a proportion from both tails. 

 

Examples 

-------- 

>>> from scipy import stats 

>>> x = np.arange(20) 

>>> stats.tmean(x) 

9.5 

>>> stats.tmean(x, (3,17)) 

10.0 

 

""" 

a = asarray(a) 

if limits is None: 

return np.mean(a, None) 

 

am = _mask_to_limits(a.ravel(), limits, inclusive) 

return am.mean(axis=axis) 

 

 

def tvar(a, limits=None, inclusive=(True, True), axis=0, ddof=1): 

""" 

Compute the trimmed variance. 

 

This function computes the sample variance of an array of values, 

while ignoring values which are outside of given `limits`. 

 

Parameters 

---------- 

a : array_like 

Array of values. 

limits : None or (lower limit, upper limit), optional 

Values in the input array less than the lower limit or greater than the 

upper limit will be ignored. When limits is None, then all values are 

used. Either of the limit values in the tuple can also be None 

representing a half-open interval. The default value is None. 

inclusive : (bool, bool), optional 

A tuple consisting of the (lower flag, upper flag). These flags 

determine whether values exactly equal to the lower or upper limits 

are included. The default value is (True, True). 

axis : int or None, optional 

Axis along which to operate. Default is 0. If None, compute over the 

whole array `a`. 

ddof : int, optional 

Delta degrees of freedom. Default is 1. 

 

Returns 

------- 

tvar : float 

Trimmed variance. 

 

Notes 

----- 

`tvar` computes the unbiased sample variance, i.e. it uses a correction 

factor ``n / (n - 1)``. 

 

Examples 

-------- 

>>> from scipy import stats 

>>> x = np.arange(20) 

>>> stats.tvar(x) 

35.0 

>>> stats.tvar(x, (3,17)) 

20.0 

 

""" 

a = asarray(a) 

a = a.astype(float).ravel() 

if limits is None: 

n = len(a) 

return a.var() * n / (n - 1.) 

am = _mask_to_limits(a, limits, inclusive) 

return np.ma.var(am, ddof=ddof, axis=axis) 

 

 

def tmin(a, lowerlimit=None, axis=0, inclusive=True, nan_policy='propagate'): 

""" 

Compute the trimmed minimum. 

 

This function finds the miminum value of an array `a` along the 

specified axis, but only considering values greater than a specified 

lower limit. 

 

Parameters 

---------- 

a : array_like 

array of values 

lowerlimit : None or float, optional 

Values in the input array less than the given limit will be ignored. 

When lowerlimit is None, then all values are used. The default value 

is None. 

axis : int or None, optional 

Axis along which to operate. Default is 0. If None, compute over the 

whole array `a`. 

inclusive : {True, False}, optional 

This flag determines whether values exactly equal to the lower limit 

are included. The default value is True. 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. 

 

Returns 

------- 

tmin : float, int or ndarray 

 

Examples 

-------- 

>>> from scipy import stats 

>>> x = np.arange(20) 

>>> stats.tmin(x) 

0 

 

>>> stats.tmin(x, 13) 

13 

 

>>> stats.tmin(x, 13, inclusive=False) 

14 

 

""" 

a, axis = _chk_asarray(a, axis) 

am = _mask_to_limits(a, (lowerlimit, None), (inclusive, False)) 

 

contains_nan, nan_policy = _contains_nan(am, nan_policy) 

 

if contains_nan and nan_policy == 'omit': 

am = ma.masked_invalid(am) 

 

res = ma.minimum.reduce(am, axis).data 

if res.ndim == 0: 

return res[()] 

return res 

 

 

def tmax(a, upperlimit=None, axis=0, inclusive=True, nan_policy='propagate'): 

""" 

Compute the trimmed maximum. 

 

This function computes the maximum value of an array along a given axis, 

while ignoring values larger than a specified upper limit. 

 

Parameters 

---------- 

a : array_like 

array of values 

upperlimit : None or float, optional 

Values in the input array greater than the given limit will be ignored. 

When upperlimit is None, then all values are used. The default value 

is None. 

axis : int or None, optional 

Axis along which to operate. Default is 0. If None, compute over the 

whole array `a`. 

inclusive : {True, False}, optional 

This flag determines whether values exactly equal to the upper limit 

are included. The default value is True. 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. 

 

Returns 

------- 

tmax : float, int or ndarray 

 

Examples 

-------- 

>>> from scipy import stats 

>>> x = np.arange(20) 

>>> stats.tmax(x) 

19 

 

>>> stats.tmax(x, 13) 

13 

 

>>> stats.tmax(x, 13, inclusive=False) 

12 

 

""" 

a, axis = _chk_asarray(a, axis) 

am = _mask_to_limits(a, (None, upperlimit), (False, inclusive)) 

 

contains_nan, nan_policy = _contains_nan(am, nan_policy) 

 

if contains_nan and nan_policy == 'omit': 

am = ma.masked_invalid(am) 

 

res = ma.maximum.reduce(am, axis).data 

if res.ndim == 0: 

return res[()] 

return res 

 

 

def tstd(a, limits=None, inclusive=(True, True), axis=0, ddof=1): 

""" 

Compute the trimmed sample standard deviation. 

 

This function finds the sample standard deviation of given values, 

ignoring values outside the given `limits`. 

 

Parameters 

---------- 

a : array_like 

array of values 

limits : None or (lower limit, upper limit), optional 

Values in the input array less than the lower limit or greater than the 

upper limit will be ignored. When limits is None, then all values are 

used. Either of the limit values in the tuple can also be None 

representing a half-open interval. The default value is None. 

inclusive : (bool, bool), optional 

A tuple consisting of the (lower flag, upper flag). These flags 

determine whether values exactly equal to the lower or upper limits 

are included. The default value is (True, True). 

axis : int or None, optional 

Axis along which to operate. Default is 0. If None, compute over the 

whole array `a`. 

ddof : int, optional 

Delta degrees of freedom. Default is 1. 

 

Returns 

------- 

tstd : float 

 

Notes 

----- 

`tstd` computes the unbiased sample standard deviation, i.e. it uses a 

correction factor ``n / (n - 1)``. 

 

Examples 

-------- 

>>> from scipy import stats 

>>> x = np.arange(20) 

>>> stats.tstd(x) 

5.9160797830996161 

>>> stats.tstd(x, (3,17)) 

4.4721359549995796 

 

""" 

return np.sqrt(tvar(a, limits, inclusive, axis, ddof)) 

 

 

def tsem(a, limits=None, inclusive=(True, True), axis=0, ddof=1): 

""" 

Compute the trimmed standard error of the mean. 

 

This function finds the standard error of the mean for given 

values, ignoring values outside the given `limits`. 

 

Parameters 

---------- 

a : array_like 

array of values 

limits : None or (lower limit, upper limit), optional 

Values in the input array less than the lower limit or greater than the 

upper limit will be ignored. When limits is None, then all values are 

used. Either of the limit values in the tuple can also be None 

representing a half-open interval. The default value is None. 

inclusive : (bool, bool), optional 

A tuple consisting of the (lower flag, upper flag). These flags 

determine whether values exactly equal to the lower or upper limits 

are included. The default value is (True, True). 

axis : int or None, optional 

Axis along which to operate. Default is 0. If None, compute over the 

whole array `a`. 

ddof : int, optional 

Delta degrees of freedom. Default is 1. 

 

Returns 

------- 

tsem : float 

 

Notes 

----- 

`tsem` uses unbiased sample standard deviation, i.e. it uses a 

correction factor ``n / (n - 1)``. 

 

Examples 

-------- 

>>> from scipy import stats 

>>> x = np.arange(20) 

>>> stats.tsem(x) 

1.3228756555322954 

>>> stats.tsem(x, (3,17)) 

1.1547005383792515 

 

""" 

a = np.asarray(a).ravel() 

if limits is None: 

return a.std(ddof=ddof) / np.sqrt(a.size) 

 

am = _mask_to_limits(a, limits, inclusive) 

sd = np.sqrt(np.ma.var(am, ddof=ddof, axis=axis)) 

return sd / np.sqrt(am.count()) 

 

 

##################################### 

# MOMENTS # 

##################################### 

 

def moment(a, moment=1, axis=0, nan_policy='propagate'): 

r""" 

Calculate the nth moment about the mean for a sample. 

 

A moment is a specific quantitative measure of the shape of a set of 

points. It is often used to calculate coefficients of skewness and kurtosis 

due to its close relationship with them. 

 

 

Parameters 

---------- 

a : array_like 

data 

moment : int or array_like of ints, optional 

order of central moment that is returned. Default is 1. 

axis : int or None, optional 

Axis along which the central moment is computed. Default is 0. 

If None, compute over the whole array `a`. 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. 

 

Returns 

------- 

n-th central moment : ndarray or float 

The appropriate moment along the given axis or over all values if axis 

is None. The denominator for the moment calculation is the number of 

observations, no degrees of freedom correction is done. 

 

See also 

-------- 

kurtosis, skew, describe 

 

Notes 

----- 

The k-th central moment of a data sample is: 

 

.. math:: 

 

m_k = \frac{1}{n} \sum_{i = 1}^n (x_i - \bar{x})^k 

 

Where n is the number of samples and x-bar is the mean. This function uses 

exponentiation by squares [1]_ for efficiency. 

 

References 

---------- 

.. [1] http://eli.thegreenplace.net/2009/03/21/efficient-integer-exponentiation-algorithms 

 

Examples 

-------- 

>>> from scipy.stats import moment 

>>> moment([1, 2, 3, 4, 5], moment=1) 

0.0 

>>> moment([1, 2, 3, 4, 5], moment=2) 

2.0 

""" 

a, axis = _chk_asarray(a, axis) 

 

contains_nan, nan_policy = _contains_nan(a, nan_policy) 

 

if contains_nan and nan_policy == 'omit': 

a = ma.masked_invalid(a) 

return mstats_basic.moment(a, moment, axis) 

 

if a.size == 0: 

# empty array, return nan(s) with shape matching `moment` 

if np.isscalar(moment): 

return np.nan 

else: 

return np.ones(np.asarray(moment).shape, dtype=np.float64) * np.nan 

 

# for array_like moment input, return a value for each. 

if not np.isscalar(moment): 

mmnt = [_moment(a, i, axis) for i in moment] 

return np.array(mmnt) 

else: 

return _moment(a, moment, axis) 

 

 

def _moment(a, moment, axis): 

if np.abs(moment - np.round(moment)) > 0: 

raise ValueError("All moment parameters must be integers") 

 

if moment == 0: 

# When moment equals 0, the result is 1, by definition. 

shape = list(a.shape) 

del shape[axis] 

if shape: 

# return an actual array of the appropriate shape 

return np.ones(shape, dtype=float) 

else: 

# the input was 1D, so return a scalar instead of a rank-0 array 

return 1.0 

 

elif moment == 1: 

# By definition the first moment about the mean is 0. 

shape = list(a.shape) 

del shape[axis] 

if shape: 

# return an actual array of the appropriate shape 

return np.zeros(shape, dtype=float) 

else: 

# the input was 1D, so return a scalar instead of a rank-0 array 

return np.float64(0.0) 

else: 

# Exponentiation by squares: form exponent sequence 

n_list = [moment] 

current_n = moment 

while current_n > 2: 

if current_n % 2: 

current_n = (current_n - 1) / 2 

else: 

current_n /= 2 

n_list.append(current_n) 

 

# Starting point for exponentiation by squares 

a_zero_mean = a - np.expand_dims(np.mean(a, axis), axis) 

if n_list[-1] == 1: 

s = a_zero_mean.copy() 

else: 

s = a_zero_mean**2 

 

# Perform multiplications 

for n in n_list[-2::-1]: 

s = s**2 

if n % 2: 

s *= a_zero_mean 

return np.mean(s, axis) 

 

 

def variation(a, axis=0, nan_policy='propagate'): 

""" 

Compute the coefficient of variation, the ratio of the biased standard 

deviation to the mean. 

 

Parameters 

---------- 

a : array_like 

Input array. 

axis : int or None, optional 

Axis along which to calculate the coefficient of variation. Default 

is 0. If None, compute over the whole array `a`. 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. 

 

Returns 

------- 

variation : ndarray 

The calculated variation along the requested axis. 

 

References 

---------- 

.. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard 

Probability and Statistics Tables and Formulae. Chapman & Hall: New 

York. 2000. 

 

Examples 

-------- 

>>> from scipy.stats import variation 

>>> variation([1, 2, 3, 4, 5]) 

0.47140452079103173 

""" 

a, axis = _chk_asarray(a, axis) 

 

contains_nan, nan_policy = _contains_nan(a, nan_policy) 

 

if contains_nan and nan_policy == 'omit': 

a = ma.masked_invalid(a) 

return mstats_basic.variation(a, axis) 

 

return a.std(axis) / a.mean(axis) 

 

 

def skew(a, axis=0, bias=True, nan_policy='propagate'): 

""" 

Compute the skewness of a data set. 

 

For normally distributed data, the skewness should be about 0. For 

unimodal continuous distributions, a skewness value > 0 means that 

there is more weight in the right tail of the distribution. The 

function `skewtest` can be used to determine if the skewness value 

is close enough to 0, statistically speaking. 

 

Parameters 

---------- 

a : ndarray 

data 

axis : int or None, optional 

Axis along which skewness is calculated. Default is 0. 

If None, compute over the whole array `a`. 

bias : bool, optional 

If False, then the calculations are corrected for statistical bias. 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. 

 

Returns 

------- 

skewness : ndarray 

The skewness of values along an axis, returning 0 where all values are 

equal. 

 

References 

---------- 

 

.. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard 

Probability and Statistics Tables and Formulae. Chapman & Hall: New 

York. 2000. 

Section 2.2.24.1 

 

Examples 

-------- 

>>> from scipy.stats import skew 

>>> skew([1, 2, 3, 4, 5]) 

0.0 

>>> skew([2, 8, 0, 4, 1, 9, 9, 0]) 

0.2650554122698573 

""" 

a, axis = _chk_asarray(a, axis) 

n = a.shape[axis] 

 

contains_nan, nan_policy = _contains_nan(a, nan_policy) 

 

if contains_nan and nan_policy == 'omit': 

a = ma.masked_invalid(a) 

return mstats_basic.skew(a, axis, bias) 

 

m2 = moment(a, 2, axis) 

m3 = moment(a, 3, axis) 

zero = (m2 == 0) 

vals = _lazywhere(~zero, (m2, m3), 

lambda m2, m3: m3 / m2**1.5, 

0.) 

if not bias: 

can_correct = (n > 2) & (m2 > 0) 

if can_correct.any(): 

m2 = np.extract(can_correct, m2) 

m3 = np.extract(can_correct, m3) 

nval = np.sqrt((n - 1.0) * n) / (n - 2.0) * m3 / m2**1.5 

np.place(vals, can_correct, nval) 

 

if vals.ndim == 0: 

return vals.item() 

 

return vals 

 

 

def kurtosis(a, axis=0, fisher=True, bias=True, nan_policy='propagate'): 

""" 

Compute the kurtosis (Fisher or Pearson) of a dataset. 

 

Kurtosis is the fourth central moment divided by the square of the 

variance. If Fisher's definition is used, then 3.0 is subtracted from 

the result to give 0.0 for a normal distribution. 

 

If bias is False then the kurtosis is calculated using k statistics to 

eliminate bias coming from biased moment estimators 

 

Use `kurtosistest` to see if result is close enough to normal. 

 

Parameters 

---------- 

a : array 

data for which the kurtosis is calculated 

axis : int or None, optional 

Axis along which the kurtosis is calculated. Default is 0. 

If None, compute over the whole array `a`. 

fisher : bool, optional 

If True, Fisher's definition is used (normal ==> 0.0). If False, 

Pearson's definition is used (normal ==> 3.0). 

bias : bool, optional 

If False, then the calculations are corrected for statistical bias. 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. 

 

Returns 

------- 

kurtosis : array 

The kurtosis of values along an axis. If all values are equal, 

return -3 for Fisher's definition and 0 for Pearson's definition. 

 

References 

---------- 

.. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard 

Probability and Statistics Tables and Formulae. Chapman & Hall: New 

York. 2000. 

 

Examples 

-------- 

>>> from scipy.stats import kurtosis 

>>> kurtosis([1, 2, 3, 4, 5]) 

-1.3 

""" 

a, axis = _chk_asarray(a, axis) 

 

contains_nan, nan_policy = _contains_nan(a, nan_policy) 

 

if contains_nan and nan_policy == 'omit': 

a = ma.masked_invalid(a) 

return mstats_basic.kurtosis(a, axis, fisher, bias) 

 

n = a.shape[axis] 

m2 = moment(a, 2, axis) 

m4 = moment(a, 4, axis) 

zero = (m2 == 0) 

olderr = np.seterr(all='ignore') 

try: 

vals = np.where(zero, 0, m4 / m2**2.0) 

finally: 

np.seterr(**olderr) 

 

if not bias: 

can_correct = (n > 3) & (m2 > 0) 

if can_correct.any(): 

m2 = np.extract(can_correct, m2) 

m4 = np.extract(can_correct, m4) 

nval = 1.0/(n-2)/(n-3) * ((n**2-1.0)*m4/m2**2.0 - 3*(n-1)**2.0) 

np.place(vals, can_correct, nval + 3.0) 

 

if vals.ndim == 0: 

vals = vals.item() # array scalar 

 

if fisher: 

return vals - 3 

else: 

return vals 

 

 

DescribeResult = namedtuple('DescribeResult', 

('nobs', 'minmax', 'mean', 'variance', 'skewness', 

'kurtosis')) 

 

 

def describe(a, axis=0, ddof=1, bias=True, nan_policy='propagate'): 

""" 

Compute several descriptive statistics of the passed array. 

 

Parameters 

---------- 

a : array_like 

Input data. 

axis : int or None, optional 

Axis along which statistics are calculated. Default is 0. 

If None, compute over the whole array `a`. 

ddof : int, optional 

Delta degrees of freedom (only for variance). Default is 1. 

bias : bool, optional 

If False, then the skewness and kurtosis calculations are corrected for 

statistical bias. 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. 

 

Returns 

------- 

nobs : int or ndarray of ints 

Number of observations (length of data along `axis`). 

When 'omit' is chosen as nan_policy, each column is counted separately. 

minmax: tuple of ndarrays or floats 

Minimum and maximum value of data array. 

mean : ndarray or float 

Arithmetic mean of data along axis. 

variance : ndarray or float 

Unbiased variance of the data along axis, denominator is number of 

observations minus one. 

skewness : ndarray or float 

Skewness, based on moment calculations with denominator equal to 

the number of observations, i.e. no degrees of freedom correction. 

kurtosis : ndarray or float 

Kurtosis (Fisher). The kurtosis is normalized so that it is 

zero for the normal distribution. No degrees of freedom are used. 

 

See Also 

-------- 

skew, kurtosis 

 

Examples 

-------- 

>>> from scipy import stats 

>>> a = np.arange(10) 

>>> stats.describe(a) 

DescribeResult(nobs=10, minmax=(0, 9), mean=4.5, variance=9.166666666666666, 

skewness=0.0, kurtosis=-1.2242424242424244) 

>>> b = [[1, 2], [3, 4]] 

>>> stats.describe(b) 

DescribeResult(nobs=2, minmax=(array([1, 2]), array([3, 4])), 

mean=array([2., 3.]), variance=array([2., 2.]), 

skewness=array([0., 0.]), kurtosis=array([-2., -2.])) 

 

""" 

a, axis = _chk_asarray(a, axis) 

 

contains_nan, nan_policy = _contains_nan(a, nan_policy) 

 

if contains_nan and nan_policy == 'omit': 

a = ma.masked_invalid(a) 

return mstats_basic.describe(a, axis, ddof, bias) 

 

if a.size == 0: 

raise ValueError("The input must not be empty.") 

n = a.shape[axis] 

mm = (np.min(a, axis=axis), np.max(a, axis=axis)) 

m = np.mean(a, axis=axis) 

v = np.var(a, axis=axis, ddof=ddof) 

sk = skew(a, axis, bias=bias) 

kurt = kurtosis(a, axis, bias=bias) 

 

return DescribeResult(n, mm, m, v, sk, kurt) 

 

##################################### 

# NORMALITY TESTS # 

##################################### 

 

 

SkewtestResult = namedtuple('SkewtestResult', ('statistic', 'pvalue')) 

 

 

def skewtest(a, axis=0, nan_policy='propagate'): 

""" 

Test whether the skew is different from the normal distribution. 

 

This function tests the null hypothesis that the skewness of 

the population that the sample was drawn from is the same 

as that of a corresponding normal distribution. 

 

Parameters 

---------- 

a : array 

The data to be tested 

axis : int or None, optional 

Axis along which statistics are calculated. Default is 0. 

If None, compute over the whole array `a`. 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. 

 

Returns 

------- 

statistic : float 

The computed z-score for this test. 

pvalue : float 

a 2-sided p-value for the hypothesis test 

 

Notes 

----- 

The sample size must be at least 8. 

 

References 

---------- 

.. [1] R. B. D'Agostino, A. J. Belanger and R. B. D'Agostino Jr., 

"A suggestion for using powerful and informative tests of 

normality", American Statistician 44, pp. 316-321, 1990. 

 

Examples 

-------- 

>>> from scipy.stats import skewtest 

>>> skewtest([1, 2, 3, 4, 5, 6, 7, 8]) 

SkewtestResult(statistic=1.0108048609177787, pvalue=0.3121098361421897) 

>>> skewtest([2, 8, 0, 4, 1, 9, 9, 0]) 

SkewtestResult(statistic=0.44626385374196975, pvalue=0.6554066631275459) 

>>> skewtest([1, 2, 3, 4, 5, 6, 7, 8000]) 

SkewtestResult(statistic=3.571773510360407, pvalue=0.0003545719905823133) 

>>> skewtest([100, 100, 100, 100, 100, 100, 100, 101]) 

SkewtestResult(statistic=3.5717766638478072, pvalue=0.000354567720281634) 

""" 

a, axis = _chk_asarray(a, axis) 

 

contains_nan, nan_policy = _contains_nan(a, nan_policy) 

 

if contains_nan and nan_policy == 'omit': 

a = ma.masked_invalid(a) 

return mstats_basic.skewtest(a, axis) 

 

if axis is None: 

a = np.ravel(a) 

axis = 0 

b2 = skew(a, axis) 

n = float(a.shape[axis]) 

if n < 8: 

raise ValueError( 

"skewtest is not valid with less than 8 samples; %i samples" 

" were given." % int(n)) 

y = b2 * math.sqrt(((n + 1) * (n + 3)) / (6.0 * (n - 2))) 

beta2 = (3.0 * (n**2 + 27*n - 70) * (n+1) * (n+3) / 

((n-2.0) * (n+5) * (n+7) * (n+9))) 

W2 = -1 + math.sqrt(2 * (beta2 - 1)) 

delta = 1 / math.sqrt(0.5 * math.log(W2)) 

alpha = math.sqrt(2.0 / (W2 - 1)) 

y = np.where(y == 0, 1, y) 

Z = delta * np.log(y / alpha + np.sqrt((y / alpha)**2 + 1)) 

 

return SkewtestResult(Z, 2 * distributions.norm.sf(np.abs(Z))) 

 

 

KurtosistestResult = namedtuple('KurtosistestResult', ('statistic', 'pvalue')) 

 

 

def kurtosistest(a, axis=0, nan_policy='propagate'): 

""" 

Test whether a dataset has normal kurtosis. 

 

This function tests the null hypothesis that the kurtosis 

of the population from which the sample was drawn is that 

of the normal distribution: ``kurtosis = 3(n-1)/(n+1)``. 

 

Parameters 

---------- 

a : array 

array of the sample data 

axis : int or None, optional 

Axis along which to compute test. Default is 0. If None, 

compute over the whole array `a`. 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. 

 

Returns 

------- 

statistic : float 

The computed z-score for this test. 

pvalue : float 

The 2-sided p-value for the hypothesis test 

 

Notes 

----- 

Valid only for n>20. The Z-score is set to 0 for bad entries. 

This function uses the method described in [1]_. 

 

References 

---------- 

.. [1] see e.g. F. J. Anscombe, W. J. Glynn, "Distribution of the kurtosis 

statistic b2 for normal samples", Biometrika, vol. 70, pp. 227-234, 1983. 

 

Examples 

-------- 

>>> from scipy.stats import kurtosistest 

>>> kurtosistest(list(range(20))) 

KurtosistestResult(statistic=-1.7058104152122062, pvalue=0.08804338332528348) 

 

>>> np.random.seed(28041990) 

>>> s = np.random.normal(0, 1, 1000) 

>>> kurtosistest(s) 

KurtosistestResult(statistic=1.2317590987707365, pvalue=0.21803908613450895) 

""" 

a, axis = _chk_asarray(a, axis) 

 

contains_nan, nan_policy = _contains_nan(a, nan_policy) 

 

if contains_nan and nan_policy == 'omit': 

a = ma.masked_invalid(a) 

return mstats_basic.kurtosistest(a, axis) 

 

n = float(a.shape[axis]) 

if n < 5: 

raise ValueError( 

"kurtosistest requires at least 5 observations; %i observations" 

" were given." % int(n)) 

if n < 20: 

warnings.warn("kurtosistest only valid for n>=20 ... continuing " 

"anyway, n=%i" % int(n)) 

b2 = kurtosis(a, axis, fisher=False) 

 

E = 3.0*(n-1) / (n+1) 

varb2 = 24.0*n*(n-2)*(n-3) / ((n+1)*(n+1.)*(n+3)*(n+5)) # [1]_ Eq. 1 

x = (b2-E) / np.sqrt(varb2) # [1]_ Eq. 4 

# [1]_ Eq. 2: 

sqrtbeta1 = 6.0*(n*n-5*n+2)/((n+7)*(n+9)) * np.sqrt((6.0*(n+3)*(n+5)) / 

(n*(n-2)*(n-3))) 

# [1]_ Eq. 3: 

A = 6.0 + 8.0/sqrtbeta1 * (2.0/sqrtbeta1 + np.sqrt(1+4.0/(sqrtbeta1**2))) 

term1 = 1 - 2/(9.0*A) 

denom = 1 + x*np.sqrt(2/(A-4.0)) 

denom = np.where(denom < 0, 99, denom) 

term2 = np.where(denom < 0, term1, np.power((1-2.0/A)/denom, 1/3.0)) 

Z = (term1 - term2) / np.sqrt(2/(9.0*A)) # [1]_ Eq. 5 

Z = np.where(denom == 99, 0, Z) 

if Z.ndim == 0: 

Z = Z[()] 

 

# zprob uses upper tail, so Z needs to be positive 

return KurtosistestResult(Z, 2 * distributions.norm.sf(np.abs(Z))) 

 

 

NormaltestResult = namedtuple('NormaltestResult', ('statistic', 'pvalue')) 

 

def normaltest(a, axis=0, nan_policy='propagate'): 

""" 

Test whether a sample differs from a normal distribution. 

 

This function tests the null hypothesis that a sample comes 

from a normal distribution. It is based on D'Agostino and 

Pearson's [1]_, [2]_ test that combines skew and kurtosis to 

produce an omnibus test of normality. 

 

 

Parameters 

---------- 

a : array_like 

The array containing the sample to be tested. 

axis : int or None, optional 

Axis along which to compute test. Default is 0. If None, 

compute over the whole array `a`. 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. 

 

Returns 

------- 

statistic : float or array 

``s^2 + k^2``, where ``s`` is the z-score returned by `skewtest` and 

``k`` is the z-score returned by `kurtosistest`. 

pvalue : float or array 

A 2-sided chi squared probability for the hypothesis test. 

 

References 

---------- 

.. [1] D'Agostino, R. B. (1971), "An omnibus test of normality for 

moderate and large sample size", Biometrika, 58, 341-348 

 

.. [2] D'Agostino, R. and Pearson, E. S. (1973), "Tests for departure from 

normality", Biometrika, 60, 613-622 

 

Examples 

-------- 

>>> from scipy import stats 

>>> pts = 1000 

>>> np.random.seed(28041990) 

>>> a = np.random.normal(0, 1, size=pts) 

>>> b = np.random.normal(2, 1, size=pts) 

>>> x = np.concatenate((a, b)) 

>>> k2, p = stats.normaltest(x) 

>>> alpha = 1e-3 

>>> print("p = {:g}".format(p)) 

p = 3.27207e-11 

>>> if p < alpha: # null hypothesis: x comes from a normal distribution 

... print("The null hypothesis can be rejected") 

... else: 

... print("The null hypothesis cannot be rejected") 

The null hypothesis can be rejected 

""" 

a, axis = _chk_asarray(a, axis) 

 

contains_nan, nan_policy = _contains_nan(a, nan_policy) 

 

if contains_nan and nan_policy == 'omit': 

a = ma.masked_invalid(a) 

return mstats_basic.normaltest(a, axis) 

 

s, _ = skewtest(a, axis) 

k, _ = kurtosistest(a, axis) 

k2 = s*s + k*k 

 

return NormaltestResult(k2, distributions.chi2.sf(k2, 2)) 

 

 

def jarque_bera(x): 

""" 

Perform the Jarque-Bera goodness of fit test on sample data. 

 

The Jarque-Bera test tests whether the sample data has the skewness and 

kurtosis matching a normal distribution. 

 

Note that this test only works for a large enough number of data samples 

(>2000) as the test statistic asymptotically has a Chi-squared distribution 

with 2 degrees of freedom. 

 

Parameters 

---------- 

x : array_like 

Observations of a random variable. 

 

Returns 

------- 

jb_value : float 

The test statistic. 

p : float 

The p-value for the hypothesis test. 

 

References 

---------- 

.. [1] Jarque, C. and Bera, A. (1980) "Efficient tests for normality, 

homoscedasticity and serial independence of regression residuals", 

6 Econometric Letters 255-259. 

 

Examples 

-------- 

>>> from scipy import stats 

>>> np.random.seed(987654321) 

>>> x = np.random.normal(0, 1, 100000) 

>>> y = np.random.rayleigh(1, 100000) 

>>> stats.jarque_bera(x) 

(4.7165707989581342, 0.09458225503041906) 

>>> stats.jarque_bera(y) 

(6713.7098548143422, 0.0) 

 

""" 

x = np.asarray(x) 

n = float(x.size) 

if n == 0: 

raise ValueError('At least one observation is required.') 

 

mu = x.mean() 

diffx = x - mu 

skewness = (1 / n * np.sum(diffx**3)) / (1 / n * np.sum(diffx**2))**(3 / 2.) 

kurtosis = (1 / n * np.sum(diffx**4)) / (1 / n * np.sum(diffx**2))**2 

jb_value = n / 6 * (skewness**2 + (kurtosis - 3)**2 / 4) 

p = 1 - distributions.chi2.cdf(jb_value, 2) 

 

return jb_value, p 

 

 

##################################### 

# FREQUENCY FUNCTIONS # 

##################################### 

 

@np.deprecate(message="`itemfreq` is deprecated and will be removed in a " 

"future version. Use instead `np.unique(..., return_counts=True)`") 

def itemfreq(a): 

""" 

Return a 2-D array of item frequencies. 

 

Parameters 

---------- 

a : (N,) array_like 

Input array. 

 

Returns 

------- 

itemfreq : (K, 2) ndarray 

A 2-D frequency table. Column 1 contains sorted, unique values from 

`a`, column 2 contains their respective counts. 

 

Examples 

-------- 

>>> from scipy import stats 

>>> a = np.array([1, 1, 5, 0, 1, 2, 2, 0, 1, 4]) 

>>> stats.itemfreq(a) 

array([[ 0., 2.], 

[ 1., 4.], 

[ 2., 2.], 

[ 4., 1.], 

[ 5., 1.]]) 

>>> np.bincount(a) 

array([2, 4, 2, 0, 1, 1]) 

 

>>> stats.itemfreq(a/10.) 

array([[ 0. , 2. ], 

[ 0.1, 4. ], 

[ 0.2, 2. ], 

[ 0.4, 1. ], 

[ 0.5, 1. ]]) 

 

""" 

items, inv = np.unique(a, return_inverse=True) 

freq = np.bincount(inv) 

return np.array([items, freq]).T 

 

 

def scoreatpercentile(a, per, limit=(), interpolation_method='fraction', 

axis=None): 

""" 

Calculate the score at a given percentile of the input sequence. 

 

For example, the score at `per=50` is the median. If the desired quantile 

lies between two data points, we interpolate between them, according to 

the value of `interpolation`. If the parameter `limit` is provided, it 

should be a tuple (lower, upper) of two values. 

 

Parameters 

---------- 

a : array_like 

A 1-D array of values from which to extract score. 

per : array_like 

Percentile(s) at which to extract score. Values should be in range 

[0,100]. 

limit : tuple, optional 

Tuple of two scalars, the lower and upper limits within which to 

compute the percentile. Values of `a` outside 

this (closed) interval will be ignored. 

interpolation_method : {'fraction', 'lower', 'higher'}, optional 

This optional parameter specifies the interpolation method to use, 

when the desired quantile lies between two data points `i` and `j` 

 

- fraction: ``i + (j - i) * fraction`` where ``fraction`` is the 

fractional part of the index surrounded by ``i`` and ``j``. 

- lower: ``i``. 

- higher: ``j``. 

 

axis : int, optional 

Axis along which the percentiles are computed. Default is None. If 

None, compute over the whole array `a`. 

 

Returns 

------- 

score : float or ndarray 

Score at percentile(s). 

 

See Also 

-------- 

percentileofscore, numpy.percentile 

 

Notes 

----- 

This function will become obsolete in the future. 

For Numpy 1.9 and higher, `numpy.percentile` provides all the functionality 

that `scoreatpercentile` provides. And it's significantly faster. 

Therefore it's recommended to use `numpy.percentile` for users that have 

numpy >= 1.9. 

 

Examples 

-------- 

>>> from scipy import stats 

>>> a = np.arange(100) 

>>> stats.scoreatpercentile(a, 50) 

49.5 

 

""" 

# adapted from NumPy's percentile function. When we require numpy >= 1.8, 

# the implementation of this function can be replaced by np.percentile. 

a = np.asarray(a) 

if a.size == 0: 

# empty array, return nan(s) with shape matching `per` 

if np.isscalar(per): 

return np.nan 

else: 

return np.ones(np.asarray(per).shape, dtype=np.float64) * np.nan 

 

if limit: 

a = a[(limit[0] <= a) & (a <= limit[1])] 

 

sorted = np.sort(a, axis=axis) 

if axis is None: 

axis = 0 

 

return _compute_qth_percentile(sorted, per, interpolation_method, axis) 

 

 

# handle sequence of per's without calling sort multiple times 

def _compute_qth_percentile(sorted, per, interpolation_method, axis): 

if not np.isscalar(per): 

score = [_compute_qth_percentile(sorted, i, interpolation_method, axis) 

for i in per] 

return np.array(score) 

 

if (per < 0) or (per > 100): 

raise ValueError("percentile must be in the range [0, 100]") 

 

indexer = [slice(None)] * sorted.ndim 

idx = per / 100. * (sorted.shape[axis] - 1) 

 

if int(idx) != idx: 

# round fractional indices according to interpolation method 

if interpolation_method == 'lower': 

idx = int(np.floor(idx)) 

elif interpolation_method == 'higher': 

idx = int(np.ceil(idx)) 

elif interpolation_method == 'fraction': 

pass # keep idx as fraction and interpolate 

else: 

raise ValueError("interpolation_method can only be 'fraction', " 

"'lower' or 'higher'") 

 

i = int(idx) 

if i == idx: 

indexer[axis] = slice(i, i + 1) 

weights = array(1) 

sumval = 1.0 

else: 

indexer[axis] = slice(i, i + 2) 

j = i + 1 

weights = array([(j - idx), (idx - i)], float) 

wshape = [1] * sorted.ndim 

wshape[axis] = 2 

weights.shape = wshape 

sumval = weights.sum() 

 

# Use np.add.reduce (== np.sum but a little faster) to coerce data type 

return np.add.reduce(sorted[indexer] * weights, axis=axis) / sumval 

 

 

def percentileofscore(a, score, kind='rank'): 

""" 

The percentile rank of a score relative to a list of scores. 

 

A `percentileofscore` of, for example, 80% means that 80% of the 

scores in `a` are below the given score. In the case of gaps or 

ties, the exact definition depends on the optional keyword, `kind`. 

 

Parameters 

---------- 

a : array_like 

Array of scores to which `score` is compared. 

score : int or float 

Score that is compared to the elements in `a`. 

kind : {'rank', 'weak', 'strict', 'mean'}, optional 

This optional parameter specifies the interpretation of the 

resulting score: 

 

- "rank": Average percentage ranking of score. In case of 

multiple matches, average the percentage rankings of 

all matching scores. 

- "weak": This kind corresponds to the definition of a cumulative 

distribution function. A percentileofscore of 80% 

means that 80% of values are less than or equal 

to the provided score. 

- "strict": Similar to "weak", except that only values that are 

strictly less than the given score are counted. 

- "mean": The average of the "weak" and "strict" scores, often used in 

testing. See 

 

http://en.wikipedia.org/wiki/Percentile_rank 

 

Returns 

------- 

pcos : float 

Percentile-position of score (0-100) relative to `a`. 

 

See Also 

-------- 

numpy.percentile 

 

Examples 

-------- 

Three-quarters of the given values lie below a given score: 

 

>>> from scipy import stats 

>>> stats.percentileofscore([1, 2, 3, 4], 3) 

75.0 

 

With multiple matches, note how the scores of the two matches, 0.6 

and 0.8 respectively, are averaged: 

 

>>> stats.percentileofscore([1, 2, 3, 3, 4], 3) 

70.0 

 

Only 2/5 values are strictly less than 3: 

 

>>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='strict') 

40.0 

 

But 4/5 values are less than or equal to 3: 

 

>>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='weak') 

80.0 

 

The average between the weak and the strict scores is 

 

>>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='mean') 

60.0 

 

""" 

if np.isnan(score): 

return np.nan 

a = np.asarray(a) 

n = len(a) 

if n == 0: 

return 100.0 

 

if kind == 'rank': 

left = np.count_nonzero(a < score) 

right = np.count_nonzero(a <= score) 

pct = (right + left + (1 if right > left else 0)) * 50.0/n 

return pct 

elif kind == 'strict': 

return np.count_nonzero(a < score) / float(n) * 100 

elif kind == 'weak': 

return np.count_nonzero(a <= score) / float(n) * 100 

elif kind == 'mean': 

pct = (np.count_nonzero(a < score) + np.count_nonzero(a <= score)) / float(n) * 50 

return pct 

else: 

raise ValueError("kind can only be 'rank', 'strict', 'weak' or 'mean'") 

 

 

HistogramResult = namedtuple('HistogramResult', 

('count', 'lowerlimit', 'binsize', 'extrapoints')) 

 

 

def _histogram(a, numbins=10, defaultlimits=None, weights=None, printextras=False): 

""" 

Separate the range into several bins and return the number of instances 

in each bin. 

 

Parameters 

---------- 

a : array_like 

Array of scores which will be put into bins. 

numbins : int, optional 

The number of bins to use for the histogram. Default is 10. 

defaultlimits : tuple (lower, upper), optional 

The lower and upper values for the range of the histogram. 

If no value is given, a range slightly larger than the range of the 

values in a is used. Specifically ``(a.min() - s, a.max() + s)``, 

where ``s = (1/2)(a.max() - a.min()) / (numbins - 1)``. 

weights : array_like, optional 

The weights for each value in `a`. Default is None, which gives each 

value a weight of 1.0 

printextras : bool, optional 

If True, if there are extra points (i.e. the points that fall outside 

the bin limits) a warning is raised saying how many of those points 

there are. Default is False. 

 

Returns 

------- 

count : ndarray 

Number of points (or sum of weights) in each bin. 

lowerlimit : float 

Lowest value of histogram, the lower limit of the first bin. 

binsize : float 

The size of the bins (all bins have the same size). 

extrapoints : int 

The number of points outside the range of the histogram. 

 

See Also 

-------- 

numpy.histogram 

 

Notes 

----- 

This histogram is based on numpy's histogram but has a larger range by 

default if default limits is not set. 

 

""" 

a = np.ravel(a) 

if defaultlimits is None: 

if a.size == 0: 

# handle empty arrays. Undetermined range, so use 0-1. 

defaultlimits = (0, 1) 

else: 

# no range given, so use values in `a` 

data_min = a.min() 

data_max = a.max() 

# Have bins extend past min and max values slightly 

s = (data_max - data_min) / (2. * (numbins - 1.)) 

defaultlimits = (data_min - s, data_max + s) 

 

# use numpy's histogram method to compute bins 

hist, bin_edges = np.histogram(a, bins=numbins, range=defaultlimits, 

weights=weights) 

# hist are not always floats, convert to keep with old output 

hist = np.array(hist, dtype=float) 

# fixed width for bins is assumed, as numpy's histogram gives 

# fixed width bins for int values for 'bins' 

binsize = bin_edges[1] - bin_edges[0] 

# calculate number of extra points 

extrapoints = len([v for v in a 

if defaultlimits[0] > v or v > defaultlimits[1]]) 

if extrapoints > 0 and printextras: 

warnings.warn("Points outside given histogram range = %s" 

% extrapoints) 

 

return HistogramResult(hist, defaultlimits[0], binsize, extrapoints) 

 

 

CumfreqResult = namedtuple('CumfreqResult', 

('cumcount', 'lowerlimit', 'binsize', 

'extrapoints')) 

 

 

def cumfreq(a, numbins=10, defaultreallimits=None, weights=None): 

""" 

Return a cumulative frequency histogram, using the histogram function. 

 

A cumulative histogram is a mapping that counts the cumulative number of 

observations in all of the bins up to the specified bin. 

 

Parameters 

---------- 

a : array_like 

Input array. 

numbins : int, optional 

The number of bins to use for the histogram. Default is 10. 

defaultreallimits : tuple (lower, upper), optional 

The lower and upper values for the range of the histogram. 

If no value is given, a range slightly larger than the range of the 

values in `a` is used. Specifically ``(a.min() - s, a.max() + s)``, 

where ``s = (1/2)(a.max() - a.min()) / (numbins - 1)``. 

weights : array_like, optional 

The weights for each value in `a`. Default is None, which gives each 

value a weight of 1.0 

 

Returns 

------- 

cumcount : ndarray 

Binned values of cumulative frequency. 

lowerlimit : float 

Lower real limit 

binsize : float 

Width of each bin. 

extrapoints : int 

Extra points. 

 

Examples 

-------- 

>>> import matplotlib.pyplot as plt 

>>> from scipy import stats 

>>> x = [1, 4, 2, 1, 3, 1] 

>>> res = stats.cumfreq(x, numbins=4, defaultreallimits=(1.5, 5)) 

>>> res.cumcount 

array([ 1., 2., 3., 3.]) 

>>> res.extrapoints 

3 

 

Create a normal distribution with 1000 random values 

 

>>> rng = np.random.RandomState(seed=12345) 

>>> samples = stats.norm.rvs(size=1000, random_state=rng) 

 

Calculate cumulative frequencies 

 

>>> res = stats.cumfreq(samples, numbins=25) 

 

Calculate space of values for x 

 

>>> x = res.lowerlimit + np.linspace(0, res.binsize*res.cumcount.size, 

... res.cumcount.size) 

 

Plot histogram and cumulative histogram 

 

>>> fig = plt.figure(figsize=(10, 4)) 

>>> ax1 = fig.add_subplot(1, 2, 1) 

>>> ax2 = fig.add_subplot(1, 2, 2) 

>>> ax1.hist(samples, bins=25) 

>>> ax1.set_title('Histogram') 

>>> ax2.bar(x, res.cumcount, width=res.binsize) 

>>> ax2.set_title('Cumulative histogram') 

>>> ax2.set_xlim([x.min(), x.max()]) 

 

>>> plt.show() 

 

""" 

h, l, b, e = _histogram(a, numbins, defaultreallimits, weights=weights) 

cumhist = np.cumsum(h * 1, axis=0) 

return CumfreqResult(cumhist, l, b, e) 

 

 

RelfreqResult = namedtuple('RelfreqResult', 

('frequency', 'lowerlimit', 'binsize', 

'extrapoints')) 

 

 

def relfreq(a, numbins=10, defaultreallimits=None, weights=None): 

""" 

Return a relative frequency histogram, using the histogram function. 

 

A relative frequency histogram is a mapping of the number of 

observations in each of the bins relative to the total of observations. 

 

Parameters 

---------- 

a : array_like 

Input array. 

numbins : int, optional 

The number of bins to use for the histogram. Default is 10. 

defaultreallimits : tuple (lower, upper), optional 

The lower and upper values for the range of the histogram. 

If no value is given, a range slightly larger than the range of the 

values in a is used. Specifically ``(a.min() - s, a.max() + s)``, 

where ``s = (1/2)(a.max() - a.min()) / (numbins - 1)``. 

weights : array_like, optional 

The weights for each value in `a`. Default is None, which gives each 

value a weight of 1.0 

 

Returns 

------- 

frequency : ndarray 

Binned values of relative frequency. 

lowerlimit : float 

Lower real limit 

binsize : float 

Width of each bin. 

extrapoints : int 

Extra points. 

 

Examples 

-------- 

>>> import matplotlib.pyplot as plt 

>>> from scipy import stats 

>>> a = np.array([2, 4, 1, 2, 3, 2]) 

>>> res = stats.relfreq(a, numbins=4) 

>>> res.frequency 

array([ 0.16666667, 0.5 , 0.16666667, 0.16666667]) 

>>> np.sum(res.frequency) # relative frequencies should add up to 1 

1.0 

 

Create a normal distribution with 1000 random values 

 

>>> rng = np.random.RandomState(seed=12345) 

>>> samples = stats.norm.rvs(size=1000, random_state=rng) 

 

Calculate relative frequencies 

 

>>> res = stats.relfreq(samples, numbins=25) 

 

Calculate space of values for x 

 

>>> x = res.lowerlimit + np.linspace(0, res.binsize*res.frequency.size, 

... res.frequency.size) 

 

Plot relative frequency histogram 

 

>>> fig = plt.figure(figsize=(5, 4)) 

>>> ax = fig.add_subplot(1, 1, 1) 

>>> ax.bar(x, res.frequency, width=res.binsize) 

>>> ax.set_title('Relative frequency histogram') 

>>> ax.set_xlim([x.min(), x.max()]) 

 

>>> plt.show() 

 

""" 

a = np.asanyarray(a) 

h, l, b, e = _histogram(a, numbins, defaultreallimits, weights=weights) 

h = h / float(a.shape[0]) 

 

return RelfreqResult(h, l, b, e) 

 

 

##################################### 

# VARIABILITY FUNCTIONS # 

##################################### 

 

def obrientransform(*args): 

""" 

Compute the O'Brien transform on input data (any number of arrays). 

 

Used to test for homogeneity of variance prior to running one-way stats. 

Each array in ``*args`` is one level of a factor. 

If `f_oneway` is run on the transformed data and found significant, 

the variances are unequal. From Maxwell and Delaney [1]_, p.112. 

 

Parameters 

---------- 

args : tuple of array_like 

Any number of arrays. 

 

Returns 

------- 

obrientransform : ndarray 

Transformed data for use in an ANOVA. The first dimension 

of the result corresponds to the sequence of transformed 

arrays. If the arrays given are all 1-D of the same length, 

the return value is a 2-D array; otherwise it is a 1-D array 

of type object, with each element being an ndarray. 

 

References 

---------- 

.. [1] S. E. Maxwell and H. D. Delaney, "Designing Experiments and 

Analyzing Data: A Model Comparison Perspective", Wadsworth, 1990. 

 

Examples 

-------- 

We'll test the following data sets for differences in their variance. 

 

>>> x = [10, 11, 13, 9, 7, 12, 12, 9, 10] 

>>> y = [13, 21, 5, 10, 8, 14, 10, 12, 7, 15] 

 

Apply the O'Brien transform to the data. 

 

>>> from scipy.stats import obrientransform 

>>> tx, ty = obrientransform(x, y) 

 

Use `scipy.stats.f_oneway` to apply a one-way ANOVA test to the 

transformed data. 

 

>>> from scipy.stats import f_oneway 

>>> F, p = f_oneway(tx, ty) 

>>> p 

0.1314139477040335 

 

If we require that ``p < 0.05`` for significance, we cannot conclude 

that the variances are different. 

""" 

TINY = np.sqrt(np.finfo(float).eps) 

 

# `arrays` will hold the transformed arguments. 

arrays = [] 

 

for arg in args: 

a = np.asarray(arg) 

n = len(a) 

mu = np.mean(a) 

sq = (a - mu)**2 

sumsq = sq.sum() 

 

# The O'Brien transform. 

t = ((n - 1.5) * n * sq - 0.5 * sumsq) / ((n - 1) * (n - 2)) 

 

# Check that the mean of the transformed data is equal to the 

# original variance. 

var = sumsq / (n - 1) 

if abs(var - np.mean(t)) > TINY: 

raise ValueError('Lack of convergence in obrientransform.') 

 

arrays.append(t) 

 

return np.array(arrays) 

 

 

def sem(a, axis=0, ddof=1, nan_policy='propagate'): 

""" 

Calculate the standard error of the mean (or standard error of 

measurement) of the values in the input array. 

 

Parameters 

---------- 

a : array_like 

An array containing the values for which the standard error is 

returned. 

axis : int or None, optional 

Axis along which to operate. Default is 0. If None, compute over 

the whole array `a`. 

ddof : int, optional 

Delta degrees-of-freedom. How many degrees of freedom to adjust 

for bias in limited samples relative to the population estimate 

of variance. Defaults to 1. 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. 

 

Returns 

------- 

s : ndarray or float 

The standard error of the mean in the sample(s), along the input axis. 

 

Notes 

----- 

The default value for `ddof` is different to the default (0) used by other 

ddof containing routines, such as np.std and np.nanstd. 

 

Examples 

-------- 

Find standard error along the first axis: 

 

>>> from scipy import stats 

>>> a = np.arange(20).reshape(5,4) 

>>> stats.sem(a) 

array([ 2.8284, 2.8284, 2.8284, 2.8284]) 

 

Find standard error across the whole array, using n degrees of freedom: 

 

>>> stats.sem(a, axis=None, ddof=0) 

1.2893796958227628 

 

""" 

a, axis = _chk_asarray(a, axis) 

 

contains_nan, nan_policy = _contains_nan(a, nan_policy) 

 

if contains_nan and nan_policy == 'omit': 

a = ma.masked_invalid(a) 

return mstats_basic.sem(a, axis, ddof) 

 

n = a.shape[axis] 

s = np.std(a, axis=axis, ddof=ddof) / np.sqrt(n) 

return s 

 

 

def zscore(a, axis=0, ddof=0): 

""" 

Calculate the z score of each value in the sample, relative to the 

sample mean and standard deviation. 

 

Parameters 

---------- 

a : array_like 

An array like object containing the sample data. 

axis : int or None, optional 

Axis along which to operate. Default is 0. If None, compute over 

the whole array `a`. 

ddof : int, optional 

Degrees of freedom correction in the calculation of the 

standard deviation. Default is 0. 

 

Returns 

------- 

zscore : array_like 

The z-scores, standardized by mean and standard deviation of 

input array `a`. 

 

Notes 

----- 

This function preserves ndarray subclasses, and works also with 

matrices and masked arrays (it uses `asanyarray` instead of 

`asarray` for parameters). 

 

Examples 

-------- 

>>> a = np.array([ 0.7972, 0.0767, 0.4383, 0.7866, 0.8091, 

... 0.1954, 0.6307, 0.6599, 0.1065, 0.0508]) 

>>> from scipy import stats 

>>> stats.zscore(a) 

array([ 1.1273, -1.247 , -0.0552, 1.0923, 1.1664, -0.8559, 0.5786, 

0.6748, -1.1488, -1.3324]) 

 

Computing along a specified axis, using n-1 degrees of freedom 

(``ddof=1``) to calculate the standard deviation: 

 

>>> b = np.array([[ 0.3148, 0.0478, 0.6243, 0.4608], 

... [ 0.7149, 0.0775, 0.6072, 0.9656], 

... [ 0.6341, 0.1403, 0.9759, 0.4064], 

... [ 0.5918, 0.6948, 0.904 , 0.3721], 

... [ 0.0921, 0.2481, 0.1188, 0.1366]]) 

>>> stats.zscore(b, axis=1, ddof=1) 

array([[-0.19264823, -1.28415119, 1.07259584, 0.40420358], 

[ 0.33048416, -1.37380874, 0.04251374, 1.00081084], 

[ 0.26796377, -1.12598418, 1.23283094, -0.37481053], 

[-0.22095197, 0.24468594, 1.19042819, -1.21416216], 

[-0.82780366, 1.4457416 , -0.43867764, -0.1792603 ]]) 

""" 

a = np.asanyarray(a) 

mns = a.mean(axis=axis) 

sstd = a.std(axis=axis, ddof=ddof) 

if axis and mns.ndim < a.ndim: 

return ((a - np.expand_dims(mns, axis=axis)) / 

np.expand_dims(sstd, axis=axis)) 

else: 

return (a - mns) / sstd 

 

 

def zmap(scores, compare, axis=0, ddof=0): 

""" 

Calculate the relative z-scores. 

 

Return an array of z-scores, i.e., scores that are standardized to 

zero mean and unit variance, where mean and variance are calculated 

from the comparison array. 

 

Parameters 

---------- 

scores : array_like 

The input for which z-scores are calculated. 

compare : array_like 

The input from which the mean and standard deviation of the 

normalization are taken; assumed to have the same dimension as 

`scores`. 

axis : int or None, optional 

Axis over which mean and variance of `compare` are calculated. 

Default is 0. If None, compute over the whole array `scores`. 

ddof : int, optional 

Degrees of freedom correction in the calculation of the 

standard deviation. Default is 0. 

 

Returns 

------- 

zscore : array_like 

Z-scores, in the same shape as `scores`. 

 

Notes 

----- 

This function preserves ndarray subclasses, and works also with 

matrices and masked arrays (it uses `asanyarray` instead of 

`asarray` for parameters). 

 

Examples 

-------- 

>>> from scipy.stats import zmap 

>>> a = [0.5, 2.0, 2.5, 3] 

>>> b = [0, 1, 2, 3, 4] 

>>> zmap(a, b) 

array([-1.06066017, 0. , 0.35355339, 0.70710678]) 

""" 

scores, compare = map(np.asanyarray, [scores, compare]) 

mns = compare.mean(axis=axis) 

sstd = compare.std(axis=axis, ddof=ddof) 

if axis and mns.ndim < compare.ndim: 

return ((scores - np.expand_dims(mns, axis=axis)) / 

np.expand_dims(sstd, axis=axis)) 

else: 

return (scores - mns) / sstd 

 

 

# Private dictionary initialized only once at module level 

# See https://en.wikipedia.org/wiki/Robust_measures_of_scale 

_scale_conversions = {'raw': 1.0, 

'normal': special.erfinv(0.5) * 2.0 * math.sqrt(2.0)} 

 

 

def iqr(x, axis=None, rng=(25, 75), scale='raw', nan_policy='propagate', 

interpolation='linear', keepdims=False): 

""" 

Compute the interquartile range of the data along the specified axis. 

 

The interquartile range (IQR) is the difference between the 75th and 

25th percentile of the data. It is a measure of the dispersion 

similar to standard deviation or variance, but is much more robust 

against outliers [2]_. 

 

The ``rng`` parameter allows this function to compute other 

percentile ranges than the actual IQR. For example, setting 

``rng=(0, 100)`` is equivalent to `numpy.ptp`. 

 

The IQR of an empty array is `np.nan`. 

 

.. versionadded:: 0.18.0 

 

Parameters 

---------- 

x : array_like 

Input array or object that can be converted to an array. 

axis : int or sequence of int, optional 

Axis along which the range is computed. The default is to 

compute the IQR for the entire array. 

rng : Two-element sequence containing floats in range of [0,100] optional 

Percentiles over which to compute the range. Each must be 

between 0 and 100, inclusive. The default is the true IQR: 

`(25, 75)`. The order of the elements is not important. 

scale : scalar or str, optional 

The numerical value of scale will be divided out of the final 

result. The following string values are recognized: 

 

'raw' : No scaling, just return the raw IQR. 

'normal' : Scale by :math:`2 \\sqrt{2} erf^{-1}(\\frac{1}{2}) \\approx 1.349`. 

 

The default is 'raw'. Array-like scale is also allowed, as long 

as it broadcasts correctly to the output such that 

``out / scale`` is a valid operation. The output dimensions 

depend on the input array, `x`, the `axis` argument, and the 

`keepdims` flag. 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' 

returns nan, 'raise' throws an error, 'omit' performs the 

calculations ignoring nan values. Default is 'propagate'. 

interpolation : {'linear', 'lower', 'higher', 'midpoint', 'nearest'}, optional 

Specifies the interpolation method to use when the percentile 

boundaries lie between two data points `i` and `j`: 

 

* 'linear' : `i + (j - i) * fraction`, where `fraction` is the 

fractional part of the index surrounded by `i` and `j`. 

* 'lower' : `i`. 

* 'higher' : `j`. 

* 'nearest' : `i` or `j` whichever is nearest. 

* 'midpoint' : `(i + j) / 2`. 

 

Default is 'linear'. 

keepdims : bool, optional 

If this is set to `True`, the reduced axes are left in the 

result as dimensions with size one. With this option, the result 

will broadcast correctly against the original array `x`. 

 

Returns 

------- 

iqr : scalar or ndarray 

If ``axis=None``, a scalar is returned. If the input contains 

integers or floats of smaller precision than ``np.float64``, then the 

output data-type is ``np.float64``. Otherwise, the output data-type is 

the same as that of the input. 

 

See Also 

-------- 

numpy.std, numpy.var 

 

Examples 

-------- 

>>> from scipy.stats import iqr 

>>> x = np.array([[10, 7, 4], [3, 2, 1]]) 

>>> x 

array([[10, 7, 4], 

[ 3, 2, 1]]) 

>>> iqr(x) 

4.0 

>>> iqr(x, axis=0) 

array([ 3.5, 2.5, 1.5]) 

>>> iqr(x, axis=1) 

array([ 3., 1.]) 

>>> iqr(x, axis=1, keepdims=True) 

array([[ 3.], 

[ 1.]]) 

 

Notes 

----- 

This function is heavily dependent on the version of `numpy` that is 

installed. Versions greater than 1.11.0b3 are highly recommended, as they 

include a number of enhancements and fixes to `numpy.percentile` and 

`numpy.nanpercentile` that affect the operation of this function. The 

following modifications apply: 

 

Below 1.10.0 : `nan_policy` is poorly defined. 

The default behavior of `numpy.percentile` is used for 'propagate'. This 

is a hybrid of 'omit' and 'propagate' that mostly yields a skewed 

version of 'omit' since NaNs are sorted to the end of the data. A 

warning is raised if there are NaNs in the data. 

Below 1.9.0: `numpy.nanpercentile` does not exist. 

This means that `numpy.percentile` is used regardless of `nan_policy` 

and a warning is issued. See previous item for a description of the 

behavior. 

Below 1.9.0: `keepdims` and `interpolation` are not supported. 

The keywords get ignored with a warning if supplied with non-default 

values. However, multiple axes are still supported. 

 

References 

---------- 

.. [1] "Interquartile range" https://en.wikipedia.org/wiki/Interquartile_range 

.. [2] "Robust measures of scale" https://en.wikipedia.org/wiki/Robust_measures_of_scale 

.. [3] "Quantile" https://en.wikipedia.org/wiki/Quantile 

""" 

x = asarray(x) 

 

# This check prevents percentile from raising an error later. Also, it is 

# consistent with `np.var` and `np.std`. 

if not x.size: 

return np.nan 

 

# An error may be raised here, so fail-fast, before doing lengthy 

# computations, even though `scale` is not used until later 

if isinstance(scale, string_types): 

scale_key = scale.lower() 

if scale_key not in _scale_conversions: 

raise ValueError("{0} not a valid scale for `iqr`".format(scale)) 

scale = _scale_conversions[scale_key] 

 

# Select the percentile function to use based on nans and policy 

contains_nan, nan_policy = _contains_nan(x, nan_policy) 

 

if contains_nan and nan_policy == 'omit': 

percentile_func = _iqr_nanpercentile 

else: 

percentile_func = _iqr_percentile 

 

if len(rng) != 2: 

raise TypeError("quantile range must be two element sequence") 

 

rng = sorted(rng) 

pct = percentile_func(x, rng, axis=axis, interpolation=interpolation, 

keepdims=keepdims, contains_nan=contains_nan) 

out = np.subtract(pct[1], pct[0]) 

 

if scale != 1.0: 

out /= scale 

 

return out 

 

 

def _iqr_percentile(x, q, axis=None, interpolation='linear', keepdims=False, contains_nan=False): 

""" 

Private wrapper that works around older versions of `numpy`. 

 

While this function is pretty much necessary for the moment, it 

should be removed as soon as the minimum supported numpy version 

allows. 

""" 

if contains_nan and NumpyVersion(np.__version__) < '1.10.0a': 

# I see no way to avoid the version check to ensure that the corrected 

# NaN behavior has been implemented except to call `percentile` on a 

# small array. 

msg = "Keyword nan_policy='propagate' not correctly supported for " \ 

"numpy versions < 1.10.x. The default behavior of " \ 

"`numpy.percentile` will be used." 

warnings.warn(msg, RuntimeWarning) 

 

try: 

# For older versions of numpy, there are two things that can cause a 

# problem here: missing keywords and non-scalar axis. The former can be 

# partially handled with a warning, the latter can be handled fully by 

# hacking in an implementation similar to numpy's function for 

# providing multi-axis functionality 

# (`numpy.lib.function_base._ureduce` for the curious). 

result = np.percentile(x, q, axis=axis, keepdims=keepdims, 

interpolation=interpolation) 

except TypeError: 

if interpolation != 'linear' or keepdims: 

# At time or writing, this means np.__version__ < 1.9.0 

warnings.warn("Keywords interpolation and keepdims not supported " 

"for your version of numpy", RuntimeWarning) 

try: 

# Special processing if axis is an iterable 

original_size = len(axis) 

except TypeError: 

# Axis is a scalar at this point 

pass 

else: 

axis = np.unique(np.asarray(axis) % x.ndim) 

if original_size > axis.size: 

# mimic numpy if axes are duplicated 

raise ValueError("duplicate value in axis") 

if axis.size == x.ndim: 

# axis includes all axes: revert to None 

axis = None 

elif axis.size == 1: 

# no rolling necessary 

axis = axis[0] 

else: 

# roll multiple axes to the end and flatten that part out 

for ax in axis[::-1]: 

x = np.rollaxis(x, ax, x.ndim) 

x = x.reshape(x.shape[:-axis.size] + 

(np.prod(x.shape[-axis.size:]),)) 

axis = -1 

result = np.percentile(x, q, axis=axis) 

 

return result 

 

 

def _iqr_nanpercentile(x, q, axis=None, interpolation='linear', keepdims=False, 

contains_nan=False): 

""" 

Private wrapper that works around the following: 

 

1. A bug in `np.nanpercentile` that was around until numpy version 

1.11.0. 

2. A bug in `np.percentile` NaN handling that was fixed in numpy 

version 1.10.0. 

3. The non-existence of `np.nanpercentile` before numpy version 

1.9.0. 

 

While this function is pretty much necessary for the moment, it 

should be removed as soon as the minimum supported numpy version 

allows. 

""" 

if hasattr(np, 'nanpercentile'): 

# At time or writing, this means np.__version__ < 1.9.0 

result = np.nanpercentile(x, q, axis=axis, 

interpolation=interpolation, 

keepdims=keepdims) 

# If non-scalar result and nanpercentile does not do proper axis roll. 

# I see no way of avoiding the version test since dimensions may just 

# happen to match in the data. 

if result.ndim > 1 and NumpyVersion(np.__version__) < '1.11.0a': 

axis = np.asarray(axis) 

if axis.size == 1: 

# If only one axis specified, reduction happens along that dimension 

if axis.ndim == 0: 

axis = axis[None] 

result = np.rollaxis(result, axis[0]) 

else: 

# If multiple axes, reduced dimeision is last 

result = np.rollaxis(result, -1) 

else: 

msg = "Keyword nan_policy='omit' not correctly supported for numpy " \ 

"versions < 1.9.x. The default behavior of numpy.percentile " \ 

"will be used." 

warnings.warn(msg, RuntimeWarning) 

result = _iqr_percentile(x, q, axis=axis) 

 

return result 

 

 

##################################### 

# TRIMMING FUNCTIONS # 

##################################### 

 

SigmaclipResult = namedtuple('SigmaclipResult', ('clipped', 'lower', 'upper')) 

 

 

def sigmaclip(a, low=4., high=4.): 

""" 

Iterative sigma-clipping of array elements. 

 

Starting from the full sample, all elements outside the critical range are 

removed, i.e. all elements of the input array `c` that satisfy either of 

the following conditions :: 

 

c < mean(c) - std(c)*low 

c > mean(c) + std(c)*high 

 

The iteration continues with the updated sample until no 

elements are outside the (updated) range. 

 

Parameters 

---------- 

a : array_like 

Data array, will be raveled if not 1-D. 

low : float, optional 

Lower bound factor of sigma clipping. Default is 4. 

high : float, optional 

Upper bound factor of sigma clipping. Default is 4. 

 

Returns 

------- 

clipped : ndarray 

Input array with clipped elements removed. 

lower : float 

Lower threshold value use for clipping. 

upper : float 

Upper threshold value use for clipping. 

 

Examples 

-------- 

>>> from scipy.stats import sigmaclip 

>>> a = np.concatenate((np.linspace(9.5, 10.5, 31), 

... np.linspace(0, 20, 5))) 

>>> fact = 1.5 

>>> c, low, upp = sigmaclip(a, fact, fact) 

>>> c 

array([ 9.96666667, 10. , 10.03333333, 10. ]) 

>>> c.var(), c.std() 

(0.00055555555555555165, 0.023570226039551501) 

>>> low, c.mean() - fact*c.std(), c.min() 

(9.9646446609406727, 9.9646446609406727, 9.9666666666666668) 

>>> upp, c.mean() + fact*c.std(), c.max() 

(10.035355339059327, 10.035355339059327, 10.033333333333333) 

 

>>> a = np.concatenate((np.linspace(9.5, 10.5, 11), 

... np.linspace(-100, -50, 3))) 

>>> c, low, upp = sigmaclip(a, 1.8, 1.8) 

>>> (c == np.linspace(9.5, 10.5, 11)).all() 

True 

 

""" 

c = np.asarray(a).ravel() 

delta = 1 

while delta: 

c_std = c.std() 

c_mean = c.mean() 

size = c.size 

critlower = c_mean - c_std * low 

critupper = c_mean + c_std * high 

c = c[(c >= critlower) & (c <= critupper)] 

delta = size - c.size 

 

return SigmaclipResult(c, critlower, critupper) 

 

 

def trimboth(a, proportiontocut, axis=0): 

""" 

Slices off a proportion of items from both ends of an array. 

 

Slices off the passed proportion of items from both ends of the passed 

array (i.e., with `proportiontocut` = 0.1, slices leftmost 10% **and** 

rightmost 10% of scores). The trimmed values are the lowest and 

highest ones. 

Slices off less if proportion results in a non-integer slice index (i.e., 

conservatively slices off`proportiontocut`). 

 

Parameters 

---------- 

a : array_like 

Data to trim. 

proportiontocut : float 

Proportion (in range 0-1) of total data set to trim of each end. 

axis : int or None, optional 

Axis along which to trim data. Default is 0. If None, compute over 

the whole array `a`. 

 

Returns 

------- 

out : ndarray 

Trimmed version of array `a`. The order of the trimmed content 

is undefined. 

 

See Also 

-------- 

trim_mean 

 

Examples 

-------- 

>>> from scipy import stats 

>>> a = np.arange(20) 

>>> b = stats.trimboth(a, 0.1) 

>>> b.shape 

(16,) 

 

""" 

a = np.asarray(a) 

 

if a.size == 0: 

return a 

 

if axis is None: 

a = a.ravel() 

axis = 0 

 

nobs = a.shape[axis] 

lowercut = int(proportiontocut * nobs) 

uppercut = nobs - lowercut 

if (lowercut >= uppercut): 

raise ValueError("Proportion too big.") 

 

atmp = np.partition(a, (lowercut, uppercut - 1), axis) 

 

sl = [slice(None)] * atmp.ndim 

sl[axis] = slice(lowercut, uppercut) 

return atmp[sl] 

 

 

def trim1(a, proportiontocut, tail='right', axis=0): 

""" 

Slices off a proportion from ONE end of the passed array distribution. 

 

If `proportiontocut` = 0.1, slices off 'leftmost' or 'rightmost' 

10% of scores. The lowest or highest values are trimmed (depending on 

the tail). 

Slices off less if proportion results in a non-integer slice index 

(i.e., conservatively slices off `proportiontocut` ). 

 

Parameters 

---------- 

a : array_like 

Input array 

proportiontocut : float 

Fraction to cut off of 'left' or 'right' of distribution 

tail : {'left', 'right'}, optional 

Defaults to 'right'. 

axis : int or None, optional 

Axis along which to trim data. Default is 0. If None, compute over 

the whole array `a`. 

 

Returns 

------- 

trim1 : ndarray 

Trimmed version of array `a`. The order of the trimmed content is 

undefined. 

 

""" 

a = np.asarray(a) 

if axis is None: 

a = a.ravel() 

axis = 0 

 

nobs = a.shape[axis] 

 

# avoid possible corner case 

if proportiontocut >= 1: 

return [] 

 

if tail.lower() == 'right': 

lowercut = 0 

uppercut = nobs - int(proportiontocut * nobs) 

 

elif tail.lower() == 'left': 

lowercut = int(proportiontocut * nobs) 

uppercut = nobs 

 

atmp = np.partition(a, (lowercut, uppercut - 1), axis) 

 

return atmp[lowercut:uppercut] 

 

 

def trim_mean(a, proportiontocut, axis=0): 

""" 

Return mean of array after trimming distribution from both tails. 

 

If `proportiontocut` = 0.1, slices off 'leftmost' and 'rightmost' 10% of 

scores. The input is sorted before slicing. Slices off less if proportion 

results in a non-integer slice index (i.e., conservatively slices off 

`proportiontocut` ). 

 

Parameters 

---------- 

a : array_like 

Input array 

proportiontocut : float 

Fraction to cut off of both tails of the distribution 

axis : int or None, optional 

Axis along which the trimmed means are computed. Default is 0. 

If None, compute over the whole array `a`. 

 

Returns 

------- 

trim_mean : ndarray 

Mean of trimmed array. 

 

See Also 

-------- 

trimboth 

tmean : compute the trimmed mean ignoring values outside given `limits`. 

 

Examples 

-------- 

>>> from scipy import stats 

>>> x = np.arange(20) 

>>> stats.trim_mean(x, 0.1) 

9.5 

>>> x2 = x.reshape(5, 4) 

>>> x2 

array([[ 0, 1, 2, 3], 

[ 4, 5, 6, 7], 

[ 8, 9, 10, 11], 

[12, 13, 14, 15], 

[16, 17, 18, 19]]) 

>>> stats.trim_mean(x2, 0.25) 

array([ 8., 9., 10., 11.]) 

>>> stats.trim_mean(x2, 0.25, axis=1) 

array([ 1.5, 5.5, 9.5, 13.5, 17.5]) 

 

""" 

a = np.asarray(a) 

 

if a.size == 0: 

return np.nan 

 

if axis is None: 

a = a.ravel() 

axis = 0 

 

nobs = a.shape[axis] 

lowercut = int(proportiontocut * nobs) 

uppercut = nobs - lowercut 

if (lowercut > uppercut): 

raise ValueError("Proportion too big.") 

 

atmp = np.partition(a, (lowercut, uppercut - 1), axis) 

 

sl = [slice(None)] * atmp.ndim 

sl[axis] = slice(lowercut, uppercut) 

return np.mean(atmp[sl], axis=axis) 

 

 

F_onewayResult = namedtuple('F_onewayResult', ('statistic', 'pvalue')) 

 

 

def f_oneway(*args): 

""" 

Performs a 1-way ANOVA. 

 

The one-way ANOVA tests the null hypothesis that two or more groups have 

the same population mean. The test is applied to samples from two or 

more groups, possibly with differing sizes. 

 

Parameters 

---------- 

sample1, sample2, ... : array_like 

The sample measurements for each group. 

 

Returns 

------- 

statistic : float 

The computed F-value of the test. 

pvalue : float 

The associated p-value from the F-distribution. 

 

Notes 

----- 

The ANOVA test has important assumptions that must be satisfied in order 

for the associated p-value to be valid. 

 

1. The samples are independent. 

2. Each sample is from a normally distributed population. 

3. The population standard deviations of the groups are all equal. This 

property is known as homoscedasticity. 

 

If these assumptions are not true for a given set of data, it may still be 

possible to use the Kruskal-Wallis H-test (`scipy.stats.kruskal`) although 

with some loss of power. 

 

The algorithm is from Heiman[2], pp.394-7. 

 

 

References 

---------- 

.. [1] Lowry, Richard. "Concepts and Applications of Inferential 

Statistics". Chapter 14. 

http://faculty.vassar.edu/lowry/ch14pt1.html 

 

.. [2] Heiman, G.W. Research Methods in Statistics. 2002. 

 

.. [3] McDonald, G. H. "Handbook of Biological Statistics", One-way ANOVA. 

http://www.biostathandbook.com/onewayanova.html 

 

Examples 

-------- 

>>> import scipy.stats as stats 

 

[3]_ Here are some data on a shell measurement (the length of the anterior 

adductor muscle scar, standardized by dividing by length) in the mussel 

Mytilus trossulus from five locations: Tillamook, Oregon; Newport, Oregon; 

Petersburg, Alaska; Magadan, Russia; and Tvarminne, Finland, taken from a 

much larger data set used in McDonald et al. (1991). 

 

>>> tillamook = [0.0571, 0.0813, 0.0831, 0.0976, 0.0817, 0.0859, 0.0735, 

... 0.0659, 0.0923, 0.0836] 

>>> newport = [0.0873, 0.0662, 0.0672, 0.0819, 0.0749, 0.0649, 0.0835, 

... 0.0725] 

>>> petersburg = [0.0974, 0.1352, 0.0817, 0.1016, 0.0968, 0.1064, 0.105] 

>>> magadan = [0.1033, 0.0915, 0.0781, 0.0685, 0.0677, 0.0697, 0.0764, 

... 0.0689] 

>>> tvarminne = [0.0703, 0.1026, 0.0956, 0.0973, 0.1039, 0.1045] 

>>> stats.f_oneway(tillamook, newport, petersburg, magadan, tvarminne) 

(7.1210194716424473, 0.00028122423145345439) 

 

""" 

args = [np.asarray(arg, dtype=float) for arg in args] 

# ANOVA on N groups, each in its own array 

num_groups = len(args) 

alldata = np.concatenate(args) 

bign = len(alldata) 

 

# Determine the mean of the data, and subtract that from all inputs to a 

# variance (via sum_of_sq / sq_of_sum) calculation. Variance is invariance 

# to a shift in location, and centering all data around zero vastly 

# improves numerical stability. 

offset = alldata.mean() 

alldata -= offset 

 

sstot = _sum_of_squares(alldata) - (_square_of_sums(alldata) / float(bign)) 

ssbn = 0 

for a in args: 

ssbn += _square_of_sums(a - offset) / float(len(a)) 

 

# Naming: variables ending in bn/b are for "between treatments", wn/w are 

# for "within treatments" 

ssbn -= (_square_of_sums(alldata) / float(bign)) 

sswn = sstot - ssbn 

dfbn = num_groups - 1 

dfwn = bign - num_groups 

msb = ssbn / float(dfbn) 

msw = sswn / float(dfwn) 

f = msb / msw 

 

prob = special.fdtrc(dfbn, dfwn, f) # equivalent to stats.f.sf 

 

return F_onewayResult(f, prob) 

 

 

def pearsonr(x, y): 

r""" 

Calculate a Pearson correlation coefficient and the p-value for testing 

non-correlation. 

 

The Pearson correlation coefficient measures the linear relationship 

between two datasets. Strictly speaking, Pearson's correlation requires 

that each dataset be normally distributed, and not necessarily zero-mean. 

Like other correlation coefficients, this one varies between -1 and +1 

with 0 implying no correlation. Correlations of -1 or +1 imply an exact 

linear relationship. Positive correlations imply that as x increases, so 

does y. Negative correlations imply that as x increases, y decreases. 

 

The p-value roughly indicates the probability of an uncorrelated system 

producing datasets that have a Pearson correlation at least as extreme 

as the one computed from these datasets. The p-values are not entirely 

reliable but are probably reasonable for datasets larger than 500 or so. 

 

Parameters 

---------- 

x : (N,) array_like 

Input 

y : (N,) array_like 

Input 

 

Returns 

------- 

r : float 

Pearson's correlation coefficient 

p-value : float 

2-tailed p-value 

 

Notes 

----- 

 

The correlation coefficient is calculated as follows: 

 

.. math:: 

 

r_{pb} = \frac{\sum (x - m_x) (y - m_y) 

}{\sqrt{\sum (x - m_x)^2 (y - m_y)^2}} 

 

where :math:`m_x` is the mean of the vector :math:`x` and :math:`m_y` is 

the mean of the vector :math:`y`. 

 

 

References 

---------- 

http://www.statsoft.com/textbook/glosp.html#Pearson%20Correlation 

 

Examples 

-------- 

>>> from scipy import stats 

>>> a = np.array([0, 0, 0, 1, 1, 1, 1]) 

>>> b = np.arange(7) 

>>> stats.pearsonr(a, b) 

(0.8660254037844386, 0.011724811003954654) 

 

>>> stats.pearsonr([1,2,3,4,5], [5,6,7,8,7]) 

(0.83205029433784372, 0.080509573298498519) 

""" 

# x and y should have same length. 

x = np.asarray(x) 

y = np.asarray(y) 

n = len(x) 

mx = x.mean() 

my = y.mean() 

xm, ym = x - mx, y - my 

r_num = np.add.reduce(xm * ym) 

r_den = np.sqrt(_sum_of_squares(xm) * _sum_of_squares(ym)) 

r = r_num / r_den 

 

# Presumably, if abs(r) > 1, then it is only some small artifact of 

# floating point arithmetic. 

r = max(min(r, 1.0), -1.0) 

df = n - 2 

if abs(r) == 1.0: 

prob = 0.0 

else: 

t_squared = r**2 * (df / ((1.0 - r) * (1.0 + r))) 

prob = _betai(0.5*df, 0.5, df/(df+t_squared)) 

 

return r, prob 

 

 

def fisher_exact(table, alternative='two-sided'): 

"""Performs a Fisher exact test on a 2x2 contingency table. 

 

Parameters 

---------- 

table : array_like of ints 

A 2x2 contingency table. Elements should be non-negative integers. 

alternative : {'two-sided', 'less', 'greater'}, optional 

Which alternative hypothesis to the null hypothesis the test uses. 

Default is 'two-sided'. 

 

Returns 

------- 

oddsratio : float 

This is prior odds ratio and not a posterior estimate. 

p_value : float 

P-value, the probability of obtaining a distribution at least as 

extreme as the one that was actually observed, assuming that the 

null hypothesis is true. 

 

See Also 

-------- 

chi2_contingency : Chi-square test of independence of variables in a 

contingency table. 

 

Notes 

----- 

The calculated odds ratio is different from the one R uses. This scipy 

implementation returns the (more common) "unconditional Maximum 

Likelihood Estimate", while R uses the "conditional Maximum Likelihood 

Estimate". 

 

For tables with large numbers, the (inexact) chi-square test implemented 

in the function `chi2_contingency` can also be used. 

 

Examples 

-------- 

Say we spend a few days counting whales and sharks in the Atlantic and 

Indian oceans. In the Atlantic ocean we find 8 whales and 1 shark, in the 

Indian ocean 2 whales and 5 sharks. Then our contingency table is:: 

 

Atlantic Indian 

whales 8 2 

sharks 1 5 

 

We use this table to find the p-value: 

 

>>> import scipy.stats as stats 

>>> oddsratio, pvalue = stats.fisher_exact([[8, 2], [1, 5]]) 

>>> pvalue 

0.0349... 

 

The probability that we would observe this or an even more imbalanced ratio 

by chance is about 3.5%. A commonly used significance level is 5%--if we 

adopt that, we can therefore conclude that our observed imbalance is 

statistically significant; whales prefer the Atlantic while sharks prefer 

the Indian ocean. 

 

""" 

hypergeom = distributions.hypergeom 

c = np.asarray(table, dtype=np.int64) # int32 is not enough for the algorithm 

if not c.shape == (2, 2): 

raise ValueError("The input `table` must be of shape (2, 2).") 

 

if np.any(c < 0): 

raise ValueError("All values in `table` must be nonnegative.") 

 

if 0 in c.sum(axis=0) or 0 in c.sum(axis=1): 

# If both values in a row or column are zero, the p-value is 1 and 

# the odds ratio is NaN. 

return np.nan, 1.0 

 

if c[1, 0] > 0 and c[0, 1] > 0: 

oddsratio = c[0, 0] * c[1, 1] / float(c[1, 0] * c[0, 1]) 

else: 

oddsratio = np.inf 

 

n1 = c[0, 0] + c[0, 1] 

n2 = c[1, 0] + c[1, 1] 

n = c[0, 0] + c[1, 0] 

 

def binary_search(n, n1, n2, side): 

"""Binary search for where to begin lower/upper halves in two-sided 

test. 

""" 

if side == "upper": 

minval = mode 

maxval = n 

else: 

minval = 0 

maxval = mode 

guess = -1 

while maxval - minval > 1: 

if maxval == minval + 1 and guess == minval: 

guess = maxval 

else: 

guess = (maxval + minval) // 2 

pguess = hypergeom.pmf(guess, n1 + n2, n1, n) 

if side == "upper": 

ng = guess - 1 

else: 

ng = guess + 1 

if pguess <= pexact < hypergeom.pmf(ng, n1 + n2, n1, n): 

break 

elif pguess < pexact: 

maxval = guess 

else: 

minval = guess 

if guess == -1: 

guess = minval 

if side == "upper": 

while guess > 0 and hypergeom.pmf(guess, n1 + n2, n1, n) < pexact * epsilon: 

guess -= 1 

while hypergeom.pmf(guess, n1 + n2, n1, n) > pexact / epsilon: 

guess += 1 

else: 

while hypergeom.pmf(guess, n1 + n2, n1, n) < pexact * epsilon: 

guess += 1 

while guess > 0 and hypergeom.pmf(guess, n1 + n2, n1, n) > pexact / epsilon: 

guess -= 1 

return guess 

 

if alternative == 'less': 

pvalue = hypergeom.cdf(c[0, 0], n1 + n2, n1, n) 

elif alternative == 'greater': 

# Same formula as the 'less' case, but with the second column. 

pvalue = hypergeom.cdf(c[0, 1], n1 + n2, n1, c[0, 1] + c[1, 1]) 

elif alternative == 'two-sided': 

mode = int(float((n + 1) * (n1 + 1)) / (n1 + n2 + 2)) 

pexact = hypergeom.pmf(c[0, 0], n1 + n2, n1, n) 

pmode = hypergeom.pmf(mode, n1 + n2, n1, n) 

 

epsilon = 1 - 1e-4 

if np.abs(pexact - pmode) / np.maximum(pexact, pmode) <= 1 - epsilon: 

return oddsratio, 1. 

 

elif c[0, 0] < mode: 

plower = hypergeom.cdf(c[0, 0], n1 + n2, n1, n) 

if hypergeom.pmf(n, n1 + n2, n1, n) > pexact / epsilon: 

return oddsratio, plower 

 

guess = binary_search(n, n1, n2, "upper") 

pvalue = plower + hypergeom.sf(guess - 1, n1 + n2, n1, n) 

else: 

pupper = hypergeom.sf(c[0, 0] - 1, n1 + n2, n1, n) 

if hypergeom.pmf(0, n1 + n2, n1, n) > pexact / epsilon: 

return oddsratio, pupper 

 

guess = binary_search(n, n1, n2, "lower") 

pvalue = pupper + hypergeom.cdf(guess, n1 + n2, n1, n) 

else: 

msg = "`alternative` should be one of {'two-sided', 'less', 'greater'}" 

raise ValueError(msg) 

 

if pvalue > 1.0: 

pvalue = 1.0 

 

return oddsratio, pvalue 

 

 

SpearmanrResult = namedtuple('SpearmanrResult', ('correlation', 'pvalue')) 

 

 

def spearmanr(a, b=None, axis=0, nan_policy='propagate'): 

""" 

Calculate a Spearman rank-order correlation coefficient and the p-value 

to test for non-correlation. 

 

The Spearman correlation is a nonparametric measure of the monotonicity 

of the relationship between two datasets. Unlike the Pearson correlation, 

the Spearman correlation does not assume that both datasets are normally 

distributed. Like other correlation coefficients, this one varies 

between -1 and +1 with 0 implying no correlation. Correlations of -1 or 

+1 imply an exact monotonic relationship. Positive correlations imply that 

as x increases, so does y. Negative correlations imply that as x 

increases, y decreases. 

 

The p-value roughly indicates the probability of an uncorrelated system 

producing datasets that have a Spearman correlation at least as extreme 

as the one computed from these datasets. The p-values are not entirely 

reliable but are probably reasonable for datasets larger than 500 or so. 

 

Parameters 

---------- 

a, b : 1D or 2D array_like, b is optional 

One or two 1-D or 2-D arrays containing multiple variables and 

observations. When these are 1-D, each represents a vector of 

observations of a single variable. For the behavior in the 2-D case, 

see under ``axis``, below. 

Both arrays need to have the same length in the ``axis`` dimension. 

axis : int or None, optional 

If axis=0 (default), then each column represents a variable, with 

observations in the rows. If axis=1, the relationship is transposed: 

each row represents a variable, while the columns contain observations. 

If axis=None, then both arrays will be raveled. 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. 

 

Returns 

------- 

correlation : float or ndarray (2-D square) 

Spearman correlation matrix or correlation coefficient (if only 2 

variables are given as parameters. Correlation matrix is square with 

length equal to total number of variables (columns or rows) in a and b 

combined. 

pvalue : float 

The two-sided p-value for a hypothesis test whose null hypothesis is 

that two sets of data are uncorrelated, has same dimension as rho. 

 

Notes 

----- 

Changes in scipy 0.8.0: rewrite to add tie-handling, and axis. 

 

References 

---------- 

 

.. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard 

Probability and Statistics Tables and Formulae. Chapman & Hall: New 

York. 2000. 

Section 14.7 

 

Examples 

-------- 

>>> from scipy import stats 

>>> stats.spearmanr([1,2,3,4,5], [5,6,7,8,7]) 

(0.82078268166812329, 0.088587005313543798) 

>>> np.random.seed(1234321) 

>>> x2n = np.random.randn(100, 2) 

>>> y2n = np.random.randn(100, 2) 

>>> stats.spearmanr(x2n) 

(0.059969996999699973, 0.55338590803773591) 

>>> stats.spearmanr(x2n[:,0], x2n[:,1]) 

(0.059969996999699973, 0.55338590803773591) 

>>> rho, pval = stats.spearmanr(x2n, y2n) 

>>> rho 

array([[ 1. , 0.05997 , 0.18569457, 0.06258626], 

[ 0.05997 , 1. , 0.110003 , 0.02534653], 

[ 0.18569457, 0.110003 , 1. , 0.03488749], 

[ 0.06258626, 0.02534653, 0.03488749, 1. ]]) 

>>> pval 

array([[ 0. , 0.55338591, 0.06435364, 0.53617935], 

[ 0.55338591, 0. , 0.27592895, 0.80234077], 

[ 0.06435364, 0.27592895, 0. , 0.73039992], 

[ 0.53617935, 0.80234077, 0.73039992, 0. ]]) 

>>> rho, pval = stats.spearmanr(x2n.T, y2n.T, axis=1) 

>>> rho 

array([[ 1. , 0.05997 , 0.18569457, 0.06258626], 

[ 0.05997 , 1. , 0.110003 , 0.02534653], 

[ 0.18569457, 0.110003 , 1. , 0.03488749], 

[ 0.06258626, 0.02534653, 0.03488749, 1. ]]) 

>>> stats.spearmanr(x2n, y2n, axis=None) 

(0.10816770419260482, 0.1273562188027364) 

>>> stats.spearmanr(x2n.ravel(), y2n.ravel()) 

(0.10816770419260482, 0.1273562188027364) 

 

>>> xint = np.random.randint(10, size=(100, 2)) 

>>> stats.spearmanr(xint) 

(0.052760927029710199, 0.60213045837062351) 

 

""" 

a, axisout = _chk_asarray(a, axis) 

 

a_contains_nan, nan_policy = _contains_nan(a, nan_policy) 

 

if a_contains_nan: 

a = ma.masked_invalid(a) 

 

if a.size <= 1: 

return SpearmanrResult(np.nan, np.nan) 

 

ar = np.apply_along_axis(rankdata, axisout, a) 

 

br = None 

if b is not None: 

b, axisout = _chk_asarray(b, axis) 

 

b_contains_nan, nan_policy = _contains_nan(b, nan_policy) 

 

if a_contains_nan or b_contains_nan: 

b = ma.masked_invalid(b) 

 

if nan_policy == 'propagate': 

rho, pval = mstats_basic.spearmanr(a, b, use_ties=True) 

return SpearmanrResult(rho * np.nan, pval * np.nan) 

 

if nan_policy == 'omit': 

return mstats_basic.spearmanr(a, b, use_ties=True) 

 

br = np.apply_along_axis(rankdata, axisout, b) 

n = a.shape[axisout] 

rs = np.corrcoef(ar, br, rowvar=axisout) 

 

olderr = np.seterr(divide='ignore') # rs can have elements equal to 1 

try: 

# clip the small negative values possibly caused by rounding 

# errors before taking the square root 

t = rs * np.sqrt(((n-2)/((rs+1.0)*(1.0-rs))).clip(0)) 

finally: 

np.seterr(**olderr) 

 

prob = 2 * distributions.t.sf(np.abs(t), n-2) 

 

if rs.shape == (2, 2): 

return SpearmanrResult(rs[1, 0], prob[1, 0]) 

else: 

return SpearmanrResult(rs, prob) 

 

 

PointbiserialrResult = namedtuple('PointbiserialrResult', 

('correlation', 'pvalue')) 

 

 

def pointbiserialr(x, y): 

r""" 

Calculate a point biserial correlation coefficient and its p-value. 

 

The point biserial correlation is used to measure the relationship 

between a binary variable, x, and a continuous variable, y. Like other 

correlation coefficients, this one varies between -1 and +1 with 0 

implying no correlation. Correlations of -1 or +1 imply a determinative 

relationship. 

 

This function uses a shortcut formula but produces the same result as 

`pearsonr`. 

 

Parameters 

---------- 

x : array_like of bools 

Input array. 

y : array_like 

Input array. 

 

Returns 

------- 

correlation : float 

R value 

pvalue : float 

2-tailed p-value 

 

Notes 

----- 

`pointbiserialr` uses a t-test with ``n-1`` degrees of freedom. 

It is equivalent to `pearsonr.` 

 

The value of the point-biserial correlation can be calculated from: 

 

.. math:: 

 

r_{pb} = \frac{\overline{Y_{1}} - 

\overline{Y_{0}}}{s_{y}}\sqrt{\frac{N_{1} N_{2}}{N (N - 1))}} 

 

Where :math:`Y_{0}` and :math:`Y_{1}` are means of the metric 

observations coded 0 and 1 respectively; :math:`N_{0}` and :math:`N_{1}` 

are number of observations coded 0 and 1 respectively; :math:`N` is the 

total number of observations and :math:`s_{y}` is the standard 

deviation of all the metric observations. 

 

A value of :math:`r_{pb}` that is significantly different from zero is 

completely equivalent to a significant difference in means between the two 

groups. Thus, an independent groups t Test with :math:`N-2` degrees of 

freedom may be used to test whether :math:`r_{pb}` is nonzero. The 

relation between the t-statistic for comparing two independent groups and 

:math:`r_{pb}` is given by: 

 

.. math:: 

 

t = \sqrt{N - 2}\frac{r_{pb}}{\sqrt{1 - r^{2}_{pb}}} 

 

References 

---------- 

.. [1] J. Lev, "The Point Biserial Coefficient of Correlation", Ann. Math. 

Statist., Vol. 20, no.1, pp. 125-126, 1949. 

 

.. [2] R.F. Tate, "Correlation Between a Discrete and a Continuous 

Variable. Point-Biserial Correlation.", Ann. Math. Statist., Vol. 25, 

np. 3, pp. 603-607, 1954. 

 

.. [3] http://onlinelibrary.wiley.com/doi/10.1002/9781118445112.stat06227/full 

 

Examples 

-------- 

>>> from scipy import stats 

>>> a = np.array([0, 0, 0, 1, 1, 1, 1]) 

>>> b = np.arange(7) 

>>> stats.pointbiserialr(a, b) 

(0.8660254037844386, 0.011724811003954652) 

>>> stats.pearsonr(a, b) 

(0.86602540378443871, 0.011724811003954626) 

>>> np.corrcoef(a, b) 

array([[ 1. , 0.8660254], 

[ 0.8660254, 1. ]]) 

 

""" 

rpb, prob = pearsonr(x, y) 

return PointbiserialrResult(rpb, prob) 

 

 

KendalltauResult = namedtuple('KendalltauResult', ('correlation', 'pvalue')) 

 

 

def kendalltau(x, y, initial_lexsort=None, nan_policy='propagate'): 

""" 

Calculate Kendall's tau, a correlation measure for ordinal data. 

 

Kendall's tau is a measure of the correspondence between two rankings. 

Values close to 1 indicate strong agreement, values close to -1 indicate 

strong disagreement. This is the 1945 "tau-b" version of Kendall's 

tau [2]_, which can account for ties and which reduces to the 1938 "tau-a" 

version [1]_ in absence of ties. 

 

Parameters 

---------- 

x, y : array_like 

Arrays of rankings, of the same shape. If arrays are not 1-D, they will 

be flattened to 1-D. 

initial_lexsort : bool, optional 

Unused (deprecated). 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. Note that if the input contains nan 

'omit' delegates to mstats_basic.kendalltau(), which has a different 

implementation. 

 

Returns 

------- 

correlation : float 

The tau statistic. 

pvalue : float 

The two-sided p-value for a hypothesis test whose null hypothesis is 

an absence of association, tau = 0. 

 

See also 

-------- 

spearmanr : Calculates a Spearman rank-order correlation coefficient. 

theilslopes : Computes the Theil-Sen estimator for a set of points (x, y). 

weightedtau : Computes a weighted version of Kendall's tau. 

 

Notes 

----- 

The definition of Kendall's tau that is used is [2]_:: 

 

tau = (P - Q) / sqrt((P + Q + T) * (P + Q + U)) 

 

where P is the number of concordant pairs, Q the number of discordant 

pairs, T the number of ties only in `x`, and U the number of ties only in 

`y`. If a tie occurs for the same pair in both `x` and `y`, it is not 

added to either T or U. 

 

References 

---------- 

.. [1] Maurice G. Kendall, "A New Measure of Rank Correlation", Biometrika 

Vol. 30, No. 1/2, pp. 81-93, 1938. 

.. [2] Maurice G. Kendall, "The treatment of ties in ranking problems", 

Biometrika Vol. 33, No. 3, pp. 239-251. 1945. 

.. [3] Gottfried E. Noether, "Elements of Nonparametric Statistics", John 

Wiley & Sons, 1967. 

.. [4] Peter M. Fenwick, "A new data structure for cumulative frequency 

tables", Software: Practice and Experience, Vol. 24, No. 3, 

pp. 327-336, 1994. 

 

Examples 

-------- 

>>> from scipy import stats 

>>> x1 = [12, 2, 1, 12, 2] 

>>> x2 = [1, 4, 7, 1, 0] 

>>> tau, p_value = stats.kendalltau(x1, x2) 

>>> tau 

-0.47140452079103173 

>>> p_value 

0.2827454599327748 

 

""" 

x = np.asarray(x).ravel() 

y = np.asarray(y).ravel() 

 

if x.size != y.size: 

raise ValueError("All inputs to `kendalltau` must be of the same size, " 

"found x-size %s and y-size %s" % (x.size, y.size)) 

elif not x.size or not y.size: 

return KendalltauResult(np.nan, np.nan) # Return NaN if arrays are empty 

 

# check both x and y 

cnx, npx = _contains_nan(x, nan_policy) 

cny, npy = _contains_nan(y, nan_policy) 

contains_nan = cnx or cny 

if npx == 'omit' or npy == 'omit': 

nan_policy = 'omit' 

 

if contains_nan and nan_policy == 'propagate': 

return KendalltauResult(np.nan, np.nan) 

 

elif contains_nan and nan_policy == 'omit': 

x = ma.masked_invalid(x) 

y = ma.masked_invalid(y) 

return mstats_basic.kendalltau(x, y) 

 

if initial_lexsort is not None: # deprecate to drop! 

warnings.warn('"initial_lexsort" is gone!') 

 

def count_rank_tie(ranks): 

cnt = np.bincount(ranks).astype('int64', copy=False) 

cnt = cnt[cnt > 1] 

return ((cnt * (cnt - 1) // 2).sum(), 

(cnt * (cnt - 1.) * (cnt - 2)).sum(), 

(cnt * (cnt - 1.) * (2*cnt + 5)).sum()) 

 

size = x.size 

perm = np.argsort(y) # sort on y and convert y to dense ranks 

x, y = x[perm], y[perm] 

y = np.r_[True, y[1:] != y[:-1]].cumsum(dtype=np.intp) 

 

# stable sort on x and convert x to dense ranks 

perm = np.argsort(x, kind='mergesort') 

x, y = x[perm], y[perm] 

x = np.r_[True, x[1:] != x[:-1]].cumsum(dtype=np.intp) 

 

dis = _kendall_dis(x, y) # discordant pairs 

 

obs = np.r_[True, (x[1:] != x[:-1]) | (y[1:] != y[:-1]), True] 

cnt = np.diff(np.where(obs)[0]).astype('int64', copy=False) 

 

ntie = (cnt * (cnt - 1) // 2).sum() # joint ties 

xtie, x0, x1 = count_rank_tie(x) # ties in x, stats 

ytie, y0, y1 = count_rank_tie(y) # ties in y, stats 

 

tot = (size * (size - 1)) // 2 

 

if xtie == tot or ytie == tot: 

return KendalltauResult(np.nan, np.nan) 

 

# Note that tot = con + dis + (xtie - ntie) + (ytie - ntie) + ntie 

# = con + dis + xtie + ytie - ntie 

con_minus_dis = tot - xtie - ytie + ntie - 2 * dis 

tau = con_minus_dis / np.sqrt(tot - xtie) / np.sqrt(tot - ytie) 

# Limit range to fix computational errors 

tau = min(1., max(-1., tau)) 

 

# con_minus_dis is approx normally distributed with this variance [3]_ 

var = (size * (size - 1) * (2.*size + 5) - x1 - y1) / 18. + ( 

2. * xtie * ytie) / (size * (size - 1)) + x0 * y0 / (9. * 

size * (size - 1) * (size - 2)) 

pvalue = special.erfc(np.abs(con_minus_dis) / np.sqrt(var) / np.sqrt(2)) 

 

# Limit range to fix computational errors 

return KendalltauResult(min(1., max(-1., tau)), pvalue) 

 

 

WeightedTauResult = namedtuple('WeightedTauResult', ('correlation', 'pvalue')) 

 

 

def weightedtau(x, y, rank=True, weigher=None, additive=True): 

r""" 

Compute a weighted version of Kendall's :math:`\tau`. 

 

The weighted :math:`\tau` is a weighted version of Kendall's 

:math:`\tau` in which exchanges of high weight are more influential than 

exchanges of low weight. The default parameters compute the additive 

hyperbolic version of the index, :math:`\tau_\mathrm h`, which has 

been shown to provide the best balance between important and 

unimportant elements [1]_. 

 

The weighting is defined by means of a rank array, which assigns a 

nonnegative rank to each element, and a weigher function, which 

assigns a weight based from the rank to each element. The weight of an 

exchange is then the sum or the product of the weights of the ranks of 

the exchanged elements. The default parameters compute 

:math:`\tau_\mathrm h`: an exchange between elements with rank 

:math:`r` and :math:`s` (starting from zero) has weight 

:math:`1/(r+1) + 1/(s+1)`. 

 

Specifying a rank array is meaningful only if you have in mind an 

external criterion of importance. If, as it usually happens, you do 

not have in mind a specific rank, the weighted :math:`\tau` is 

defined by averaging the values obtained using the decreasing 

lexicographical rank by (`x`, `y`) and by (`y`, `x`). This is the 

behavior with default parameters. 

 

Note that if you are computing the weighted :math:`\tau` on arrays of 

ranks, rather than of scores (i.e., a larger value implies a lower 

rank) you must negate the ranks, so that elements of higher rank are 

associated with a larger value. 

 

Parameters 

---------- 

x, y : array_like 

Arrays of scores, of the same shape. If arrays are not 1-D, they will 

be flattened to 1-D. 

rank: array_like of ints or bool, optional 

A nonnegative rank assigned to each element. If it is None, the 

decreasing lexicographical rank by (`x`, `y`) will be used: elements of 

higher rank will be those with larger `x`-values, using `y`-values to 

break ties (in particular, swapping `x` and `y` will give a different 

result). If it is False, the element indices will be used 

directly as ranks. The default is True, in which case this 

function returns the average of the values obtained using the 

decreasing lexicographical rank by (`x`, `y`) and by (`y`, `x`). 

weigher : callable, optional 

The weigher function. Must map nonnegative integers (zero 

representing the most important element) to a nonnegative weight. 

The default, None, provides hyperbolic weighing, that is, 

rank :math:`r` is mapped to weight :math:`1/(r+1)`. 

additive : bool, optional 

If True, the weight of an exchange is computed by adding the 

weights of the ranks of the exchanged elements; otherwise, the weights 

are multiplied. The default is True. 

 

Returns 

------- 

correlation : float 

The weighted :math:`\tau` correlation index. 

pvalue : float 

Presently ``np.nan``, as the null statistics is unknown (even in the 

additive hyperbolic case). 

 

See also 

-------- 

kendalltau : Calculates Kendall's tau. 

spearmanr : Calculates a Spearman rank-order correlation coefficient. 

theilslopes : Computes the Theil-Sen estimator for a set of points (x, y). 

 

Notes 

----- 

This function uses an :math:`O(n \log n)`, mergesort-based algorithm 

[1]_ that is a weighted extension of Knight's algorithm for Kendall's 

:math:`\tau` [2]_. It can compute Shieh's weighted :math:`\tau` [3]_ 

between rankings without ties (i.e., permutations) by setting 

`additive` and `rank` to False, as the definition given in [1]_ is a 

generalization of Shieh's. 

 

NaNs are considered the smallest possible score. 

 

.. versionadded:: 0.19.0 

 

References 

---------- 

.. [1] Sebastiano Vigna, "A weighted correlation index for rankings with 

ties", Proceedings of the 24th international conference on World 

Wide Web, pp. 1166-1176, ACM, 2015. 

.. [2] W.R. Knight, "A Computer Method for Calculating Kendall's Tau with 

Ungrouped Data", Journal of the American Statistical Association, 

Vol. 61, No. 314, Part 1, pp. 436-439, 1966. 

.. [3] Grace S. Shieh. "A weighted Kendall's tau statistic", Statistics & 

Probability Letters, Vol. 39, No. 1, pp. 17-24, 1998. 

 

Examples 

-------- 

>>> from scipy import stats 

>>> x = [12, 2, 1, 12, 2] 

>>> y = [1, 4, 7, 1, 0] 

>>> tau, p_value = stats.weightedtau(x, y) 

>>> tau 

-0.56694968153682723 

>>> p_value 

nan 

>>> tau, p_value = stats.weightedtau(x, y, additive=False) 

>>> tau 

-0.62205716951801038 

 

NaNs are considered the smallest possible score: 

 

>>> x = [12, 2, 1, 12, 2] 

>>> y = [1, 4, 7, 1, np.nan] 

>>> tau, _ = stats.weightedtau(x, y) 

>>> tau 

-0.56694968153682723 

 

This is exactly Kendall's tau: 

 

>>> x = [12, 2, 1, 12, 2] 

>>> y = [1, 4, 7, 1, 0] 

>>> tau, _ = stats.weightedtau(x, y, weigher=lambda x: 1) 

>>> tau 

-0.47140452079103173 

 

>>> x = [12, 2, 1, 12, 2] 

>>> y = [1, 4, 7, 1, 0] 

>>> stats.weightedtau(x, y, rank=None) 

WeightedTauResult(correlation=-0.4157652301037516, pvalue=nan) 

>>> stats.weightedtau(y, x, rank=None) 

WeightedTauResult(correlation=-0.7181341329699028, pvalue=nan) 

 

""" 

x = np.asarray(x).ravel() 

y = np.asarray(y).ravel() 

 

if x.size != y.size: 

raise ValueError("All inputs to `weightedtau` must be of the same size, " 

"found x-size %s and y-size %s" % (x.size, y.size)) 

if not x.size: 

return WeightedTauResult(np.nan, np.nan) # Return NaN if arrays are empty 

 

# If there are NaNs we apply _toint64() 

if np.isnan(np.sum(x)): 

x = _toint64(x) 

if np.isnan(np.sum(x)): 

y = _toint64(y) 

 

# Reduce to ranks unsupported types 

if x.dtype != y.dtype: 

if x.dtype != np.int64: 

x = _toint64(x) 

if y.dtype != np.int64: 

y = _toint64(y) 

else: 

if x.dtype not in (np.int32, np.int64, np.float32, np.float64): 

x = _toint64(x) 

y = _toint64(y) 

 

if rank is True: 

return WeightedTauResult(( 

_weightedrankedtau(x, y, None, weigher, additive) + 

_weightedrankedtau(y, x, None, weigher, additive) 

) / 2, np.nan) 

 

if rank is False: 

rank = np.arange(x.size, dtype=np.intp) 

elif rank is not None: 

rank = np.asarray(rank).ravel() 

if rank.size != x.size: 

raise ValueError("All inputs to `weightedtau` must be of the same size, " 

"found x-size %s and rank-size %s" % (x.size, rank.size)) 

 

return WeightedTauResult(_weightedrankedtau(x, y, rank, weigher, additive), np.nan) 

 

 

##################################### 

# INFERENTIAL STATISTICS # 

##################################### 

 

Ttest_1sampResult = namedtuple('Ttest_1sampResult', ('statistic', 'pvalue')) 

 

 

def ttest_1samp(a, popmean, axis=0, nan_policy='propagate'): 

""" 

Calculate the T-test for the mean of ONE group of scores. 

 

This is a two-sided test for the null hypothesis that the expected value 

(mean) of a sample of independent observations `a` is equal to the given 

population mean, `popmean`. 

 

Parameters 

---------- 

a : array_like 

sample observation 

popmean : float or array_like 

expected value in null hypothesis. If array_like, then it must have the 

same shape as `a` excluding the axis dimension 

axis : int or None, optional 

Axis along which to compute test. If None, compute over the whole 

array `a`. 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. 

 

Returns 

------- 

statistic : float or array 

t-statistic 

pvalue : float or array 

two-tailed p-value 

 

Examples 

-------- 

>>> from scipy import stats 

 

>>> np.random.seed(7654567) # fix seed to get the same result 

>>> rvs = stats.norm.rvs(loc=5, scale=10, size=(50,2)) 

 

Test if mean of random sample is equal to true mean, and different mean. 

We reject the null hypothesis in the second case and don't reject it in 

the first case. 

 

>>> stats.ttest_1samp(rvs,5.0) 

(array([-0.68014479, -0.04323899]), array([ 0.49961383, 0.96568674])) 

>>> stats.ttest_1samp(rvs,0.0) 

(array([ 2.77025808, 4.11038784]), array([ 0.00789095, 0.00014999])) 

 

Examples using axis and non-scalar dimension for population mean. 

 

>>> stats.ttest_1samp(rvs,[5.0,0.0]) 

(array([-0.68014479, 4.11038784]), array([ 4.99613833e-01, 1.49986458e-04])) 

>>> stats.ttest_1samp(rvs.T,[5.0,0.0],axis=1) 

(array([-0.68014479, 4.11038784]), array([ 4.99613833e-01, 1.49986458e-04])) 

>>> stats.ttest_1samp(rvs,[[5.0],[0.0]]) 

(array([[-0.68014479, -0.04323899], 

[ 2.77025808, 4.11038784]]), array([[ 4.99613833e-01, 9.65686743e-01], 

[ 7.89094663e-03, 1.49986458e-04]])) 

 

""" 

a, axis = _chk_asarray(a, axis) 

 

contains_nan, nan_policy = _contains_nan(a, nan_policy) 

 

if contains_nan and nan_policy == 'omit': 

a = ma.masked_invalid(a) 

return mstats_basic.ttest_1samp(a, popmean, axis) 

 

n = a.shape[axis] 

df = n - 1 

 

d = np.mean(a, axis) - popmean 

v = np.var(a, axis, ddof=1) 

denom = np.sqrt(v / float(n)) 

 

with np.errstate(divide='ignore', invalid='ignore'): 

t = np.divide(d, denom) 

t, prob = _ttest_finish(df, t) 

 

return Ttest_1sampResult(t, prob) 

 

 

def _ttest_finish(df, t): 

"""Common code between all 3 t-test functions.""" 

prob = distributions.t.sf(np.abs(t), df) * 2 # use np.abs to get upper tail 

if t.ndim == 0: 

t = t[()] 

 

return t, prob 

 

 

def _ttest_ind_from_stats(mean1, mean2, denom, df): 

 

d = mean1 - mean2 

with np.errstate(divide='ignore', invalid='ignore'): 

t = np.divide(d, denom) 

t, prob = _ttest_finish(df, t) 

 

return (t, prob) 

 

 

def _unequal_var_ttest_denom(v1, n1, v2, n2): 

vn1 = v1 / n1 

vn2 = v2 / n2 

with np.errstate(divide='ignore', invalid='ignore'): 

df = (vn1 + vn2)**2 / (vn1**2 / (n1 - 1) + vn2**2 / (n2 - 1)) 

 

# If df is undefined, variances are zero (assumes n1 > 0 & n2 > 0). 

# Hence it doesn't matter what df is as long as it's not NaN. 

df = np.where(np.isnan(df), 1, df) 

denom = np.sqrt(vn1 + vn2) 

return df, denom 

 

 

def _equal_var_ttest_denom(v1, n1, v2, n2): 

df = n1 + n2 - 2.0 

svar = ((n1 - 1) * v1 + (n2 - 1) * v2) / df 

denom = np.sqrt(svar * (1.0 / n1 + 1.0 / n2)) 

return df, denom 

 

 

Ttest_indResult = namedtuple('Ttest_indResult', ('statistic', 'pvalue')) 

 

 

def ttest_ind_from_stats(mean1, std1, nobs1, mean2, std2, nobs2, 

equal_var=True): 

""" 

T-test for means of two independent samples from descriptive statistics. 

 

This is a two-sided test for the null hypothesis that two independent 

samples have identical average (expected) values. 

 

Parameters 

---------- 

mean1 : array_like 

The mean(s) of sample 1. 

std1 : array_like 

The standard deviation(s) of sample 1. 

nobs1 : array_like 

The number(s) of observations of sample 1. 

mean2 : array_like 

The mean(s) of sample 2 

std2 : array_like 

The standard deviations(s) of sample 2. 

nobs2 : array_like 

The number(s) of observations of sample 2. 

equal_var : bool, optional 

If True (default), perform a standard independent 2 sample test 

that assumes equal population variances [1]_. 

If False, perform Welch's t-test, which does not assume equal 

population variance [2]_. 

 

Returns 

------- 

statistic : float or array 

The calculated t-statistics 

pvalue : float or array 

The two-tailed p-value. 

 

See Also 

-------- 

scipy.stats.ttest_ind 

 

Notes 

----- 

 

.. versionadded:: 0.16.0 

 

References 

---------- 

.. [1] http://en.wikipedia.org/wiki/T-test#Independent_two-sample_t-test 

 

.. [2] http://en.wikipedia.org/wiki/Welch%27s_t_test 

 

Examples 

-------- 

Suppose we have the summary data for two samples, as follows:: 

 

Sample Sample 

Size Mean Variance 

Sample 1 13 15.0 87.5 

Sample 2 11 12.0 39.0 

 

Apply the t-test to this data (with the assumption that the population 

variances are equal): 

 

>>> from scipy.stats import ttest_ind_from_stats 

>>> ttest_ind_from_stats(mean1=15.0, std1=np.sqrt(87.5), nobs1=13, 

... mean2=12.0, std2=np.sqrt(39.0), nobs2=11) 

Ttest_indResult(statistic=0.9051358093310269, pvalue=0.3751996797581487) 

 

For comparison, here is the data from which those summary statistics 

were taken. With this data, we can compute the same result using 

`scipy.stats.ttest_ind`: 

 

>>> a = np.array([1, 3, 4, 6, 11, 13, 15, 19, 22, 24, 25, 26, 26]) 

>>> b = np.array([2, 4, 6, 9, 11, 13, 14, 15, 18, 19, 21]) 

>>> from scipy.stats import ttest_ind 

>>> ttest_ind(a, b) 

Ttest_indResult(statistic=0.905135809331027, pvalue=0.3751996797581486) 

 

""" 

if equal_var: 

df, denom = _equal_var_ttest_denom(std1**2, nobs1, std2**2, nobs2) 

else: 

df, denom = _unequal_var_ttest_denom(std1**2, nobs1, 

std2**2, nobs2) 

 

res = _ttest_ind_from_stats(mean1, mean2, denom, df) 

return Ttest_indResult(*res) 

 

 

def ttest_ind(a, b, axis=0, equal_var=True, nan_policy='propagate'): 

""" 

Calculate the T-test for the means of *two independent* samples of scores. 

 

This is a two-sided test for the null hypothesis that 2 independent samples 

have identical average (expected) values. This test assumes that the 

populations have identical variances by default. 

 

Parameters 

---------- 

a, b : array_like 

The arrays must have the same shape, except in the dimension 

corresponding to `axis` (the first, by default). 

axis : int or None, optional 

Axis along which to compute test. If None, compute over the whole 

arrays, `a`, and `b`. 

equal_var : bool, optional 

If True (default), perform a standard independent 2 sample test 

that assumes equal population variances [1]_. 

If False, perform Welch's t-test, which does not assume equal 

population variance [2]_. 

 

.. versionadded:: 0.11.0 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. 

 

 

Returns 

------- 

statistic : float or array 

The calculated t-statistic. 

pvalue : float or array 

The two-tailed p-value. 

 

Notes 

----- 

We can use this test, if we observe two independent samples from 

the same or different population, e.g. exam scores of boys and 

girls or of two ethnic groups. The test measures whether the 

average (expected) value differs significantly across samples. If 

we observe a large p-value, for example larger than 0.05 or 0.1, 

then we cannot reject the null hypothesis of identical average scores. 

If the p-value is smaller than the threshold, e.g. 1%, 5% or 10%, 

then we reject the null hypothesis of equal averages. 

 

References 

---------- 

.. [1] http://en.wikipedia.org/wiki/T-test#Independent_two-sample_t-test 

 

.. [2] http://en.wikipedia.org/wiki/Welch%27s_t_test 

 

Examples 

-------- 

>>> from scipy import stats 

>>> np.random.seed(12345678) 

 

Test with sample with identical means: 

 

>>> rvs1 = stats.norm.rvs(loc=5,scale=10,size=500) 

>>> rvs2 = stats.norm.rvs(loc=5,scale=10,size=500) 

>>> stats.ttest_ind(rvs1,rvs2) 

(0.26833823296239279, 0.78849443369564776) 

>>> stats.ttest_ind(rvs1,rvs2, equal_var = False) 

(0.26833823296239279, 0.78849452749500748) 

 

`ttest_ind` underestimates p for unequal variances: 

 

>>> rvs3 = stats.norm.rvs(loc=5, scale=20, size=500) 

>>> stats.ttest_ind(rvs1, rvs3) 

(-0.46580283298287162, 0.64145827413436174) 

>>> stats.ttest_ind(rvs1, rvs3, equal_var = False) 

(-0.46580283298287162, 0.64149646246569292) 

 

When n1 != n2, the equal variance t-statistic is no longer equal to the 

unequal variance t-statistic: 

 

>>> rvs4 = stats.norm.rvs(loc=5, scale=20, size=100) 

>>> stats.ttest_ind(rvs1, rvs4) 

(-0.99882539442782481, 0.3182832709103896) 

>>> stats.ttest_ind(rvs1, rvs4, equal_var = False) 

(-0.69712570584654099, 0.48716927725402048) 

 

T-test with different means, variance, and n: 

 

>>> rvs5 = stats.norm.rvs(loc=8, scale=20, size=100) 

>>> stats.ttest_ind(rvs1, rvs5) 

(-1.4679669854490653, 0.14263895620529152) 

>>> stats.ttest_ind(rvs1, rvs5, equal_var = False) 

(-0.94365973617132992, 0.34744170334794122) 

 

""" 

a, b, axis = _chk2_asarray(a, b, axis) 

 

# check both a and b 

cna, npa = _contains_nan(a, nan_policy) 

cnb, npb = _contains_nan(b, nan_policy) 

contains_nan = cna or cnb 

if npa == 'omit' or npb == 'omit': 

nan_policy = 'omit' 

 

if contains_nan and nan_policy == 'omit': 

a = ma.masked_invalid(a) 

b = ma.masked_invalid(b) 

return mstats_basic.ttest_ind(a, b, axis, equal_var) 

 

if a.size == 0 or b.size == 0: 

return Ttest_indResult(np.nan, np.nan) 

 

v1 = np.var(a, axis, ddof=1) 

v2 = np.var(b, axis, ddof=1) 

n1 = a.shape[axis] 

n2 = b.shape[axis] 

 

if equal_var: 

df, denom = _equal_var_ttest_denom(v1, n1, v2, n2) 

else: 

df, denom = _unequal_var_ttest_denom(v1, n1, v2, n2) 

 

res = _ttest_ind_from_stats(np.mean(a, axis), np.mean(b, axis), denom, df) 

 

return Ttest_indResult(*res) 

 

 

Ttest_relResult = namedtuple('Ttest_relResult', ('statistic', 'pvalue')) 

 

 

def ttest_rel(a, b, axis=0, nan_policy='propagate'): 

""" 

Calculate the T-test on TWO RELATED samples of scores, a and b. 

 

This is a two-sided test for the null hypothesis that 2 related or 

repeated samples have identical average (expected) values. 

 

Parameters 

---------- 

a, b : array_like 

The arrays must have the same shape. 

axis : int or None, optional 

Axis along which to compute test. If None, compute over the whole 

arrays, `a`, and `b`. 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. 

 

Returns 

------- 

statistic : float or array 

t-statistic 

pvalue : float or array 

two-tailed p-value 

 

Notes 

----- 

Examples for the use are scores of the same set of student in 

different exams, or repeated sampling from the same units. The 

test measures whether the average score differs significantly 

across samples (e.g. exams). If we observe a large p-value, for 

example greater than 0.05 or 0.1 then we cannot reject the null 

hypothesis of identical average scores. If the p-value is smaller 

than the threshold, e.g. 1%, 5% or 10%, then we reject the null 

hypothesis of equal averages. Small p-values are associated with 

large t-statistics. 

 

References 

---------- 

https://en.wikipedia.org/wiki/T-test#Dependent_t-test_for_paired_samples 

 

Examples 

-------- 

>>> from scipy import stats 

>>> np.random.seed(12345678) # fix random seed to get same numbers 

 

>>> rvs1 = stats.norm.rvs(loc=5,scale=10,size=500) 

>>> rvs2 = (stats.norm.rvs(loc=5,scale=10,size=500) + 

... stats.norm.rvs(scale=0.2,size=500)) 

>>> stats.ttest_rel(rvs1,rvs2) 

(0.24101764965300962, 0.80964043445811562) 

>>> rvs3 = (stats.norm.rvs(loc=8,scale=10,size=500) + 

... stats.norm.rvs(scale=0.2,size=500)) 

>>> stats.ttest_rel(rvs1,rvs3) 

(-3.9995108708727933, 7.3082402191726459e-005) 

 

""" 

a, b, axis = _chk2_asarray(a, b, axis) 

 

cna, npa = _contains_nan(a, nan_policy) 

cnb, npb = _contains_nan(b, nan_policy) 

contains_nan = cna or cnb 

if npa == 'omit' or npb == 'omit': 

nan_policy = 'omit' 

 

if contains_nan and nan_policy == 'omit': 

a = ma.masked_invalid(a) 

b = ma.masked_invalid(b) 

m = ma.mask_or(ma.getmask(a), ma.getmask(b)) 

aa = ma.array(a, mask=m, copy=True) 

bb = ma.array(b, mask=m, copy=True) 

return mstats_basic.ttest_rel(aa, bb, axis) 

 

if a.shape[axis] != b.shape[axis]: 

raise ValueError('unequal length arrays') 

 

if a.size == 0 or b.size == 0: 

return np.nan, np.nan 

 

n = a.shape[axis] 

df = float(n - 1) 

 

d = (a - b).astype(np.float64) 

v = np.var(d, axis, ddof=1) 

dm = np.mean(d, axis) 

denom = np.sqrt(v / float(n)) 

 

with np.errstate(divide='ignore', invalid='ignore'): 

t = np.divide(dm, denom) 

t, prob = _ttest_finish(df, t) 

 

return Ttest_relResult(t, prob) 

 

 

KstestResult = namedtuple('KstestResult', ('statistic', 'pvalue')) 

 

 

def kstest(rvs, cdf, args=(), N=20, alternative='two-sided', mode='approx'): 

""" 

Perform the Kolmogorov-Smirnov test for goodness of fit. 

 

This performs a test of the distribution G(x) of an observed 

random variable against a given distribution F(x). Under the null 

hypothesis the two distributions are identical, G(x)=F(x). The 

alternative hypothesis can be either 'two-sided' (default), 'less' 

or 'greater'. The KS test is only valid for continuous distributions. 

 

Parameters 

---------- 

rvs : str, array or callable 

If a string, it should be the name of a distribution in `scipy.stats`. 

If an array, it should be a 1-D array of observations of random 

variables. 

If a callable, it should be a function to generate random variables; 

it is required to have a keyword argument `size`. 

cdf : str or callable 

If a string, it should be the name of a distribution in `scipy.stats`. 

If `rvs` is a string then `cdf` can be False or the same as `rvs`. 

If a callable, that callable is used to calculate the cdf. 

args : tuple, sequence, optional 

Distribution parameters, used if `rvs` or `cdf` are strings. 

N : int, optional 

Sample size if `rvs` is string or callable. Default is 20. 

alternative : {'two-sided', 'less','greater'}, optional 

Defines the alternative hypothesis (see explanation above). 

Default is 'two-sided'. 

mode : 'approx' (default) or 'asymp', optional 

Defines the distribution used for calculating the p-value. 

 

- 'approx' : use approximation to exact distribution of test statistic 

- 'asymp' : use asymptotic distribution of test statistic 

 

Returns 

------- 

statistic : float 

KS test statistic, either D, D+ or D-. 

pvalue : float 

One-tailed or two-tailed p-value. 

 

Notes 

----- 

In the one-sided test, the alternative is that the empirical 

cumulative distribution function of the random variable is "less" 

or "greater" than the cumulative distribution function F(x) of the 

hypothesis, ``G(x)<=F(x)``, resp. ``G(x)>=F(x)``. 

 

Examples 

-------- 

>>> from scipy import stats 

 

>>> x = np.linspace(-15, 15, 9) 

>>> stats.kstest(x, 'norm') 

(0.44435602715924361, 0.038850142705171065) 

 

>>> np.random.seed(987654321) # set random seed to get the same result 

>>> stats.kstest('norm', False, N=100) 

(0.058352892479417884, 0.88531190944151261) 

 

The above lines are equivalent to: 

 

>>> np.random.seed(987654321) 

>>> stats.kstest(stats.norm.rvs(size=100), 'norm') 

(0.058352892479417884, 0.88531190944151261) 

 

*Test against one-sided alternative hypothesis* 

 

Shift distribution to larger values, so that ``cdf_dgp(x) < norm.cdf(x)``: 

 

>>> np.random.seed(987654321) 

>>> x = stats.norm.rvs(loc=0.2, size=100) 

>>> stats.kstest(x,'norm', alternative = 'less') 

(0.12464329735846891, 0.040989164077641749) 

 

Reject equal distribution against alternative hypothesis: less 

 

>>> stats.kstest(x,'norm', alternative = 'greater') 

(0.0072115233216311081, 0.98531158590396395) 

 

Don't reject equal distribution against alternative hypothesis: greater 

 

>>> stats.kstest(x,'norm', mode='asymp') 

(0.12464329735846891, 0.08944488871182088) 

 

*Testing t distributed random variables against normal distribution* 

 

With 100 degrees of freedom the t distribution looks close to the normal 

distribution, and the K-S test does not reject the hypothesis that the 

sample came from the normal distribution: 

 

>>> np.random.seed(987654321) 

>>> stats.kstest(stats.t.rvs(100,size=100),'norm') 

(0.072018929165471257, 0.67630062862479168) 

 

With 3 degrees of freedom the t distribution looks sufficiently different 

from the normal distribution, that we can reject the hypothesis that the 

sample came from the normal distribution at the 10% level: 

 

>>> np.random.seed(987654321) 

>>> stats.kstest(stats.t.rvs(3,size=100),'norm') 

(0.131016895759829, 0.058826222555312224) 

 

""" 

if isinstance(rvs, string_types): 

if (not cdf) or (cdf == rvs): 

cdf = getattr(distributions, rvs).cdf 

rvs = getattr(distributions, rvs).rvs 

else: 

raise AttributeError("if rvs is string, cdf has to be the " 

"same distribution") 

 

if isinstance(cdf, string_types): 

cdf = getattr(distributions, cdf).cdf 

if callable(rvs): 

kwds = {'size': N} 

vals = np.sort(rvs(*args, **kwds)) 

else: 

vals = np.sort(rvs) 

N = len(vals) 

cdfvals = cdf(vals, *args) 

 

# to not break compatibility with existing code 

if alternative == 'two_sided': 

alternative = 'two-sided' 

 

if alternative in ['two-sided', 'greater']: 

Dplus = (np.arange(1.0, N + 1)/N - cdfvals).max() 

if alternative == 'greater': 

return KstestResult(Dplus, distributions.ksone.sf(Dplus, N)) 

 

if alternative in ['two-sided', 'less']: 

Dmin = (cdfvals - np.arange(0.0, N)/N).max() 

if alternative == 'less': 

return KstestResult(Dmin, distributions.ksone.sf(Dmin, N)) 

 

if alternative == 'two-sided': 

D = np.max([Dplus, Dmin]) 

if mode == 'asymp': 

return KstestResult(D, distributions.kstwobign.sf(D * np.sqrt(N))) 

if mode == 'approx': 

pval_two = distributions.kstwobign.sf(D * np.sqrt(N)) 

if N > 2666 or pval_two > 0.80 - N*0.3/1000: 

return KstestResult(D, pval_two) 

else: 

return KstestResult(D, 2 * distributions.ksone.sf(D, N)) 

 

 

# Map from names to lambda_ values used in power_divergence(). 

_power_div_lambda_names = { 

"pearson": 1, 

"log-likelihood": 0, 

"freeman-tukey": -0.5, 

"mod-log-likelihood": -1, 

"neyman": -2, 

"cressie-read": 2/3, 

} 

 

 

def _count(a, axis=None): 

""" 

Count the number of non-masked elements of an array. 

 

This function behaves like np.ma.count(), but is much faster 

for ndarrays. 

""" 

if hasattr(a, 'count'): 

num = a.count(axis=axis) 

if isinstance(num, np.ndarray) and num.ndim == 0: 

# In some cases, the `count` method returns a scalar array (e.g. 

# np.array(3)), but we want a plain integer. 

num = int(num) 

else: 

if axis is None: 

num = a.size 

else: 

num = a.shape[axis] 

return num 

 

 

Power_divergenceResult = namedtuple('Power_divergenceResult', 

('statistic', 'pvalue')) 

 

def power_divergence(f_obs, f_exp=None, ddof=0, axis=0, lambda_=None): 

""" 

Cressie-Read power divergence statistic and goodness of fit test. 

 

This function tests the null hypothesis that the categorical data 

has the given frequencies, using the Cressie-Read power divergence 

statistic. 

 

Parameters 

---------- 

f_obs : array_like 

Observed frequencies in each category. 

f_exp : array_like, optional 

Expected frequencies in each category. By default the categories are 

assumed to be equally likely. 

ddof : int, optional 

"Delta degrees of freedom": adjustment to the degrees of freedom 

for the p-value. The p-value is computed using a chi-squared 

distribution with ``k - 1 - ddof`` degrees of freedom, where `k` 

is the number of observed frequencies. The default value of `ddof` 

is 0. 

axis : int or None, optional 

The axis of the broadcast result of `f_obs` and `f_exp` along which to 

apply the test. If axis is None, all values in `f_obs` are treated 

as a single data set. Default is 0. 

lambda_ : float or str, optional 

`lambda_` gives the power in the Cressie-Read power divergence 

statistic. The default is 1. For convenience, `lambda_` may be 

assigned one of the following strings, in which case the 

corresponding numerical value is used:: 

 

String Value Description 

"pearson" 1 Pearson's chi-squared statistic. 

In this case, the function is 

equivalent to `stats.chisquare`. 

"log-likelihood" 0 Log-likelihood ratio. Also known as 

the G-test [3]_. 

"freeman-tukey" -1/2 Freeman-Tukey statistic. 

"mod-log-likelihood" -1 Modified log-likelihood ratio. 

"neyman" -2 Neyman's statistic. 

"cressie-read" 2/3 The power recommended in [5]_. 

 

Returns 

------- 

statistic : float or ndarray 

The Cressie-Read power divergence test statistic. The value is 

a float if `axis` is None or if` `f_obs` and `f_exp` are 1-D. 

pvalue : float or ndarray 

The p-value of the test. The value is a float if `ddof` and the 

return value `stat` are scalars. 

 

See Also 

-------- 

chisquare 

 

Notes 

----- 

This test is invalid when the observed or expected frequencies in each 

category are too small. A typical rule is that all of the observed 

and expected frequencies should be at least 5. 

 

When `lambda_` is less than zero, the formula for the statistic involves 

dividing by `f_obs`, so a warning or error may be generated if any value 

in `f_obs` is 0. 

 

Similarly, a warning or error may be generated if any value in `f_exp` is 

zero when `lambda_` >= 0. 

 

The default degrees of freedom, k-1, are for the case when no parameters 

of the distribution are estimated. If p parameters are estimated by 

efficient maximum likelihood then the correct degrees of freedom are 

k-1-p. If the parameters are estimated in a different way, then the 

dof can be between k-1-p and k-1. However, it is also possible that 

the asymptotic distribution is not a chisquare, in which case this 

test is not appropriate. 

 

This function handles masked arrays. If an element of `f_obs` or `f_exp` 

is masked, then data at that position is ignored, and does not count 

towards the size of the data set. 

 

.. versionadded:: 0.13.0 

 

References 

---------- 

.. [1] Lowry, Richard. "Concepts and Applications of Inferential 

Statistics". Chapter 8. http://faculty.vassar.edu/lowry/ch8pt1.html 

.. [2] "Chi-squared test", http://en.wikipedia.org/wiki/Chi-squared_test 

.. [3] "G-test", http://en.wikipedia.org/wiki/G-test 

.. [4] Sokal, R. R. and Rohlf, F. J. "Biometry: the principles and 

practice of statistics in biological research", New York: Freeman 

(1981) 

.. [5] Cressie, N. and Read, T. R. C., "Multinomial Goodness-of-Fit 

Tests", J. Royal Stat. Soc. Series B, Vol. 46, No. 3 (1984), 

pp. 440-464. 

 

Examples 

-------- 

 

(See `chisquare` for more examples.) 

 

When just `f_obs` is given, it is assumed that the expected frequencies 

are uniform and given by the mean of the observed frequencies. Here we 

perform a G-test (i.e. use the log-likelihood ratio statistic): 

 

>>> from scipy.stats import power_divergence 

>>> power_divergence([16, 18, 16, 14, 12, 12], lambda_='log-likelihood') 

(2.006573162632538, 0.84823476779463769) 

 

The expected frequencies can be given with the `f_exp` argument: 

 

>>> power_divergence([16, 18, 16, 14, 12, 12], 

... f_exp=[16, 16, 16, 16, 16, 8], 

... lambda_='log-likelihood') 

(3.3281031458963746, 0.6495419288047497) 

 

When `f_obs` is 2-D, by default the test is applied to each column. 

 

>>> obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T 

>>> obs.shape 

(6, 2) 

>>> power_divergence(obs, lambda_="log-likelihood") 

(array([ 2.00657316, 6.77634498]), array([ 0.84823477, 0.23781225])) 

 

By setting ``axis=None``, the test is applied to all data in the array, 

which is equivalent to applying the test to the flattened array. 

 

>>> power_divergence(obs, axis=None) 

(23.31034482758621, 0.015975692534127565) 

>>> power_divergence(obs.ravel()) 

(23.31034482758621, 0.015975692534127565) 

 

`ddof` is the change to make to the default degrees of freedom. 

 

>>> power_divergence([16, 18, 16, 14, 12, 12], ddof=1) 

(2.0, 0.73575888234288467) 

 

The calculation of the p-values is done by broadcasting the 

test statistic with `ddof`. 

 

>>> power_divergence([16, 18, 16, 14, 12, 12], ddof=[0,1,2]) 

(2.0, array([ 0.84914504, 0.73575888, 0.5724067 ])) 

 

`f_obs` and `f_exp` are also broadcast. In the following, `f_obs` has 

shape (6,) and `f_exp` has shape (2, 6), so the result of broadcasting 

`f_obs` and `f_exp` has shape (2, 6). To compute the desired chi-squared 

statistics, we must use ``axis=1``: 

 

>>> power_divergence([16, 18, 16, 14, 12, 12], 

... f_exp=[[16, 16, 16, 16, 16, 8], 

... [8, 20, 20, 16, 12, 12]], 

... axis=1) 

(array([ 3.5 , 9.25]), array([ 0.62338763, 0.09949846])) 

 

""" 

# Convert the input argument `lambda_` to a numerical value. 

if isinstance(lambda_, string_types): 

if lambda_ not in _power_div_lambda_names: 

names = repr(list(_power_div_lambda_names.keys()))[1:-1] 

raise ValueError("invalid string for lambda_: {0!r}. Valid strings " 

"are {1}".format(lambda_, names)) 

lambda_ = _power_div_lambda_names[lambda_] 

elif lambda_ is None: 

lambda_ = 1 

 

f_obs = np.asanyarray(f_obs) 

 

if f_exp is not None: 

f_exp = np.atleast_1d(np.asanyarray(f_exp)) 

else: 

# Compute the equivalent of 

# f_exp = f_obs.mean(axis=axis, keepdims=True) 

# Older versions of numpy do not have the 'keepdims' argument, so 

# we have to do a little work to achieve the same result. 

# Ignore 'invalid' errors so the edge case of a data set with length 0 

# is handled without spurious warnings. 

with np.errstate(invalid='ignore'): 

f_exp = np.atleast_1d(f_obs.mean(axis=axis)) 

if axis is not None: 

reduced_shape = list(f_obs.shape) 

reduced_shape[axis] = 1 

f_exp.shape = reduced_shape 

 

# `terms` is the array of terms that are summed along `axis` to create 

# the test statistic. We use some specialized code for a few special 

# cases of lambda_. 

if lambda_ == 1: 

# Pearson's chi-squared statistic 

terms = (f_obs - f_exp)**2 / f_exp 

elif lambda_ == 0: 

# Log-likelihood ratio (i.e. G-test) 

terms = 2.0 * special.xlogy(f_obs, f_obs / f_exp) 

elif lambda_ == -1: 

# Modified log-likelihood ratio 

terms = 2.0 * special.xlogy(f_exp, f_exp / f_obs) 

else: 

# General Cressie-Read power divergence. 

terms = f_obs * ((f_obs / f_exp)**lambda_ - 1) 

terms /= 0.5 * lambda_ * (lambda_ + 1) 

 

stat = terms.sum(axis=axis) 

 

num_obs = _count(terms, axis=axis) 

ddof = asarray(ddof) 

p = distributions.chi2.sf(stat, num_obs - 1 - ddof) 

 

return Power_divergenceResult(stat, p) 

 

 

def chisquare(f_obs, f_exp=None, ddof=0, axis=0): 

""" 

Calculate a one-way chi square test. 

 

The chi square test tests the null hypothesis that the categorical data 

has the given frequencies. 

 

Parameters 

---------- 

f_obs : array_like 

Observed frequencies in each category. 

f_exp : array_like, optional 

Expected frequencies in each category. By default the categories are 

assumed to be equally likely. 

ddof : int, optional 

"Delta degrees of freedom": adjustment to the degrees of freedom 

for the p-value. The p-value is computed using a chi-squared 

distribution with ``k - 1 - ddof`` degrees of freedom, where `k` 

is the number of observed frequencies. The default value of `ddof` 

is 0. 

axis : int or None, optional 

The axis of the broadcast result of `f_obs` and `f_exp` along which to 

apply the test. If axis is None, all values in `f_obs` are treated 

as a single data set. Default is 0. 

 

Returns 

------- 

chisq : float or ndarray 

The chi-squared test statistic. The value is a float if `axis` is 

None or `f_obs` and `f_exp` are 1-D. 

p : float or ndarray 

The p-value of the test. The value is a float if `ddof` and the 

return value `chisq` are scalars. 

 

See Also 

-------- 

power_divergence 

mstats.chisquare 

 

Notes 

----- 

This test is invalid when the observed or expected frequencies in each 

category are too small. A typical rule is that all of the observed 

and expected frequencies should be at least 5. 

 

The default degrees of freedom, k-1, are for the case when no parameters 

of the distribution are estimated. If p parameters are estimated by 

efficient maximum likelihood then the correct degrees of freedom are 

k-1-p. If the parameters are estimated in a different way, then the 

dof can be between k-1-p and k-1. However, it is also possible that 

the asymptotic distribution is not a chisquare, in which case this 

test is not appropriate. 

 

References 

---------- 

.. [1] Lowry, Richard. "Concepts and Applications of Inferential 

Statistics". Chapter 8. http://faculty.vassar.edu/lowry/ch8pt1.html 

.. [2] "Chi-squared test", http://en.wikipedia.org/wiki/Chi-squared_test 

 

Examples 

-------- 

When just `f_obs` is given, it is assumed that the expected frequencies 

are uniform and given by the mean of the observed frequencies. 

 

>>> from scipy.stats import chisquare 

>>> chisquare([16, 18, 16, 14, 12, 12]) 

(2.0, 0.84914503608460956) 

 

With `f_exp` the expected frequencies can be given. 

 

>>> chisquare([16, 18, 16, 14, 12, 12], f_exp=[16, 16, 16, 16, 16, 8]) 

(3.5, 0.62338762774958223) 

 

When `f_obs` is 2-D, by default the test is applied to each column. 

 

>>> obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T 

>>> obs.shape 

(6, 2) 

>>> chisquare(obs) 

(array([ 2. , 6.66666667]), array([ 0.84914504, 0.24663415])) 

 

By setting ``axis=None``, the test is applied to all data in the array, 

which is equivalent to applying the test to the flattened array. 

 

>>> chisquare(obs, axis=None) 

(23.31034482758621, 0.015975692534127565) 

>>> chisquare(obs.ravel()) 

(23.31034482758621, 0.015975692534127565) 

 

`ddof` is the change to make to the default degrees of freedom. 

 

>>> chisquare([16, 18, 16, 14, 12, 12], ddof=1) 

(2.0, 0.73575888234288467) 

 

The calculation of the p-values is done by broadcasting the 

chi-squared statistic with `ddof`. 

 

>>> chisquare([16, 18, 16, 14, 12, 12], ddof=[0,1,2]) 

(2.0, array([ 0.84914504, 0.73575888, 0.5724067 ])) 

 

`f_obs` and `f_exp` are also broadcast. In the following, `f_obs` has 

shape (6,) and `f_exp` has shape (2, 6), so the result of broadcasting 

`f_obs` and `f_exp` has shape (2, 6). To compute the desired chi-squared 

statistics, we use ``axis=1``: 

 

>>> chisquare([16, 18, 16, 14, 12, 12], 

... f_exp=[[16, 16, 16, 16, 16, 8], [8, 20, 20, 16, 12, 12]], 

... axis=1) 

(array([ 3.5 , 9.25]), array([ 0.62338763, 0.09949846])) 

 

""" 

return power_divergence(f_obs, f_exp=f_exp, ddof=ddof, axis=axis, 

lambda_="pearson") 

 

 

Ks_2sampResult = namedtuple('Ks_2sampResult', ('statistic', 'pvalue')) 

 

 

def ks_2samp(data1, data2): 

""" 

Compute the Kolmogorov-Smirnov statistic on 2 samples. 

 

This is a two-sided test for the null hypothesis that 2 independent samples 

are drawn from the same continuous distribution. 

 

Parameters 

---------- 

data1, data2 : sequence of 1-D ndarrays 

two arrays of sample observations assumed to be drawn from a continuous 

distribution, sample sizes can be different 

 

Returns 

------- 

statistic : float 

KS statistic 

pvalue : float 

two-tailed p-value 

 

Notes 

----- 

This tests whether 2 samples are drawn from the same distribution. Note 

that, like in the case of the one-sample K-S test, the distribution is 

assumed to be continuous. 

 

This is the two-sided test, one-sided tests are not implemented. 

The test uses the two-sided asymptotic Kolmogorov-Smirnov distribution. 

 

If the K-S statistic is small or the p-value is high, then we cannot 

reject the hypothesis that the distributions of the two samples 

are the same. 

 

Examples 

-------- 

>>> from scipy import stats 

>>> np.random.seed(12345678) #fix random seed to get the same result 

>>> n1 = 200 # size of first sample 

>>> n2 = 300 # size of second sample 

 

For a different distribution, we can reject the null hypothesis since the 

pvalue is below 1%: 

 

>>> rvs1 = stats.norm.rvs(size=n1, loc=0., scale=1) 

>>> rvs2 = stats.norm.rvs(size=n2, loc=0.5, scale=1.5) 

>>> stats.ks_2samp(rvs1, rvs2) 

(0.20833333333333337, 4.6674975515806989e-005) 

 

For a slightly different distribution, we cannot reject the null hypothesis 

at a 10% or lower alpha since the p-value at 0.144 is higher than 10% 

 

>>> rvs3 = stats.norm.rvs(size=n2, loc=0.01, scale=1.0) 

>>> stats.ks_2samp(rvs1, rvs3) 

(0.10333333333333333, 0.14498781825751686) 

 

For an identical distribution, we cannot reject the null hypothesis since 

the p-value is high, 41%: 

 

>>> rvs4 = stats.norm.rvs(size=n2, loc=0.0, scale=1.0) 

>>> stats.ks_2samp(rvs1, rvs4) 

(0.07999999999999996, 0.41126949729859719) 

 

""" 

data1 = np.sort(data1) 

data2 = np.sort(data2) 

n1 = data1.shape[0] 

n2 = data2.shape[0] 

data_all = np.concatenate([data1, data2]) 

cdf1 = np.searchsorted(data1, data_all, side='right') / (1.0*n1) 

cdf2 = np.searchsorted(data2, data_all, side='right') / (1.0*n2) 

d = np.max(np.absolute(cdf1 - cdf2)) 

# Note: d absolute not signed distance 

en = np.sqrt(n1 * n2 / float(n1 + n2)) 

try: 

prob = distributions.kstwobign.sf((en + 0.12 + 0.11 / en) * d) 

except: 

prob = 1.0 

 

return Ks_2sampResult(d, prob) 

 

 

def tiecorrect(rankvals): 

""" 

Tie correction factor for ties in the Mann-Whitney U and 

Kruskal-Wallis H tests. 

 

Parameters 

---------- 

rankvals : array_like 

A 1-D sequence of ranks. Typically this will be the array 

returned by `stats.rankdata`. 

 

Returns 

------- 

factor : float 

Correction factor for U or H. 

 

See Also 

-------- 

rankdata : Assign ranks to the data 

mannwhitneyu : Mann-Whitney rank test 

kruskal : Kruskal-Wallis H test 

 

References 

---------- 

.. [1] Siegel, S. (1956) Nonparametric Statistics for the Behavioral 

Sciences. New York: McGraw-Hill. 

 

Examples 

-------- 

>>> from scipy.stats import tiecorrect, rankdata 

>>> tiecorrect([1, 2.5, 2.5, 4]) 

0.9 

>>> ranks = rankdata([1, 3, 2, 4, 5, 7, 2, 8, 4]) 

>>> ranks 

array([ 1. , 4. , 2.5, 5.5, 7. , 8. , 2.5, 9. , 5.5]) 

>>> tiecorrect(ranks) 

0.9833333333333333 

 

""" 

arr = np.sort(rankvals) 

idx = np.nonzero(np.r_[True, arr[1:] != arr[:-1], True])[0] 

cnt = np.diff(idx).astype(np.float64) 

 

size = np.float64(arr.size) 

return 1.0 if size < 2 else 1.0 - (cnt**3 - cnt).sum() / (size**3 - size) 

 

 

MannwhitneyuResult = namedtuple('MannwhitneyuResult', ('statistic', 'pvalue')) 

 

def mannwhitneyu(x, y, use_continuity=True, alternative=None): 

""" 

Compute the Mann-Whitney rank test on samples x and y. 

 

Parameters 

---------- 

x, y : array_like 

Array of samples, should be one-dimensional. 

use_continuity : bool, optional 

Whether a continuity correction (1/2.) should be taken into 

account. Default is True. 

alternative : None (deprecated), 'less', 'two-sided', or 'greater' 

Whether to get the p-value for the one-sided hypothesis ('less' 

or 'greater') or for the two-sided hypothesis ('two-sided'). 

Defaults to None, which results in a p-value half the size of 

the 'two-sided' p-value and a different U statistic. The 

default behavior is not the same as using 'less' or 'greater': 

it only exists for backward compatibility and is deprecated. 

 

Returns 

------- 

statistic : float 

The Mann-Whitney U statistic, equal to min(U for x, U for y) if 

`alternative` is equal to None (deprecated; exists for backward 

compatibility), and U for y otherwise. 

pvalue : float 

p-value assuming an asymptotic normal distribution. One-sided or 

two-sided, depending on the choice of `alternative`. 

 

Notes 

----- 

Use only when the number of observation in each sample is > 20 and 

you have 2 independent samples of ranks. Mann-Whitney U is 

significant if the u-obtained is LESS THAN or equal to the critical 

value of U. 

 

This test corrects for ties and by default uses a continuity correction. 

 

References 

---------- 

.. [1] https://en.wikipedia.org/wiki/Mann-Whitney_U_test 

 

.. [2] H.B. Mann and D.R. Whitney, "On a Test of Whether one of Two Random 

Variables is Stochastically Larger than the Other," The Annals of 

Mathematical Statistics, vol. 18, no. 1, pp. 50-60, 1947. 

 

""" 

if alternative is None: 

warnings.warn("Calling `mannwhitneyu` without specifying " 

"`alternative` is deprecated.", DeprecationWarning) 

 

x = np.asarray(x) 

y = np.asarray(y) 

n1 = len(x) 

n2 = len(y) 

ranked = rankdata(np.concatenate((x, y))) 

rankx = ranked[0:n1] # get the x-ranks 

u1 = n1*n2 + (n1*(n1+1))/2.0 - np.sum(rankx, axis=0) # calc U for x 

u2 = n1*n2 - u1 # remainder is U for y 

T = tiecorrect(ranked) 

if T == 0: 

raise ValueError('All numbers are identical in mannwhitneyu') 

sd = np.sqrt(T * n1 * n2 * (n1+n2+1) / 12.0) 

 

meanrank = n1*n2/2.0 + 0.5 * use_continuity 

if alternative is None or alternative == 'two-sided': 

bigu = max(u1, u2) 

elif alternative == 'less': 

bigu = u1 

elif alternative == 'greater': 

bigu = u2 

else: 

raise ValueError("alternative should be None, 'less', 'greater' " 

"or 'two-sided'") 

 

z = (bigu - meanrank) / sd 

if alternative is None: 

# This behavior, equal to half the size of the two-sided 

# p-value, is deprecated. 

p = distributions.norm.sf(abs(z)) 

elif alternative == 'two-sided': 

p = 2 * distributions.norm.sf(abs(z)) 

else: 

p = distributions.norm.sf(z) 

 

u = u2 

# This behavior is deprecated. 

if alternative is None: 

u = min(u1, u2) 

return MannwhitneyuResult(u, p) 

 

 

RanksumsResult = namedtuple('RanksumsResult', ('statistic', 'pvalue')) 

 

 

def ranksums(x, y): 

""" 

Compute the Wilcoxon rank-sum statistic for two samples. 

 

The Wilcoxon rank-sum test tests the null hypothesis that two sets 

of measurements are drawn from the same distribution. The alternative 

hypothesis is that values in one sample are more likely to be 

larger than the values in the other sample. 

 

This test should be used to compare two samples from continuous 

distributions. It does not handle ties between measurements 

in x and y. For tie-handling and an optional continuity correction 

see `scipy.stats.mannwhitneyu`. 

 

Parameters 

---------- 

x,y : array_like 

The data from the two samples 

 

Returns 

------- 

statistic : float 

The test statistic under the large-sample approximation that the 

rank sum statistic is normally distributed 

pvalue : float 

The two-sided p-value of the test 

 

References 

---------- 

.. [1] http://en.wikipedia.org/wiki/Wilcoxon_rank-sum_test 

 

""" 

x, y = map(np.asarray, (x, y)) 

n1 = len(x) 

n2 = len(y) 

alldata = np.concatenate((x, y)) 

ranked = rankdata(alldata) 

x = ranked[:n1] 

s = np.sum(x, axis=0) 

expected = n1 * (n1+n2+1) / 2.0 

z = (s - expected) / np.sqrt(n1*n2*(n1+n2+1)/12.0) 

prob = 2 * distributions.norm.sf(abs(z)) 

 

return RanksumsResult(z, prob) 

 

 

KruskalResult = namedtuple('KruskalResult', ('statistic', 'pvalue')) 

 

 

def kruskal(*args, **kwargs): 

""" 

Compute the Kruskal-Wallis H-test for independent samples 

 

The Kruskal-Wallis H-test tests the null hypothesis that the population 

median of all of the groups are equal. It is a non-parametric version of 

ANOVA. The test works on 2 or more independent samples, which may have 

different sizes. Note that rejecting the null hypothesis does not 

indicate which of the groups differs. Post-hoc comparisons between 

groups are required to determine which groups are different. 

 

Parameters 

---------- 

sample1, sample2, ... : array_like 

Two or more arrays with the sample measurements can be given as 

arguments. 

nan_policy : {'propagate', 'raise', 'omit'}, optional 

Defines how to handle when input contains nan. 'propagate' returns nan, 

'raise' throws an error, 'omit' performs the calculations ignoring nan 

values. Default is 'propagate'. 

 

Returns 

------- 

statistic : float 

The Kruskal-Wallis H statistic, corrected for ties 

pvalue : float 

The p-value for the test using the assumption that H has a chi 

square distribution 

 

See Also 

-------- 

f_oneway : 1-way ANOVA 

mannwhitneyu : Mann-Whitney rank test on two samples. 

friedmanchisquare : Friedman test for repeated measurements 

 

Notes 

----- 

Due to the assumption that H has a chi square distribution, the number 

of samples in each group must not be too small. A typical rule is 

that each sample must have at least 5 measurements. 

 

References 

---------- 

.. [1] W. H. Kruskal & W. W. Wallis, "Use of Ranks in 

One-Criterion Variance Analysis", Journal of the American Statistical 

Association, Vol. 47, Issue 260, pp. 583-621, 1952. 

.. [2] http://en.wikipedia.org/wiki/Kruskal-Wallis_one-way_analysis_of_variance 

 

Examples 

-------- 

>>> from scipy import stats 

>>> x = [1, 3, 5, 7, 9] 

>>> y = [2, 4, 6, 8, 10] 

>>> stats.kruskal(x, y) 

KruskalResult(statistic=0.2727272727272734, pvalue=0.6015081344405895) 

 

>>> x = [1, 1, 1] 

>>> y = [2, 2, 2] 

>>> z = [2, 2] 

>>> stats.kruskal(x, y, z) 

KruskalResult(statistic=7.0, pvalue=0.0301973834223185) 

 

""" 

args = list(map(np.asarray, args)) 

num_groups = len(args) 

if num_groups < 2: 

raise ValueError("Need at least two groups in stats.kruskal()") 

 

for arg in args: 

if arg.size == 0: 

return KruskalResult(np.nan, np.nan) 

n = np.asarray(list(map(len, args))) 

 

if 'nan_policy' in kwargs.keys(): 

if kwargs['nan_policy'] not in ('propagate', 'raise', 'omit'): 

raise ValueError("nan_policy must be 'propagate', " 

"'raise' or'omit'") 

else: 

nan_policy = kwargs['nan_policy'] 

else: 

nan_policy = 'propagate' 

 

contains_nan = False 

for arg in args: 

cn = _contains_nan(arg, nan_policy) 

if cn[0]: 

contains_nan = True 

break 

 

if contains_nan and nan_policy == 'omit': 

for a in args: 

a = ma.masked_invalid(a) 

return mstats_basic.kruskal(*args) 

 

if contains_nan and nan_policy == 'propagate': 

return KruskalResult(np.nan, np.nan) 

 

alldata = np.concatenate(args) 

ranked = rankdata(alldata) 

ties = tiecorrect(ranked) 

if ties == 0: 

raise ValueError('All numbers are identical in kruskal') 

 

# Compute sum^2/n for each group and sum 

j = np.insert(np.cumsum(n), 0, 0) 

ssbn = 0 

for i in range(num_groups): 

ssbn += _square_of_sums(ranked[j[i]:j[i+1]]) / float(n[i]) 

 

totaln = np.sum(n) 

h = 12.0 / (totaln * (totaln + 1)) * ssbn - 3 * (totaln + 1) 

df = num_groups - 1 

h /= ties 

 

return KruskalResult(h, distributions.chi2.sf(h, df)) 

 

 

FriedmanchisquareResult = namedtuple('FriedmanchisquareResult', 

('statistic', 'pvalue')) 

 

 

def friedmanchisquare(*args): 

""" 

Compute the Friedman test for repeated measurements 

 

The Friedman test tests the null hypothesis that repeated measurements of 

the same individuals have the same distribution. It is often used 

to test for consistency among measurements obtained in different ways. 

For example, if two measurement techniques are used on the same set of 

individuals, the Friedman test can be used to determine if the two 

measurement techniques are consistent. 

 

Parameters 

---------- 

measurements1, measurements2, measurements3... : array_like 

Arrays of measurements. All of the arrays must have the same number 

of elements. At least 3 sets of measurements must be given. 

 

Returns 

------- 

statistic : float 

the test statistic, correcting for ties 

pvalue : float 

the associated p-value assuming that the test statistic has a chi 

squared distribution 

 

Notes 

----- 

Due to the assumption that the test statistic has a chi squared 

distribution, the p-value is only reliable for n > 10 and more than 

6 repeated measurements. 

 

References 

---------- 

.. [1] http://en.wikipedia.org/wiki/Friedman_test 

 

""" 

k = len(args) 

if k < 3: 

raise ValueError('Less than 3 levels. Friedman test not appropriate.') 

 

n = len(args[0]) 

for i in range(1, k): 

if len(args[i]) != n: 

raise ValueError('Unequal N in friedmanchisquare. Aborting.') 

 

# Rank data 

data = np.vstack(args).T 

data = data.astype(float) 

for i in range(len(data)): 

data[i] = rankdata(data[i]) 

 

# Handle ties 

ties = 0 

for i in range(len(data)): 

replist, repnum = find_repeats(array(data[i])) 

for t in repnum: 

ties += t * (t*t - 1) 

c = 1 - ties / float(k*(k*k - 1)*n) 

 

ssbn = np.sum(data.sum(axis=0)**2) 

chisq = (12.0 / (k*n*(k+1)) * ssbn - 3*n*(k+1)) / c 

 

return FriedmanchisquareResult(chisq, distributions.chi2.sf(chisq, k - 1)) 

 

 

def combine_pvalues(pvalues, method='fisher', weights=None): 

""" 

Methods for combining the p-values of independent tests bearing upon the 

same hypothesis. 

 

Parameters 

---------- 

pvalues : array_like, 1-D 

Array of p-values assumed to come from independent tests. 

method : {'fisher', 'stouffer'}, optional 

Name of method to use to combine p-values. The following methods are 

available: 

 

- "fisher": Fisher's method (Fisher's combined probability test), 

the default. 

- "stouffer": Stouffer's Z-score method. 

weights : array_like, 1-D, optional 

Optional array of weights used only for Stouffer's Z-score method. 

 

Returns 

------- 

statistic: float 

The statistic calculated by the specified method: 

- "fisher": The chi-squared statistic 

- "stouffer": The Z-score 

pval: float 

The combined p-value. 

 

Notes 

----- 

Fisher's method (also known as Fisher's combined probability test) [1]_ uses 

a chi-squared statistic to compute a combined p-value. The closely related 

Stouffer's Z-score method [2]_ uses Z-scores rather than p-values. The 

advantage of Stouffer's method is that it is straightforward to introduce 

weights, which can make Stouffer's method more powerful than Fisher's 

method when the p-values are from studies of different size [3]_ [4]_. 

 

Fisher's method may be extended to combine p-values from dependent tests 

[5]_. Extensions such as Brown's method and Kost's method are not currently 

implemented. 

 

.. versionadded:: 0.15.0 

 

References 

---------- 

.. [1] https://en.wikipedia.org/wiki/Fisher%27s_method 

.. [2] http://en.wikipedia.org/wiki/Fisher's_method#Relation_to_Stouffer.27s_Z-score_method 

.. [3] Whitlock, M. C. "Combining probability from independent tests: the 

weighted Z-method is superior to Fisher's approach." Journal of 

Evolutionary Biology 18, no. 5 (2005): 1368-1373. 

.. [4] Zaykin, Dmitri V. "Optimally weighted Z-test is a powerful method 

for combining probabilities in meta-analysis." Journal of 

Evolutionary Biology 24, no. 8 (2011): 1836-1841. 

.. [5] https://en.wikipedia.org/wiki/Extensions_of_Fisher%27s_method 

 

""" 

pvalues = np.asarray(pvalues) 

if pvalues.ndim != 1: 

raise ValueError("pvalues is not 1-D") 

 

if method == 'fisher': 

Xsq = -2 * np.sum(np.log(pvalues)) 

pval = distributions.chi2.sf(Xsq, 2 * len(pvalues)) 

return (Xsq, pval) 

elif method == 'stouffer': 

if weights is None: 

weights = np.ones_like(pvalues) 

elif len(weights) != len(pvalues): 

raise ValueError("pvalues and weights must be of the same size.") 

 

weights = np.asarray(weights) 

if weights.ndim != 1: 

raise ValueError("weights is not 1-D") 

 

Zi = distributions.norm.isf(pvalues) 

Z = np.dot(weights, Zi) / np.linalg.norm(weights) 

pval = distributions.norm.sf(Z) 

 

return (Z, pval) 

else: 

raise ValueError( 

"Invalid method '%s'. Options are 'fisher' or 'stouffer'", method) 

 

##################################### 

# PROBABILITY CALCULATIONS # 

##################################### 

 

 

def _betai(a, b, x): 

x = np.asarray(x) 

x = np.where(x < 1.0, x, 1.0) # if x > 1 then return 1.0 

return special.betainc(a, b, x) 

 

 

##################################### 

# STATISTICAL DISTANCES # 

##################################### 

 

def wasserstein_distance(u_values, v_values, u_weights=None, v_weights=None): 

r""" 

Compute the first Wasserstein distance between two 1D distributions. 

 

This distance is also known as the earth mover's distance, since it can be 

seen as the minimum amount of "work" required to transform :math:`u` into 

:math:`v`, where "work" is measured as the amount of distribution weight 

that must be moved, multiplied by the distance it has to be moved. 

 

.. versionadded:: 1.0.0 

 

Parameters 

---------- 

u_values, v_values : array_like 

Values observed in the (empirical) distribution. 

u_weights, v_weights : array_like, optional 

Weight for each value. If unspecified, each value is assigned the same 

weight. 

`u_weights` (resp. `v_weights`) must have the same length as 

`u_values` (resp. `v_values`). If the weight sum differs from 1, it 

must still be positive and finite so that the weights can be normalized 

to sum to 1. 

 

Returns 

------- 

distance : float 

The computed distance between the distributions. 

 

Notes 

----- 

The first Wasserstein distance between the distributions :math:`u` and 

:math:`v` is: 

 

.. math:: 

 

l_1 (u, v) = \inf_{\pi \in \Gamma (u, v)} \int_{\mathbb{R} \times 

\mathbb{R}} |x-y| \mathrm{d} \pi (x, y) 

 

where :math:`\Gamma (u, v)` is the set of (probability) distributions on 

:math:`\mathbb{R} \times \mathbb{R}` whose marginals are :math:`u` and 

:math:`v` on the first and second factors respectively. 

 

If :math:`U` and :math:`V` are the respective CDFs of :math:`u` and 

:math:`v`, this distance also equals to: 

 

.. math:: 

 

l_1(u, v) = \int_{-\infty}^{+\infty} |U-V| 

 

See [2]_ for a proof of the equivalence of both definitions. 

 

The input distributions can be empirical, therefore coming from samples 

whose values are effectively inputs of the function, or they can be seen as 

generalized functions, in which case they are weighted sums of Dirac delta 

functions located at the specified values. 

 

References 

---------- 

.. [1] "Wasserstein metric", http://en.wikipedia.org/wiki/Wasserstein_metric 

.. [2] Ramdas, Garcia, Cuturi "On Wasserstein Two Sample Testing and Related 

Families of Nonparametric Tests" (2015). :arXiv:`1509.02237`. 

 

Examples 

-------- 

>>> from scipy.stats import wasserstein_distance 

>>> wasserstein_distance([0, 1, 3], [5, 6, 8]) 

5.0 

>>> wasserstein_distance([0, 1], [0, 1], [3, 1], [2, 2]) 

0.25 

>>> wasserstein_distance([3.4, 3.9, 7.5, 7.8], [4.5, 1.4], 

... [1.4, 0.9, 3.1, 7.2], [3.2, 3.5]) 

4.0781331438047861 

""" 

return _cdf_distance(1, u_values, v_values, u_weights, v_weights) 

 

 

def energy_distance(u_values, v_values, u_weights=None, v_weights=None): 

r""" 

Compute the energy distance between two 1D distributions. 

 

.. versionadded:: 1.0.0 

 

Parameters 

---------- 

u_values, v_values : array_like 

Values observed in the (empirical) distribution. 

u_weights, v_weights : array_like, optional 

Weight for each value. If unspecified, each value is assigned the same 

weight. 

`u_weights` (resp. `v_weights`) must have the same length as 

`u_values` (resp. `v_values`). If the weight sum differs from 1, it 

must still be positive and finite so that the weights can be normalized 

to sum to 1. 

 

Returns 

------- 

distance : float 

The computed distance between the distributions. 

 

Notes 

----- 

The energy distance between two distributions :math:`u` and :math:`v`, whose 

respective CDFs are :math:`U` and :math:`V`, equals to: 

 

.. math:: 

 

D(u, v) = \left( 2\mathbb E|X - Y| - \mathbb E|X - X'| - 

\mathbb E|Y - Y'| \right)^{1/2} 

 

where :math:`X` and :math:`X'` (resp. :math:`Y` and :math:`Y'`) are 

independent random variables whose probability distribution is :math:`u` 

(resp. :math:`v`). 

 

As shown in [2]_, for one-dimensional real-valued variables, the energy 

distance is linked to the non-distribution-free version of the Cramer-von 

Mises distance: 

 

.. math:: 

 

D(u, v) = \sqrt{2} l_2(u, v) = \left( 2 \int_{-\infty}^{+\infty} (U-V)^2 

\right)^{1/2} 

 

Note that the common Cramer-von Mises criterion uses the distribution-free 

version of the distance. See [2]_ (section 2), for more details about both 

versions of the distance. 

 

The input distributions can be empirical, therefore coming from samples 

whose values are effectively inputs of the function, or they can be seen as 

generalized functions, in which case they are weighted sums of Dirac delta 

functions located at the specified values. 

 

References 

---------- 

.. [1] "Energy distance", https://en.wikipedia.org/wiki/Energy_distance 

.. [2] Szekely "E-statistics: The energy of statistical samples." Bowling 

Green State University, Department of Mathematics and Statistics, 

Technical Report 02-16 (2002). 

.. [3] Rizzo, Szekely "Energy distance." Wiley Interdisciplinary Reviews: 

Computational Statistics, 8(1):27-38 (2015). 

.. [4] Bellemare, Danihelka, Dabney, Mohamed, Lakshminarayanan, Hoyer, 

Munos "The Cramer Distance as a Solution to Biased Wasserstein 

Gradients" (2017). :arXiv:`1705.10743`. 

 

Examples 

-------- 

>>> from scipy.stats import energy_distance 

>>> energy_distance([0], [2]) 

2.0000000000000004 

>>> energy_distance([0, 8], [0, 8], [3, 1], [2, 2]) 

1.0000000000000002 

>>> energy_distance([0.7, 7.4, 2.4, 6.8], [1.4, 8. ], 

... [2.1, 4.2, 7.4, 8. ], [7.6, 8.8]) 

0.88003340976158217 

""" 

return np.sqrt(2) * _cdf_distance(2, u_values, v_values, 

u_weights, v_weights) 

 

 

def _cdf_distance(p, u_values, v_values, u_weights=None, v_weights=None): 

r""" 

Compute, between two one-dimensional distributions :math:`u` and 

:math:`v`, whose respective CDFs are :math:`U` and :math:`V`, the 

statistical distance that is defined as: 

 

.. math:: 

 

l_p(u, v) = \left( \int_{-\infty}^{+\infty} |U-V|^p \right)^{1/p} 

 

p is a positive parameter; p = 1 gives the Wasserstein distance, p = 2 

gives the energy distance. 

 

Parameters 

---------- 

u_values, v_values : array_like 

Values observed in the (empirical) distribution. 

u_weights, v_weights : array_like, optional 

Weight for each value. If unspecified, each value is assigned the same 

weight. 

`u_weights` (resp. `v_weights`) must have the same length as 

`u_values` (resp. `v_values`). If the weight sum differs from 1, it 

must still be positive and finite so that the weights can be normalized 

to sum to 1. 

 

Returns 

------- 

distance : float 

The computed distance between the distributions. 

 

Notes 

----- 

The input distributions can be empirical, therefore coming from samples 

whose values are effectively inputs of the function, or they can be seen as 

generalized functions, in which case they are weighted sums of Dirac delta 

functions located at the specified values. 

 

References 

---------- 

.. [1] Bellemare, Danihelka, Dabney, Mohamed, Lakshminarayanan, Hoyer, 

Munos "The Cramer Distance as a Solution to Biased Wasserstein 

Gradients" (2017). :arXiv:`1705.10743`. 

""" 

u_values, u_weights = _validate_distribution(u_values, u_weights) 

v_values, v_weights = _validate_distribution(v_values, v_weights) 

 

u_sorter = np.argsort(u_values) 

v_sorter = np.argsort(v_values) 

 

all_values = np.concatenate((u_values, v_values)) 

all_values.sort(kind='mergesort') 

 

# Compute the differences between pairs of successive values of u and v. 

deltas = np.diff(all_values) 

 

# Get the respective positions of the values of u and v among the values of 

# both distributions. 

u_cdf_indices = u_values[u_sorter].searchsorted(all_values[:-1], 'right') 

v_cdf_indices = v_values[v_sorter].searchsorted(all_values[:-1], 'right') 

 

# Calculate the CDFs of u and v using their weights, if specified. 

if u_weights is None: 

u_cdf = u_cdf_indices / u_values.size 

else: 

u_sorted_cumweights = np.concatenate(([0], 

np.cumsum(u_weights[u_sorter]))) 

u_cdf = u_sorted_cumweights[u_cdf_indices] / u_sorted_cumweights[-1] 

 

if v_weights is None: 

v_cdf = v_cdf_indices / v_values.size 

else: 

v_sorted_cumweights = np.concatenate(([0], 

np.cumsum(v_weights[v_sorter]))) 

v_cdf = v_sorted_cumweights[v_cdf_indices] / v_sorted_cumweights[-1] 

 

# Compute the value of the integral based on the CDFs. 

# If p = 1 or p = 2, we avoid using np.power, which introduces an overhead 

# of about 15%. 

if p == 1: 

return np.sum(np.multiply(np.abs(u_cdf - v_cdf), deltas)) 

if p == 2: 

return np.sqrt(np.sum(np.multiply(np.square(u_cdf - v_cdf), deltas))) 

return np.power(np.sum(np.multiply(np.power(np.abs(u_cdf - v_cdf), p), 

deltas)), 1/p) 

 

 

def _validate_distribution(values, weights): 

""" 

Validate the values and weights from a distribution input of `cdf_distance` 

and return them as ndarray objects. 

 

Parameters 

---------- 

values : array_like 

Values observed in the (empirical) distribution. 

weights : array_like 

Weight for each value. 

 

Returns 

------- 

values : ndarray 

Values as ndarray. 

weights : ndarray 

Weights as ndarray. 

""" 

# Validate the value array. 

values = np.asarray(values, dtype=float) 

if len(values) == 0: 

raise ValueError("Distribution can't be empty.") 

 

# Validate the weight array, if specified. 

if weights is not None: 

weights = np.asarray(weights, dtype=float) 

if len(weights) != len(values): 

raise ValueError('Value and weight array-likes for the same ' 

'empirical distribution must be of the same size.') 

if np.any(weights < 0): 

raise ValueError('All weights must be non-negative.') 

if not 0 < np.sum(weights) < np.inf: 

raise ValueError('Weight array-like sum must be positive and ' 

'finite. Set as None for an equal distribution of ' 

'weight.') 

 

return values, weights 

 

return values, None 

 

 

##################################### 

# SUPPORT FUNCTIONS # 

##################################### 

 

RepeatedResults = namedtuple('RepeatedResults', ('values', 'counts')) 

 

 

def find_repeats(arr): 

""" 

Find repeats and repeat counts. 

 

Parameters 

---------- 

arr : array_like 

Input array. This is cast to float64. 

 

Returns 

------- 

values : ndarray 

The unique values from the (flattened) input that are repeated. 

 

counts : ndarray 

Number of times the corresponding 'value' is repeated. 

 

Notes 

----- 

In numpy >= 1.9 `numpy.unique` provides similar functionality. The main 

difference is that `find_repeats` only returns repeated values. 

 

Examples 

-------- 

>>> from scipy import stats 

>>> stats.find_repeats([2, 1, 2, 3, 2, 2, 5]) 

RepeatedResults(values=array([2.]), counts=array([4])) 

 

>>> stats.find_repeats([[10, 20, 1, 2], [5, 5, 4, 4]]) 

RepeatedResults(values=array([4., 5.]), counts=array([2, 2])) 

 

""" 

# Note: always copies. 

return RepeatedResults(*_find_repeats(np.array(arr, dtype=np.float64))) 

 

 

def _sum_of_squares(a, axis=0): 

""" 

Square each element of the input array, and return the sum(s) of that. 

 

Parameters 

---------- 

a : array_like 

Input array. 

axis : int or None, optional 

Axis along which to calculate. Default is 0. If None, compute over 

the whole array `a`. 

 

Returns 

------- 

sum_of_squares : ndarray 

The sum along the given axis for (a**2). 

 

See also 

-------- 

_square_of_sums : The square(s) of the sum(s) (the opposite of 

`_sum_of_squares`). 

""" 

a, axis = _chk_asarray(a, axis) 

return np.sum(a*a, axis) 

 

 

def _square_of_sums(a, axis=0): 

""" 

Sum elements of the input array, and return the square(s) of that sum. 

 

Parameters 

---------- 

a : array_like 

Input array. 

axis : int or None, optional 

Axis along which to calculate. Default is 0. If None, compute over 

the whole array `a`. 

 

Returns 

------- 

square_of_sums : float or ndarray 

The square of the sum over `axis`. 

 

See also 

-------- 

_sum_of_squares : The sum of squares (the opposite of `square_of_sums`). 

""" 

a, axis = _chk_asarray(a, axis) 

s = np.sum(a, axis) 

if not np.isscalar(s): 

return s.astype(float) * s 

else: 

return float(s) * s 

 

 

def rankdata(a, method='average'): 

""" 

Assign ranks to data, dealing with ties appropriately. 

 

Ranks begin at 1. The `method` argument controls how ranks are assigned 

to equal values. See [1]_ for further discussion of ranking methods. 

 

Parameters 

---------- 

a : array_like 

The array of values to be ranked. The array is first flattened. 

method : str, optional 

The method used to assign ranks to tied elements. 

The options are 'average', 'min', 'max', 'dense' and 'ordinal'. 

 

'average': 

The average of the ranks that would have been assigned to 

all the tied values is assigned to each value. 

'min': 

The minimum of the ranks that would have been assigned to all 

the tied values is assigned to each value. (This is also 

referred to as "competition" ranking.) 

'max': 

The maximum of the ranks that would have been assigned to all 

the tied values is assigned to each value. 

'dense': 

Like 'min', but the rank of the next highest element is assigned 

the rank immediately after those assigned to the tied elements. 

'ordinal': 

All values are given a distinct rank, corresponding to the order 

that the values occur in `a`. 

 

The default is 'average'. 

 

Returns 

------- 

ranks : ndarray 

An array of length equal to the size of `a`, containing rank 

scores. 

 

References 

---------- 

.. [1] "Ranking", http://en.wikipedia.org/wiki/Ranking 

 

Examples 

-------- 

>>> from scipy.stats import rankdata 

>>> rankdata([0, 2, 3, 2]) 

array([ 1. , 2.5, 4. , 2.5]) 

>>> rankdata([0, 2, 3, 2], method='min') 

array([ 1, 2, 4, 2]) 

>>> rankdata([0, 2, 3, 2], method='max') 

array([ 1, 3, 4, 3]) 

>>> rankdata([0, 2, 3, 2], method='dense') 

array([ 1, 2, 3, 2]) 

>>> rankdata([0, 2, 3, 2], method='ordinal') 

array([ 1, 2, 4, 3]) 

""" 

if method not in ('average', 'min', 'max', 'dense', 'ordinal'): 

raise ValueError('unknown method "{0}"'.format(method)) 

 

arr = np.ravel(np.asarray(a)) 

algo = 'mergesort' if method == 'ordinal' else 'quicksort' 

sorter = np.argsort(arr, kind=algo) 

 

inv = np.empty(sorter.size, dtype=np.intp) 

inv[sorter] = np.arange(sorter.size, dtype=np.intp) 

 

if method == 'ordinal': 

return inv + 1 

 

arr = arr[sorter] 

obs = np.r_[True, arr[1:] != arr[:-1]] 

dense = obs.cumsum()[inv] 

 

if method == 'dense': 

return dense 

 

# cumulative counts of each unique value 

count = np.r_[np.nonzero(obs)[0], len(obs)] 

 

if method == 'max': 

return count[dense] 

 

if method == 'min': 

return count[dense - 1] + 1 

 

# average method 

return .5 * (count[dense] + count[dense - 1] + 1)