591 lines
20 KiB
Python
591 lines
20 KiB
Python
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"""General linear model
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author: Yichuan Liu
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"""
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import numpy as np
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from numpy.linalg import eigvals, inv, solve, matrix_rank, pinv, svd
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from scipy import stats
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import pandas as pd
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from patsy import DesignInfo
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from statsmodels.compat.pandas import Substitution
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from statsmodels.base.model import Model
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from statsmodels.iolib import summary2
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__docformat__ = 'restructuredtext en'
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_hypotheses_doc = \
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"""hypotheses : list[tuple]
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Hypothesis `L*B*M = C` to be tested where B is the parameters in
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regression Y = X*B. Each element is a tuple of length 2, 3, or 4:
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* (name, contrast_L)
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* (name, contrast_L, transform_M)
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* (name, contrast_L, transform_M, constant_C)
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containing a string `name`, the contrast matrix L, the transform
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matrix M (for transforming dependent variables), and right-hand side
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constant matrix constant_C, respectively.
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contrast_L : 2D array or an array of strings
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Left-hand side contrast matrix for hypotheses testing.
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If 2D array, each row is an hypotheses and each column is an
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independent variable. At least 1 row
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(1 by k_exog, the number of independent variables) is required.
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If an array of strings, it will be passed to
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patsy.DesignInfo().linear_constraint.
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transform_M : 2D array or an array of strings or None, optional
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Left hand side transform matrix.
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If `None` or left out, it is set to a k_endog by k_endog
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identity matrix (i.e. do not transform y matrix).
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If an array of strings, it will be passed to
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patsy.DesignInfo().linear_constraint.
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constant_C : 2D array or None, optional
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Right-hand side constant matrix.
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if `None` or left out it is set to a matrix of zeros
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Must has the same number of rows as contrast_L and the same
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number of columns as transform_M
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If `hypotheses` is None: 1) the effect of each independent variable
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on the dependent variables will be tested. Or 2) if model is created
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using a formula, `hypotheses` will be created according to
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`design_info`. 1) and 2) is equivalent if no additional variables
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are created by the formula (e.g. dummy variables for categorical
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variables and interaction terms)
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"""
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def _multivariate_ols_fit(endog, exog, method='svd', tolerance=1e-8):
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"""
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Solve multivariate linear model y = x * params
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where y is dependent variables, x is independent variables
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Parameters
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----------
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endog : array_like
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each column is a dependent variable
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exog : array_like
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each column is a independent variable
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method : str
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'svd' - Singular value decomposition
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'pinv' - Moore-Penrose pseudoinverse
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tolerance : float, a small positive number
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Tolerance for eigenvalue. Values smaller than tolerance is considered
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zero.
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Returns
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-------
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a tuple of matrices or values necessary for hypotheses testing
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.. [*] https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_introreg_sect012.htm
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Notes
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-----
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Status: experimental and incomplete
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"""
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y = endog
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x = exog
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nobs, k_endog = y.shape
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nobs1, k_exog= x.shape
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if nobs != nobs1:
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raise ValueError('x(n=%d) and y(n=%d) should have the same number of '
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'rows!' % (nobs1, nobs))
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# Calculate the matrices necessary for hypotheses testing
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df_resid = nobs - k_exog
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if method == 'pinv':
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# Regression coefficients matrix
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pinv_x = pinv(x)
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params = pinv_x.dot(y)
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# inverse of x'x
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inv_cov = pinv_x.dot(pinv_x.T)
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if matrix_rank(inv_cov,tol=tolerance) < k_exog:
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raise ValueError('Covariance of x singular!')
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# Sums of squares and cross-products of residuals
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# Y'Y - (X * params)'B * params
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t = x.dot(params)
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sscpr = np.subtract(y.T.dot(y), t.T.dot(t))
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return (params, df_resid, inv_cov, sscpr)
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elif method == 'svd':
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u, s, v = svd(x, 0)
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if (s > tolerance).sum() < len(s):
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raise ValueError('Covariance of x singular!')
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invs = 1. / s
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params = v.T.dot(np.diag(invs)).dot(u.T).dot(y)
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inv_cov = v.T.dot(np.diag(np.power(invs, 2))).dot(v)
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t = np.diag(s).dot(v).dot(params)
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sscpr = np.subtract(y.T.dot(y), t.T.dot(t))
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return (params, df_resid, inv_cov, sscpr)
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else:
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raise ValueError('%s is not a supported method!' % method)
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def multivariate_stats(eigenvals,
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r_err_sscp,
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r_contrast, df_resid, tolerance=1e-8):
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"""
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For multivariate linear model Y = X * B
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Testing hypotheses
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L*B*M = 0
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where L is contrast matrix, B is the parameters of the
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multivariate linear model and M is dependent variable transform matrix.
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T = L*inv(X'X)*L'
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H = M'B'L'*inv(T)*LBM
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E = M'(Y'Y - B'X'XB)M
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Parameters
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----------
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eigenvals : ndarray
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The eigenvalues of inv(E + H)*H
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r_err_sscp : int
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Rank of E + H
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r_contrast : int
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Rank of T matrix
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df_resid : int
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Residual degree of freedom (n_samples minus n_variables of X)
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tolerance : float
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smaller than which eigenvalue is considered 0
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Returns
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-------
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A DataFrame
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References
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----------
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.. [*] https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_introreg_sect012.htm
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"""
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v = df_resid
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p = r_err_sscp
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q = r_contrast
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s = np.min([p, q])
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ind = eigenvals > tolerance
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n_e = ind.sum()
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eigv2 = eigenvals[ind]
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eigv1 = np.array([i / (1 - i) for i in eigv2])
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m = (np.abs(p - q) - 1) / 2
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n = (v - p - 1) / 2
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cols = ['Value', 'Num DF', 'Den DF', 'F Value', 'Pr > F']
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index = ["Wilks' lambda", "Pillai's trace",
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"Hotelling-Lawley trace", "Roy's greatest root"]
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results = pd.DataFrame(columns=cols,
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index=index)
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def fn(x):
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return np.real([x])[0]
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results.loc["Wilks' lambda", 'Value'] = fn(np.prod(1 - eigv2))
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results.loc["Pillai's trace", 'Value'] = fn(eigv2.sum())
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results.loc["Hotelling-Lawley trace", 'Value'] = fn(eigv1.sum())
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results.loc["Roy's greatest root", 'Value'] = fn(eigv1.max())
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r = v - (p - q + 1)/2
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u = (p*q - 2) / 4
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df1 = p * q
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if p*p + q*q - 5 > 0:
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t = np.sqrt((p*p*q*q - 4) / (p*p + q*q - 5))
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else:
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t = 1
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df2 = r*t - 2*u
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lmd = results.loc["Wilks' lambda", 'Value']
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lmd = np.power(lmd, 1 / t)
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F = (1 - lmd) / lmd * df2 / df1
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results.loc["Wilks' lambda", 'Num DF'] = df1
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results.loc["Wilks' lambda", 'Den DF'] = df2
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results.loc["Wilks' lambda", 'F Value'] = F
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pval = stats.f.sf(F, df1, df2)
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results.loc["Wilks' lambda", 'Pr > F'] = pval
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V = results.loc["Pillai's trace", 'Value']
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df1 = s * (2*m + s + 1)
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df2 = s * (2*n + s + 1)
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F = df2 / df1 * V / (s - V)
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results.loc["Pillai's trace", 'Num DF'] = df1
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results.loc["Pillai's trace", 'Den DF'] = df2
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results.loc["Pillai's trace", 'F Value'] = F
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pval = stats.f.sf(F, df1, df2)
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results.loc["Pillai's trace", 'Pr > F'] = pval
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U = results.loc["Hotelling-Lawley trace", 'Value']
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if n > 0:
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b = (p + 2*n) * (q + 2*n) / 2 / (2*n + 1) / (n - 1)
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df1 = p * q
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df2 = 4 + (p*q + 2) / (b - 1)
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c = (df2 - 2) / 2 / n
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F = df2 / df1 * U / c
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else:
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df1 = s * (2*m + s + 1)
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df2 = s * (s*n + 1)
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F = df2 / df1 / s * U
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results.loc["Hotelling-Lawley trace", 'Num DF'] = df1
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results.loc["Hotelling-Lawley trace", 'Den DF'] = df2
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results.loc["Hotelling-Lawley trace", 'F Value'] = F
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pval = stats.f.sf(F, df1, df2)
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results.loc["Hotelling-Lawley trace", 'Pr > F'] = pval
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sigma = results.loc["Roy's greatest root", 'Value']
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r = np.max([p, q])
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df1 = r
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df2 = v - r + q
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F = df2 / df1 * sigma
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results.loc["Roy's greatest root", 'Num DF'] = df1
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results.loc["Roy's greatest root", 'Den DF'] = df2
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results.loc["Roy's greatest root", 'F Value'] = F
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pval = stats.f.sf(F, df1, df2)
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results.loc["Roy's greatest root", 'Pr > F'] = pval
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return results
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def _multivariate_ols_test(hypotheses, fit_results, exog_names,
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endog_names):
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def fn(L, M, C):
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# .. [1] https://support.sas.com/documentation/cdl/en/statug/63033
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# /HTML/default/viewer.htm#statug_introreg_sect012.htm
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params, df_resid, inv_cov, sscpr = fit_results
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# t1 = (L * params)M
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t1 = L.dot(params).dot(M) - C
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# H = t1'L(X'X)^L't1
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t2 = L.dot(inv_cov).dot(L.T)
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q = matrix_rank(t2)
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H = t1.T.dot(inv(t2)).dot(t1)
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# E = M'(Y'Y - B'(X'X)B)M
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E = M.T.dot(sscpr).dot(M)
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return E, H, q, df_resid
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return _multivariate_test(hypotheses, exog_names, endog_names, fn)
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@Substitution(hypotheses_doc=_hypotheses_doc)
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def _multivariate_test(hypotheses, exog_names, endog_names, fn):
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"""
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Multivariate linear model hypotheses testing
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For y = x * params, where y are the dependent variables and x are the
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independent variables, testing L * params * M = 0 where L is the contrast
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matrix for hypotheses testing and M is the transformation matrix for
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transforming the dependent variables in y.
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Algorithm:
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T = L*inv(X'X)*L'
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H = M'B'L'*inv(T)*LBM
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E = M'(Y'Y - B'X'XB)M
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where H and E correspond to the numerator and denominator of a univariate
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F-test. Then find the eigenvalues of inv(H + E)*H from which the
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multivariate test statistics are calculated.
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.. [*] https://support.sas.com/documentation/cdl/en/statug/63033/HTML
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/default/viewer.htm#statug_introreg_sect012.htm
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Parameters
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----------
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%(hypotheses_doc)s
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k_xvar : int
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The number of independent variables
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k_yvar : int
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The number of dependent variables
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fn : function
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a function fn(contrast_L, transform_M) that returns E, H, q, df_resid
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where q is the rank of T matrix
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Returns
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-------
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results : MANOVAResults
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"""
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k_xvar = len(exog_names)
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k_yvar = len(endog_names)
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results = {}
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for hypo in hypotheses:
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if len(hypo) ==2:
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name, L = hypo
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M = None
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C = None
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elif len(hypo) == 3:
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name, L, M = hypo
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C = None
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elif len(hypo) == 4:
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name, L, M, C = hypo
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else:
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raise ValueError('hypotheses must be a tuple of length 2, 3 or 4.'
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' len(hypotheses)=%d' % len(hypo))
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if any(isinstance(j, str) for j in L):
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L = DesignInfo(exog_names).linear_constraint(L).coefs
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else:
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if not isinstance(L, np.ndarray) or len(L.shape) != 2:
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raise ValueError('Contrast matrix L must be a 2-d array!')
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if L.shape[1] != k_xvar:
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raise ValueError('Contrast matrix L should have the same '
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'number of columns as exog! %d != %d' %
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(L.shape[1], k_xvar))
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if M is None:
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M = np.eye(k_yvar)
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elif any(isinstance(j, str) for j in M):
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M = DesignInfo(endog_names).linear_constraint(M).coefs.T
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else:
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if M is not None:
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if not isinstance(M, np.ndarray) or len(M.shape) != 2:
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raise ValueError('Transform matrix M must be a 2-d array!')
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if M.shape[0] != k_yvar:
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raise ValueError('Transform matrix M should have the same '
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'number of rows as the number of columns '
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'of endog! %d != %d' %
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(M.shape[0], k_yvar))
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if C is None:
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C = np.zeros([L.shape[0], M.shape[1]])
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elif not isinstance(C, np.ndarray):
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raise ValueError('Constant matrix C must be a 2-d array!')
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if C.shape[0] != L.shape[0]:
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raise ValueError('contrast L and constant C must have the same '
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'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
|
||
|
|
|
||
|
|
|
||
|
|
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().__init__(endog, exog, missing=missing,
|
||
|
|
hasconst=hasconst, **kwargs)
|
||
|
|
|
||
|
|
def fit(self, method='svd'):
|
||
|
|
self._fittedmod = _multivariate_ols_fit(
|
||
|
|
self.endog, self.exog, method=method)
|
||
|
|
return _MultivariateOLSResults(self)
|
||
|
|
|
||
|
|
|
||
|
|
class _MultivariateOLSResults:
|
||
|
|
"""
|
||
|
|
_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__()
|
||
|
|
|
||
|
|
@Substitution(hypotheses_doc=_hypotheses_doc)
|
||
|
|
def mv_test(self, hypotheses=None, skip_intercept_test=False):
|
||
|
|
"""
|
||
|
|
Linear hypotheses testing
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
%(hypotheses_doc)s
|
||
|
|
skip_intercept_test : bool
|
||
|
|
If true, then testing the intercept is skipped, the model is not
|
||
|
|
changed.
|
||
|
|
Note: If a term has a numerically insignificant effect, then
|
||
|
|
an exception because of emtpy arrays may be raised. This can
|
||
|
|
happen for the intercept if the data has been demeaned.
|
||
|
|
|
||
|
|
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:
|
||
|
|
if skip_intercept_test and key == 'Intercept':
|
||
|
|
continue
|
||
|
|
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)
|
||
|
|
|
||
|
|
def summary(self):
|
||
|
|
raise NotImplementedError
|
||
|
|
|
||
|
|
|
||
|
|
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
|
||
|
|
|
||
|
|
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 = list(df.columns)
|
||
|
|
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
|