3 Copyright (C) 2005 Free Software Foundation, Inc.
4 Written by Jason H. Stover.
6 This program is free software; you can redistribute it and/or modify
7 it under the terms of the GNU General Public License as published by
8 the Free Software Foundation; either version 2 of the License, or (at
9 your option) any later version.
11 This program is distributed in the hope that it will be useful, but
12 WITHOUT ANY WARRANTY; without even the implied warranty of
13 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
14 General Public License for more details.
16 You should have received a copy of the GNU General Public License
17 along with this program; if not, write to the Free Software
18 Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
23 Find the least-squares estimate of b for the linear model:
27 where Y is an n-by-1 column vector, X is an n-by-p matrix of
28 independent variables, b is a p-by-1 vector of regression coefficients,
29 and Z is an n-by-1 normally-distributed random vector with independent
30 identically distributed components with mean 0.
32 This estimate is found via the sweep operator or singular-value
33 decomposition with gsl.
38 1. Matrix Computations, third edition. GH Golub and CF Van Loan.
39 The Johns Hopkins University Press. 1996. ISBN 0-8018-5414-8.
41 2. Numerical Analysis for Statisticians. K Lange. Springer. 1999.
44 3. Numerical Linear Algebra for Applications in Statistics. JE Gentle.
45 Springer. 1998. ISBN 0-387-98542-5.
48 #include "pspp_linreg.h"
49 #include <gsl/gsl_errno.h>
51 Get the mean and standard deviation of a vector
52 of doubles via a form of the Kalman filter as
53 described on page 32 of [3].
56 linreg_mean_std (gsl_vector_const_view v, double *mp, double *sp, double *ssp)
65 mean = gsl_vector_get (&v.vector, 0);
67 for (i = 1; i < v.vector.size; i++)
70 tmp = gsl_vector_get (&v.vector, i);
73 variance += j * (j - 1.0) * d * d;
76 *sp = sqrt (variance / (j - 1.0));
83 Allocate a pspp_linreg_cache and return a pointer
84 to it. n is the number of cases, p is the number of
85 independent variables.
88 pspp_linreg_cache_alloc (size_t n, size_t p)
92 c = (pspp_linreg_cache *) malloc (sizeof (pspp_linreg_cache));
93 c->param_estimates = gsl_vector_alloc (p + 1);
94 c->indep_means = gsl_vector_alloc (p);
95 c->indep_std = gsl_vector_alloc (p);
96 c->ssx = gsl_vector_alloc (p); /* Sums of squares for the independent
99 c->ss_indeps = gsl_vector_alloc (p); /* Sums of squares for the model
102 c->cov = gsl_matrix_alloc (p + 1, p + 1); /* Covariance matrix. */
108 c->method = PSPP_LINREG_SWEEP;
114 pspp_linreg_cache_free (pspp_linreg_cache * c)
116 gsl_vector_free (c->param_estimates);
117 gsl_vector_free (c->indep_means);
118 gsl_vector_free (c->indep_std);
119 gsl_vector_free (c->ss_indeps);
120 gsl_matrix_free (c->cov);
126 Fit the linear model via least squares. All pointers passed to pspp_linreg
127 are assumed to be allocated to the correct size and initialized to the
128 values as indicated by opts.
131 pspp_linreg (const gsl_vector * Y, const gsl_matrix * X,
132 const pspp_linreg_opts * opts, pspp_linreg_cache * cache)
138 gsl_matrix_view xmxtx;
154 if (opts->get_depvar_mean_std)
156 linreg_mean_std (gsl_vector_const_subvector (Y, 0, Y->size),
158 cache->depvar_mean = m;
159 cache->depvar_std = s;
162 for (i = 0; i < cache->n_indeps; i++)
164 if (opts->get_indep_mean_std[i])
166 linreg_mean_std (gsl_matrix_const_column (X, i), &m, &s, &ss);
167 gsl_vector_set (cache->indep_means, i, m);
168 gsl_vector_set (cache->indep_std, i, s);
169 gsl_vector_set (cache->ssx, i, ss);
172 cache->dft = cache->n_obs - 1;
173 cache->dfm = cache->n_indeps;
174 cache->dfe = cache->dft - cache->dfm;
175 if (cache->method == PSPP_LINREG_SWEEP)
179 Subtract the means to improve the condition of the design
180 matrix. This requires copying X and Y. We do not divide by the
181 standard deviations of the independent variables here since doing
182 so would cause a miscalculation of the residual sums of
183 squares. Dividing by the standard deviation is done GSL's linear
184 regression functions, so if the design matrix has a poor
185 condition, use QR decomposition.
187 The design matrix here does not include a column for the intercept
188 (i.e., a column of 1's). If using PSPP_LINREG_QR, we need that column,
189 so design is allocated here when sweeping, or below if using QR.
191 design = gsl_matrix_alloc (X->size1, X->size2);
192 for (i = 0; i < X->size2; i++)
194 m = gsl_vector_get (cache->indep_means, i);
195 for (j = 0; j < X->size1; j++)
197 tmp = (gsl_matrix_get (X, j, i) - m);
198 gsl_matrix_set (design, j, i, tmp);
201 sw = gsl_matrix_calloc (cache->n_indeps + 1, cache->n_indeps + 1);
202 xtx = gsl_matrix_submatrix (sw, 0, 0, cache->n_indeps, cache->n_indeps);
204 for (i = 0; i < xtx.matrix.size1; i++)
206 tmp = gsl_vector_get (cache->ssx, i);
207 gsl_matrix_set (&(xtx.matrix), i, i, tmp);
208 xi = gsl_matrix_column (design, i);
209 for (j = (i + 1); j < xtx.matrix.size2; j++)
211 xj = gsl_matrix_column (design, j);
212 gsl_blas_ddot (&(xi.vector), &(xj.vector), &tmp);
213 gsl_matrix_set (&(xtx.matrix), i, j, tmp);
217 gsl_matrix_set (sw, cache->n_indeps, cache->n_indeps, cache->sst);
218 xty = gsl_matrix_column (sw, cache->n_indeps);
220 This loop starts at 1, with i=0 outside the loop, so we can get
221 the model sum of squares due to the first independent variable.
223 xi = gsl_matrix_column (design, 0);
224 gsl_blas_ddot (&(xi.vector), Y, &tmp);
225 gsl_vector_set (&(xty.vector), 0, tmp);
226 tmp *= tmp / gsl_vector_get (cache->ssx, 0);
227 gsl_vector_set (cache->ss_indeps, 0, tmp);
228 for (i = 1; i < cache->n_indeps; i++)
230 xi = gsl_matrix_column (design, i);
231 gsl_blas_ddot (&(xi.vector), Y, &tmp);
232 gsl_vector_set (&(xty.vector), i, tmp);
236 Sweep on the matrix sw, which contains XtX, XtY and YtY.
239 cache->sse = gsl_matrix_get (sw, cache->n_indeps, cache->n_indeps);
240 cache->mse = cache->sse / cache->dfe;
244 m = cache->depvar_mean;
245 for (i = 0; i < cache->n_indeps; i++)
247 tmp = gsl_matrix_get (sw, i, cache->n_indeps);
248 cache->coeff[i + 1].estimate = tmp;
249 gsl_vector_set (cache->param_estimates, i + 1, tmp);
250 m -= tmp * gsl_vector_get (cache->indep_means, i);
253 Get the covariance matrix of the parameter estimates.
254 Only the upper triangle is necessary.
258 The loops below do not compute the entries related
259 to the estimated intercept.
261 for (i = 0; i < cache->n_indeps; i++)
262 for (j = i; j < cache->n_indeps; j++)
264 tmp = -1.0 * cache->mse * gsl_matrix_get (sw, i, j);
265 gsl_matrix_set (cache->cov, i + 1, j + 1, tmp);
268 Get the covariances related to the intercept.
270 xtx = gsl_matrix_submatrix (sw, 0, 0, cache->n_indeps, cache->n_indeps);
271 xmxtx = gsl_matrix_submatrix (cache->cov, 0, 1, 1, cache->n_indeps);
272 xm = gsl_matrix_view_vector (cache->indep_means, 1, cache->n_indeps);
273 rc = gsl_blas_dsymm (CblasRight, CblasUpper, cache->mse,
274 &xtx.matrix, &xm.matrix, 0.0, &xmxtx.matrix);
275 if (rc == GSL_SUCCESS)
277 tmp = cache->mse / cache->n_obs;
278 for (i = 1; i < 1 + cache->n_indeps; i++)
280 tmp -= gsl_matrix_get (cache->cov, 0, i)
281 * gsl_vector_get (cache->indep_means, i - 1);
283 gsl_matrix_set (cache->cov, 0, 0, tmp);
285 gsl_vector_set (cache->param_estimates, 0, m);
289 fprintf (stderr, "%s:%d:gsl_blas_dsymm: %s\n",
290 __FILE__, __LINE__, gsl_strerror (rc));
293 gsl_matrix_free (sw);
298 Use QR decomposition via GSL.
300 design = gsl_matrix_alloc (X->size1, 1 + X->size2);
302 for (j = 0; j < X->size1; j++)
304 gsl_matrix_set (design, j, 0, 1.0);
305 for (i = 0; i < X->size2; i++)
307 tmp = gsl_matrix_get (X, j, i);
308 gsl_matrix_set (design, j, i + 1, tmp);
311 gsl_multifit_linear_workspace *wk =
312 gsl_multifit_linear_alloc (design->size1, design->size2);
313 rc = gsl_multifit_linear (design, Y, cache->param_estimates,
314 cache->cov, &(cache->sse), wk);
315 if (rc == GSL_SUCCESS)
317 gsl_multifit_linear_free (wk);
321 fprintf (stderr, "%s:%d: gsl_multifit_linear returned %d\n",
322 __FILE__, __LINE__, rc);
327 cache->ssm = cache->sst - cache->sse;
329 Get the remaining sums of squares for the independent
333 for (i = 1; i < cache->n_indeps; i++)
336 m += gsl_vector_get (cache->ss_indeps, j);
337 tmp = cache->ssm - m;
338 gsl_vector_set (cache->ss_indeps, i, tmp);
341 gsl_matrix_free (design);