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+/* fit/linear.c
+ *
+ * Copyright (C) 2000 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 2 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 c00, const double c01, const double c11,
+ double *y, double *y_err)
+{
+ *y = c0 + c1 * x;
+ *y_err = sqrt (c00 + x * (2 * c01 + c11 * 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 c11,
+ double *y, double *y_err)
+{
+ *y = c1 * x;
+ *y_err = sqrt (c11) * fabs (x);
+ return GSL_SUCCESS;
+}
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