Multi-parameter fitting - GNU Scientific Library -- Reference Manual
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### 37.4 Multi-parameter fitting

The functions described in this section perform least-squares fits to a general linear model, y = X c where y is a vector of n observations, X is an n by p matrix of predictor variables, and the elements of the vector c are the p unknown best-fit parameters which are to be estimated. The chi-squared value is given by \chi^2 = \sum_i w_i (y_i - \sum_j X_{ij} c_j)^2.

This formulation can be used for fits to any number of functions and/or variables by preparing the n-by-p matrix X appropriately. For example, to fit to a p-th order polynomial in x, use the following matrix,

         X_{ij} = x_i^j

where the index i runs over the observations and the index j runs from 0 to p-1.

To fit to a set of p sinusoidal functions with fixed frequencies \omega_1, \omega_2, ..., \omega_p, use,

         X_{ij} = sin(\omega_j x_i)

To fit to p independent variables x_1, x_2, ..., x_p, use,

         X_{ij} = x_j(i)

where x_j(i) is the i-th value of the predictor variable x_j.

The functions described in this section are declared in the header file `gsl_multifit.h`.

The solution of the general linear least-squares system requires an additional working space for intermediate results, such as the singular value decomposition of the matrix X.

— Function: gsl_multifit_linear_workspace * gsl_multifit_linear_alloc (size_t n, size_t p)

> This function allocates a workspace for fitting a model to n observations using p parameters.

— Function: void gsl_multifit_linear_free (gsl_multifit_linear_workspace * work)

> This function frees the memory associated with the workspace w.

— Function: 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)

> This function computes the best-fit parameters c of the model y = X c for the observations y and the matrix of predictor variables X, using the preallocated workspace provided in work. The variance-covariance matrix of the model parameters cov is estimated from the scatter of the observations about the best-fit. The sum of squares of the residuals from the best-fit, \chi^2, is returned in chisq. If the coefficient of determination is desired, it can be computed from the expression R^2 = 1 - \chi^2 / TSS, where the total sum of squares (TSS) of the observations y may be computed from `gsl_stats_tss`.

> The best-fit is found by singular value decomposition of the matrix X using the modified Golub-Reinsch SVD algorithm, with column scaling to improve the accuracy of the singular values. Any components which have zero singular value (to machine precision) are discarded from the fit.

— Function: int gsl_multifit_wlinear (const gsl_matrix * X, const gsl_vector * w, const gsl_vector * y, gsl_vector * c, gsl_matrix * cov, double * chisq, gsl_multifit_linear_workspace * work)

> This function computes the best-fit parameters c of the weighted model y = X c for the observations y with weights w and the matrix of predictor variables X, using the preallocated workspace provided in work. The covariance matrix of the model parameters cov is computed with the given weights. The weighted sum of squares of the residuals from the best-fit, \chi^2, is returned in chisq. If the coefficient of determination is desired, it can be computed from the expression R^2 = 1 - \chi^2 / WTSS, where the weighted total sum of squares (WTSS) of the observations y may be computed from `gsl_stats_wtss`.

— Function: int gsl_multifit_linear_svd (const gsl_matrix * X, const gsl_vector * y, double tol, size_t * rank, gsl_vector * c, gsl_matrix * cov, double * chisq, gsl_multifit_linear_workspace * work)
— Function: int gsl_multifit_wlinear_svd (const gsl_matrix * X, const gsl_vector * w, const gsl_vector * y, double tol, size_t * rank, gsl_vector * c, gsl_matrix * cov, double * chisq, gsl_multifit_linear_workspace * work)

> In these functions components of the fit are discarded if the ratio of singular values s_i/s_0 falls below the user-specified tolerance tol, and the effective rank is returned in rank.

— Function: int gsl_multifit_linear_usvd (const gsl_matrix * X, const gsl_vector * y, double tol, size_t * rank, gsl_vector * c, gsl_matrix * cov, double * chisq, gsl_multifit_linear_workspace * work)
— Function: int gsl_multifit_wlinear_usvd (const gsl_matrix * X, const gsl_vector * w, const gsl_vector * y, double tol, size_t * rank, gsl_vector * c, gsl_matrix * cov, double * chisq, gsl_multifit_linear_workspace * work)

> These functions compute the fit using an SVD without column scaling.

— Function: int gsl_multifit_linear_est (const gsl_vector * x, const gsl_vector * c, const gsl_matrix * cov, double * y, double * y_err)

> This function uses the best-fit multilinear regression coefficients c and their covariance matrix cov to compute the fitted function value y and its standard deviation y_err for the model y = x.c at the point x.

— Function: int gsl_multifit_linear_residuals (const gsl_matrix * X, const gsl_vector * y, const gsl_vector * c, gsl_vector * r)

> This function computes the vector of residuals r = y - X c for the observations y, coefficients c and matrix of predictor variables X.

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