'''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.misc import comb
## start moment helpers
[docs]def mc2mnc(mc):
'''convert central to non-central moments, uses recursive formula
optionally adjusts first moment to return mean
'''
n = len(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=1) * mc[k] * mean**(n-k)
return mnc[1:]
[docs]def mnc2mc(mnc, wmean = True):
'''convert non-central to central moments, uses recursive formula
optionally adjusts first moment to return mean
'''
n = len(mnc)
mean = mnc[0]
mnc = [1] + list(mnc) # add zero moment = 1
mu = [] #np.zeros(n+1)
for n,m in enumerate(mnc):
mu.append(0)
#[comb(n-1,k,exact=1) for k in range(n)]
for k in range(n+1):
mu[n] += (-1)**(n-k) * comb(n,k,exact=1) * mnc[k] * mean**(n-k)
if wmean:
mu[1] = mean
return mu[1:]
[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
(http://books.google.ca/books?id=gm7kwttyRT0C&pg=PA40&lpg=PA40&dq=convert+cumulants+to+moments&source=web&ots=qyIaY6oaWH&sig=cShTDWl-YrWAzV7NlcMTRQV6y0A&hl=en&sa=X&oi=book_result&resnum=1&ct=result)
'''
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=1) * kappa[n-k]*mc[k]
mc[1] = kappa0 # insert mean as first moments by convention
return mc[1:]
[docs]def mnc2cum(mnc):
'''convert non-central moments to cumulants
recursive formula produces as many cumulants as moments
http://en.wikipedia.org/wiki/Cumulant#Cumulants_and_moments
'''
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):
kappa[n] -= comb(n-1,k-1,exact=1) * kappa[k]*mnc[n-k]
return kappa[1:]
def mc2cum(mc):
'''just chained because I have still the test case
'''
return mnc2cum(mc2mnc(mc))
[docs]def mvsk2mc(args):
'''convert mean, variance, skew, kurtosis to central moments'''
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)
[docs]def mvsk2mnc(args):
'''convert mean, variance, skew, kurtosis to non-central moments'''
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)
[docs]def mc2mvsk(args):
'''convert central moments to mean, variance, skew, kurtosis
'''
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)
[docs]def mnc2mvsk(args):
'''convert central moments to mean, variance, skew, kurtosis
'''
#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))
#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))
