Source code for statsmodels.stats.moment_helpers

"""helper functions conversion between moments

contains:

* conversion between central and non-central moments, skew, kurtosis and
  cummulants
* cov2corr : convert covariance matrix to correlation matrix


Author: Josef Perktold
License: BSD-3

"""
from statsmodels.compat.python import range

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 """ 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, m 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 """ X = _convert_to_multidim(mnc) def _local_counts(mnc): mean = mnc[0] mnc = [1] + list(mnc) # add zero moment = 1 mu = [] for n, m 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 non-central moments to cumulants recursive formula produces as many cumulants as 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, m 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 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): """ just chained because I have still the test case """ 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""" 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""" 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""" 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 central moments to mean, variance, skew, 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 Returns ------- corr : ndarray (subclass) correlation matrix return_std : bool If this is true then the standard deviation is also returned. By default only the correlation matrix is returned. 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))