# -*- coding: utf-8 -*-
"""Canonical correlation analysis
author: Yichuan Liu
"""
from __future__ import division
import numpy as np
from numpy.linalg import svd
import scipy
import pandas as pd
from statsmodels.base.model import Model
from statsmodels.iolib import summary2
from .multivariate_ols import multivariate_stats
[docs]class CanCorr(Model):
"""
Canonical correlation analysis using singluar value decomposition
For matrices exog=x and endog=y, find projections x_cancoef and y_cancoef
such that:
x1 = x * x_cancoef, x1' * x1 is identity matrix
y1 = y * y_cancoef, y1' * y1 is identity matrix
and the correlation between x1 and y1 is maximized.
Attributes
-----------
endog : array
See Parameters.
exog : array
See Parameters.
cancorr : array
The canonical correlation values
y_cancoeff: array
The canonical coeefficients for endog
x_cancoeff: array
The canonical coefficients for exog
References
----------
.. [*] http://numerical.recipes/whp/notes/CanonCorrBySVD.pdf
.. [*] http://www.csun.edu/~ata20315/psy524/docs/Psy524%20Lecture%208%20CC.pdf
.. [*] http://www.mathematica-journal.com/2014/06/canonical-correlation-analysis/
"""
def __init__(self, endog, exog, tolerance=1e-8, missing='none', hasconst=None, **kwargs):
super(CanCorr, self).__init__(endog, exog, missing=missing,
hasconst=hasconst, **kwargs)
self._fit(tolerance)
def _fit(self, tolerance=1e-8):
"""Fit the model
A ValueError is raised if there are singular values smaller than the
tolerance. The treatment of singular arrays might change in future.
Parameters
----------
tolerance : float
eigenvalue tolerance, values smaller than which is considered 0
"""
nobs, k_yvar = self.endog.shape
nobs, k_xvar = self.exog.shape
k = np.min([k_yvar, k_xvar])
x = np.array(self.exog)
x = x - x.mean(0)
y = np.array(self.endog)
y = y - y.mean(0)
ux, sx, vx = svd(x, 0)
# vx_ds = vx.T divided by sx
vx_ds = vx.T
mask = sx > tolerance
if mask.sum() < len(mask):
raise ValueError('exog is collinear.')
vx_ds[:, mask] /= sx[mask]
uy, sy, vy = svd(y, 0)
# vy_ds = vy.T divided by sy
vy_ds = vy.T
mask = sy > tolerance
if mask.sum() < len(mask):
raise ValueError('endog is collinear.')
vy_ds[:, mask] /= sy[mask]
u, s, v = svd(ux.T.dot(uy), 0)
# Correct any roundoff
self.cancorr = np.array([max(0, min(s[i], 1)) for i in range(len(s))])
self.x_cancoef = vx_ds.dot(u[:, :k])
self.y_cancoef = vy_ds.dot(v.T[:, :k])
[docs] def corr_test(self):
"""Approximate F test
Perform multivariate statistical tests of the hypothesis that
there is no canonical correlation between endog and exog.
For each canonical correlation, testing its significance based on
Wilks' lambda.
Returns
-------
CanCorrTestResults instance
"""
nobs, k_yvar = self.endog.shape
nobs, k_xvar = self.exog.shape
eigenvals = np.power(self.cancorr, 2)
stats = pd.DataFrame(columns=['Canonical Correlation', "Wilks' lambda",
'Num DF','Den DF', 'F Value','Pr > F'],
index=list(range(len(eigenvals) - 1, -1, -1)))
prod = 1
for i in range(len(eigenvals) - 1, -1, -1):
prod *= 1 - eigenvals[i]
p = k_yvar - i
q = k_xvar - i
r = (nobs - k_yvar - 1) - (p - q + 1) / 2
u = (p * q - 2) / 4
df1 = p * q
if p ** 2 + q ** 2 - 5 > 0:
t = np.sqrt(((p * q) ** 2 - 4) / (p ** 2 + q ** 2 - 5))
else:
t = 1
df2 = r * t - 2 * u
lmd = np.power(prod, 1 / t)
F = (1 - lmd) / lmd * df2 / df1
stats.loc[i, 'Canonical Correlation'] = self.cancorr[i]
stats.loc[i, "Wilks' lambda"] = prod
stats.loc[i, 'Num DF'] = df1
stats.loc[i, 'Den DF'] = df2
stats.loc[i, 'F Value'] = F
pval = scipy.stats.f.sf(F, df1, df2)
stats.loc[i, 'Pr > F'] = pval
'''
# Wilk's Chi square test of each canonical correlation
df = (p - i + 1) * (q - i + 1)
chi2 = a * np.log(prod)
pval = stats.chi2.sf(chi2, df)
stats.loc[i, 'Canonical correlation'] = self.cancorr[i]
stats.loc[i, 'Chi-square'] = chi2
stats.loc[i, 'DF'] = df
stats.loc[i, 'Pr > ChiSq'] = pval
'''
ind = stats.index.values[::-1]
stats = stats.loc[ind, :]
# Multivariate tests (remember x has mean removed)
stats_mv = multivariate_stats(eigenvals,
k_yvar, k_xvar, nobs - k_xvar - 1)
return CanCorrTestResults(stats, stats_mv)
class CanCorrTestResults(object):
"""
Canonical correlation results class
Attributes
-----------
stats : DataFrame
Contain statistical tests results for each canonical correlation
stats_mv : DataFrame
Contain the multivariate statistical tests results
"""
def __init__(self, stats, stats_mv):
self.stats = stats
self.stats_mv = stats_mv
def __str__(self):
return self.summary().__str__()
def summary(self):
summ = summary2.Summary()
summ.add_title('Cancorr results')
summ.add_df(self.stats)
summ.add_dict({'': ''})
summ.add_dict({'Multivariate Statistics and F Approximations': ''})
summ.add_df(self.stats_mv)
return summ
