# -*- coding: utf-8 -*-
"""General linear model
author: Yichuan Liu
"""
from __future__ import division
import numpy as np
from numpy.linalg import eigvals, inv, solve, matrix_rank, pinv, svd
from scipy import stats
import pandas as pd
from patsy import DesignInfo
from statsmodels.compat import string_types
from statsmodels.base.model import Model
from statsmodels.iolib import summary2
__docformat__ = 'restructuredtext en'
_hypotheses_doc = \
"""hypotheses: A list of tuples
Hypothesis `L*B*M = C` to be tested where B is the parameters in
regression Y = X*B. Each element is a tuple of length 2, 3, or 4:
* (name, contrast_L)
* (name, contrast_L, transform_M)
* (name, contrast_L, transform_M, constant_C)
containing a string `name`, the contrast matrix L, the transform
matrix M (for transforming dependent variables), and right-hand side
constant matrix constant_C, respectively.
contrast_L : 2D array or an array of strings
Left-hand side contrast matrix for hypotheses testing.
If 2D array, each row is an hypotheses and each column is an
independent variable. At least 1 row
(1 by k_exog, the number of independent variables) is required.
If an array of strings, it will be passed to
patsy.DesignInfo().linear_constraint.
transform_M : 2D array or an array of strings or None, optional
Left hand side transform matrix.
If `None` or left out, it is set to a k_endog by k_endog
identity matrix (i.e. do not transform y matrix).
If an array of strings, it will be passed to
patsy.DesignInfo().linear_constraint.
constant_C : 2D array or None, optional
Right-hand side constant matrix.
if `None` or left out it is set to a matrix of zeros
Must has the same number of rows as contrast_L and the same
number of columns as transform_M
If `hypotheses` is None: 1) the effect of each independent variable
on the dependent variables will be tested. Or 2) if model is created
using a formula, `hypotheses` will be created according to
`design_info`. 1) and 2) is equivalent if no additional variables
are created by the formula (e.g. dummy variables for categorical
variables and interaction terms)
"""
def _multivariate_ols_fit(endog, exog, method='svd', tolerance=1e-8):
"""
Solve multivariate linear model y = x * params
where y is dependent variables, x is independent variables
Parameters
----------
endog : array_like
each column is a dependent variable
exog : array_like
each column is a independent variable
method : string
'svd' - Singular value decomposition
'pinv' - Moore-Penrose pseudoinverse
tolerance : float, a small positive number
Tolerance for eigenvalue. Values smaller than tolerance is considered
zero.
Returns
-------
a tuple of matrices or values necessary for hypotheses testing
.. [*] https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_introreg_sect012.htm
Notes
-----
Status: experimental and incomplete
"""
y = endog
x = exog
nobs, k_endog = y.shape
nobs1, k_exog= x.shape
if nobs != nobs1:
raise ValueError('x(n=%d) and y(n=%d) should have the same number of '
'rows!' % (nobs1, nobs))
# Calculate the matrices necessary for hypotheses testing
df_resid = nobs - k_exog
if method == 'pinv':
# Regression coefficients matrix
pinv_x = pinv(x)
params = pinv_x.dot(y)
# inverse of x'x
inv_cov = pinv_x.dot(pinv_x.T)
if matrix_rank(inv_cov,tol=tolerance) < k_exog:
raise ValueError('Covariance of x singular!')
# Sums of squares and cross-products of residuals
# Y'Y - (X * params)'B * params
t = x.dot(params)
sscpr = np.subtract(y.T.dot(y), t.T.dot(t))
return (params, df_resid, inv_cov, sscpr)
elif method == 'svd':
u, s, v = svd(x, 0)
if (s > tolerance).sum() < len(s):
raise ValueError('Covariance of x singular!')
invs = 1. / s
params = v.T.dot(np.diag(invs)).dot(u.T).dot(y)
inv_cov = v.T.dot(np.diag(np.power(invs, 2))).dot(v)
t = np.diag(s).dot(v).dot(params)
sscpr = np.subtract(y.T.dot(y), t.T.dot(t))
return (params, df_resid, inv_cov, sscpr)
else:
raise ValueError('%s is not a supported method!' % method)
def multivariate_stats(eigenvals,
r_err_sscp,
r_contrast, df_resid, tolerance=1e-8):
"""
For multivariate linear model Y = X * B
Testing hypotheses
L*B*M = 0
where L is contrast matrix, B is the parameters of the
multivariate linear model and M is dependent variable transform matrix.
T = L*inv(X'X)*L'
H = M'B'L'*inv(T)*LBM
E = M'(Y'Y - B'X'XB)M
Parameters
----------
eigenvals : array
The eigenvalues of inv(E + H)*H
r_err_sscp : int
Rank of E + H
r_contrast : int
Rank of T matrix
df_resid : int
Residual degree of freedom (n_samples minus n_variables of X)
tolerance : float
smaller than which eigenvalue is considered 0
Returns
-------
A DataFrame
References
----------
.. [*] https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_introreg_sect012.htm
"""
v = df_resid
p = r_err_sscp
q = r_contrast
s = np.min([p, q])
ind = eigenvals > tolerance
n_e = ind.sum()
eigv2 = eigenvals[ind]
eigv1 = np.array([i / (1 - i) for i in eigv2])
m = (np.abs(p - q) - 1) / 2
n = (v - p - 1) / 2
cols = ['Value', 'Num DF', 'Den DF', 'F Value', 'Pr > F']
index = ["Wilks' lambda", "Pillai's trace",
"Hotelling-Lawley trace", "Roy's greatest root"]
results = pd.DataFrame(columns=cols,
index=index)
def fn(x):
return np.real([x])[0]
results.loc["Wilks' lambda", 'Value'] = fn(np.prod(1 - eigv2))
results.loc["Pillai's trace", 'Value'] = fn(eigv2.sum())
results.loc["Hotelling-Lawley trace", 'Value'] = fn(eigv1.sum())
results.loc["Roy's greatest root", 'Value'] = fn(eigv1.max())
r = v - (p - q + 1)/2
u = (p*q - 2) / 4
df1 = p * q
if p*p + q*q - 5 > 0:
t = np.sqrt((p*p*q*q - 4) / (p*p + q*q - 5))
else:
t = 1
df2 = r*t - 2*u
lmd = results.loc["Wilks' lambda", 'Value']
lmd = np.power(lmd, 1 / t)
F = (1 - lmd) / lmd * df2 / df1
results.loc["Wilks' lambda", 'Num DF'] = df1
results.loc["Wilks' lambda", 'Den DF'] = df2
results.loc["Wilks' lambda", 'F Value'] = F
pval = stats.f.sf(F, df1, df2)
results.loc["Wilks' lambda", 'Pr > F'] = pval
V = results.loc["Pillai's trace", 'Value']
df1 = s * (2*m + s + 1)
df2 = s * (2*n + s + 1)
F = df2 / df1 * V / (s - V)
results.loc["Pillai's trace", 'Num DF'] = df1
results.loc["Pillai's trace", 'Den DF'] = df2
results.loc["Pillai's trace", 'F Value'] = F
pval = stats.f.sf(F, df1, df2)
results.loc["Pillai's trace", 'Pr > F'] = pval
U = results.loc["Hotelling-Lawley trace", 'Value']
if n > 0:
b = (p + 2*n) * (q + 2*n) / 2 / (2*n + 1) / (n - 1)
df1 = p * q
df2 = 4 + (p*q + 2) / (b - 1)
c = (df2 - 2) / 2 / n
F = df2 / df1 * U / c
else:
df1 = s * (2*m + s + 1)
df2 = s * (s*n + 1)
F = df2 / df1 / s * U
results.loc["Hotelling-Lawley trace", 'Num DF'] = df1
results.loc["Hotelling-Lawley trace", 'Den DF'] = df2
results.loc["Hotelling-Lawley trace", 'F Value'] = F
pval = stats.f.sf(F, df1, df2)
results.loc["Hotelling-Lawley trace", 'Pr > F'] = pval
sigma = results.loc["Roy's greatest root", 'Value']
r = np.max([p, q])
df1 = r
df2 = v - r + q
F = df2 / df1 * sigma
results.loc["Roy's greatest root", 'Num DF'] = df1
results.loc["Roy's greatest root", 'Den DF'] = df2
results.loc["Roy's greatest root", 'F Value'] = F
pval = stats.f.sf(F, df1, df2)
results.loc["Roy's greatest root", 'Pr > F'] = pval
return results
def _multivariate_ols_test(hypotheses, fit_results, exog_names,
endog_names):
def fn(L, M, C):
# .. [1] https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_introreg_sect012.htm
params, df_resid, inv_cov, sscpr = fit_results
# t1 = (L * params)M
t1 = L.dot(params).dot(M) - C
# H = t1'L(X'X)^L't1
t2 = L.dot(inv_cov).dot(L.T)
q = matrix_rank(t2)
H = t1.T.dot(inv(t2)).dot(t1)
# E = M'(Y'Y - B'(X'X)B)M
E = M.T.dot(sscpr).dot(M)
return E, H, q, df_resid
return _multivariate_test(hypotheses, exog_names, endog_names, fn)
def _multivariate_test(hypotheses, exog_names, endog_names, fn):
k_xvar = len(exog_names)
k_yvar = len(endog_names)
results = {}
for hypo in hypotheses:
if len(hypo) ==2:
name, L = hypo
M = None
C = None
elif len(hypo) == 3:
name, L, M = hypo
C = None
elif len(hypo) == 4:
name, L, M, C = hypo
else:
raise ValueError('hypotheses must be a tuple of length 2, 3 or 4.'
' len(hypotheses)=%d' % len(hypo))
if any(isinstance(j, string_types) for j in L):
L = DesignInfo(exog_names).linear_constraint(L).coefs
else:
if not isinstance(L, np.ndarray) or len(L.shape) != 2:
raise ValueError('Contrast matrix L must be a 2-d array!')
if L.shape[1] != k_xvar:
raise ValueError('Contrast matrix L should have the same '
'number of columns as exog! %d != %d' %
(L.shape[1], k_xvar))
if M is None:
M = np.eye(k_yvar)
elif any(isinstance(j, string_types) for j in M):
M = DesignInfo(endog_names).linear_constraint(M).coefs.T
else:
if M is not None:
if not isinstance(M, np.ndarray) or len(M.shape) != 2:
raise ValueError('Transform matrix M must be a 2-d array!')
if M.shape[0] != k_yvar:
raise ValueError('Transform matrix M should have the same '
'number of rows as the number of columns '
'of endog! %d != %d' %
(M.shape[0], k_yvar))
if C is None:
C = np.zeros([L.shape[0], M.shape[1]])
elif not isinstance(C, np.ndarray):
raise ValueError('Constant matrix C must be a 2-d array!')
if C.shape[0] != L.shape[0]:
raise ValueError('contrast L and constant C must have the same '
'number of rows! %d!=%d'
% (L.shape[0], C.shape[0]))
if C.shape[1] != M.shape[1]:
raise ValueError('transform M and constant C must have the same '
'number of columns! %d!=%d'
% (M.shape[1], C.shape[1]))
E, H, q, df_resid = fn(L, M, C)
EH = np.add(E, H)
p = matrix_rank(EH)
# eigenvalues of inv(E + H)H
eigv2 = np.sort(eigvals(solve(EH, H)))
stat_table = multivariate_stats(eigv2, p, q, df_resid)
results[name] = {'stat':stat_table, 'contrast_L':L,
'transform_M':M, 'constant_C':C}
return results
_multivariate_test.__doc__ = (
"""
Multivariate linear model hypotheses testing
For y = x * params, where y are the dependent variables and x are the
independent variables, testing L * params * M = 0 where L is the contrast
matrix for hypotheses testing and M is the transformation matrix for
transforming the dependent variables in y.
Algorithm:
T = L*inv(X'X)*L'
H = M'B'L'*inv(T)*LBM
E = M'(Y'Y - B'X'XB)M
And then finding the eigenvalues of inv(H + E)*H
.. [*] https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_introreg_sect012.htm
Parameters
----------
""" + _hypotheses_doc +
"""
k_xvar : int
The number of independent variables
k_yvar : int
The number of dependent variables
fn : function
a function fn(contrast_L, transform_M) that returns E, H, q, df_resid
where q is the rank of T matrix
Returns
-------
results : MANOVAResults
""")
[docs]class _MultivariateOLS(Model):
"""
Multivariate linear model via least squares
Parameters
----------
endog : array_like
Dependent variables. A nobs x k_endog array where nobs is
the number of observations and k_endog is the number of dependent
variables
exog : array_like
Independent variables. A nobs x k_exog array where nobs is the
number of observations and k_exog is the number of independent
variables. An intercept is not included by default and should be added
by the user (models specified using a formula include an intercept by
default)
Attributes
----------
endog : array
See Parameters.
exog : array
See Parameters.
"""
def __init__(self, endog, exog, missing='none', hasconst=None, **kwargs):
if len(endog.shape) == 1 or endog.shape[1] == 1:
raise ValueError('There must be more than one dependent variable'
' to fit multivariate OLS!')
super(_MultivariateOLS, self).__init__(endog, exog, missing=missing,
hasconst=hasconst, **kwargs)
[docs] def fit(self, method='svd'):
self._fittedmod = _multivariate_ols_fit(
self.endog, self.exog, method=method)
return _MultivariateOLSResults(self)
[docs]class _MultivariateOLSResults(object):
"""
_MultivariateOLS results class
"""
def __init__(self, fitted_mv_ols):
if (hasattr(fitted_mv_ols, 'data') and
hasattr(fitted_mv_ols.data, 'design_info')):
self.design_info = fitted_mv_ols.data.design_info
else:
self.design_info = None
self.exog_names = fitted_mv_ols.exog_names
self.endog_names = fitted_mv_ols.endog_names
self._fittedmod = fitted_mv_ols._fittedmod
def __str__(self):
return self.summary().__str__()
[docs] def mv_test(self, hypotheses=None):
k_xvar = len(self.exog_names)
if hypotheses is None:
if self.design_info is not None:
terms = self.design_info.term_name_slices
hypotheses = []
for key in terms:
L_contrast = np.eye(k_xvar)[terms[key], :]
hypotheses.append([key, L_contrast, None])
else:
hypotheses = []
for i in range(k_xvar):
name = 'x%d' % (i)
L = np.zeros([1, k_xvar])
L[i] = 1
hypotheses.append([name, L, None])
results = _multivariate_ols_test(hypotheses, self._fittedmod,
self.exog_names, self.endog_names)
return MultivariateTestResults(results,
self.endog_names,
self.exog_names)
mv_test.__doc__ = ("""
Linear hypotheses testing
Parameters
----------
""" + _hypotheses_doc + """
Returns
-------
results: _MultivariateOLSResults
Notes
-----
Tests hypotheses of the form
L * params * M = C
where `params` is the regression coefficient matrix for the
linear model y = x * params, `L` is the contrast matrix, `M` is the
dependent variable transform matrix and C is the constant matrix.
""")
[docs] def summary(self):
raise NotImplementedError
[docs]class MultivariateTestResults(object):
""" Multivariate test results class
Returned by `mv_test` method of `_MultivariateOLSResults` class
Attributes
----------
results : dict
For hypothesis name `key`:
results[key]['stat'] contains the multivaraite test results
results[key]['contrast_L'] contains the contrast_L matrix
results[key]['transform_M'] contains the transform_M matrix
results[key]['constant_C'] contains the constant_C matrix
endog_names : string
exog_names : string
summary_frame : multiindex dataframe
Returns results as a multiindex dataframe
"""
def __init__(self, mv_test_df, endog_names, exog_names):
self.results = mv_test_df
self.endog_names = endog_names
self.exog_names = exog_names
def __str__(self):
return self.summary().__str__()
def __getitem__(self, item):
return self.results[item]
@property
def summary_frame(self):
"""
Return results as a multiindex dataframe
"""
df = []
for key in self.results:
tmp = self.results[key]['stat'].copy()
tmp.loc[:, 'Effect'] = key
df.append(tmp.reset_index())
df = pd.concat(df, axis=0)
df = df.set_index(['Effect', 'index'])
df.index.set_names(['Effect', 'Statistic'], inplace=True)
return df
[docs] def summary(self, show_contrast_L=False, show_transform_M=False,
show_constant_C=False):
"""
Parameters
----------
contrast_L : True or False
Whether to show contrast_L matrix
transform_M : True or False
Whether to show transform_M matrix
"""
summ = summary2.Summary()
summ.add_title('Multivariate linear model')
for key in self.results:
summ.add_dict({'':''})
df = self.results[key]['stat'].copy()
df = df.reset_index()
c = df.columns.values
c[0] = key
df.columns = c
df.index = ['', '', '', '']
summ.add_df(df)
if show_contrast_L:
summ.add_dict({key:' contrast L='})
df = pd.DataFrame(self.results[key]['contrast_L'],
columns=self.exog_names)
summ.add_df(df)
if show_transform_M:
summ.add_dict({key:' transform M='})
df = pd.DataFrame(self.results[key]['transform_M'],
index=self.endog_names)
summ.add_df(df)
if show_constant_C:
summ.add_dict({key:' constant C='})
df = pd.DataFrame(self.results[key]['constant_C'])
summ.add_df(df)
return summ
