/* multifit/multilinear.c
*
* Copyright (C) 2000, 2007, 2010 Brian Gough
* Copyright (C) 2013, 2015 Patrick Alken
*
* 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, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
#include <config.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_multifit.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_linalg.h>
#include "linear_common.c"
static int multifit_linear_svd (const gsl_matrix * X,
const int balance,
gsl_multifit_linear_workspace * work);
int
gsl_multifit_linear (const gsl_matrix * X,
const gsl_vector * y,
gsl_vector * c,
gsl_matrix * cov,
double *chisq, gsl_multifit_linear_workspace * work)
{
size_t rank;
int status = gsl_multifit_linear_tsvd(X, y, GSL_DBL_EPSILON, c, cov, chisq, &rank, work);
return status;
}
/*
gsl_multifit_linear_tsvd()
Solve linear least squares system with truncated SVD
Inputs: X - least squares matrix, n-by-p
y - right hand side vector, n-by-1
tol - tolerance for singular value truncation; if
s_j <= tol * s_0
then it is discarded from series expansion
c - (output) solution vector, p-by-1
cov - (output) covariance matrix, p-by-p
chisq - (output) cost function chi^2
rank - (output) effective rank (number of singular values
used in solution)
work - workspace
*/
int
gsl_multifit_linear_tsvd (const gsl_matrix * X,
const gsl_vector * y,
const double tol,
gsl_vector * c,
gsl_matrix * cov,
double * chisq,
size_t * rank,
gsl_multifit_linear_workspace * work)
{
const size_t n = X->size1;
const size_t p = X->size2;
if (y->size != n)
{
GSL_ERROR("number of observations in y does not match matrix",
GSL_EBADLEN);
}
else if (p != c->size)
{
GSL_ERROR ("number of parameters c does not match matrix",
GSL_EBADLEN);
}
else if (tol <= 0)
{
GSL_ERROR ("tolerance must be positive", GSL_EINVAL);
}
else
{
int status;
double rnorm = 0.0, snorm;
/* compute balanced SVD */
status = gsl_multifit_linear_bsvd (X, work);
if (status)
return status;
status = multifit_linear_solve (X, y, tol, -1.0, rank,
c, &rnorm, &snorm, work);
*chisq = rnorm * rnorm;
/* variance-covariance matrix cov = s2 * (Q S^-1) (Q S^-1)^T */
{
double r2 = rnorm * rnorm;
double s2 = r2 / (double)(n - *rank);
size_t i, j;
gsl_matrix_view QSI = gsl_matrix_submatrix(work->QSI, 0, 0, p, p);
gsl_vector_view D = gsl_vector_subvector(work->D, 0, p);
for (i = 0; i < p; i++)
{
gsl_vector_view row_i = gsl_matrix_row (&QSI.matrix, i);
double d_i = gsl_vector_get (&D.vector, i);
for (j = i; j < p; j++)
{
gsl_vector_view row_j = gsl_matrix_row (&QSI.matrix, j);
double d_j = gsl_vector_get (&D.vector, j);
double s;
gsl_blas_ddot (&row_i.vector, &row_j.vector, &s);
gsl_matrix_set (cov, i, j, s * s2 / (d_i * d_j));
gsl_matrix_set (cov, j, i, s * s2 / (d_i * d_j));
}
}
}
return status;
}
}
/*
gsl_multifit_linear_svd()
Perform SVD decomposition of the matrix X and store in work without
balancing
*/
int
gsl_multifit_linear_svd (const gsl_matrix * X,
gsl_multifit_linear_workspace * work)
{
/* do not balance by default */
int status = multifit_linear_svd(X, 0, work);
return status;
}
/*
gsl_multifit_linear_bsvd()
Perform SVD decomposition of the matrix X and store in work with
balancing
*/
int
gsl_multifit_linear_bsvd (const gsl_matrix * X,
gsl_multifit_linear_workspace * work)
{
int status = multifit_linear_svd(X, 1, work);
return status;
}
size_t
gsl_multifit_linear_rank(const double tol, const gsl_multifit_linear_workspace * work)
{
double s0 = gsl_vector_get (work->S, 0);
size_t rank = 0;
size_t j;
for (j = 0; j < work->p; j++)
{
double sj = gsl_vector_get (work->S, j);
if (sj > tol * s0)
++rank;
}
return rank;
}
/* Estimation of values for given x */
int
gsl_multifit_linear_est (const gsl_vector * x,
const gsl_vector * c,
const gsl_matrix * cov, double *y, double *y_err)
{
if (x->size != c->size)
{
GSL_ERROR ("number of parameters c does not match number of observations x",
GSL_EBADLEN);
}
else if (cov->size1 != cov->size2)
{
GSL_ERROR ("covariance matrix is not square", GSL_ENOTSQR);
}
else if (c->size != cov->size1)
{
GSL_ERROR ("number of parameters c does not match size of covariance matrix cov",
GSL_EBADLEN);
}
else
{
size_t i, j;
double var = 0;
gsl_blas_ddot(x, c, y); /* y = x.c */
/* var = x' cov x */
for (i = 0; i < x->size; i++)
{
const double xi = gsl_vector_get (x, i);
var += xi * xi * gsl_matrix_get (cov, i, i);
for (j = 0; j < i; j++)
{
const double xj = gsl_vector_get (x, j);
var += 2 * xi * xj * gsl_matrix_get (cov, i, j);
}
}
*y_err = sqrt (var);
return GSL_SUCCESS;
}
}
/*
gsl_multifit_linear_rcond()
Return reciprocal condition number of LS matrix;
gsl_multifit_linear_svd() must first be called to
compute the SVD of X and its reciprocal condition number
*/
double
gsl_multifit_linear_rcond (const gsl_multifit_linear_workspace * w)
{
return w->rcond;
}
/*
gsl_multifit_linear_residuals()
Compute vector of residuals from fit
Inputs: X - design matrix
y - rhs vector
c - fit coefficients
r - (output) where to store residuals
*/
int
gsl_multifit_linear_residuals (const gsl_matrix *X, const gsl_vector *y,
const gsl_vector *c, gsl_vector *r)
{
if (X->size1 != y->size)
{
GSL_ERROR
("number of observations in y does not match rows of matrix X",
GSL_EBADLEN);
}
else if (X->size2 != c->size)
{
GSL_ERROR ("number of parameters c does not match columns of matrix X",
GSL_EBADLEN);
}
else if (y->size != r->size)
{
GSL_ERROR ("number of observations in y does not match number of residuals",
GSL_EBADLEN);
}
else
{
/* r = y - X c */
gsl_vector_memcpy(r, y);
gsl_blas_dgemv(CblasNoTrans, -1.0, X, c, 1.0, r);
return GSL_SUCCESS;
}
} /* gsl_multifit_linear_residuals() */
/* Perform a SVD decomposition on the least squares matrix X = U S Q^T
*
* Inputs: X - least squares matrix
* balance - 1 to perform column balancing
* work - workspace
*
* Notes:
* 1) On output,
* work->A contains the matrix U
* work->Q contains the matrix Q
* work->S contains the vector of singular values
* 2) The matrix X may have smaller dimensions than the workspace
* in the case of stdform2() - but the dimensions cannot be larger
* 3) On output, work->n and work->p are set to the dimensions of X
* 4) On output, work->rcond is set to the reciprocal condition number of X
*/
static int
multifit_linear_svd (const gsl_matrix * X,
const int balance,
gsl_multifit_linear_workspace * work)
{
const size_t n = X->size1;
const size_t p = X->size2;
if (n > work->nmax || p > work->pmax)
{
GSL_ERROR("observation matrix larger than workspace", GSL_EBADLEN);
}
else
{
gsl_matrix_view A = gsl_matrix_submatrix(work->A, 0, 0, n, p);
gsl_matrix_view Q = gsl_matrix_submatrix(work->Q, 0, 0, p, p);
gsl_matrix_view QSI = gsl_matrix_submatrix(work->QSI, 0, 0, p, p);
gsl_vector_view S = gsl_vector_subvector(work->S, 0, p);
gsl_vector_view xt = gsl_vector_subvector(work->xt, 0, p);
gsl_vector_view D = gsl_vector_subvector(work->D, 0, p);
/* Copy X to workspace, A <= X */
gsl_matrix_memcpy (&A.matrix, X);
/* Balance the columns of the matrix A if requested */
if (balance)
{
gsl_linalg_balance_columns (&A.matrix, &D.vector);
}
else
{
gsl_vector_set_all (&D.vector, 1.0);
}
/* decompose A into U S Q^T */
gsl_linalg_SV_decomp_mod (&A.matrix, &QSI.matrix, &Q.matrix,
&S.vector, &xt.vector);
/* compute reciprocal condition number rcond = smin / smax */
{
double smin, smax;
gsl_vector_minmax(&S.vector, &smin, &smax);
work->rcond = smin / smax;
}
work->n = n;
work->p = p;
return GSL_SUCCESS;
}
}