Source code for dask.array.stats

"""
Statistical functions and tests, following scipy.stats.

Some differences

- We don't handle missing values at all

"""
# This is lightly adapted from scipy.stats 0.19
# https://github.com/scipy/scipy/blob/v0.19.0/scipy/stats/stats.py
# The original copyright notice follows:

# 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.
import math

import numpy as np

import dask.array as da
from dask import delayed
from dask.array.ufunc import wrap_elemwise
from dask.utils import derived_from

try:
    import scipy.stats
except ImportError as e:
    raise ImportError("`dask.array.stats` requires `scipy` to be installed.") from e
from scipy import special
from scipy.stats import distributions
from scipy.stats.stats import (
    F_onewayResult,
    KurtosistestResult,
    NormaltestResult,
    Power_divergenceResult,
    SkewtestResult,
    Ttest_1sampResult,
    Ttest_indResult,
    Ttest_relResult,
)

__all__ = [
    "ttest_ind",
    "ttest_1samp",
    "ttest_rel",
    "chisquare",
    "power_divergence",
    "skew",
    "skewtest",
    "kurtosis",
    "kurtosistest",
    "normaltest",
    "f_oneway",
    "moment",
]

# -----------------
# Statistical Tests
# -----------------


[docs]@derived_from(scipy.stats) def ttest_ind(a, b, axis=0, equal_var=True): v1 = da.var(a, axis, ddof=1) # XXX: np -> da v2 = da.var(b, axis, ddof=1) # XXX: np -> da 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(da.mean(a, axis), da.mean(b, axis), denom, df) return delayed(Ttest_indResult, nout=2)(*res)
[docs]@derived_from(scipy.stats) def ttest_1samp(a, popmean, axis=0, nan_policy="propagate"): if nan_policy != "propagate": raise NotImplementedError( "`nan_policy` other than 'propagate' have not been implemented." ) n = a.shape[axis] df = n - 1 d = da.mean(a, axis) - popmean v = da.var(a, axis, ddof=1) denom = da.sqrt(v / float(n)) with np.errstate(divide="ignore", invalid="ignore"): t = da.divide(d, denom) t, prob = _ttest_finish(df, t) return delayed(Ttest_1sampResult, nout=2)(t, prob)
[docs]@derived_from(scipy.stats) def ttest_rel(a, b, axis=0, nan_policy="propagate"): if nan_policy != "propagate": raise NotImplementedError( "`nan_policy` other than 'propagate' have not been implemented." ) n = a.shape[axis] df = float(n - 1) d = (a - b).astype(np.float64) v = da.var(d, axis, ddof=1) dm = da.mean(d, axis) denom = da.sqrt(v / float(n)) with np.errstate(divide="ignore", invalid="ignore"): t = da.divide(dm, denom) t, prob = _ttest_finish(df, t) return delayed(Ttest_relResult, nout=2)(t, prob)
[docs]@derived_from(scipy.stats) def chisquare(f_obs, f_exp=None, ddof=0, axis=0): return power_divergence(f_obs, f_exp=f_exp, ddof=ddof, axis=axis, lambda_="pearson")
[docs]@derived_from(scipy.stats) def power_divergence(f_obs, f_exp=None, ddof=0, axis=0, lambda_=None): if isinstance(lambda_, str): # TODO: public api if lambda_ not in scipy.stats.stats._power_div_lambda_names: names = repr(list(scipy.stats.stats._power_div_lambda_names.keys()))[1:-1] raise ValueError( f"invalid string for lambda_: {lambda_!r}. " f"Valid strings are {names}" ) lambda_ = scipy.stats.stats._power_div_lambda_names[lambda_] elif lambda_ is None: lambda_ = 1 if f_exp is not None: # f_exp = np.atleast_1d(np.asanyarray(f_exp)) pass else: f_exp = f_obs.mean(axis=axis, keepdims=True) # `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 * _xlogy(f_obs, f_obs / f_exp) elif lambda_ == -1: # Modified log-likelihood ratio terms = 2.0 * _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 = delayed(distributions.chi2.sf)(stat, num_obs - 1 - ddof) return delayed(Power_divergenceResult, nout=2)(stat, p)
[docs]@derived_from(scipy.stats) def skew(a, axis=0, bias=True, nan_policy="propagate"): if nan_policy != "propagate": raise NotImplementedError( "`nan_policy` other than 'propagate' have not been implemented." ) n = a.shape[axis] # noqa; for bias m2 = moment(a, 2, axis) m3 = moment(a, 3, axis) zero = m2 == 0 vals = da.where(~zero, m3 / m2 ** 1.5, 0.0) # vals = da.where(~zero, (m2, m3), # lambda m2, m3: m3 / m2**1.5, # 0.) if not bias: # Need a version of np.place raise NotImplementedError("bias=False is not implemented.") if vals.ndim == 0: # TODO: scalar, min is a workaround return vals.min() return vals
[docs]@derived_from(scipy.stats) def skewtest(a, axis=0, nan_policy="propagate"): if nan_policy != "propagate": raise NotImplementedError( "`nan_policy` other than 'propagate' have not been implemented." ) 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 delayed(SkewtestResult, nout=2)(Z, 2 * distributions.norm.sf(np.abs(Z)))
[docs]@derived_from(scipy.stats) def kurtosis(a, axis=0, fisher=True, bias=True, nan_policy="propagate"): if nan_policy != "propagate": raise NotImplementedError( "`nan_policy` other than 'propagate' have not been implemented." ) n = a.shape[axis] # noqa; for bias m2 = moment(a, 2, axis) m4 = moment(a, 4, axis) zero = m2 == 0 olderr = np.seterr(all="ignore") try: vals = da.where(zero, 0, m4 / m2 ** 2.0) finally: np.seterr(**olderr) if not bias: # need a version of np.place raise NotImplementedError("bias=False is not implemented.") if fisher: return vals - 3 else: if vals.ndim == 0: # TODO: scalar, min is a workaround return vals.min() return vals
[docs]@derived_from(scipy.stats) def kurtosistest(a, axis=0, nan_policy="propagate"): if nan_policy != "propagate": raise NotImplementedError( "`nan_policy` other than 'propagate' have not been implemented." ) n = float(a.shape[axis]) b2 = kurtosis(a, axis, fisher=False) E = 3.0 * (n - 1) / (n + 1) varb2 = ( 24.0 * n * (n - 2) * (n - 3) / ((n + 1) * (n + 1.0) * (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 delayed(KurtosistestResult, nout=2)(Z, 2 * distributions.norm.sf(np.abs(Z)))
[docs]@derived_from(scipy.stats) def normaltest(a, axis=0, nan_policy="propagate"): if nan_policy != "propagate": raise NotImplementedError( "`nan_policy` other than 'propagate' have not been implemented." ) s, _ = skewtest(a, axis) k, _ = kurtosistest(a, axis) k2 = s * s + k * k return delayed(NormaltestResult, nout=2)(k2, delayed(distributions.chi2.sf)(k2, 2))
[docs]@derived_from(scipy.stats) def f_oneway(*args): # args = [np.asarray(arg, dtype=float) for arg in args] # ANOVA on N groups, each in its own array num_groups = len(args) alldata = da.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 = _fdtrc(dfbn, dfwn, f) # equivalent to stats.f.sf return delayed(F_onewayResult, nout=2)(f, prob)
[docs]@derived_from(scipy.stats) def moment(a, moment=1, axis=0, nan_policy="propagate"): if nan_policy != "propagate": raise NotImplementedError( "`nan_policy` other than 'propagate' have not been implemented." ) return da.moment(a, moment, axis=axis)
# ------- # Helpers # ------- # Don't really want to do all of scipy.special (or do we?) _xlogy = wrap_elemwise(special.xlogy, source=special) _fdtrc = wrap_elemwise(special.fdtrc, source=special) def _equal_var_ttest_denom(v1, n1, v2, n2): df = n1 + n2 - 2.0 svar = ((n1 - 1) * v1 + (n2 - 1) * v2) / df denom = da.sqrt(svar * (1.0 / n1 + 1.0 / n2)) # XXX: np -> da return df, denom def _unequal_var_ttest_denom(v1, n1, v2, n2): vn1 = v1 / n1 vn2 = v2 / n2 with np.errstate(divide="ignore", invalid="ignore"): df = (vn1 + vn2) ** 2 / (vn1 ** 2 / (n1 - 1) + vn2 ** 2 / (n2 - 1)) # If df is undefined, variances are zero (assumes n1 > 0 & n2 > 0). # Hence it doesn't matter what df is as long as it's not NaN. df = da.where(da.isnan(df), 1, df) # XXX: np -> da denom = da.sqrt(vn1 + vn2) return df, denom def _ttest_ind_from_stats(mean1, mean2, denom, df): d = mean1 - mean2 with np.errstate(divide="ignore", invalid="ignore"): t = da.divide(d, denom) t, prob = _ttest_finish(df, t) return (t, prob) def _ttest_finish(df, t): """Common code between all 3 t-test functions.""" # XXX: np.abs -> da.absolute # XXX: delayed(distributions.t.sf) prob = ( delayed(distributions.t.sf)(da.absolute(t), df) * 2 ) # use np.abs to get upper tail if t.ndim == 0: t = t[()] return t, prob def _count(x, axis=None): if axis is None: return x.size else: return x.shape[axis] def _sum_of_squares(a, axis=0): """ Squares each element of the input array, and returns 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`). """ return da.sum(a * a, axis) def _square_of_sums(a, axis=0): """ Sums elements of the input array, and returns 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`). """ s = da.sum(a, axis) return s * s