1 /* PSPP - a program for statistical analysis.
2 Copyright (C) 2006 Free Software Foundation, Inc.
4 This program is free software: you can redistribute it and/or modify
5 it under the terms of the GNU General Public License as published by
6 the Free Software Foundation, either version 3 of the License, or
7 (at your option) any later version.
9 This program is distributed in the hope that it will be useful,
10 but WITHOUT ANY WARRANTY; without even the implied warranty of
11 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
12 GNU General Public License for more details.
14 You should have received a copy of the GNU General Public License
15 along with this program. If not, see <http://www.gnu.org/licenses/>. */
18 Find preliminary ARMA coefficients via the innovations algorithm.
19 Also compute the sample mean and covariance matrix for each series.
23 P. J. Brockwell and R. A. Davis. Time Series: Theory and
24 Methods. Second edition. Springer. New York. 1991. ISBN
25 0-387-97429-6. Sections 5.2, 8.3 and 8.4.
29 #include <gsl/gsl_matrix.h>
30 #include <gsl/gsl_vector.h>
31 #include <gsl/gsl_math.h>
33 #include <libpspp/alloc.h>
34 #include <libpspp/compiler.h>
35 #include <math/coefficient.h>
36 #include <math/ts/innovations.h>
39 get_mean (const gsl_matrix *data,
40 struct innovations_estimate **est)
48 for (n = 0; n < data->size2; n++)
53 for (i = 0; i < data->size1; i++)
55 for (n = 0; n < data->size2; n++)
57 tmp = gsl_matrix_get (data, i, n);
61 d = (tmp - est[n]->mean) / est[n]->n_obs;
68 update_cov (struct innovations_estimate **est, gsl_vector_const_view x,
69 gsl_vector_const_view y, size_t lag)
75 for (j = 0; j < x.vector.size; j++)
77 xj = gsl_vector_get (&x.vector, j);
78 yj = gsl_vector_get (&y.vector, j);
85 *(est[j]->cov + lag) += xj * yj;
91 get_covariance (const gsl_matrix *data,
92 struct innovations_estimate **est, size_t max_lag)
99 assert (data != NULL);
100 assert (est != NULL);
102 for (j = 0; j < data->size2; j++)
104 for (lag = 0; lag <= max_lag; lag++)
106 *(est[j]->cov + lag) = 0.0;
110 The rows are in the outer loop because a gsl_matrix is stored in
113 for (i = 0; i < data->size1; i++)
115 for (lag = 0; lag <= max_lag && lag < data->size1 - i; lag++)
117 update_cov (est, gsl_matrix_const_row (data, i),
118 gsl_matrix_const_row (data, i + lag), lag);
121 for (j = 0; j < data->size2; j++)
123 for (lag = 0; lag <= max_lag; lag++)
125 *(est[j]->cov + lag) /= est[j]->n_obs;
133 innovations_convolve (double *x, double *y, struct innovations_estimate *est,
139 assert (x != NULL && y != NULL);
140 assert (est != NULL);
141 assert (est->scale != NULL);
143 for (k = 0; k < i; k++)
145 result += x[k] * y[k] * est->scale[i-k-1];
150 innovations_update_scale (struct innovations_estimate *est, double *theta,
157 if (i < (size_t) est->max_lag)
159 result = est->cov[0];
160 for (j = 0; j < i; j++)
163 result -= theta[k] * theta[k] * est->scale[j];
165 est->scale[i] = result;
169 init_theta (double **theta, size_t max_lag)
174 for (i = 0; i < max_lag; i++)
176 for (j = 0; j <= i; j++)
183 innovations_update_coeff (double **theta, struct innovations_estimate *est,
190 for (i = 0; i < max_lag; i++)
192 theta[i][i] = est->cov[i+1] / est->scale[0];
193 for (j = 1; j <= i; j++)
196 theta[i][k] = (est->cov[k+1] -
197 innovations_convolve (theta[i] + k + 1, theta[j - 1], est, j))
200 innovations_update_scale (est, theta[i], i + 1);
204 get_coef (const gsl_matrix *data,
205 struct innovations_estimate **est, size_t max_lag)
211 theta = xnmalloc (max_lag, sizeof (*theta));
212 for (i = 0; i < max_lag; i++)
214 theta[i] = xnmalloc (max_lag, sizeof (**(theta + i)));
217 for (n = 0; n < data->size2; n++)
219 init_theta (theta, max_lag);
220 innovations_update_scale (est[n], theta[0], 0);
221 innovations_update_coeff (theta, est[n], max_lag);
222 /* Copy the final row of coefficients into EST->COEFF.*/
223 for (i = 0; i < max_lag; i++)
226 The order of storage here means that the best predicted value
227 for the time series is computed as follows:
229 Let X[m], X[m-1],... denote the original series.
230 Let X_hat[0] denote the best predicted value of X[0],
231 X_hat[1] denote the projection of X[1] onto the subspace
232 spanned by {X[0] - X_hat[0]}. Let X_hat[m] denote the
233 projection of X[m] onto the subspace spanned by {X[m-1] - X_hat[m-1],
234 X[m-2] - X_hat[m-2],...,X[0] - X_hat[0]}.
236 Then X_hat[m] = est->coeff[m-1] * (X[m-1] - X_hat[m-1])
237 + est->coeff[m-1] * (X[m-2] - X_hat[m-2])
239 + est->coeff[m-max_lag] * (X[m - max_lag] - X_hat[m - max_lag])
241 pspp_coeff_set_estimate (est[n]->coeff[i], theta[max_lag - 1][i]);
245 for (i = 0; i < max_lag; i++)
253 innovations_struct_init (struct innovations_estimate *est,
254 const struct design_matrix *dm,
260 /* COV[0] stores the lag 0 covariance (i.e., the variance), COV[1]
261 holds the lag-1 covariance, etc.
263 est->cov = xnmalloc (lag + 1, sizeof (*est->cov));
264 est->scale = xnmalloc (lag + 1, sizeof (*est->scale));
265 est->coeff = xnmalloc (lag, sizeof (*est->coeff)); /* No intercept. */
268 The loop below is an unusual use of PSPP_COEFF_INIT(). In a
269 typical model, one column of a DESIGN_MATRIX has one
270 coefficient. But in a time-series model, one column has many
273 for (j = 0; j < lag; j++)
275 pspp_coeff_init (est->coeff + j, dm);
277 est->max_lag = (double) lag;
280 The mean is subtracted from the original data before computing the
281 coefficients. The mean is NOT added back, so if you want to predict
282 a new value, you must add the mean to X_hat[m] to get the correct
286 subtract_mean (gsl_matrix *m, struct innovations_estimate **est)
292 for (i = 0; i < m->size1; i++)
294 for (j = 0; j < m->size2; j++)
296 tmp = gsl_matrix_get (m, i, j) - est[j]->mean;
297 gsl_matrix_set (m, i, j, tmp);
301 struct innovations_estimate **
302 pspp_innovations (const struct design_matrix *dm, size_t lag)
304 struct innovations_estimate **est;
307 est = xnmalloc (dm->m->size2, sizeof *est);
308 for (i = 0; i < dm->m->size2; i++)
310 est[i] = xmalloc (sizeof *est[i]);
311 /* est[i]->variable = vars[i]; */
312 innovations_struct_init (est[i], dm, lag);
315 get_mean (dm->m, est);
316 subtract_mean (dm->m, est);
317 get_covariance (dm->m, est, lag);
318 get_coef (dm->m, est, lag);
324 pspp_innovations_free_one (struct innovations_estimate *est)
328 assert (est != NULL);
329 for (i = 0; i < (size_t) est->max_lag; i++)
331 pspp_coeff_free (est->coeff[i]);
338 void pspp_innovations_free (struct innovations_estimate **est, size_t n)
342 assert (est != NULL);
343 for (i = 0; i < n; i++)
345 pspp_innovations_free_one (est[i]);