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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.
"""
'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']
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
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
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)
""" 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))
""" 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")
""" 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)
"""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
""" 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)
""" 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)
""" 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
""" 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
""" 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))
""" 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 # #####################################
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)
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)
""" 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)
""" 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
""" 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
('nobs', 'minmax', 'mean', 'variance', 'skewness', 'kurtosis'))
""" 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 # #####################################
""" 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)))
""" 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)))
""" 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))
""" 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 # #####################################
"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
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 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
""" 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'")
('count', 'lowerlimit', 'binsize', 'extrapoints'))
""" 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)
('cumcount', 'lowerlimit', 'binsize', 'extrapoints'))
""" 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)
('frequency', 'lowerlimit', 'binsize', 'extrapoints'))
""" 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 # #####################################
""" 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)
""" 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
""" 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
""" 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 'normal': special.erfinv(0.5) * 2.0 * math.sqrt(2.0)}
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
""" 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
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 # #####################################
""" 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)
""" 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]
""" 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]
""" 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)
""" 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)
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
"""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
""" 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)
('correlation', 'pvalue'))
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)
""" 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)
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 # #####################################
""" 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)
"""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
d = mean1 - mean2 with np.errstate(divide='ignore', invalid='ignore'): t = np.divide(d, denom) t, prob = _ttest_finish(df, t)
return (t, prob)
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
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
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)
""" 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)
""" 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)
""" 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(). "pearson": 1, "log-likelihood": 0, "freeman-tukey": -0.5, "mod-log-likelihood": -1, "neyman": -2, "cressie-read": 2/3, }
""" 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
('statistic', 'pvalue'))
""" 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)
""" 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")
""" 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)
""" 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)
""" 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)
""" 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)
""" 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))
('statistic', 'pvalue'))
""" 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))
""" 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 # #####################################
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 # #####################################
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)
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)
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)
""" 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 # #####################################
""" 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)))
""" 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)
""" 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
""" 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) |