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MeteoInfo/meteoinfo-lab/pylib/mipylib/numeric/stats/stats.py
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# coding=utf-8
#-----------------------------------------------------
# Author: Yaqiang Wang
# Date: 2017-3-9
# Purpose: MeteoInfoLab stats module
# Note: Jython
#-----------------------------------------------------
from org.meteoinfo.math.stats import StatsUtil
from org.meteoinfo.ndarray.math import ArrayMath
from org.meteoinfo.ndarray import Array
from .. import core as np
from collections import namedtuple
import warnings
__all__ = [
'chi2_contingency','chisquare','covariance','cov','pearsonr','spearmanr','kendalltau','kurtosis',
'linregress','mlinregress','percentile','skew','ttest_1samp', 'ttest_ind','ttest_rel','taylor_stats'
]
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()
contains_nan = a.contains_nan()
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
# Moment with optional pre-computed mean, equal to a.mean(axis, keepdims=True)
def _moment(a, moment, axis, mean=None):
if moment == 0 or moment == 1:
# By definition the zeroth moment about the mean is 1, and the first
# moment is 0.
shape = list(a.shape)
del shape[axis]
dtype = a.dtype
if len(shape) == 0:
return dtype(1.0 if moment == 0 else 0.0)
else:
return (np.ones(shape, dtype=dtype) if moment == 0
else np.zeros(shape, dtype=dtype))
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
mean = a.mean(axis, keepdims=True) if mean is None else mean
a_zero_mean = a - mean
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 skew(a, axis=0, bias=True):
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.
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 numeric.stats import skew
>>> skew([1, 2, 3, 4, 5])
0.0
>>> skew([2, 8, 0, 4, 1, 9, 9, 0])
0.2650554122698573
"""
a = np.asanyarray(a)
n = a.shape[axis]
mean = a.mean(axis, keepdims=True)
m2 = _moment(a, 2, axis, mean=mean)
m3 = _moment(a, 3, axis, mean=mean)
g1 = m3 / m2 ** 1.5
if not bias:
g1 = np.sqrt((n - 1.0) * n) / (n - 2.0) * g1
return g1
def kurtosis(a, axis=0, fisher=True, bias=True):
"""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.
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 mipylib.numeric.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 = np.asanyarray(a)
n = a.shape[axis]
mean = a.mean(axis, keepdims=True)
m2 = _moment(a, 2, axis, mean=mean)
m4 = _moment(a, 4, axis, mean=mean)
g1 = m4 / m2 ** 2.0
if not bias:
g1 = 1.0/(n-2)/(n-3) * ((n**2-1.0)*m4/m2**2.0 - 3*(n-1)**2.0)
return g1 - 3 if fisher else g1
def covariance(x, y, bias=False):
"""
Calculate covariance of two array.
:param x: (*array_like*) A 1-D array containing multiple variables and observations.
:param y: (*array_like*) An additional set of variables and observations. y has the same form as
that of x.
:param bias: (*boolean*) Default normalization (False) is by (N - 1), where N is the number of observations
given (unbiased estimate). If bias is True, then normalization is by N.
returns: Covariance
"""
if isinstance(x, (list, tuple)):
x = np.array(x)
if isinstance(y, (list, tuple)):
y = np.array(y)
r = StatsUtil.covariance(x.asarray(), y.asarray(), bias)
return r
def cov(m, y=None, rowvar=True, bias=False):
"""
Estimate a covariance matrix.
:param m: (*array_like*) A 1-D or 2-D array containing multiple variables and observations.
:param y: (*array_like*) Optional. An additional set of variables and observations. y has the same form as
that of m.
:param rowvar: (*boolean*) If ``rowvar`` is True (default), then each row represents a variable, with
observations in the columns. Otherwise, the relationship is transposed: each column represents a
variable, while the rows contain observations.
:param bias: (*boolean*) Default normalization (False) is by (N - 1), where N is the number of observations
given (unbiased estimate). If bias is True, then normalization is by N.
:returns: Covariance.
"""
if isinstance(m, list):
m = np.array(m)
if rowvar == True and m.ndim == 2:
m = m.T
if y is None:
r = StatsUtil.cov(m.asarray(), not bias)
if isinstance(r, Array):
return np.array(r)
else:
return r
else:
if isinstance(y, list):
y = np.array(y)
if rowvar == True and y.ndim == 2:
y = y.T
r = StatsUtil.cov(m.asarray(), y.asarray(), not bias)
return np.array(r)
def pearsonr(x, y, axis=None):
"""
Calculates a Pearson correlation coefficient and the p-value for testing non-correlation.
The Pearson correlation coefficient measures the linear relationship between two datasets.
Strictly speaking, Pearson’s correlation requires that each dataset be normally distributed,
and not necessarily zero-mean. Like other correlation coefficients, this one varies between
-1 and +1 with 0 implying no correlation. Correlations of -1 or +1 imply an exact linear
relationship. Positive correlations imply that as x increases, so does y. Negative
correlations imply that as x increases, y decreases.
The p-value roughly indicates the probability of an uncorrelated system producing datasets
that have a Pearson correlation at least as extreme as the one computed from these datasets.
The p-values are not entirely reliable but are probably reasonable for datasets larger than
500 or so.
:param x: (*array_like*) x data array.
:param y: (*array_like*) y data array.
:param axis: (*int*) By default, the index is into the flattened array, otherwise
along the specified axis.
:returns: Pearson’s correlation coefficient and 2-tailed p-value.
"""
if isinstance(x, list):
x = np.array(x)
if isinstance(y, list):
y = np.array(y)
if axis is None:
r = StatsUtil.pearsonr(x.asarray(), y.asarray())
return r[0], r[1]
else:
r = StatsUtil.pearsonr(x.asarray(), y.asarray(), axis)
return np.array(r[0]), np.array(r[1])
KendalltauResult = namedtuple('KendalltauResult', ('correlation', 'pvalue'))
def kendalltau(x, y, nan_policy='propagate', method='auto', variant='b'):
"""
Calculates Kendall's tau, a correlation measure for ordinal data.
Kendall's tau is a measure of the correspondence between two rankings.
Values close to 1 indicate strong agreement, values close to -1 indicate
strong disagreement. This is the 1945 "tau-b" version of Kendall's
tau [2]_, which can account for ties and which reduces to the 1938 "tau-a"
version [1]_ in absence of ties.
:param x: (*array_like*) x data array.
:param y: (*array_like*) y data array.
:param non_policy: (*str*) {'auto', 'asymptotic', 'exact'}, optional
Defines which method is used to calculate the p-value [5]_.
The following options are available (default is 'auto'):
* 'auto': selects the appropriate method based on a trade-off
between speed and accuracy
* 'asymptotic': uses a normal approximation valid for large samples
* 'exact': computes the exact p-value, but can only be used if no ties
are present. As the sample size increases, the 'exact' computation
time may grow and the result may lose some precision.
:param variant: (*str*) {'b', 'c'}, optional
Defines which variant of Kendall's tau is returned. Default is 'b'.
:returns: Correlation.
Notes
-----
The definition of Kendall's tau that is used is [2]_::
tau = (P - Q) / sqrt((P + Q + T) * (P + Q + U))
where P is the number of concordant pairs, Q the number of discordant
pairs, T the number of ties only in `x`, and U the number of ties only in
`y`. If a tie occurs for the same pair in both `x` and `y`, it is not
added to either T or U.
References
----------
.. [1] Maurice G. Kendall, "A New Measure of Rank Correlation", Biometrika
Vol. 30, No. 1/2, pp. 81-93, 1938.
.. [2] Maurice G. Kendall, "The treatment of ties in ranking problems",
Biometrika Vol. 33, No. 3, pp. 239-251. 1945.
.. [3] Gottfried E. Noether, "Elements of Nonparametric Statistics", John
Wiley & Sons, 1967.
.. [4] Peter M. Fenwick, "A new data structure for cumulative frequency
tables", Software: Practice and Experience, Vol. 24, No. 3,
pp. 327-336, 1994.
.. [5] Maurice G. Kendall, "Rank Correlation Methods" (4th Edition),
Charles Griffin & Co., 1970.
"""
x = np.asarray(x).ravel()
y = np.asarray(y).ravel()
# if x.size != y.size:
# raise ValueError("All inputs to `kendalltau` must be of the same "
# f"size, found x-size {x.size} and y-size {y.size}")
# elif not x.size or not y.size:
# # Return NaN if arrays are empty
# return KendalltauResult(np.nan, np.nan)
#
# # check both x and y
# cnx, npx = _contains_nan(x, nan_policy)
# cny, npy = _contains_nan(y, nan_policy)
# contains_nan = cnx or cny
# if npx == 'omit' or npy == 'omit':
# nan_policy = 'omit'
# elif contains_nan and nan_policy == 'omit':
# x = ma.masked_invalid(x)
# y = ma.masked_invalid(y)
# if variant == 'b':
# return mstats_basic.kendalltau(x, y, method=method, use_ties=True)
# else:
# raise ValueError("Only variant 'b' is supported for masked arrays")
#
# if isinstance(x, list):
# x = np.array(x)
# if isinstance(y, list):
# y = np.array(y)
r = StatsUtil.kendalltau(x.asarray(), y.asarray())
return r[0], r[1]
def spearmanr(m, y=None, axis=0):
"""
Calculates a Spearman rank-order correlation coefficient.
The Spearman correlation is a nonparametric measure of the monotonicity of the relationship
between two datasets. Unlike the Pearson correlation, the Spearman correlation does not
assume that both datasets are normally distributed. Like other correlation coefficients,
this one varies between -1 and +1 with 0 implying no correlation. Correlations of -1 or +1
imply an exact monotonic relationship. Positive correlations imply that as x increases, so
does y. Negative correlations imply that as x increases, y decreases.
:param m: (*array_like*) A 1-D or 2-D array containing multiple variables and observations.
:param y: (*array_like*) Optional. An additional set of variables and observations. y has the same form as
that of m.
:param axis: (*int*) If axis=0 (default), then each column represents a variable, with
observations in the rows. If axis=1, the relationship is transposed: each row represents
a variable, while the columns contain observations..
:returns: Spearman correlation and pvalue.
"""
if isinstance(m, list):
m = np.array(m)
if axis == 1 and m.ndim == 2:
m = m.T
if y is None:
r = StatsUtil.spearmanr(m.asarray())
if isinstance(r, Array):
return np.array(r)
else:
return r
else:
if isinstance(y, list):
y = np.array(y)
if axis == 1 and y.ndim == 2:
y = y.T
r = StatsUtil.spearmanr(m.asarray(), y.asarray())
return r[0], r[1]
def linregress(x, y, outvdn=False):
"""
Calculate a linear least-squares regression for two sets of measurements.
:param x, y: (*array_like*) Two sets of measurements. Both arrays should have the same length.
:param outvdn: (*boolean*) Output validate data number or not. Default is False.
:returns: Result slope, intercept, relative coefficient, two-sided p-value for a hypothesis test
whose null hypothesis is that the slope is zero, standard error of the estimated gradient,
validate data number (remove NaN values).
"""
if isinstance(x, list):
x = np.array(x)
if isinstance(y, list):
y = np.array(y)
r = ArrayMath.lineRegress(x.asarray(), y.asarray())
if outvdn:
return r[0], r[1], r[2], r[3], r[4], r[5]
else:
return r[0], r[1], r[2], r[3], r[4]
def mlinregress(y, x):
"""
Implements ordinary least squares (OLS) to estimate the parameters of a multiple linear
regression model.
:param y: (*array_like*) Y sample data - one dimension array.
:param x: (*array_like*) X sample data - two dimension array.
:returns: Estimated regression parameters and residuals.
"""
if isinstance(x, list):
x = np.array(x)
if isinstance(y, list):
y = np.array(y)
r = StatsUtil.multipleLineRegress_OLS(y.asarray(), x.asarray())
return np.array(r[0]), np.array(r[1])
def percentile(a, q, axis=None):
"""
Compute the qth percentile of the data along the specified axis.
:param a: (*array_like*) Input array.
:param q: (*float*) float in range of [0,100].
Percentile to compute, which must be between 0 and 100 inclusive.
:param axis: (*int*) Axis or axes along which the percentiles are computed. The default is
to compute the percentile along a flattened version of the array.
:returns: (*float*) qth percentile value.
"""
if isinstance(a, list):
a = np.array(x)
if axis is None:
r = StatsUtil.percentile(a.asarray(), q)
else:
r = StatsUtil.percentile(a.asarray(), q, axis)
r = np.array(r)
return r
def ttest_1samp(a, popmean):
"""
Calculate the T-test for the mean of ONE group of scores.
This is a two-sided test for the null hypothesis that the expected value (mean) of
a sample of independent observations a is equal to the given population mean, popmean.
:param a: (*array_like*) Sample observation.
:param popmean: (*float*) Expected value in null hypothesis.
:returns: t-statistic and p-value
"""
if isinstance(a, list):
a = np.array(x)
r = StatsUtil.tTest(a.asarray(), popmean)
return r[0], r[1]
def ttest_rel(a, b):
"""
Calculates the T-test on TWO RELATED samples of scores, a and b.
This is a two-sided test for the null hypothesis that 2 related or repeated samples
have identical average (expected) values.
:param a: (*array_like*) Sample data a.
:param b: (*array_like*) Sample data b.
:returns: t-statistic and p-value
"""
if isinstance(a, list):
a = np.array(a)
if isinstance(b, list):
b = np.array(b)
r = StatsUtil.pairedTTest(a.asarray(), b.asarray())
return r[0], r[1]
def ttest_ind(a, b):
"""
Calculates the T-test for the means of TWO INDEPENDENT samples of scores.
This is a two-sided test for the null hypothesis that 2 independent samples have
identical average (expected) values. This test assumes that the populations have
identical variances.
:param a: (*array_like*) Sample data a.
:param b: (*array_like*) Sample data b.
:returns: t-statistic and p-value
"""
if isinstance(a, list):
a = np.array(a)
if isinstance(b, list):
b = np.array(b)
r = StatsUtil.tTest(a.asarray(), b.asarray())
return r[0], r[1]
def chisquare(f_obs, f_exp=None):
"""
Calculates a one-way chi square test.
The chi square test tests the null hypothesis that the categorical data has the
given frequencies.
:param f_obs: (*array_like*) Observed frequencies in each category.
:param f_exp: (*array_like*) Expected frequencies in each category. By default the categories
are assumed to be equally likely.
:returns: Chi-square statistic and p-value
"""
if isinstance(f_obs, list):
f_obs = np.array(f_obs)
if f_exp is None:
n = len(f_obs)
f_exp = np.ones(n) / n * f_obs.sum()
elif isinstance(f_exp, list):
f_exp = np.array(f_exp)
r = StatsUtil.chiSquareTest(f_exp.asarray(), f_obs.asarray())
return r[0], r[1]
def chi2_contingency(observed):
"""
Chi-square test of independence of variables in a contingency table.
This function computes the chi-square statistic and p-value for the hypothesis test of
independence of the observed frequencies in the contingency table observed.
:param observed: (*array_like*) The contingency table. The table contains the observed
frequencies (i.e. number of occurrences) in each category. In the two-dimensional case,
the table is often described as an `R x C table`.
:returns: Chi-square statistic and p-value
"""
if isinstance(observed, list):
observed = np.array(observed)
r = StatsUtil.chiSquareTest(observed.asarray())
return r[0], r[1]
def taylor_stats(p, r):
"""
Calculates the statistics needed to create a Taylor diagram as
described in Taylor (2001) using the data provided in the predicted
field (PREDICTED) and the reference field (REFERENCE).
The statistics are returned in the STATS dictionary.
If a dictionary is provided for PREDICTED or REFERENCE, then
the name of the field must be supplied in FIELD.
The function currently supports dictionaries, lists, and np.ndarray,
types for the PREDICTED and REFERENCE variables.
Input:
p : predicted field
r : reference field
NORM : logical flag specifying statistics are to be normalized
with respect to standard deviation of reference field
= True, statistics are normalized
= False, statistics are not normalized
Output:
STATS : dictionary containing statistics
STATS['ccoef'] : correlation coefficients (R)
STATS['crmsd'] : centered root-mean-square (RMS) differences (E')
STATS['sdev'] : standard deviations
Each of these outputs are one-dimensional with the same length.
First index corresponds to the reference series for the diagram.
For example SDEV[1] is the standard deviation of the reference
series (sigma_r) and SDEV[2:N] are the standard deviations of the
other (predicted) series.
Reference:
Taylor, K. E. (2001), Summarizing multiple aspects of model
performance in a single diagram, J. Geophys. Res., 106(D7),
7183-7192, doi:10.1029/2000JD900719.
Author: Peter A. Rochford
Symplectic, LLC
www.thesymplectic.com
prochford@thesymplectic.com
Created on Dec 3, 2016
"""
# Check that dimensions of predicted and reference fields match
pdims= p.shape
rdims= r.shape
if pdims != rdims:
message = 'predicted and reference field dimensions do not' + \
' match.\n' + \
'shape(predicted)= ' + str(pdims) + ', ' + \
'shape(reference)= ' + str(rdims) + \
'\npredicted type: ' + str(type(p))
raise ValueError(message)
# Calculate correlation coefficient
ccoef = np.corrcoef(p, r)
ccoef = ccoef[0]
# Calculate centered root-mean-square (RMS) difference (E')^2
# Calculate means
pmean = np.mean(p)
rmean = np.mean(r)
# Calculate (E')^2
crmsd = np.square((p - pmean) - (r - rmean))
crmsd = np.sum(crmsd)/p.size
crmsd = np.sqrt(crmsd)
crmsd = [0.0, crmsd]
# Calculate standard deviation of predicted field w.r.t N (sigma_p)
sdevp = np.std(p)
# Calculate standard deviation of reference field w.r.t N (sigma_r)
sdevr = np.std(r)
sdev = [sdevr, sdevp]
# Store statistics in a dictionary
stats = {'ccoef': ccoef, 'crmsd': crmsd, 'sdev': sdev}
return stats
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