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* BUG: Fix random state bug multiscale_graphcorr

* BUG: Fix index error in multiscale_graphcorr

* BUG: fix nonetype exception

* ENH: Remove workers dependence for random_state in multiscale_graphcorr

* MAIN: Move random state keyword to end of parameter list

* MAIN: change list comprehension so it is more clear

* MAIN: fix typo in seed parameter

* ENH: Change seed generation in multiscale_graphcorr

* DOC: Use random_state keyword in example

* MAIN: Add whitespace around bitwise operator
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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 gstd iqr median_absolute_deviation Trimming Functions ------------------ .. autosummary:: :toctree: generated/ trimboth trim1 Correlation Functions --------------------- .. autosummary:: :toctree: generated/ pearsonr fisher_exact spearmanr pointbiserialr kendalltau weightedtau linregress theilslopes multiscale_graphcorr Inferential Stats ----------------- .. autosummary:: :toctree: generated/ ttest_1samp ttest_ind ttest_ind_from_stats ttest_rel chisquare power_divergence ks_2samp epps_singleton_2samp mannwhitneyu ranksums wilcoxon kruskal friedmanchisquare brunnermunzel combine_pvalues Statistical Distances --------------------- .. autosummary:: :toctree: generated/ wasserstein_distance energy_distance ANOVA Functions --------------- .. autosummary:: :toctree: generated/ f_oneway Support Functions ----------------- .. autosummary:: :toctree: generated/ rankdata rvs_ratio_uniforms References ---------- .. [CRCProbStat2000] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000. """ from __future__ import division, print_function, absolute_import import warnings import sys import math if sys.version_info >= (3, 5): from math import gcd else: from fractions import gcd from collections import namedtuple import numpy as np from numpy import array, asarray, ma from scipy._lib.six import callable, string_types from scipy.spatial.distance import cdist from scipy.ndimage import measurements from scipy._lib._version import NumpyVersion from scipy._lib._util import _lazywhere, check_random_state, MapWrapper import scipy.special as special from scipy import linalg from . import distributions from . import mstats_basic from ._stats_mstats_common import (_find_repeats, linregress, theilslopes, siegelslopes) from ._stats import (_kendall_dis, _toint64, _weightedrankedtau, _local_correlations) from ._rvs_sampling import rvs_ratio_uniforms from ._hypotests import epps_singleton_2samp __all__ = ['find_repeats', 'gmean', 'hmean', 'mode', 'tmean', 'tvar', 'tmin', 'tmax', 'tstd', 'tsem', 'moment', 'variation', 'skew', 'kurtosis', 'describe', 'skewtest', 'kurtosistest', 'normaltest', 'jarque_bera', 'itemfreq', 'scoreatpercentile', 'percentileofscore', 'cumfreq', 'relfreq', 'obrientransform', 'sem', 'zmap', 'zscore', 'iqr', 'gstd', 'median_absolute_deviation', 'sigmaclip', 'trimboth', 'trim1', 'trim_mean', 'f_oneway', 'PearsonRConstantInputWarning', 'PearsonRNearConstantInputWarning', 'pearsonr', 'fisher_exact', 'spearmanr', 'pointbiserialr', 'kendalltau', 'weightedtau', 'multiscale_graphcorr', 'linregress', 'siegelslopes', 'theilslopes', 'ttest_1samp', 'ttest_ind', 'ttest_ind_from_stats', 'ttest_rel', 'kstest', 'chisquare', 'power_divergence', 'ks_2samp', 'mannwhitneyu', 'tiecorrect', 'ranksums', 'kruskal', 'friedmanchisquare', 'rankdata', 'rvs_ratio_uniforms', 'combine_pvalues', 'wasserstein_distance', 'energy_distance', 'brunnermunzel', 'epps_singleton_2samp'] def _chk_asarray(a, axis): if axis is None: a = np.ravel(a) outaxis = 0 else: a = np.asarray(a) outaxis = axis if a.ndim == 0: a = np.atleast_1d(a) return a, outaxis def _chk2_asarray(a, b, axis): if axis is None: a = np.ravel(a) b = np.ravel(b) outaxis = 0 else: a = np.asarray(a) b = np.asarray(b) outaxis = axis if a.ndim == 0: a = np.atleast_1d(a) if b.ndim == 0: b = np.atleast_1d(b) return a, b, outaxis def _contains_nan(a, nan_policy='propagate'): policies = ['propagate', 'raise', 'omit'] if nan_policy not in policies: raise ValueError("nan_policy must be one of {%s}" % ', '.join("'%s'" % s for s in policies)) try: # Calling np.sum to avoid creating a huge array into memory # e.g. np.isnan(a).any() with np.errstate(invalid='ignore'): contains_nan = np.isnan(np.sum(a)) except TypeError: # This can happen when attempting to sum things which are not # numbers (e.g. as in the function mode). Try an alternative method: try: contains_nan = np.nan in set(a.ravel()) except TypeError: # Don't know what to do. Fall back to omitting nan values and # issue a warning. contains_nan = False nan_policy = 'omit' warnings.warn("The input array could not be properly checked for nan " "values. nan values will be ignored.", RuntimeWarning) if contains_nan and nan_policy == 'raise': raise ValueError("The input contains nan values") return (contains_nan, nan_policy) def gmean(a, axis=0, dtype=None): """ Compute the geometric mean along the specified axis. Return the geometric average of the array elements. That is: n-th root of (x1 * x2 * ... * xn) Parameters ---------- a : array_like Input array or object that can be converted to an array. axis : int or None, optional Axis along which the geometric mean is computed. Default is 0. If None, compute over the whole array a. dtype : dtype, optional Type of the returned array and of the accumulator in which the elements are summed. If dtype is not specified, it defaults to the dtype of a, unless a has an integer dtype with a precision less than that of the default platform integer. In that case, the default platform integer is used. Returns ------- gmean : ndarray See dtype parameter above. See Also -------- numpy.mean : Arithmetic average numpy.average : Weighted average hmean : Harmonic mean Notes ----- The geometric average is computed over a single dimension of the input array, axis=0 by default, or all values in the array if axis=None. float64 intermediate and return values are used for integer inputs. Use masked arrays to ignore any non-finite values in the input or that arise in the calculations such as Not a Number and infinity because masked arrays automatically mask any non-finite values. Examples -------- >>> from scipy.stats import gmean >>> gmean([1, 4]) 2.0 >>> gmean([1, 2, 3, 4, 5, 6, 7]) 3.3800151591412964 """ if not isinstance(a, np.ndarray): # if not an ndarray object attempt to convert it log_a = np.log(np.array(a, dtype=dtype)) elif dtype: # Must change the default dtype allowing array type if isinstance(a, np.ma.MaskedArray): log_a = np.log(np.ma.asarray(a, dtype=dtype)) else: log_a = np.log(np.asarray(a, dtype=dtype)) else: log_a = np.log(a) return np.exp(log_a.mean(axis=axis)) def hmean(a, axis=0, dtype=None): """ Calculate the harmonic mean along the specified axis. That is: n / (1/x1 + 1/x2 + ... + 1/xn) Parameters ---------- a : array_like Input array, masked array or object that can be converted to an array. axis : int or None, optional Axis along which the harmonic mean is computed. Default is 0. If None, compute over the whole array a. dtype : dtype, optional Type of the returned array and of the accumulator in which the elements are summed. If dtype is not specified, it defaults to the dtype of a, unless a has an integer dtype with a precision less than that of the default platform integer. In that case, the default platform integer is used. Returns ------- hmean : ndarray See dtype parameter above. See Also -------- numpy.mean : Arithmetic average numpy.average : Weighted average gmean : Geometric mean Notes ----- The harmonic mean is computed over a single dimension of the input array, axis=0 by default, or all values in the array if axis=None. float64 intermediate and return values are used for integer inputs. Use masked arrays to ignore any non-finite values in the input or that arise in the calculations such as Not a Number and infinity. Examples -------- >>> from scipy.stats import hmean >>> hmean([1, 4]) 1.6000000000000001 >>> hmean([1, 2, 3, 4, 5, 6, 7]) 2.6997245179063363 """ if not isinstance(a, np.ndarray): a = np.array(a, dtype=dtype) if np.all(a >= 0): # Harmonic mean only defined if greater than or equal to to 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] with np.errstate(divide='ignore'): return size / np.sum(1.0 / a, axis=axis, dtype=dtype) else: raise ValueError("Harmonic mean only defined if all elements greater " "than or equal to zero") ModeResult = namedtuple('ModeResult', ('mode', 'count')) def mode(a, axis=0, nan_policy='propagate'): """ Return an array of the modal (most common) value in the passed array. If there is more than one such value, only the smallest is returned. The bin-count for the modal bins is also returned. Parameters ---------- a : array_like n-dimensional array of which to find mode(s). axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array a. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values 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) if a.dtype == object and np.nan in set(a.ravel()): # Fall back to a slower method since np.unique does not work with NaN scores = set(np.ravel(a)) # get ALL unique values testshape = list(a.shape) testshape[axis] = 1 oldmostfreq = np.zeros(testshape, dtype=a.dtype) oldcounts = np.zeros(testshape, dtype=int) for score in scores: template = (a == score) counts = np.expand_dims(np.sum(template, axis), axis) mostfrequent = np.where(counts > oldcounts, score, oldmostfreq) oldcounts = np.maximum(counts, oldcounts) oldmostfreq = mostfrequent return ModeResult(mostfrequent, oldcounts) def _mode1D(a): vals, cnts = np.unique(a, return_counts=True) return vals[cnts.argmax()], cnts.max() # np.apply_along_axis will convert the _mode1D tuples to a numpy array, casting types in the process # This recreates the results without that issue # View of a, rotated so the requested axis is last in_dims = list(range(a.ndim)) a_view = np.transpose(a, in_dims[:axis] + in_dims[axis+1:] + [axis]) inds = np.ndindex(a_view.shape[:-1]) modes = np.empty(a_view.shape[:-1], dtype=a.dtype) counts = np.zeros(a_view.shape[:-1], dtype=np.int) for ind in inds: modes[ind], counts[ind] = _mode1D(a_view[ind]) newshape = list(a.shape) newshape[axis] = 1 return ModeResult(modes.reshape(newshape), counts.reshape(newshape)) def _mask_to_limits(a, limits, inclusive): """Mask an array for values outside of given limits. This is primarily a utility function. Parameters ---------- a : array limits : (float or None, float or None) A tuple consisting of the (lower limit, upper limit). Values in the input array less than the lower limit or greater than the upper limit will be masked out. None implies no limit. inclusive : (bool, bool) A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to lower or upper are allowed. Returns ------- A MaskedArray. Raises ------ A ValueError if there are no values within the given limits. """ lower_limit, upper_limit = limits lower_include, upper_include = inclusive am = ma.MaskedArray(a) if lower_limit is not None: if lower_include: am = ma.masked_less(am, lower_limit) else: am = ma.masked_less_equal(am, lower_limit) if upper_limit is not None: if upper_include: am = ma.masked_greater(am, upper_limit) else: am = ma.masked_greater_equal(am, upper_limit) if am.count() == 0: raise ValueError("No array values within given limits") return am def tmean(a, limits=None, inclusive=(True, True), axis=None): """ Compute the trimmed mean. This function finds the arithmetic mean of given values, ignoring values outside the given limits. Parameters ---------- a : array_like Array of values. limits : None or (lower limit, upper limit), optional Values in the input array less than the lower limit or greater than the upper limit will be ignored. When limits is None (default), then all values are used. Either of the limit values in the tuple can also be None representing a half-open interval. inclusive : (bool, bool), optional A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to the lower or upper limits are included. The default value is (True, True). axis : int or None, optional Axis along which to compute test. Default is None. Returns ------- tmean : float Trimmed mean. See Also -------- trim_mean : Returns mean after trimming a proportion from both tails. Examples -------- >>> from scipy import stats >>> x = np.arange(20) >>> stats.tmean(x) 9.5 >>> stats.tmean(x, (3,17)) 10.0 """ a = asarray(a) if limits is None: return np.mean(a, None) am = _mask_to_limits(a.ravel(), limits, inclusive) return am.mean(axis=axis) def tvar(a, limits=None, inclusive=(True, True), axis=0, ddof=1): """ Compute the trimmed variance. This function computes the sample variance of an array of values, while ignoring values which are outside of given limits. Parameters ---------- a : array_like Array of values. limits : None or (lower limit, upper limit), optional Values in the input array less than the lower limit or greater than the upper limit will be ignored. When limits is None, then all values are used. Either of the limit values in the tuple can also be None representing a half-open interval. The default value is None. inclusive : (bool, bool), optional A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to the lower or upper limits are included. The default value is (True, True). axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array a. ddof : int, optional Delta degrees of freedom. Default is 1. Returns ------- tvar : float Trimmed variance. Notes ----- tvar computes the unbiased sample variance, i.e. it uses a correction factor n / (n - 1). Examples -------- >>> from scipy import stats >>> x = np.arange(20) >>> stats.tvar(x) 35.0 >>> stats.tvar(x, (3,17)) 20.0 """ a = asarray(a) a = a.astype(float) if limits is None: return a.var(ddof=ddof, axis=axis) am = _mask_to_limits(a, limits, inclusive) amnan = am.filled(fill_value=np.nan) return np.nanvar(amnan, ddof=ddof, axis=axis) def tmin(a, lowerlimit=None, axis=0, inclusive=True, nan_policy='propagate'): """ Compute the trimmed minimum. This function finds the miminum value of an array a along the specified axis, but only considering values greater than a specified lower limit. Parameters ---------- a : array_like Array of values. lowerlimit : None or float, optional Values in the input array less than the given limit will be ignored. When lowerlimit is None, then all values are used. The default value is None. axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array a. inclusive : {True, False}, optional This flag determines whether values exactly equal to the lower limit are included. The default value is True. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- tmin : float, int or ndarray Trimmed minimum. Examples -------- >>> from scipy import stats >>> x = np.arange(20) >>> stats.tmin(x) 0 >>> stats.tmin(x, 13) 13 >>> stats.tmin(x, 13, inclusive=False) 14 """ a, axis = _chk_asarray(a, axis) am = _mask_to_limits(a, (lowerlimit, None), (inclusive, False)) contains_nan, nan_policy = _contains_nan(am, nan_policy) if contains_nan and nan_policy == 'omit': am = ma.masked_invalid(am) res = ma.minimum.reduce(am, axis).data if res.ndim == 0: return res[()] return res def tmax(a, upperlimit=None, axis=0, inclusive=True, nan_policy='propagate'): """ Compute the trimmed maximum. This function computes the maximum value of an array along a given axis, while ignoring values larger than a specified upper limit. Parameters ---------- a : array_like Array of values. upperlimit : None or float, optional Values in the input array greater than the given limit will be ignored. When upperlimit is None, then all values are used. The default value is None. axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array a. inclusive : {True, False}, optional This flag determines whether values exactly equal to the upper limit are included. The default value is True. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- tmax : float, int or ndarray Trimmed maximum. Examples -------- >>> from scipy import stats >>> x = np.arange(20) >>> stats.tmax(x) 19 >>> stats.tmax(x, 13) 13 >>> stats.tmax(x, 13, inclusive=False) 12 """ a, axis = _chk_asarray(a, axis) am = _mask_to_limits(a, (None, upperlimit), (False, inclusive)) contains_nan, nan_policy = _contains_nan(am, nan_policy) if contains_nan and nan_policy == 'omit': am = ma.masked_invalid(am) res = ma.maximum.reduce(am, axis).data if res.ndim == 0: return res[()] return res def tstd(a, limits=None, inclusive=(True, True), axis=0, ddof=1): """ Compute the trimmed sample standard deviation. This function finds the sample standard deviation of given values, ignoring values outside the given limits. Parameters ---------- a : array_like Array of values. limits : None or (lower limit, upper limit), optional Values in the input array less than the lower limit or greater than the upper limit will be ignored. When limits is None, then all values are used. Either of the limit values in the tuple can also be None representing a half-open interval. The default value is None. inclusive : (bool, bool), optional A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to the lower or upper limits are included. The default value is (True, True). axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array a. ddof : int, optional Delta degrees of freedom. Default is 1. Returns ------- tstd : float Trimmed sample standard deviation. Notes ----- tstd computes the unbiased sample standard deviation, i.e. it uses a correction factor n / (n - 1). Examples -------- >>> from scipy import stats >>> x = np.arange(20) >>> stats.tstd(x) 5.9160797830996161 >>> stats.tstd(x, (3,17)) 4.4721359549995796 """ return np.sqrt(tvar(a, limits, inclusive, axis, ddof)) def tsem(a, limits=None, inclusive=(True, True), axis=0, ddof=1): """ Compute the trimmed standard error of the mean. This function finds the standard error of the mean for given values, ignoring values outside the given limits. Parameters ---------- a : array_like Array of values. limits : None or (lower limit, upper limit), optional Values in the input array less than the lower limit or greater than the upper limit will be ignored. When limits is None, then all values are used. Either of the limit values in the tuple can also be None representing a half-open interval. The default value is None. inclusive : (bool, bool), optional A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to the lower or upper limits are included. The default value is (True, True). axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array a. ddof : int, optional Delta degrees of freedom. Default is 1. Returns ------- tsem : float Trimmed standard error of the mean. Notes ----- tsem uses unbiased sample standard deviation, i.e. it uses a correction factor n / (n - 1). Examples -------- >>> from scipy import stats >>> x = np.arange(20) >>> stats.tsem(x) 1.3228756555322954 >>> stats.tsem(x, (3,17)) 1.1547005383792515 """ a = np.asarray(a).ravel() if limits is None: return a.std(ddof=ddof) / np.sqrt(a.size) am = _mask_to_limits(a, limits, inclusive) sd = np.sqrt(np.ma.var(am, ddof=ddof, axis=axis)) return sd / np.sqrt(am.count()) ##################################### # MOMENTS # ##################################### def moment(a, moment=1, axis=0, nan_policy='propagate'): r""" Calculate the nth moment about the mean for a sample. A moment is a specific quantitative measure of the shape of a set of points. It is often used to calculate coefficients of skewness and kurtosis due to its close relationship with them. Parameters ---------- a : array_like Input array. 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. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values 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] https://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.full(np.asarray(moment).shape, np.nan, dtype=np.float64) # for array_like moment input, return a value for each. if not np.isscalar(moment): mmnt = [_moment(a, i, axis) for i in moment] return np.array(mmnt) else: return _moment(a, moment, axis) def _moment(a, moment, axis): if np.abs(moment - np.round(moment)) > 0: raise ValueError("All moment parameters must be integers") if moment == 0: # When moment equals 0, the result is 1, by definition. shape = list(a.shape) del shape[axis] if shape: # return an actual array of the appropriate shape return np.ones(shape, dtype=float) else: # the input was 1D, so return a scalar instead of a rank-0 array return 1.0 elif moment == 1: # By definition the first moment about the mean is 0. shape = list(a.shape) del shape[axis] if shape: # return an actual array of the appropriate shape return np.zeros(shape, dtype=float) else: # the input was 1D, so return a scalar instead of a rank-0 array return np.float64(0.0) else: # Exponentiation by squares: form exponent sequence n_list = [moment] current_n = moment while current_n > 2: if current_n % 2: current_n = (current_n - 1) / 2 else: current_n /= 2 n_list.append(current_n) # Starting point for exponentiation by squares a_zero_mean = a - np.expand_dims(np.mean(a, axis), axis) if n_list[-1] == 1: s = a_zero_mean.copy() else: s = a_zero_mean**2 # Perform multiplications for n in n_list[-2::-1]: s = s**2 if n % 2: s *= a_zero_mean return np.mean(s, axis) def variation(a, axis=0, nan_policy='propagate'): """ Compute the coefficient of variation. The coefficient of variation is 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. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- variation : ndarray The calculated variation along the requested axis. References ---------- .. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000. Examples -------- >>> from scipy.stats import variation >>> variation([1, 2, 3, 4, 5]) 0.47140452079103173 """ a, axis = _chk_asarray(a, axis) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': a = ma.masked_invalid(a) return mstats_basic.variation(a, axis) return a.std(axis) / a.mean(axis) def skew(a, axis=0, bias=True, nan_policy='propagate'): r""" Compute the sample skewness of a data set. For normally distributed data, the skewness should be about zero. For unimodal continuous distributions, a skewness value greater than zero 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 zero, statistically speaking. Parameters ---------- a : ndarray Input array. 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. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- skewness : ndarray The skewness of values along an axis, returning 0 where all values are equal. Notes ----- The sample skewness is computed as the Fisher-Pearson coefficient of skewness, i.e. .. math:: g_1=\frac{m_3}{m_2^{3/2}} where .. math:: m_i=\frac{1}{N}\sum_{n=1}^N(x[n]-\bar{x})^i is the biased sample :math:i\texttt{th} central moment, and :math:\bar{x} is the sample mean. If bias is False, the calculations are corrected for bias and the value computed is the adjusted Fisher-Pearson standardized moment coefficient, i.e. .. math:: G_1=\frac{k_3}{k_2^{3/2}}= \frac{\sqrt{N(N-1)}}{N-2}\frac{m_3}{m_2^{3/2}}. References ---------- .. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000. Section 2.2.24.1 Examples -------- >>> from scipy.stats import skew >>> skew([1, 2, 3, 4, 5]) 0.0 >>> skew([2, 8, 0, 4, 1, 9, 9, 0]) 0.2650554122698573 """ a, axis = _chk_asarray(a, axis) n = a.shape[axis] contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': a = ma.masked_invalid(a) return mstats_basic.skew(a, axis, bias) m2 = moment(a, 2, axis) m3 = moment(a, 3, axis) zero = (m2 == 0) vals = _lazywhere(~zero, (m2, m3), lambda m2, m3: m3 / m2**1.5, 0.) if not bias: can_correct = (n > 2) & (m2 > 0) if can_correct.any(): m2 = np.extract(can_correct, m2) m3 = np.extract(can_correct, m3) nval = np.sqrt((n - 1.0) * n) / (n - 2.0) * m3 / m2**1.5 np.place(vals, can_correct, nval) if vals.ndim == 0: return vals.item() return vals def kurtosis(a, axis=0, fisher=True, bias=True, nan_policy='propagate'): """ Compute the kurtosis (Fisher or Pearson) of a dataset. Kurtosis is the fourth central moment divided by the square of the variance. If Fisher's definition is used, then 3.0 is subtracted from the result to give 0.0 for a normal distribution. If bias is False then the kurtosis is calculated using k statistics to eliminate bias coming from biased moment estimators Use kurtosistest to see if result is close enough to normal. Parameters ---------- a : array Data for which the kurtosis is calculated. axis : int or None, optional Axis along which the kurtosis is calculated. Default is 0. If None, compute over the whole array a. fisher : bool, optional If True, Fisher's definition is used (normal ==> 0.0). If False, Pearson's definition is used (normal ==> 3.0). bias : bool, optional If False, then the calculations are corrected for statistical bias. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. 'propagate' returns nan, 'raise' throws an error, 'omit' performs the calculations ignoring nan values. Default is 'propagate'. Returns ------- kurtosis : array The kurtosis of values along an axis. If all values are equal, return -3 for Fisher's definition and 0 for Pearson's definition. References ---------- .. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000. Examples -------- In Fisher's definiton, the kurtosis of the normal distribution is zero. In the following example, the kurtosis is close to zero, because it was calculated from the dataset, not from the continuous distribution. >>> from scipy.stats import norm, kurtosis >>> data = norm.rvs(size=1000, random_state=3) >>> kurtosis(data) -0.06928694200380558 The distribution with a higher kurtosis has a heavier tail. The zero valued kurtosis of the normal distribution in Fisher's definition can serve as a reference point. >>> import matplotlib.pyplot as plt >>> import scipy.stats as stats >>> from scipy.stats import kurtosis >>> x = np.linspace(-5, 5, 100) >>> ax = plt.subplot() >>> distnames = ['laplace', 'norm', 'uniform'] >>> for distname in distnames: ... if distname == 'uniform': ... dist = getattr(stats, distname)(loc=-2, scale=4) ... else: ... dist = getattr(stats, distname) ... data = dist.rvs(size=1000) ... kur = kurtosis(data, fisher=True) ... y = dist.pdf(x) ... ax.plot(x, y, label="{}, {}".format(distname, round(kur, 3))) ... ax.legend() The Laplace distribution has a heavier tail than the normal distribution. The uniform distribution (which has negative kurtosis) has the thinnest tail. """ 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 return vals - 3 if fisher else vals DescribeResult = namedtuple('DescribeResult', ('nobs', 'minmax', 'mean', 'variance', 'skewness', 'kurtosis')) def describe(a, axis=0, ddof=1, bias=True, nan_policy='propagate'): """ Compute several descriptive statistics of the passed array. Parameters ---------- a : array_like Input data. axis : int or None, optional Axis along which statistics are calculated. Default is 0. If None, compute over the whole array a. ddof : int, optional Delta degrees of freedom (only for variance). Default is 1. bias : bool, optional If False, then the skewness and kurtosis calculations are corrected for statistical bias. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- nobs : int or ndarray of ints Number of observations (length of data along axis). When 'omit' is chosen as nan_policy, each column is counted separately. minmax: tuple of ndarrays or floats Minimum and maximum value of data array. mean : ndarray or float Arithmetic mean of data along axis. variance : ndarray or float Unbiased variance of the data along axis, denominator is number of observations minus one. skewness : ndarray or float Skewness, based on moment calculations with denominator equal to the number of observations, i.e. no degrees of freedom correction. kurtosis : ndarray or float Kurtosis (Fisher). The kurtosis is normalized so that it is zero for the normal distribution. No degrees of freedom are used. See Also -------- skew, kurtosis Examples -------- >>> from scipy import stats >>> a = np.arange(10) >>> stats.describe(a) DescribeResult(nobs=10, minmax=(0, 9), mean=4.5, variance=9.166666666666666, skewness=0.0, kurtosis=-1.2242424242424244) >>> b = [[1, 2], [3, 4]] >>> stats.describe(b) DescribeResult(nobs=2, minmax=(array([1, 2]), array([3, 4])), mean=array([2., 3.]), variance=array([2., 2.]), skewness=array([0., 0.]), kurtosis=array([-2., -2.])) """ a, axis = _chk_asarray(a, axis) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': a = ma.masked_invalid(a) return mstats_basic.describe(a, axis, ddof, bias) if a.size == 0: raise ValueError("The input must not be empty.") n = a.shape[axis] mm = (np.min(a, axis=axis), np.max(a, axis=axis)) m = np.mean(a, axis=axis) v = np.var(a, axis=axis, ddof=ddof) sk = skew(a, axis, bias=bias) kurt = kurtosis(a, axis, bias=bias) return DescribeResult(n, mm, m, v, sk, kurt) ##################################### # NORMALITY TESTS # ##################################### SkewtestResult = namedtuple('SkewtestResult', ('statistic', 'pvalue')) def skewtest(a, axis=0, nan_policy='propagate'): """ Test whether the skew is different from the normal distribution. This function tests the null hypothesis that the skewness of the population that the sample was drawn from is the same as that of a corresponding normal distribution. Parameters ---------- a : array The data to be tested. axis : int or None, optional Axis along which statistics are calculated. Default is 0. If None, compute over the whole array a. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- statistic : float The computed z-score for this test. pvalue : float Two-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 = a.shape[axis] if n < 8: raise ValueError( "skewtest is not valid with less than 8 samples; %i samples" " were given." % int(n)) y = b2 * math.sqrt(((n + 1) * (n + 3)) / (6.0 * (n - 2))) beta2 = (3.0 * (n**2 + 27*n - 70) * (n+1) * (n+3) / ((n-2.0) * (n+5) * (n+7) * (n+9))) W2 = -1 + math.sqrt(2 * (beta2 - 1)) delta = 1 / math.sqrt(0.5 * math.log(W2)) alpha = math.sqrt(2.0 / (W2 - 1)) y = np.where(y == 0, 1, y) Z = delta * np.log(y / alpha + np.sqrt((y / alpha)**2 + 1)) return SkewtestResult(Z, 2 * distributions.norm.sf(np.abs(Z))) KurtosistestResult = namedtuple('KurtosistestResult', ('statistic', 'pvalue')) def kurtosistest(a, axis=0, nan_policy='propagate'): """ Test whether a dataset has normal kurtosis. This function tests the null hypothesis that the kurtosis of the population from which the sample was drawn is that of the normal distribution: kurtosis = 3(n-1)/(n+1). Parameters ---------- a : array Array of the sample data. axis : int or None, optional Axis along which to compute test. Default is 0. If None, compute over the whole array a. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- statistic : float The computed z-score for this test. pvalue : float The two-sided p-value for the hypothesis test. Notes ----- Valid only for n>20. 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 = 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)) term2 = np.sign(denom) * np.where(denom == 0.0, np.nan, np.power((1-2.0/A)/np.abs(denom), 1/3.0)) if np.any(denom == 0): msg = "Test statistic not defined in some cases due to division by " \ "zero. Return nan in that case..." warnings.warn(msg, RuntimeWarning) Z = (term1 - term2) / np.sqrt(2/(9.0*A)) # [1]_ Eq. 5 if Z.ndim == 0: Z = Z[()] # zprob uses upper tail, so Z needs to be positive return KurtosistestResult(Z, 2 * distributions.norm.sf(np.abs(Z))) NormaltestResult = namedtuple('NormaltestResult', ('statistic', 'pvalue')) def normaltest(a, axis=0, nan_policy='propagate'): """ Test whether a sample differs from a normal distribution. This function tests the null hypothesis that a sample comes from a normal distribution. It is based on D'Agostino and Pearson's [1]_, [2]_ test that combines skew and kurtosis to produce an omnibus test of normality. Parameters ---------- a : array_like The array containing the sample to be tested. axis : int or None, optional Axis along which to compute test. Default is 0. If None, compute over the whole array a. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- statistic : float or array s^2 + k^2, where s is the z-score returned by skewtest and k is the z-score returned by kurtosistest. pvalue : float or array A 2-sided chi squared probability for the hypothesis test. References ---------- .. [1] D'Agostino, R. B. (1971), "An omnibus test of normality for moderate and large sample size", Biometrika, 58, 341-348 .. [2] D'Agostino, R. and Pearson, E. S. (1973), "Tests for departure from normality", Biometrika, 60, 613-622 Examples -------- >>> from scipy import stats >>> pts = 1000 >>> np.random.seed(28041990) >>> a = np.random.normal(0, 1, size=pts) >>> b = np.random.normal(2, 1, size=pts) >>> x = np.concatenate((a, b)) >>> k2, p = stats.normaltest(x) >>> alpha = 1e-3 >>> print("p = {:g}".format(p)) p = 3.27207e-11 >>> if p < alpha: # null hypothesis: x comes from a normal distribution ... print("The null hypothesis can be rejected") ... else: ... print("The null hypothesis cannot be rejected") The null hypothesis can be rejected """ a, axis = _chk_asarray(a, axis) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': a = ma.masked_invalid(a) return mstats_basic.normaltest(a, axis) s, _ = skewtest(a, axis) k, _ = kurtosistest(a, axis) k2 = s*s + k*k return NormaltestResult(k2, distributions.chi2.sf(k2, 2)) def jarque_bera(x): """ Perform the Jarque-Bera goodness of fit test on sample data. The Jarque-Bera test tests whether the sample data has the skewness and kurtosis matching a normal distribution. Note that this test only works for a large enough number of data samples (>2000) as the test statistic asymptotically has a Chi-squared distribution with 2 degrees of freedom. Parameters ---------- x : array_like Observations of a random variable. Returns ------- jb_value : float The test statistic. p : float The p-value for the hypothesis test. References ---------- .. [1] Jarque, C. and Bera, A. (1980) "Efficient tests for normality, homoscedasticity and serial independence of regression residuals", 6 Econometric Letters 255-259. Examples -------- >>> from scipy import stats >>> np.random.seed(987654321) >>> x = np.random.normal(0, 1, 100000) >>> y = np.random.rayleigh(1, 100000) >>> stats.jarque_bera(x) (4.7165707989581342, 0.09458225503041906) >>> stats.jarque_bera(y) (6713.7098548143422, 0.0) """ x = np.asarray(x) n = x.size if n == 0: raise ValueError('At least one observation is required.') mu = x.mean() diffx = x - mu skewness = (1 / n * np.sum(diffx**3)) / (1 / n * np.sum(diffx**2))**(3 / 2.) kurtosis = (1 / n * np.sum(diffx**4)) / (1 / n * np.sum(diffx**2))**2 jb_value = n / 6 * (skewness**2 + (kurtosis - 3)**2 / 4) p = 1 - distributions.chi2.cdf(jb_value, 2) return jb_value, p ##################################### # FREQUENCY FUNCTIONS # ##################################### @np.deprecate(message="itemfreq is deprecated and will be removed in a " "future version. Use instead np.unique(..., return_counts=True)") def itemfreq(a): """ Return a 2-D array of item frequencies. Parameters ---------- a : (N,) array_like Input array. Returns ------- itemfreq : (K, 2) ndarray A 2-D frequency table. Column 1 contains sorted, unique values from a, column 2 contains their respective counts. Examples -------- >>> from scipy import stats >>> a = np.array([1, 1, 5, 0, 1, 2, 2, 0, 1, 4]) >>> stats.itemfreq(a) array([[ 0., 2.], [ 1., 4.], [ 2., 2.], [ 4., 1.], [ 5., 1.]]) >>> np.bincount(a) array([2, 4, 2, 0, 1, 1]) >>> stats.itemfreq(a/10.) array([[ 0. , 2. ], [ 0.1, 4. ], [ 0.2, 2. ], [ 0.4, 1. ], [ 0.5, 1. ]]) """ items, inv = np.unique(a, return_inverse=True) freq = np.bincount(inv) return np.array([items, freq]).T def scoreatpercentile(a, per, limit=(), interpolation_method='fraction', axis=None): """ Calculate the score at a given percentile of the input sequence. For example, the score at per=50 is the median. If the desired quantile lies between two data points, we interpolate between them, according to the value of interpolation. If the parameter limit is provided, it should be a tuple (lower, upper) of two values. Parameters ---------- a : array_like A 1-D array of values from which to extract score. per : array_like Percentile(s) at which to extract score. Values should be in range [0,100]. limit : tuple, optional Tuple of two scalars, the lower and upper limits within which to compute the percentile. Values of a outside this (closed) interval will be ignored. interpolation_method : {'fraction', 'lower', 'higher'}, optional Specifies the interpolation method to use, when the desired quantile lies between two data points i and j The following options are available (default is 'fraction'): * '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.full(np.asarray(per).shape, np.nan, dtype=np.float64) if limit: a = a[(limit[0] <= a) & (a <= limit[1])] sorted_ = np.sort(a, axis=axis) if axis is None: axis = 0 return _compute_qth_percentile(sorted_, per, interpolation_method, axis) # handle sequence of per's without calling sort multiple times def _compute_qth_percentile(sorted_, per, interpolation_method, axis): if not np.isscalar(per): score = [_compute_qth_percentile(sorted_, i, interpolation_method, axis) for i in per] return np.array(score) if not (0 <= 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_[tuple(indexer)] * weights, axis=axis) / sumval def percentileofscore(a, score, kind='rank'): """ Compute 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 Specifies the interpretation of the resulting score. The following options are available (default is 'rank'): * '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 https://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) / n * 100 elif kind == 'weak': return np.count_nonzero(a <= score) / n * 100 elif kind == 'mean': pct = (np.count_nonzero(a < score) + np.count_nonzero(a <= score)) / n * 50 return pct else: raise ValueError("kind can only be 'rank', 'strict', 'weak' or 'mean'") HistogramResult = namedtuple('HistogramResult', ('count', 'lowerlimit', 'binsize', 'extrapoints')) def _histogram(a, numbins=10, defaultlimits=None, weights=None, printextras=False): """ Create a histogram. Separate the range into several bins and return the number of instances in each bin. Parameters ---------- a : array_like Array of scores which will be put into bins. numbins : int, optional The number of bins to use for the histogram. Default is 10. defaultlimits : tuple (lower, upper), optional The lower and upper values for the range of the histogram. If no value is given, a range slightly larger than the range of the values in a is used. Specifically (a.min() - s, a.max() + s), where s = (1/2)(a.max() - a.min()) / (numbins - 1). weights : array_like, optional The weights for each value in a. Default is None, which gives each value a weight of 1.0 printextras : bool, optional If True, if there are extra points (i.e. the points that fall outside the bin limits) a warning is raised saying how many of those points there are. Default is False. Returns ------- count : ndarray Number of points (or sum of weights) in each bin. lowerlimit : float Lowest value of histogram, the lower limit of the first bin. binsize : float The size of the bins (all bins have the same size). extrapoints : int The number of points outside the range of the histogram. See Also -------- numpy.histogram Notes ----- This histogram is based on numpy's histogram but has a larger range by default if default limits is not set. """ a = np.ravel(a) if defaultlimits is None: if a.size == 0: # handle empty arrays. Undetermined range, so use 0-1. defaultlimits = (0, 1) else: # no range given, so use values in a data_min = a.min() data_max = a.max() # Have bins extend past min and max values slightly s = (data_max - data_min) / (2. * (numbins - 1.)) defaultlimits = (data_min - s, data_max + s) # use numpy's histogram method to compute bins hist, bin_edges = np.histogram(a, bins=numbins, range=defaultlimits, weights=weights) # hist are not always floats, convert to keep with old output hist = np.array(hist, dtype=float) # fixed width for bins is assumed, as numpy's histogram gives # fixed width bins for int values for 'bins' binsize = bin_edges[1] - bin_edges[0] # calculate number of extra points extrapoints = len([v for v in a if defaultlimits[0] > v or v > defaultlimits[1]]) if extrapoints > 0 and printextras: warnings.warn("Points outside given histogram range = %s" % extrapoints) return HistogramResult(hist, defaultlimits[0], binsize, extrapoints) CumfreqResult = namedtuple('CumfreqResult', ('cumcount', 'lowerlimit', 'binsize', 'extrapoints')) def cumfreq(a, numbins=10, defaultreallimits=None, weights=None): """ Return a cumulative frequency histogram, using the histogram function. A cumulative histogram is a mapping that counts the cumulative number of observations in all of the bins up to the specified bin. Parameters ---------- a : array_like Input array. numbins : int, optional The number of bins to use for the histogram. Default is 10. defaultreallimits : tuple (lower, upper), optional The lower and upper values for the range of the histogram. If no value is given, a range slightly larger than the range of the values in a is used. Specifically (a.min() - s, a.max() + s), where s = (1/2)(a.max() - a.min()) / (numbins - 1). weights : array_like, optional The weights for each value in a. Default is None, which gives each value a weight of 1.0 Returns ------- cumcount : ndarray Binned values of cumulative frequency. lowerlimit : float Lower real limit binsize : float Width of each bin. extrapoints : int Extra points. Examples -------- >>> import matplotlib.pyplot as plt >>> from scipy import stats >>> x = [1, 4, 2, 1, 3, 1] >>> res = stats.cumfreq(x, numbins=4, defaultreallimits=(1.5, 5)) >>> res.cumcount array([ 1., 2., 3., 3.]) >>> res.extrapoints 3 Create a normal distribution with 1000 random values >>> rng = np.random.RandomState(seed=12345) >>> samples = stats.norm.rvs(size=1000, random_state=rng) Calculate cumulative frequencies >>> res = stats.cumfreq(samples, numbins=25) Calculate space of values for x >>> x = res.lowerlimit + np.linspace(0, res.binsize*res.cumcount.size, ... res.cumcount.size) Plot histogram and cumulative histogram >>> fig = plt.figure(figsize=(10, 4)) >>> ax1 = fig.add_subplot(1, 2, 1) >>> ax2 = fig.add_subplot(1, 2, 2) >>> ax1.hist(samples, bins=25) >>> ax1.set_title('Histogram') >>> ax2.bar(x, res.cumcount, width=res.binsize) >>> ax2.set_title('Cumulative histogram') >>> ax2.set_xlim([x.min(), x.max()]) >>> plt.show() """ h, l, b, e = _histogram(a, numbins, defaultreallimits, weights=weights) cumhist = np.cumsum(h * 1, axis=0) return CumfreqResult(cumhist, l, b, e) RelfreqResult = namedtuple('RelfreqResult', ('frequency', 'lowerlimit', 'binsize', 'extrapoints')) def relfreq(a, numbins=10, defaultreallimits=None, weights=None): """ Return a relative frequency histogram, using the histogram function. A relative frequency histogram is a mapping of the number of observations in each of the bins relative to the total of observations. Parameters ---------- a : array_like Input array. numbins : int, optional The number of bins to use for the histogram. Default is 10. defaultreallimits : tuple (lower, upper), optional The lower and upper values for the range of the histogram. If no value is given, a range slightly larger than the range of the values in a is used. Specifically (a.min() - s, a.max() + s), where s = (1/2)(a.max() - a.min()) / (numbins - 1). weights : array_like, optional The weights for each value in a. Default is None, which gives each value a weight of 1.0 Returns ------- frequency : ndarray Binned values of relative frequency. lowerlimit : float Lower real limit. binsize : float Width of each bin. extrapoints : int Extra points. Examples -------- >>> import matplotlib.pyplot as plt >>> from scipy import stats >>> a = np.array([2, 4, 1, 2, 3, 2]) >>> res = stats.relfreq(a, numbins=4) >>> res.frequency array([ 0.16666667, 0.5 , 0.16666667, 0.16666667]) >>> np.sum(res.frequency) # relative frequencies should add up to 1 1.0 Create a normal distribution with 1000 random values >>> rng = np.random.RandomState(seed=12345) >>> samples = stats.norm.rvs(size=1000, random_state=rng) Calculate relative frequencies >>> res = stats.relfreq(samples, numbins=25) Calculate space of values for x >>> x = res.lowerlimit + np.linspace(0, res.binsize*res.frequency.size, ... res.frequency.size) Plot relative frequency histogram >>> fig = plt.figure(figsize=(5, 4)) >>> ax = fig.add_subplot(1, 1, 1) >>> ax.bar(x, res.frequency, width=res.binsize) >>> ax.set_title('Relative frequency histogram') >>> ax.set_xlim([x.min(), x.max()]) >>> plt.show() """ a = np.asanyarray(a) h, l, b, e = _histogram(a, numbins, defaultreallimits, weights=weights) h = h / a.shape[0] return RelfreqResult(h, l, b, e) ##################################### # VARIABILITY FUNCTIONS # ##################################### def obrientransform(*args): """ Compute the O'Brien transform on input data (any number of arrays). Used to test for homogeneity of variance prior to running one-way stats. Each array in *args is one level of a factor. If f_oneway is run on the transformed data and found significant, the variances are unequal. From Maxwell and Delaney [1]_, p.112. Parameters ---------- args : tuple of array_like Any number of arrays. Returns ------- obrientransform : ndarray Transformed data for use in an ANOVA. The first dimension of the result corresponds to the sequence of transformed arrays. If the arrays given are all 1-D of the same length, the return value is a 2-D array; otherwise it is a 1-D array of type object, with each element being an ndarray. References ---------- .. [1] S. E. Maxwell and H. D. Delaney, "Designing Experiments and Analyzing Data: A Model Comparison Perspective", Wadsworth, 1990. Examples -------- We'll test the following data sets for differences in their variance. >>> x = [10, 11, 13, 9, 7, 12, 12, 9, 10] >>> y = [13, 21, 5, 10, 8, 14, 10, 12, 7, 15] Apply the O'Brien transform to the data. >>> from scipy.stats import obrientransform >>> tx, ty = obrientransform(x, y) Use scipy.stats.f_oneway to apply a one-way ANOVA test to the transformed data. >>> from scipy.stats import f_oneway >>> F, p = f_oneway(tx, ty) >>> p 0.1314139477040335 If we require that p < 0.05 for significance, we cannot conclude that the variances are different. """ TINY = np.sqrt(np.finfo(float).eps) # arrays will hold the transformed arguments. arrays = [] for arg in args: a = np.asarray(arg) n = len(a) mu = np.mean(a) sq = (a - mu)**2 sumsq = sq.sum() # The O'Brien transform. t = ((n - 1.5) * n * sq - 0.5 * sumsq) / ((n - 1) * (n - 2)) # Check that the mean of the transformed data is equal to the # original variance. var = sumsq / (n - 1) if abs(var - np.mean(t)) > TINY: raise ValueError('Lack of convergence in obrientransform.') arrays.append(t) return np.array(arrays) def sem(a, axis=0, ddof=1, nan_policy='propagate'): """ Compute standard error of the mean. 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. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- s : ndarray or float The standard error of the mean in the sample(s), along the input axis. Notes ----- The default value for ddof is different to the default (0) used by other ddof containing routines, such as np.std and np.nanstd. Examples -------- Find standard error along the first axis: >>> from scipy import stats >>> a = np.arange(20).reshape(5,4) >>> stats.sem(a) array([ 2.8284, 2.8284, 2.8284, 2.8284]) Find standard error across the whole array, using n degrees of freedom: >>> stats.sem(a, axis=None, ddof=0) 1.2893796958227628 """ a, axis = _chk_asarray(a, axis) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': a = ma.masked_invalid(a) return mstats_basic.sem(a, axis, ddof) n = a.shape[axis] s = np.std(a, axis=axis, ddof=ddof) / np.sqrt(n) return s def zscore(a, axis=0, ddof=0, nan_policy='propagate'): """ Compute the z score. Compute 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. 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 ------- 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) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': mns = np.nanmean(a=a, axis=axis, keepdims=True) sstd = np.nanstd(a=a, axis=axis, ddof=ddof, keepdims=True) else: mns = a.mean(axis=axis, keepdims=True) sstd = a.std(axis=axis, ddof=ddof, keepdims=True) return (a - mns) / sstd def zmap(scores, compare, axis=0, ddof=0): """ Calculate the relative z-scores. Return an array of z-scores, i.e., scores that are standardized to zero mean and unit variance, where mean and variance are calculated from the comparison array. Parameters ---------- scores : array_like The input for which z-scores are calculated. compare : array_like The input from which the mean and standard deviation of the normalization are taken; assumed to have the same dimension as scores. axis : int or None, optional Axis over which mean and variance of compare are calculated. Default is 0. If None, compute over the whole array scores. ddof : int, optional Degrees of freedom correction in the calculation of the standard deviation. Default is 0. Returns ------- zscore : array_like Z-scores, in the same shape as scores. Notes ----- This function preserves ndarray subclasses, and works also with matrices and masked arrays (it uses asanyarray instead of asarray for parameters). Examples -------- >>> from scipy.stats import zmap >>> a = [0.5, 2.0, 2.5, 3] >>> b = [0, 1, 2, 3, 4] >>> zmap(a, b) array([-1.06066017, 0. , 0.35355339, 0.70710678]) """ scores, compare = map(np.asanyarray, [scores, compare]) mns = compare.mean(axis=axis, keepdims=True) sstd = compare.std(axis=axis, ddof=ddof, keepdims=True) return (scores - mns) / sstd def gstd(a, axis=0, ddof=1): """ Calculate the geometric standard deviation of an array. The geometric standard deviation describes the spread of a set of numbers where the geometric mean is preferred. It is a multiplicative factor, and so a dimensionless quantity. It is defined as the exponent of the standard deviation of log(a). Mathematically the population geometric standard deviation can be evaluated as:: gstd = exp(std(log(a))) .. versionadded:: 1.3.0 Parameters ---------- a : array_like An array like object containing the sample data. axis : int, tuple or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array a. ddof : int, optional Degree of freedom correction in the calculation of the geometric standard deviation. Default is 1. Returns ------- ndarray or float An array of the geometric standard deviation. If axis is None or a is a 1d array a float is returned. Notes ----- As the calculation requires the use of logarithms the geometric standard deviation only supports strictly positive values. Any non-positive or infinite values will raise a ValueError. The geometric standard deviation is sometimes confused with the exponent of the standard deviation, exp(std(a)). Instead the geometric standard deviation is exp(std(log(a))). The default value for ddof is different to the default value (0) used by other ddof containing functions, such as np.std and np.nanstd. Examples -------- Find the geometric standard deviation of a log-normally distributed sample. Note that the standard deviation of the distribution is one, on a log scale this evaluates to approximately exp(1). >>> from scipy.stats import gstd >>> np.random.seed(123) >>> sample = np.random.lognormal(mean=0, sigma=1, size=1000) >>> gstd(sample) 2.7217860664589946 Compute the geometric standard deviation of a multidimensional array and of a given axis. >>> a = np.arange(1, 25).reshape(2, 3, 4) >>> gstd(a, axis=None) 2.2944076136018947 >>> gstd(a, axis=2) array([[1.82424757, 1.22436866, 1.13183117], [1.09348306, 1.07244798, 1.05914985]]) >>> gstd(a, axis=(1,2)) array([2.12939215, 1.22120169]) The geometric standard deviation further handles masked arrays. >>> a = np.arange(1, 25).reshape(2, 3, 4) >>> ma = np.ma.masked_where(a > 16, a) >>> ma masked_array( data=[[[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]], [[13, 14, 15, 16], [--, --, --, --], [--, --, --, --]]], mask=[[[False, False, False, False], [False, False, False, False], [False, False, False, False]], [[False, False, False, False], [ True, True, True, True], [ True, True, True, True]]], fill_value=999999) >>> gstd(ma, axis=2) masked_array( data=[[1.8242475707663655, 1.2243686572447428, 1.1318311657788478], [1.0934830582350938, --, --]], mask=[[False, False, False], [False, True, True]], fill_value=999999) """ a = np.asanyarray(a) log = ma.log if isinstance(a, ma.MaskedArray) else np.log try: with warnings.catch_warnings(): warnings.simplefilter("error", RuntimeWarning) return np.exp(np.std(log(a), axis=axis, ddof=ddof)) except RuntimeWarning as w: if np.isinf(a).any(): raise ValueError( 'Infinite value encountered. The geometric standard deviation ' 'is defined for strictly positive values only.') a_nan = np.isnan(a) a_nan_any = a_nan.any() # exclude NaN's from negativity check, but # avoid expensive masking for arrays with no NaN if ((a_nan_any and np.less_equal(np.nanmin(a), 0)) or (not a_nan_any and np.less_equal(a, 0).any())): raise ValueError( 'Non positive value encountered. The geometric standard ' 'deviation is defined for strictly positive values only.') elif 'Degrees of freedom <= 0 for slice' == str(w): raise ValueError(w) else: # Remaining warnings don't need to be exceptions. return np.exp(np.std(log(a, where=~a_nan), axis=axis, ddof=ddof)) except TypeError: raise ValueError( 'Invalid array input. The inputs could not be ' 'safely coerced to any supported types') # Private dictionary initialized only once at module level # See https://en.wikipedia.org/wiki/Robust_measures_of_scale _scale_conversions = {'raw': 1.0, 'normal': special.erfinv(0.5) * 2.0 * math.sqrt(2.0)} def iqr(x, axis=None, rng=(25, 75), scale='raw', nan_policy='propagate', interpolation='linear', keepdims=False): r""" 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. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values 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. The following options are available (default is 'linear'): * '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. 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 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 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.]]) """ 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") if np.isnan(rng).any(): raise ValueError("range must not contain NaNs") rng = sorted(rng) pct = percentile_func(x, rng, axis=axis, interpolation=interpolation, keepdims=keepdims, contains_nan=contains_nan) out = np.subtract(pct[1], pct[0]) if scale != 1.0: out /= scale return out def median_absolute_deviation(x, axis=0, center=np.median, scale=1.4826, nan_policy='propagate'): """ Compute the median absolute deviation of the data along the given axis. The median absolute deviation (MAD, [1]_) computes the median over the absolute deviations from the median. It is a measure of dispersion similar to the standard deviation but more robust to outliers [2]_. The MAD of an empty array is np.nan. .. versionadded:: 1.3.0 Parameters ---------- x : array_like Input array or object that can be converted to an array. axis : int or None, optional Axis along which the range is computed. Default is 0. If None, compute the MAD over the entire array. center : callable, optional A function that will return the central value. The default is to use np.median. Any user defined function used will need to have the function signature func(arr, axis). scale : int, optional The scaling factor applied to the MAD. The default scale (1.4826) ensures consistency with the standard deviation for normally distributed data. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- mad : 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, numpy.median, scipy.stats.iqr, scipy.stats.tmean, scipy.stats.tstd, scipy.stats.tvar Notes ----- The center argument only affects the calculation of the central value around which the MAD is calculated. That is, passing in center=np.mean will calculate the MAD around the mean - it will not calculate the *mean* absolute deviation. References ---------- .. [1] "Median absolute deviation" https://en.wikipedia.org/wiki/Median_absolute_deviation .. [2] "Robust measures of scale" https://en.wikipedia.org/wiki/Robust_measures_of_scale Examples -------- When comparing the behavior of median_absolute_deviation with np.std, the latter is affected when we change a single value of an array to have an outlier value while the MAD hardly changes: >>> from scipy import stats >>> x = stats.norm.rvs(size=100, scale=1, random_state=123456) >>> x.std() 0.9973906394005013 >>> stats.median_absolute_deviation(x) 1.2280762773108278 >>> x[0] = 345.6 >>> x.std() 34.42304872314415 >>> stats.median_absolute_deviation(x) 1.2340335571164334 Axis handling example: >>> x = np.array([[10, 7, 4], [3, 2, 1]]) >>> x array([[10, 7, 4], [ 3, 2, 1]]) >>> stats.median_absolute_deviation(x) array([5.1891, 3.7065, 2.2239]) >>> stats.median_absolute_deviation(x, axis=None) 2.9652 """ x = asarray(x) # Consistent with np.var and np.std. if not x.size: return np.nan contains_nan, nan_policy = _contains_nan(x, nan_policy) if contains_nan and nan_policy == 'propagate': return np.nan if contains_nan and nan_policy == 'omit': # Way faster than carrying the masks around arr = ma.masked_invalid(x).compressed() else: arr = x if axis is None: med = center(arr) mad = np.median(np.abs(arr - med)) else: med = np.apply_over_axes(center, arr, axis) mad = np.median(np.abs(arr - med), axis=axis) return scale * mad def _iqr_percentile(x, q, axis=None, interpolation='linear', keepdims=False, contains_nan=False): """ Private wrapper that works around older versions of numpy. While this function is pretty much necessary for the moment, it should be removed as soon as the minimum supported numpy version allows. """ if contains_nan and NumpyVersion(np.__version__) < '1.10.0a': # I see no way to avoid the version check to ensure that the corrected # NaN behavior has been implemented except to call percentile on a # small array. msg = "Keyword nan_policy='propagate' not correctly supported for " \ "numpy versions < 1.10.x. The default behavior of " \ "numpy.percentile will be used." warnings.warn(msg, RuntimeWarning) try: # For older versions of numpy, there are two things that can cause a # problem here: missing keywords and non-scalar axis. The former can be # partially handled with a warning, the latter can be handled fully by # hacking in an implementation similar to numpy's function for # providing multi-axis functionality # (numpy.lib.function_base._ureduce for the curious). result = np.percentile(x, q, axis=axis, keepdims=keepdims, interpolation=interpolation) except TypeError: if interpolation != 'linear' or keepdims: # At time or writing, this means np.__version__ < 1.9.0 warnings.warn("Keywords interpolation and keepdims not supported " "for your version of numpy", RuntimeWarning) try: # Special processing if axis is an iterable original_size = len(axis) except TypeError: # Axis is a scalar at this point pass else: axis = np.unique(np.asarray(axis) % x.ndim) if original_size > axis.size: # mimic numpy if axes are duplicated raise ValueError("duplicate value in axis") if axis.size == x.ndim: # axis includes all axes: revert to None axis = None elif axis.size == 1: # no rolling necessary axis = axis[0] else: # roll multiple axes to the end and flatten that part out for ax in axis[::-1]: x = np.rollaxis(x, ax, x.ndim) x = x.reshape(x.shape[:-axis.size] + (np.prod(x.shape[-axis.size:]),)) axis = -1 result = np.percentile(x, q, axis=axis) return result def _iqr_nanpercentile(x, q, axis=None, interpolation='linear', keepdims=False, contains_nan=False): """ Private wrapper that works around the following: 1. A bug in np.nanpercentile that was around until numpy version 1.11.0. 2. A bug in np.percentile NaN handling that was fixed in numpy version 1.10.0. 3. The non-existence of np.nanpercentile before numpy version 1.9.0. While this function is pretty much necessary for the moment, it should be removed as soon as the minimum supported numpy version allows. """ if hasattr(np, 'nanpercentile'): # At time or writing, this means np.__version__ < 1.9.0 result = np.nanpercentile(x, q, axis=axis, interpolation=interpolation, keepdims=keepdims) # If non-scalar result and nanpercentile does not do proper axis roll. # I see no way of avoiding the version test since dimensions may just # happen to match in the data. if result.ndim > 1 and NumpyVersion(np.__version__) < '1.11.0a': axis = np.asarray(axis) if axis.size == 1: # If only one axis specified, reduction happens along that dimension if axis.ndim == 0: axis = axis[None] result = np.rollaxis(result, axis[0]) else: # If multiple axes, reduced dimeision is last result = np.rollaxis(result, -1) else: msg = "Keyword nan_policy='omit' not correctly supported for numpy " \ "versions < 1.9.x. The default behavior of numpy.percentile " \ "will be used." warnings.warn(msg, RuntimeWarning) result = _iqr_percentile(x, q, axis=axis) return result ##################################### # TRIMMING FUNCTIONS # ##################################### SigmaclipResult = namedtuple('SigmaclipResult', ('clipped', 'lower', 'upper')) def sigmaclip(a, low=4., high=4.): """ Perform 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.