--- /dev/null
+/* PSPP - a program for statistical analysis.
+ Copyright (C) 2005 Free Software Foundation, Inc.
+
+ This program is free software: you can redistribute it and/or modify
+ it under the terms of the GNU General Public License as published by
+ the Free Software Foundation, either version 3 of the License, or
+ (at your option) any later version.
+
+ This program is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ GNU General Public License for more details.
+
+ You should have received a copy of the GNU General Public License
+ along with this program. If not, see <http://www.gnu.org/licenses/>. */
+
+#include <config.h>
+#include <gsl/gsl_blas.h>
+#include <gsl/gsl_cblas.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_fit.h>
+#include <gsl/gsl_multifit.h>
+#include <linreg/sweep.h>
+#include <math/coefficient.h>
+#include <math/linreg.h>
+#include <math/coefficient.h>
+#include <src/data/variable.h>
+#include <src/data/value.h>
+#include <gl/xalloc.h>
+
+/*
+ Find the least-squares estimate of b for the linear model:
+
+ Y = Xb + Z
+
+ where Y is an n-by-1 column vector, X is an n-by-p matrix of
+ independent variables, b is a p-by-1 vector of regression coefficients,
+ and Z is an n-by-1 normally-distributed random vector with independent
+ identically distributed components with mean 0.
+
+ This estimate is found via the sweep operator or singular-value
+ decomposition with gsl.
+
+
+ References:
+
+ 1. Matrix Computations, third edition. GH Golub and CF Van Loan.
+ The Johns Hopkins University Press. 1996. ISBN 0-8018-5414-8.
+
+ 2. Numerical Analysis for Statisticians. K Lange. Springer. 1999.
+ ISBN 0-387-94979-8.
+
+ 3. Numerical Linear Algebra for Applications in Statistics. JE Gentle.
+ Springer. 1998. ISBN 0-387-98542-5.
+*/
+
+
+/*
+ Get the mean and standard deviation of a vector
+ of doubles via a form of the Kalman filter as
+ described on page 32 of [3].
+ */
+static int
+linreg_mean_std (gsl_vector_const_view v, double *mp, double *sp, double *ssp)
+{
+ size_t i;
+ double j = 0.0;
+ double d;
+ double tmp;
+ double mean;
+ double variance;
+
+ mean = gsl_vector_get (&v.vector, 0);
+ variance = 0;
+ for (i = 1; i < v.vector.size; i++)
+ {
+ j = (double) i + 1.0;
+ tmp = gsl_vector_get (&v.vector, i);
+ d = (tmp - mean) / j;
+ mean += d;
+ variance += j * (j - 1.0) * d * d;
+ }
+ *mp = mean;
+ *sp = sqrt (variance / (j - 1.0));
+ *ssp = variance;
+
+ return GSL_SUCCESS;
+}
+
+/*
+ Set V to contain an array of pointers to the variables
+ used in the model. V must be at least C->N_COEFFS in length.
+ The return value is the number of distinct variables found.
+ */
+int
+pspp_linreg_get_vars (const void *c_, const struct variable **v)
+{
+ const pspp_linreg_cache *c = c_;
+ const struct variable *tmp;
+ int i;
+ int j;
+ int result = 0;
+
+ /*
+ Make sure the caller doesn't try to sneak a variable
+ into V that is not in the model.
+ */
+ for (i = 0; i < c->n_coeffs; i++)
+ {
+ v[i] = NULL;
+ }
+ for (j = 0; j < c->n_coeffs; j++)
+ {
+ tmp = pspp_coeff_get_var (c->coeff[j], 0);
+ assert (tmp != NULL);
+ /* Repeated variables are likely to bunch together, at the end
+ of the array. */
+ i = result - 1;
+ while (i >= 0 && v[i] != tmp)
+ {
+ i--;
+ }
+ if (i < 0 && result < c->n_coeffs)
+ {
+ v[result] = tmp;
+ result++;
+ }
+ }
+ return result;
+}
+
+/*
+ Allocate a pspp_linreg_cache and return a pointer
+ to it. n is the number of cases, p is the number of
+ independent variables.
+ */
+pspp_linreg_cache *
+pspp_linreg_cache_alloc (size_t n, size_t p)
+{
+ pspp_linreg_cache *c;
+
+ c = (pspp_linreg_cache *) malloc (sizeof (pspp_linreg_cache));
+ c->depvar = NULL;
+ c->indep_means = gsl_vector_alloc (p);
+ c->indep_std = gsl_vector_alloc (p);
+ c->ssx = gsl_vector_alloc (p); /* Sums of squares for the
+ independent variables.
+ */
+ c->ss_indeps = gsl_vector_alloc (p); /* Sums of squares for the
+ model parameters.
+ */
+ c->cov = gsl_matrix_alloc (p + 1, p + 1); /* Covariance matrix. */
+ c->n_obs = n;
+ c->n_indeps = p;
+ /*
+ Default settings.
+ */
+ c->method = PSPP_LINREG_SWEEP;
+ c->predict = pspp_linreg_predict;
+ c->residual = pspp_linreg_residual; /* The procedure to compute my
+ residuals. */
+ c->get_vars = pspp_linreg_get_vars; /* The procedure that returns
+ pointers to model
+ variables. */
+ c->resid = NULL; /* The variable storing my residuals. */
+ c->pred = NULL; /* The variable storing my predicted values. */
+
+ return c;
+}
+
+bool
+pspp_linreg_cache_free (void *m)
+{
+ int i;
+
+ pspp_linreg_cache *c = m;
+ if (c != NULL)
+ {
+ gsl_vector_free (c->indep_means);
+ gsl_vector_free (c->indep_std);
+ gsl_vector_free (c->ss_indeps);
+ gsl_matrix_free (c->cov);
+ gsl_vector_free (c->ssx);
+ for (i = 0; i < c->n_coeffs; i++)
+ {
+ pspp_coeff_free (c->coeff[i]);
+ }
+ free (c->coeff);
+ free (c);
+ }
+ return true;
+}
+
+/*
+ Fit the linear model via least squares. All pointers passed to pspp_linreg
+ are assumed to be allocated to the correct size and initialized to the
+ values as indicated by opts.
+ */
+int
+pspp_linreg (const gsl_vector * Y, const gsl_matrix * X,
+ const pspp_linreg_opts * opts, pspp_linreg_cache * cache)
+{
+ int rc;
+ gsl_matrix *design = NULL;
+ gsl_matrix_view xtx;
+ gsl_matrix_view xm;
+ gsl_matrix_view xmxtx;
+ gsl_vector_view xty;
+ gsl_vector_view xi;
+ gsl_vector_view xj;
+ gsl_vector *param_estimates;
+
+ size_t i;
+ size_t j;
+ double tmp;
+ double m;
+ double s;
+ double ss;
+
+ if (cache == NULL)
+ {
+ return GSL_EFAULT;
+ }
+ if (opts->get_depvar_mean_std)
+ {
+ linreg_mean_std (gsl_vector_const_subvector (Y, 0, Y->size),
+ &m, &s, &ss);
+ cache->depvar_mean = m;
+ cache->depvar_std = s;
+ cache->sst = ss;
+ }
+ for (i = 0; i < cache->n_indeps; i++)
+ {
+ if (opts->get_indep_mean_std[i])
+ {
+ linreg_mean_std (gsl_matrix_const_column (X, i), &m, &s, &ss);
+ gsl_vector_set (cache->indep_means, i, m);
+ gsl_vector_set (cache->indep_std, i, s);
+ gsl_vector_set (cache->ssx, i, ss);
+ }
+ }
+ cache->dft = cache->n_obs - 1;
+ cache->dfm = cache->n_indeps;
+ cache->dfe = cache->dft - cache->dfm;
+ cache->n_coeffs = X->size2;
+ cache->intercept = 0.0;
+
+ if (cache->method == PSPP_LINREG_SWEEP)
+ {
+ gsl_matrix *sw;
+ /*
+ Subtract the means to improve the condition of the design
+ matrix. This requires copying X and Y. We do not divide by the
+ standard deviations of the independent variables here since doing
+ so would cause a miscalculation of the residual sums of
+ squares. Dividing by the standard deviation is done GSL's linear
+ regression functions, so if the design matrix has a poor
+ condition, use QR decomposition.
+
+ The design matrix here does not include a column for the intercept
+ (i.e., a column of 1's). If using PSPP_LINREG_QR, we need that column,
+ so design is allocated here when sweeping, or below if using QR.
+ */
+ design = gsl_matrix_alloc (X->size1, X->size2);
+ for (i = 0; i < X->size2; i++)
+ {
+ m = gsl_vector_get (cache->indep_means, i);
+ for (j = 0; j < X->size1; j++)
+ {
+ tmp = (gsl_matrix_get (X, j, i) - m);
+ gsl_matrix_set (design, j, i, tmp);
+ }
+ }
+ sw = gsl_matrix_calloc (cache->n_indeps + 1, cache->n_indeps + 1);
+ xtx = gsl_matrix_submatrix (sw, 0, 0, cache->n_indeps, cache->n_indeps);
+
+ for (i = 0; i < xtx.matrix.size1; i++)
+ {
+ tmp = gsl_vector_get (cache->ssx, i);
+ gsl_matrix_set (&(xtx.matrix), i, i, tmp);
+ xi = gsl_matrix_column (design, i);
+ for (j = (i + 1); j < xtx.matrix.size2; j++)
+ {
+ xj = gsl_matrix_column (design, j);
+ gsl_blas_ddot (&(xi.vector), &(xj.vector), &tmp);
+ gsl_matrix_set (&(xtx.matrix), i, j, tmp);
+ }
+ }
+
+ gsl_matrix_set (sw, cache->n_indeps, cache->n_indeps, cache->sst);
+ xty = gsl_matrix_column (sw, cache->n_indeps);
+ /*
+ This loop starts at 1, with i=0 outside the loop, so we can get
+ the model sum of squares due to the first independent variable.
+ */
+ xi = gsl_matrix_column (design, 0);
+ gsl_blas_ddot (&(xi.vector), Y, &tmp);
+ gsl_vector_set (&(xty.vector), 0, tmp);
+ tmp *= tmp / gsl_vector_get (cache->ssx, 0);
+ gsl_vector_set (cache->ss_indeps, 0, tmp);
+ for (i = 1; i < cache->n_indeps; i++)
+ {
+ xi = gsl_matrix_column (design, i);
+ gsl_blas_ddot (&(xi.vector), Y, &tmp);
+ gsl_vector_set (&(xty.vector), i, tmp);
+ }
+
+ /*
+ Sweep on the matrix sw, which contains XtX, XtY and YtY.
+ */
+ reg_sweep (sw);
+ cache->sse = gsl_matrix_get (sw, cache->n_indeps, cache->n_indeps);
+ cache->mse = cache->sse / cache->dfe;
+ /*
+ Get the intercept.
+ */
+ m = cache->depvar_mean;
+ for (i = 0; i < cache->n_indeps; i++)
+ {
+ tmp = gsl_matrix_get (sw, i, cache->n_indeps);
+ cache->coeff[i]->estimate = tmp;
+ m -= tmp * gsl_vector_get (cache->indep_means, i);
+ }
+ /*
+ Get the covariance matrix of the parameter estimates.
+ Only the upper triangle is necessary.
+ */
+
+ /*
+ The loops below do not compute the entries related
+ to the estimated intercept.
+ */
+ for (i = 0; i < cache->n_indeps; i++)
+ for (j = i; j < cache->n_indeps; j++)
+ {
+ tmp = -1.0 * cache->mse * gsl_matrix_get (sw, i, j);
+ gsl_matrix_set (cache->cov, i + 1, j + 1, tmp);
+ }
+ /*
+ Get the covariances related to the intercept.
+ */
+ xtx = gsl_matrix_submatrix (sw, 0, 0, cache->n_indeps, cache->n_indeps);
+ xmxtx = gsl_matrix_submatrix (cache->cov, 0, 1, 1, cache->n_indeps);
+ xm = gsl_matrix_view_vector (cache->indep_means, 1, cache->n_indeps);
+ rc = gsl_blas_dsymm (CblasRight, CblasUpper, cache->mse,
+ &xtx.matrix, &xm.matrix, 0.0, &xmxtx.matrix);
+ if (rc == GSL_SUCCESS)
+ {
+ tmp = cache->mse / cache->n_obs;
+ for (i = 1; i < 1 + cache->n_indeps; i++)
+ {
+ tmp -= gsl_matrix_get (cache->cov, 0, i)
+ * gsl_vector_get (cache->indep_means, i - 1);
+ }
+ gsl_matrix_set (cache->cov, 0, 0, tmp);
+
+ cache->intercept = m;
+ }
+ else
+ {
+ fprintf (stderr, "%s:%d:gsl_blas_dsymm: %s\n",
+ __FILE__, __LINE__, gsl_strerror (rc));
+ exit (rc);
+ }
+ gsl_matrix_free (sw);
+ }
+ else if (cache->method == PSPP_LINREG_CONDITIONAL_INVERSE)
+ {
+ /*
+ Use the SVD of X^T X to find a conditional inverse of X^TX. If
+ the SVD is X^T X = U D V^T, then set the conditional inverse
+ to (X^T X)^c = V D^- U^T. D^- is defined as follows: If entry
+ (i, i) has value sigma_i, then entry (i, i) of D^- is 1 /
+ sigma_i if sigma_i > 0, and 0 otherwise. Then solve the normal
+ equations by setting the estimated parameter vector to
+ (X^TX)^c X^T Y.
+ */
+ }
+ else
+ {
+ gsl_multifit_linear_workspace *wk;
+ /*
+ Use QR decomposition via GSL.
+ */
+
+ param_estimates = gsl_vector_alloc (1 + X->size2);
+ design = gsl_matrix_alloc (X->size1, 1 + X->size2);
+
+ for (j = 0; j < X->size1; j++)
+ {
+ gsl_matrix_set (design, j, 0, 1.0);
+ for (i = 0; i < X->size2; i++)
+ {
+ tmp = gsl_matrix_get (X, j, i);
+ gsl_matrix_set (design, j, i + 1, tmp);
+ }
+ }
+
+ wk = gsl_multifit_linear_alloc (design->size1, design->size2);
+ rc = gsl_multifit_linear (design, Y, param_estimates,
+ cache->cov, &(cache->sse), wk);
+ for (i = 0; i < cache->n_coeffs; i++)
+ {
+ cache->coeff[i]->estimate = gsl_vector_get (param_estimates, i + 1);
+ }
+ cache->intercept = gsl_vector_get (param_estimates, 0);
+ if (rc == GSL_SUCCESS)
+ {
+ gsl_multifit_linear_free (wk);
+ gsl_vector_free (param_estimates);
+ }
+ else
+ {
+ fprintf (stderr, "%s:%d: gsl_multifit_linear returned %d\n",
+ __FILE__, __LINE__, rc);
+ }
+ }
+
+
+ cache->ssm = cache->sst - cache->sse;
+ /*
+ Get the remaining sums of squares for the independent
+ variables.
+ */
+ m = 0;
+ for (i = 1; i < cache->n_indeps; i++)
+ {
+ j = i - 1;
+ m += gsl_vector_get (cache->ss_indeps, j);
+ tmp = cache->ssm - m;
+ gsl_vector_set (cache->ss_indeps, i, tmp);
+ }
+
+ gsl_matrix_free (design);
+ return GSL_SUCCESS;
+}
+
+/*
+ Is the coefficient COEF contained in the list of coefficients
+ COEF_LIST?
+ */
+static int
+has_coefficient (const struct pspp_coeff **coef_list, const struct pspp_coeff *coef,
+ size_t n)
+{
+ size_t i = 0;
+
+ while (i < n)
+ {
+ if (coef_list[i] == coef)
+ {
+ return 1;
+ }
+ i++;
+ }
+ return 0;
+}
+/*
+ Predict the value of the dependent variable with the
+ new set of predictors. PREDICTORS must point to a list
+ of variables, each of whose values are stored in VALS,
+ in the same order.
+ */
+double
+pspp_linreg_predict (const struct variable **predictors,
+ const union value **vals, const void *c_, int n_vals)
+{
+ const pspp_linreg_cache *c = c_;
+ int j;
+ size_t next_coef = 0;
+ const struct pspp_coeff **coef_list;
+ const struct pspp_coeff *coe;
+ double result;
+ double tmp;
+
+ if (predictors == NULL || vals == NULL || c == NULL)
+ {
+ return GSL_NAN;
+ }
+ if (c->coeff == NULL)
+ {
+ /* The stupid model: just guess the mean. */
+ return c->depvar_mean;
+ }
+ coef_list = xnmalloc (c->n_coeffs, sizeof (*coef_list));
+ result = c->intercept;
+
+ /*
+ The loops guard against the possibility that the caller passed us
+ inadequate information, such as too few or too many values, or
+ a redundant list of variable names.
+ */
+ for (j = 0; j < n_vals; j++)
+ {
+ coe = pspp_linreg_get_coeff (c, predictors[j], vals[j]);
+ if (!has_coefficient (coef_list, coe, next_coef))
+ {
+ tmp = pspp_coeff_get_est (coe);
+ if (var_is_numeric (predictors[j]))
+ {
+ tmp *= vals[j]->f;
+ }
+ result += tmp;
+ coef_list[next_coef++] = coe;
+ }
+ }
+ free (coef_list);
+
+ return result;
+}
+
+double
+pspp_linreg_residual (const struct variable **predictors,
+ const union value **vals,
+ const union value *obs, const void *c, int n_vals)
+{
+ double pred;
+ double result;
+
+ if (predictors == NULL || vals == NULL || c == NULL || obs == NULL)
+ {
+ return GSL_NAN;
+ }
+ pred = pspp_linreg_predict (predictors, vals, c, n_vals);
+
+ result = gsl_isnan (pred) ? GSL_NAN : (obs->f - pred);
+ return result;
+}
+
+/*
+ Which coefficient is associated with V? The VAL argument is relevant
+ only to categorical variables.
+ */
+const struct pspp_coeff *
+pspp_linreg_get_coeff (const pspp_linreg_cache * c,
+ const struct variable *v, const union value *val)
+{
+ int i;
+ struct pspp_coeff *result = NULL;
+ const struct variable *tmp = NULL;
+
+ if (c == NULL)
+ {
+ return NULL;
+ }
+ if (c->coeff == NULL || c->n_indeps == 0 || v == NULL)
+ {
+ return NULL;
+ }
+ i = 0;
+ result = c->coeff[0];
+ tmp = pspp_coeff_get_var (result, 0);
+ while (tmp != v && i < c->n_coeffs)
+ {
+ result = c->coeff[i];
+ tmp = pspp_coeff_get_var (result, 0);
+ i++;
+ }
+ if (tmp != v)
+ {
+ /*
+ Not found.
+ */
+ return NULL;
+ }
+ if (var_is_numeric (v))
+ {
+ return result;
+ }
+ else if (val != NULL)
+ {
+ /*
+ If v is categorical, we need to ensure the coefficient
+ matches the VAL.
+ */
+ while (tmp != v && i < c->n_coeffs
+ && compare_values (pspp_coeff_get_value (result, tmp),
+ val, var_get_width (v)))
+ { /* FIX THIS */
+ i++;
+ result = c->coeff[i];
+ tmp = pspp_coeff_get_var (result, 0);
+ }
+ if (i == c->n_coeffs && tmp != v)
+ {
+ return NULL;
+ }
+ return result;
+ }
+ return NULL;
+}
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