Source code for statsmodels.multivariate.multivariate_ols

# -*- coding: utf-8 -*-

"""General linear model

author: Yichuan Liu
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.pandas import Substitution
from statsmodels.base.model import Model
from statsmodels.iolib import summary2
__docformat__ = 'restructuredtext en'

_hypotheses_doc = \
"""hypotheses : list[tuple]
    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

    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

    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

    endog : array_like
        each column is a dependent variable
    exog : array_like
        each column is a independent variable
    method : str
        'svd' - Singular value decomposition
        'pinv' - Moore-Penrose pseudoinverse
    tolerance : float, a small positive number
        Tolerance for eigenvalue. Values smaller than tolerance is considered
    a tuple of matrices or values necessary for hypotheses testing

    .. [*]
    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 =

        # inverse of x'x
        inv_cov =
        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 =
        sscpr = np.subtract(,
        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 =
        inv_cov =, 2))).dot(v)
        t = np.diag(s).dot(v).dot(params)
        sscpr = np.subtract(,
        return (params, df_resid, inv_cov, sscpr)
        raise ValueError('%s is not a supported method!' % method)

def multivariate_stats(eigenvals,
                       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

    eigenvals : ndarray
        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

    A DataFrame

    .. [*]
    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,

    def fn(x):
        return np.real([x])[0]

    results.loc["Wilks' lambda", 'Value'] = fn( - 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))
        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
        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,
    def fn(L, M, C):
        # .. [1]
        #        /HTML/default/viewer.htm#statug_introreg_sect012.htm
        params, df_resid, inv_cov, sscpr = fit_results
        # t1 = (L * params)M
        t1 = - C
        # H = t1'L(X'X)^L't1
        t2 =
        q = matrix_rank(t2)
        H =

        # E = M'(Y'Y - B'(X'X)B)M
        E =
        return E, H, q, df_resid

    return _multivariate_test(hypotheses, exog_names, endog_names, fn)

def _multivariate_test(hypotheses, exog_names, endog_names, fn):
    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.

        T = L*inv(X'X)*L'
        H = M'B'L'*inv(T)*LBM
        E =  M'(Y'Y - B'X'XB)M
    where H and E correspond to the numerator and denominator of a univariate
    F-test. Then find the eigenvalues of inv(H + E)*H from which the
    multivariate test statistics are calculated.

    .. [*]

    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

    results : MANOVAResults

    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
            raise ValueError('hypotheses must be a tuple of length 2, 3 or 4.'
                             ' len(hypotheses)=%d' % len(hypo))
        if any(isinstance(j, str) for j in L):
            L = DesignInfo(exog_names).linear_constraint(L).coefs
            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, str) for j in M):
            M = DesignInfo(endog_names).linear_constraint(M).coefs.T
            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,
                         'E': E, 'H': H}
    return results

[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 : ndarray See Parameters. exog : ndarray See Parameters. """ _formula_max_endog = None 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: """ _MultivariateOLS results class """ def __init__(self, fitted_mv_ols): if (hasattr(fitted_mv_ols, 'data') and hasattr(, 'design_info')): self.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] @Substitution(hypotheses_doc=_hypotheses_doc) def mv_test(self, hypotheses=None): """ Linear hypotheses testing Parameters ---------- %(hypotheses_doc)s 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. """ 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)
[docs] def summary(self): raise NotImplementedError
[docs]class MultivariateTestResults: """ Multivariate test results class Returned by `mv_test` method of `_MultivariateOLSResults` class Parameters ---------- results : dict[str, dict] Dictionary containing test results. See the description below for the expected format. endog_names : sequence[str] A list or other sequence of endogenous variables names exog_names : sequence[str] A list of other sequence of exogenous variables names Attributes ---------- results : dict Each hypothesis is contained in a single`key`. Each test must have the following keys: * 'stat' - contains the multivariate test results * 'contrast_L' - contains the contrast_L matrix * 'transform_M' - contains the transform_M matrix * 'constant_C' - contains the constant_C matrix * 'H' - contains an intermediate Hypothesis matrix, or the between groups sums of squares and cross-products matrix, corresponding to the numerator of the univariate F test. * 'E' - contains an intermediate Error matrix, corresponding to the denominator of the univariate F test. The Hypotheses and Error matrices can be used to calculate the same test statistics in 'stat', as well as to calculate the discriminant function (canonical correlates) from the eigenvectors of inv(E)H. endog_names : list[str] The endogenous names exog_names : list[str] The exogenous names summary_frame : DataFrame Returns results as a MultiIndex DataFrame """ def __init__(self, results, endog_names, exog_names): self.results = results self.endog_names = list(endog_names) self.exog_names = list(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): """ Summary of test results Parameters ---------- show_contrast_L : bool Whether to show contrast_L matrix show_transform_M : bool Whether to show transform_M matrix show_constant_C : bool Whether to show the constant_C """ 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