"""
Helper functions for conversion between moments
Contains:
* conversion between central and non-central moments, skew, kurtosis and
cumulants
* cov2corr : convert covariance matrix to correlation matrix
Author: Josef Perktold
License: BSD-3
"""
import numpy as np
from scipy.special import comb
def _convert_to_multidim(x):
if any([isinstance(x, list), isinstance(x, tuple)]):
return np.array(x)
elif isinstance(x, np.ndarray):
return x
else:
# something strange was passed and the function probably
# will fall, maybe insert an exception?
return x
def _convert_from_multidim(x, totype=list):
if len(x.shape) < 2:
return totype(x)
return x.T
[docs]
def mc2mnc(mc):
"""
Convert central to non-central moments, uses recursive formula
Optionally adjusts first moment to return mean.
Parameters
----------
mc : array_like
Central moments, with the first element equal to the mean.
Returns
-------
mnc : ndarray or list
Non-central moments.
"""
x = _convert_to_multidim(mc)
def _local_counts(mc):
mean = mc[0]
mc = [1, *list(mc)] # add zero moment = 1
mc[1] = 0 # define central mean as zero for formula
mnc = [1, mean] # zero and first raw moments
for nn, _ in enumerate(mc[2:]):
n = nn + 2
mnc.append(0)
for k in range(n + 1):
mnc[n] += comb(n, k, exact=True) * mc[k] * mean ** (n - k)
return mnc[1:]
res = np.apply_along_axis(_local_counts, 0, x)
# for backward compatibility convert 1-dim output to list/tuple
return _convert_from_multidim(res)
[docs]
def mnc2mc(mnc, wmean=True):
"""
Convert non-central to central moments, uses recursive formula
Optionally adjusts first moment to return mean.
Parameters
----------
mnc : array_like
Non-central moments, with the first element equal to the mean.
wmean : bool
If True (default), the first returned moment is the mean instead
of zero.
Returns
-------
mu : ndarray or list
Central moments.
"""
X = _convert_to_multidim(mnc)
def _local_counts(mnc):
mean = mnc[0]
mnc = [1, *list(mnc)] # add zero moment = 1
mu = []
for n, _ in enumerate(mnc):
mu.append(0)
for k in range(n + 1):
sgn_comb = (-1) ** (n - k) * comb(n, k, exact=True)
mu[n] += sgn_comb * mnc[k] * mean ** (n - k)
if wmean:
mu[1] = mean
return mu[1:]
res = np.apply_along_axis(_local_counts, 0, X)
# for backward compatibility convert 1-dim output to list/tuple
return _convert_from_multidim(res)
[docs]
def cum2mc(kappa):
"""
Convert cumulants to central moments
Recursive formula produces as many central moments as cumulants.
Parameters
----------
kappa : array_like
Cumulants.
Returns
-------
mc : ndarray or list
Central moments.
References
----------
Kenneth Lange: Numerical Analysis for Statisticians, page 40
"""
X = _convert_to_multidim(kappa)
def _local_counts(kappa):
mc = [1, 0.0] # _kappa[0]] # insert 0-moment and mean
kappa0 = kappa[0]
kappa = [1, *list(kappa)]
for nn, _ in enumerate(kappa[2:]):
n = nn + 2
mc.append(0)
for k in range(n - 1):
mc[n] += comb(n - 1, k, exact=True) * kappa[n - k] * mc[k]
mc[1] = kappa0 # insert mean as first moments by convention
return mc[1:]
res = np.apply_along_axis(_local_counts, 0, X)
# for backward compatibility convert 1-dim output to list/tuple
return _convert_from_multidim(res)
[docs]
def mnc2cum(mnc):
"""
Convert non-central moments to cumulants
Recursive formula produces as many cumulants as moments.
Parameters
----------
mnc : array_like
Non-central moments.
Returns
-------
kappa : ndarray or list
Cumulants.
References
----------
https://en.wikipedia.org/wiki/Cumulant#Cumulants_and_moments
"""
X = _convert_to_multidim(mnc)
def _local_counts(mnc):
mnc = [1, *list(mnc)]
kappa = [1]
for nn, m in enumerate(mnc[1:]):
n = nn + 1
kappa.append(m)
for k in range(1, n):
num_ways = comb(n - 1, k - 1, exact=True)
kappa[n] -= num_ways * kappa[k] * mnc[n - k]
return kappa[1:]
res = np.apply_along_axis(_local_counts, 0, X)
# for backward compatibility convert 1-dim output to list/tuple
return _convert_from_multidim(res)
def mc2cum(mc):
"""
Convert central moments to cumulants
Just chained because there is still a test case for it.
Parameters
----------
mc : array_like
Central moments.
Returns
-------
kappa : ndarray or list
Cumulants.
"""
first_step = mc2mnc(mc)
if isinstance(first_step, np.ndarray):
first_step = first_step.T
return mnc2cum(first_step)
# return np.apply_along_axis(lambda x: mnc2cum(mc2mnc(x)), 0, mc)
[docs]
def mvsk2mc(args):
"""
Convert mean, variance, skew, kurtosis to central moments
Parameters
----------
args : array_like
Sequence containing the mean, variance, skew, and kurtosis, in
that order.
Returns
-------
mc : tuple or ndarray
Central moments (mean, variance, 3rd and 4th central moments).
"""
X = _convert_to_multidim(args)
def _local_counts(args):
mu, sig2, sk, kur = args
cnt = [None] * 4
cnt[0] = mu
cnt[1] = sig2
cnt[2] = sk * sig2**1.5
cnt[3] = (kur + 3.0) * sig2**2.0
return tuple(cnt)
res = np.apply_along_axis(_local_counts, 0, X)
# for backward compatibility convert 1-dim output to list/tuple
return _convert_from_multidim(res, tuple)
[docs]
def mvsk2mnc(args):
"""
Convert mean, variance, skew, kurtosis to non-central moments
Parameters
----------
args : array_like
Sequence containing the mean, variance, skew, and kurtosis, in
that order.
Returns
-------
mnc : tuple or ndarray
Non-central moments.
"""
X = _convert_to_multidim(args)
def _local_counts(args):
mc, mc2, skew, kurt = args
mnc = mc
mnc2 = mc2 + mc * mc
mc3 = skew * (mc2**1.5) # 3rd central moment
mnc3 = mc3 + 3 * mc * mc2 + mc**3 # 3rd non-central moment
mc4 = (kurt + 3.0) * (mc2**2.0) # 4th central moment
mnc4 = mc4 + 4 * mc * mc3 + 6 * mc * mc * mc2 + mc**4
return (mnc, mnc2, mnc3, mnc4)
res = np.apply_along_axis(_local_counts, 0, X)
# for backward compatibility convert 1-dim output to list/tuple
return _convert_from_multidim(res, tuple)
[docs]
def mc2mvsk(args):
"""
Convert central moments to mean, variance, skew, kurtosis
Parameters
----------
args : array_like
Sequence containing the central moments (mean, variance, 3rd and
4th central moments), in that order.
Returns
-------
res : tuple or ndarray
Mean, variance, skew, and kurtosis.
"""
X = _convert_to_multidim(args)
def _local_counts(args):
mc, mc2, mc3, mc4 = args
skew = np.divide(mc3, mc2**1.5)
kurt = np.divide(mc4, mc2**2.0) - 3.0
return (mc, mc2, skew, kurt)
res = np.apply_along_axis(_local_counts, 0, X)
# for backward compatibility convert 1-dim output to list/tuple
return _convert_from_multidim(res, tuple)
[docs]
def mnc2mvsk(args):
"""
Convert non-central moments to mean, variance, skew, kurtosis
Parameters
----------
args : array_like
Sequence containing the non-central moments, in order.
Returns
-------
res : tuple or ndarray
Mean, variance, skew, and kurtosis.
"""
X = _convert_to_multidim(args)
def _local_counts(args):
# convert four non-central moments to central moments
mnc, mnc2, mnc3, mnc4 = args
mc = mnc
mc2 = mnc2 - mnc * mnc
mc3 = mnc3 - (3 * mc * mc2 + mc**3) # 3rd central moment
mc4 = mnc4 - (4 * mc * mc3 + 6 * mc * mc * mc2 + mc**4)
return mc2mvsk((mc, mc2, mc3, mc4))
res = np.apply_along_axis(_local_counts, 0, X)
# for backward compatibility convert 1-dim output to list/tuple
return _convert_from_multidim(res, tuple)
# def mnc2mc(args):
# """convert four non-central moments to central moments
# """
# mnc, mnc2, mnc3, mnc4 = args
# mc = mnc
# mc2 = mnc2 - mnc*mnc
# mc3 = mnc3 - (3*mc*mc2+mc**3) # 3rd central moment
# mc4 = mnc4 - (4*mc*mc3+6*mc*mc*mc2+mc**4)
# return mc, mc2, mc
# TODO: no return, did it get lost in cut-paste?
[docs]
def cov2corr(cov, return_std=False):
"""
Convert covariance matrix to correlation matrix
Parameters
----------
cov : array_like, 2d
Covariance matrix, see Notes.
return_std : bool
If this is true then the standard deviation is also returned.
By default only the correlation matrix is returned.
Returns
-------
corr : ndarray (subclass)
Correlation matrix.
std_ : ndarray
Standard deviation from the diagonal of cov, returned only if
return_std is True.
Notes
-----
This function does not convert subclasses of ndarrays. This requires that
division is defined elementwise. np.ma.array and np.matrix are allowed.
"""
cov = np.asanyarray(cov)
std_ = np.sqrt(np.diag(cov))
corr = cov / np.outer(std_, std_)
if return_std:
return corr, std_
else:
return corr
[docs]
def corr2cov(corr, std):
"""
Convert correlation matrix to covariance matrix given standard deviation
Parameters
----------
corr : array_like, 2d
Correlation matrix, see Notes.
std : array_like, 1d
Standard deviation.
Returns
-------
cov : ndarray (subclass)
Covariance matrix.
Notes
-----
This function does not convert subclasses of ndarrays. This requires
that multiplication is defined elementwise. np.ma.array are allowed, but
not matrices.
"""
corr = np.asanyarray(corr)
std_ = np.asanyarray(std)
cov = corr * np.outer(std_, std_)
return cov
[docs]
def se_cov(cov):
"""
Get standard deviation from covariance matrix
Just a shorthand function np.sqrt(np.diag(cov)).
Parameters
----------
cov : array_like, square
Covariance matrix.
Returns
-------
std : ndarray
Standard deviation from diagonal of cov.
"""
return np.sqrt(np.diag(cov))
