/* fit/linear.c
*
* Copyright (C) 2000, 2007 Brian Gough
*
* 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_fit.h>
/* Fit the data (x_i, y_i) to the linear relationship
Y = c0 + c1 x
returning,
c0, c1 -- coefficients
cov00, cov01, cov11 -- variance-covariance matrix of c0 and c1,
sumsq -- sum of squares of residuals
This fit can be used in the case where the errors for the data are
uknown, but assumed equal for all points. The resulting
variance-covariance matrix estimates the error in the coefficients
from the observed variance of the points around the best fit line.
*/
int
gsl_fit_linear (const double *x, const size_t xstride,
const double *y, const size_t ystride,
const size_t n,
double *c0, double *c1,
double *cov_00, double *cov_01, double *cov_11, double *sumsq)
{
double m_x = 0, m_y = 0, m_dx2 = 0, m_dxdy = 0;
size_t i;
for (i = 0; i < n; i++)
{
m_x += (x[i * xstride] - m_x) / (i + 1.0);
m_y += (y[i * ystride] - m_y) / (i + 1.0);
}
for (i = 0; i < n; i++)
{
const double dx = x[i * xstride] - m_x;
const double dy = y[i * ystride] - m_y;
m_dx2 += (dx * dx - m_dx2) / (i + 1.0);
m_dxdy += (dx * dy - m_dxdy) / (i + 1.0);
}
/* In terms of y = a + b x */
{
double s2 = 0, d2 = 0;
double b = m_dxdy / m_dx2;
double a = m_y - m_x * b;
*c0 = a;
*c1 = b;
/* Compute chi^2 = \sum (y_i - (a + b * x_i))^2 */
for (i = 0; i < n; i++)
{
const double dx = x[i * xstride] - m_x;
const double dy = y[i * ystride] - m_y;
const double d = dy - b * dx;
d2 += d * d;
}
s2 = d2 / (n - 2.0); /* chisq per degree of freedom */
*cov_00 = s2 * (1.0 / n) * (1 + m_x * m_x / m_dx2);
*cov_11 = s2 * 1.0 / (n * m_dx2);
*cov_01 = s2 * (-m_x) / (n * m_dx2);
*sumsq = d2;
}
return GSL_SUCCESS;
}
/* Fit the weighted data (x_i, w_i, y_i) to the linear relationship
Y = c0 + c1 x
returning,
c0, c1 -- coefficients
s0, s1 -- the standard deviations of c0 and c1,
r -- the correlation coefficient between c0 and c1,
chisq -- weighted sum of squares of residuals */
int
gsl_fit_wlinear (const double *x, const size_t xstride,
const double *w, const size_t wstride,
const double *y, const size_t ystride,
const size_t n,
double *c0, double *c1,
double *cov_00, double *cov_01, double *cov_11,
double *chisq)
{
/* compute the weighted means and weighted deviations from the means */
/* wm denotes a "weighted mean", wm(f) = (sum_i w_i f_i) / (sum_i w_i) */
double W = 0, wm_x = 0, wm_y = 0, wm_dx2 = 0, wm_dxdy = 0;
size_t i;
for (i = 0; i < n; i++)
{
const double wi = w[i * wstride];
if (wi > 0)
{
W += wi;
wm_x += (x[i * xstride] - wm_x) * (wi / W);
wm_y += (y[i * ystride] - wm_y) * (wi / W);
}
}
W = 0; /* reset the total weight */
for (i = 0; i < n; i++)
{
const double wi = w[i * wstride];
if (wi > 0)
{
const double dx = x[i * xstride] - wm_x;
const double dy = y[i * ystride] - wm_y;
W += wi;
wm_dx2 += (dx * dx - wm_dx2) * (wi / W);
wm_dxdy += (dx * dy - wm_dxdy) * (wi / W);
}
}
/* In terms of y = a + b x */
{
double d2 = 0;
double b = wm_dxdy / wm_dx2;
double a = wm_y - wm_x * b;
*c0 = a;
*c1 = b;
*cov_00 = (1 / W) * (1 + wm_x * wm_x / wm_dx2);
*cov_11 = 1 / (W * wm_dx2);
*cov_01 = -wm_x / (W * wm_dx2);
/* Compute chi^2 = \sum w_i (y_i - (a + b * x_i))^2 */
for (i = 0; i < n; i++)
{
const double wi = w[i * wstride];
if (wi > 0)
{
const double dx = x[i * xstride] - wm_x;
const double dy = y[i * ystride] - wm_y;
const double d = dy - b * dx;
d2 += wi * d * d;
}
}
*chisq = d2;
}
return GSL_SUCCESS;
}
int
gsl_fit_linear_est (const double x,
const double c0, const double c1,
const double cov00, const double cov01, const double cov11,
double *y, double *y_err)
{
*y = c0 + c1 * x;
*y_err = sqrt (cov00 + x * (2 * cov01 + cov11 * x));
return GSL_SUCCESS;
}
int
gsl_fit_mul (const double *x, const size_t xstride,
const double *y, const size_t ystride,
const size_t n,
double *c1, double *cov_11, double *sumsq)
{
double m_x = 0, m_y = 0, m_dx2 = 0, m_dxdy = 0;
size_t i;
for (i = 0; i < n; i++)
{
m_x += (x[i * xstride] - m_x) / (i + 1.0);
m_y += (y[i * ystride] - m_y) / (i + 1.0);
}
for (i = 0; i < n; i++)
{
const double dx = x[i * xstride] - m_x;
const double dy = y[i * ystride] - m_y;
m_dx2 += (dx * dx - m_dx2) / (i + 1.0);
m_dxdy += (dx * dy - m_dxdy) / (i + 1.0);
}
/* In terms of y = b x */
{
double s2 = 0, d2 = 0;
double b = (m_x * m_y + m_dxdy) / (m_x * m_x + m_dx2);
*c1 = b;
/* Compute chi^2 = \sum (y_i - b * x_i)^2 */
for (i = 0; i < n; i++)
{
const double dx = x[i * xstride] - m_x;
const double dy = y[i * ystride] - m_y;
const double d = (m_y - b * m_x) + dy - b * dx;
d2 += d * d;
}
s2 = d2 / (n - 1.0); /* chisq per degree of freedom */
*cov_11 = s2 * 1.0 / (n * (m_x * m_x + m_dx2));
*sumsq = d2;
}
return GSL_SUCCESS;
}
int
gsl_fit_wmul (const double *x, const size_t xstride,
const double *w, const size_t wstride,
const double *y, const size_t ystride,
const size_t n,
double *c1, double *cov_11, double *chisq)
{
/* compute the weighted means and weighted deviations from the means */
/* wm denotes a "weighted mean", wm(f) = (sum_i w_i f_i) / (sum_i w_i) */
double W = 0, wm_x = 0, wm_y = 0, wm_dx2 = 0, wm_dxdy = 0;
size_t i;
for (i = 0; i < n; i++)
{
const double wi = w[i * wstride];
if (wi > 0)
{
W += wi;
wm_x += (x[i * xstride] - wm_x) * (wi / W);
wm_y += (y[i * ystride] - wm_y) * (wi / W);
}
}
W = 0; /* reset the total weight */
for (i = 0; i < n; i++)
{
const double wi = w[i * wstride];
if (wi > 0)
{
const double dx = x[i * xstride] - wm_x;
const double dy = y[i * ystride] - wm_y;
W += wi;
wm_dx2 += (dx * dx - wm_dx2) * (wi / W);
wm_dxdy += (dx * dy - wm_dxdy) * (wi / W);
}
}
/* In terms of y = b x */
{
double d2 = 0;
double b = (wm_x * wm_y + wm_dxdy) / (wm_x * wm_x + wm_dx2);
*c1 = b;
*cov_11 = 1 / (W * (wm_x * wm_x + wm_dx2));
/* Compute chi^2 = \sum w_i (y_i - b * x_i)^2 */
for (i = 0; i < n; i++)
{
const double wi = w[i * wstride];
if (wi > 0)
{
const double dx = x[i * xstride] - wm_x;
const double dy = y[i * ystride] - wm_y;
const double d = (wm_y - b * wm_x) + (dy - b * dx);
d2 += wi * d * d;
}
}
*chisq = d2;
}
return GSL_SUCCESS;
}
int
gsl_fit_mul_est (const double x,
const double c1, const double cov11,
double *y, double *y_err)
{
*y = c1 * x;
*y_err = sqrt (cov11) * fabs (x);
return GSL_SUCCESS;
}