Gwyddion – Free SPM (AFM, SNOM/NSOM, STM, MFM, …) data analysis software

GwyNLFitter

GwyNLFitter — Marquardt-Levenberg nonlinear least square fitter

Functions

gdouble	(*GwyNLFitFunc) ()
void	(*GwyNLFitDerFunc) ()
gdouble	(*GwyNLFitIdxFunc) ()
void	(*GwyNLFitIdxDiffFunc) ()
GwyNLFitter *	gwy_math_nlfit_new ()
GwyNLFitter *	gwy_math_nlfit_new_idx ()
void	gwy_math_nlfit_free ()
GwyNLFitter *	gwy_math_nlfit_copy ()
gdouble	gwy_math_nlfit_fit ()
gdouble	gwy_math_nlfit_fit_full ()
gdouble	gwy_math_nlfit_fit_idx ()
gdouble	gwy_math_nlfit_fit_idx_full ()
gint	gwy_math_nlfit_get_max_iterations ()
void	gwy_math_nlfit_set_max_iterations ()
gboolean	gwy_math_nlfit_get_approx_geometric ()
void	gwy_math_nlfit_set_approx_geometric ()
gboolean	gwy_math_nlfit_succeeded ()
gdouble	gwy_math_nlfit_get_dispersion ()
gdouble	gwy_math_nlfit_get_correlations ()
gboolean	gwy_math_nlfit_get_eval ()
const gdouble *	gwy_math_nlfit_get_covar ()
gint	gwy_math_nlfit_get_nparam ()
gdouble	gwy_math_nlfit_get_sigma ()
void	gwy_math_nlfit_set_callbacks ()
void	gwy_math_nlfit_diff ()
void	gwy_math_nlfit_derive ()
void	gwy_math_nlfit_diff_idx ()
gint	gwy_math_nlfit_map_free_params ()

Types and Values

struct

GwyNLFitter

Includes

#include <libgwyddion/gwyddion.h>

Description

A new Marquardt-Levenberg nonlinear least square fitter can be created with gwy_math_nlfit_new(), specifying the function to fit (as GwyNLFitFunc) and its derivative (as GwyNLFitDerFunc). For functions for whose analytic derivative is not available or very impractical, gwy_math_nlfit_derive() (computing the derivative numerically) can be used instead.

A fitter can be then repeatedly used on different data either in gwy_math_nlfit_fit(), or gwy_math_nlfit_fit_full() when there are some fixed or linked parameters. Arbitrary additional (non-fitting) parameters can be passed to the fited function in user_data .

After a successfull fit additional fit information can be obtained with gwy_math_nlfit_get_dispersion(), gwy_math_nlfit_get_correlations(), gwy_math_nlfit_get_sigma(). Note these functions may be used only after a successfull fit. When a fitter is no longer needed, it should be freed with gwy_math_nlfit_free().

Several common functions are also available as fitting presets that can be fitted with gwy_nlfit_preset_fit(). See GwyNLFitPreset for details.

Functions

GwyNLFitFunc ()

gdouble
(*GwyNLFitFunc) (gdouble x,
                 gint nparam,
                 const gdouble *param,
                 gpointer user_data,
                 gboolean *success);

Fitting function type for real-valued independent variables.

Parameters

x	The value to compute the function at.
nparam	The number of parameters (size of `param` ).
param	Parameters.
user_data	User data as passed to `gwy_math_nlfit_fit()`.
success	Set to `TRUE` if succeeds, `FALSE` on failure.

Returns

The function value at x .

GwyNLFitDerFunc ()

void
(*GwyNLFitDerFunc) (gdouble x,
                    gint nparam,
                    const gdouble *param,
                    const gboolean *fixed_param,
                    GwyNLFitFunc func,
                    gpointer user_data,
                    gdouble *der,
                    gboolean *success);

Fitting function partial derivative type for real-valued independent variables.

Parameters

x	x-data as passed to `gwy_math_nlfit_fit()`.
nparam	The number of parameters (size of `param` ).
param	Parameters.
fixed_param	Which parameters should be treated as fixed (corresponding entries are set to `TRUE`).
func	The fitted function.
user_data	User data as passed to `gwy_math_nlfit_fit()`.
der	Array where the `nparam` partial derivatives by each parameter are to be stored. Derivatives by fixed parameters are ignored so the function may set corresponding elements or not, whichever is more convenient.
success	Set to `TRUE` if succeeds, `FALSE` on failure.

GwyNLFitIdxFunc ()

gdouble
(*GwyNLFitIdxFunc) (guint i,
                    const gdouble *param,
                    gpointer user_data,
                    gboolean *success);

Fitting function type for opaque indexed data.

Note that unlike GwyNLFitFunc which returns just the function value this function must return the difference between the function value and fitted data value (that is function minus data point). When opaque data are fitted the fitter does not know what the data values are.

The function must take care of weighting. The difference should be multiplied by the inverse of the (unsquared) estimated error of the i -th data point. Not multiplying by anything correspond to using the default unit weights.

Parameters

i	Data index from the set {0, 1, 2, ..., ndata-1}.
param	Parameters.
user_data	User data as passed to `gwy_math_nlfit_fit()`.
success	Set to `TRUE` if succeeds, `FALSE` on failure.

Returns

Difference between the function value and data in the i -th data point.

Since: 2.46

GwyNLFitIdxDiffFunc ()

void
(*GwyNLFitIdxDiffFunc) (guint i,
                        const gdouble *param,
                        const gboolean *fixed_param,
                        GwyNLFitIdxFunc func,
                        gpointer user_data,
                        gdouble *der,
                        gboolean *success);

Fitting function partial derivatives type for opaque indexed data.

The function must take care of weighting. The derivatives should be multiplied by the inverse of the (unsquared) estimated error of the i -th data point. Not multiplying by anything correspond to using the default unit weights.

Parameters

i	Data index from the set {0, 1, 2, ..., ndata-1}.
param	Parameters.
fixed_param	Which parameters should be treated as fixed (corresponding entries are set to `TRUE`). It may be `NULL` is no parameters are fixed. The function must set derivatives by fixed parameters to zero.
func	The fitted function.
user_data	User data as passed to `gwy_math_nlfit_fit()`.
der	Array where the `nparam` partial derivatives by each parameter are to be stored. Derivatives by fixed parameters are ignored so the function may set corresponding elements or not, whichever is more convenient.
success	Set to `TRUE` if succeeds, `FALSE` on failure.

Since: 2.46

gwy_math_nlfit_new ()

GwyNLFitter *
gwy_math_nlfit_new (GwyNLFitFunc func,
                    GwyNLFitDerFunc diff);

Creates a new Marquardt-Levenberg nonlinear fitter for function with a real-valued independent variable.

See gwy_math_nlfit_new_idx() for more complex scenarios.

You can use gwy_math_nlfit_diff() computing the derivative numerically, when you do not know the derivatives explicitely. Since 2.46 passing NULL as diff has the same effect, i.e. the fitter will automatically use numerical differentiation.

Parameters

func	The fitted function.
diff	The derivative of fitted function.

Returns

The newly created fitter.

gwy_math_nlfit_new_idx ()

GwyNLFitter *
gwy_math_nlfit_new_idx (GwyNLFitIdxFunc func,
                        GwyNLFitIdxDiffFunc diff);

Creates a new Marquardt-Levenberg nonlinear fitter for opaque indexed data.

As only the data index is passed to the functions, using this interface permits fitting more complex functions. The abscissa can be arbitrary, for instance a vector, as it is not seen by the fitter. Similarly, vector-valued functions can be emulated by mapping tuples of indices to the vector components.

You can pass NULL as diff to use automatically numerical differentiation when you do not know the derivatives explicitely. Note that this means you cannot use weighting. If you want weighting you need to pass your own diff function that performs the weighting (it can utilise the gwy_math_nlfit_diff_idx() helper).

Parameters

func	The fitted function.
diff	The derivative of fitted function.

Returns

The newly created fitter.

Since: 2.46

gwy_math_nlfit_free ()

void
gwy_math_nlfit_free (GwyNLFitter *nlfit);

Completely frees a Marquardt-Levenberg nonlinear fitter.

Parameters

nlfit

A Marquardt-Levenberg nonlinear fitter.

gwy_math_nlfit_copy ()

GwyNLFitter *
gwy_math_nlfit_copy (GwyNLFitter *nlfit);

Creates a copy of a nonlinear least squares fitter.

This function is mostly usefil for language bindings.

Parameters

nlfit

A Marquardt-Levenberg nonlinear fitter.

Returns

The newly created fitter.

Since: 2.47

gwy_math_nlfit_fit ()

gdouble
gwy_math_nlfit_fit (GwyNLFitter *nlfit,
                    gint ndata,
                    const gdouble *x,
                    const gdouble *y,
                    gint nparam,
                    gdouble *param,
                    gpointer user_data);

Performs a nonlinear fit of simple function on data.

Parameters

nlfit	A Marquardt-Levenberg nonlinear fitter.
ndata	The number of data points in `x` , `y` .
x	Array of independent variable values.
y	Array of dependent variable values.
nparam	The nuber of parameters.
param	Array of parameters (of size `nparam` ). Note the parameters must be initialized to reasonably near values.
user_data	Pointer that will be passed to the function and derivative as `user_data` .

Returns

The final residual sum, a negative number in the case of failure.

gwy_math_nlfit_fit_full ()

gdouble
gwy_math_nlfit_fit_full (GwyNLFitter *nlfit,
                         gint ndata,
                         const gdouble *x,
                         const gdouble *y,
                         const gdouble *weight,
                         gint nparam,
                         gdouble *param,
                         const gboolean *fixed_param,
                         const gint *link_map,
                         gpointer user_data);

Performs a nonlinear fit of simple function on data, allowing some fixed parameters.

Initial values of linked (dependent) parameters are overwritten by master values, their fixed_param property is ignored and master's property controls whether all are fixed or all variable.

Parameters

nlfit	A Marquardt-Levenberg nonlinear fitter.
ndata	The number of data points in `x` , `y` , `weight` .
x	Array of independent variable values.
y	Array of dependent variable values.
weight	Array of weights associated to each data point (usually equal to inverse squares errors). Can be `NULL`, unit weight is then used for all data.
nparam	The nuber of parameters.
param	Array of parameters (of size `nparam` ). Note the parameters must be initialized to reasonably near values.
fixed_param	Which parameters should be treated as fixed (set corresponding element to `TRUE` for them). May be `NULL` if all parameters are variable.
link_map	Map of linked parameters. One of linked parameters is master, Values in this array are indices of corresponding master parameter for each parameter (for independent parameters set `link_map` [i] == i). May be `NULL` if all parameter are independent.
user_data	Pointer that will be passed to the function and derivative

Returns

The final residual sum, a negative number in the case of failure.

gwy_math_nlfit_fit_idx ()

gdouble
gwy_math_nlfit_fit_idx (GwyNLFitter *nlfit,
                        guint ndata,
                        guint nparam,
                        gdouble *param,
                        gpointer user_data);

Performs a nonlinear fit of function on opaque indexed data.

Parameters

nlfit	A Marquardt-Levenberg nonlinear fitter.
ndata	The number of data points in `x` , `y` , `weight` .
nparam	The nuber of parameters.
param	Array of parameters (of size `nparam` ). Note the parameters must be initialized to reasonably near values.
user_data	Pointer that will be passed to the function and derivative

Returns

The final residual sum, a negative number in the case of failure.

Since: 2.46

gwy_math_nlfit_fit_idx_full ()

gdouble
gwy_math_nlfit_fit_idx_full (GwyNLFitter *nlfit,
                             guint ndata,
                             guint nparam,
                             gdouble *param,
                             const gboolean *fixed_param,
                             const gint *link_map,
                             gpointer user_data);

Performs a nonlinear fit of function on opaque indexed data, allowing some fixed parameters.

Initial values of linked (dependent) parameters are overwritten by master values, their fixed_param property is ignored and master's property controls whether all are fixed or all variable.

Parameters

nlfit	A Marquardt-Levenberg nonlinear fitter.
ndata	The number of data points in `x` , `y` , `weight` .
nparam	The nuber of parameters.
param	Array of parameters (of size `nparam` ). Note the parameters must be initialized to reasonably near values.
fixed_param	Which parameters should be treated as fixed (set corresponding element to `TRUE` for them). May be `NULL` if all parameters are variable.
link_map	Map of linked parameters. One of linked parameters is master, Values in this array are indices of corresponding master parameter for each parameter (for independent parameters set `link_map` [i] == i). May be `NULL` if all parameter are independent.
user_data	Pointer that will be passed to the function and derivative

Returns

The final residual sum, a negative number in the case of failure.

Since: 2.46

gwy_math_nlfit_get_max_iterations ()

gint
gwy_math_nlfit_get_max_iterations (GwyNLFitter *nlfit);

Returns the maximum number of iterations of nonlinear fitter nlfit .

Parameters

nlfit

A Marquardt-Levenberg nonlinear fitter.

Returns

The maximum number of iterations.

gwy_math_nlfit_set_max_iterations ()

void
gwy_math_nlfit_set_max_iterations (GwyNLFitter *nlfit,
                                   gint maxiter);

Sets the maximum number of iterations for nonlinear fitter nlfit .

Parameters

nlfit	A Marquardt-Levenberg nonlinear fitter.
maxiter	The maximum number of iterations.

gwy_math_nlfit_get_approx_geometric ()

gboolean
gwy_math_nlfit_get_approx_geometric (GwyNLFitter *nlfit);

Reports if a non-linear fitter performs approximately orthogonal fitting.

This function does not do anything useful.

Parameters

nlfit

A Marquardt-Levenberg nonlinear fitter.

Returns

TRUE if the fitting is approximately orthogonal; FALSE for ordinary least squares.

Since: 2.47

gwy_math_nlfit_set_approx_geometric ()

void
gwy_math_nlfit_set_approx_geometric (GwyNLFitter *nlfit,
                                     gboolean setting);

Sets a non-linear fitter to perform ordinary or approximately orthogonal fitting.

This function does not do anything useful. Since 2.56 it is no-op.

Parameters

nlfit	A Marquardt-Levenberg nonlinear fitter.
setting	`TRUE` to approximately orthogonal (geometrical) fitting.

Since: 2.47

gwy_math_nlfit_succeeded ()

gboolean
gwy_math_nlfit_succeeded (GwyNLFitter *nlfit);

Obtains the status of the last fitting.

Fitting failure can be (and usually should be) also determined by checking for negative return value of gwy_math_nlfit_fit() or gwy_math_nlfit_fit_full(). This function allows to test it later.

Parameters

nlfit

A Marquardt-Levenberg nonlinear fitter.

Returns

TRUE if the last fitting suceeded, FALSE if it failed.

Since: 2.7

gwy_math_nlfit_get_dispersion ()

gdouble
gwy_math_nlfit_get_dispersion (GwyNLFitter *nlfit);

Returns the residual sum divided by the number of degrees of freedom.

This function can be used only after a successful fit.

Parameters

nlfit

A Marquardt-Levenberg nonlinear fitter.

Returns

The dispersion.

gwy_math_nlfit_get_correlations ()

gdouble
gwy_math_nlfit_get_correlations (GwyNLFitter *nlfit,
                                 gint par1,
                                 gint par2);

Returns the correlation coefficient between par1 -th and par2 -th parameter.

This function can be used only after a successful fit.

Parameters

nlfit	A Marquardt-Levenberg nonlinear fitter.
par1	First parameter index.
par2	Second parameter index.

Returns

The correlation coefficient.

gwy_math_nlfit_get_eval ()

gboolean
gwy_math_nlfit_get_eval (GwyNLFitter *nlfit);

Gets the state of a nonlinear fitter.

Parameters

nlfit

A Marquardt-Levenberg nonlinear fitter.

Returns

TRUE if the last fit succeeded, FALSE if it failed.

Since: 2.47

gwy_math_nlfit_get_covar ()

const gdouble *
gwy_math_nlfit_get_covar (GwyNLFitter *nlfit);

Gets the covariance matrix of a nonlinear fitter.

The returned matrix may be only used between fits. Any fitting can free and reallocate it.

Parameters

nlfit

A Marquardt-Levenberg nonlinear fitter.

Returns

The covariance matrix, as lower triangular matrix, owned by the fitter. The return value will be NULL if the last fit did not succeed.

Since: 2.47

gwy_math_nlfit_get_nparam ()

gint
gwy_math_nlfit_get_nparam (GwyNLFitter *nlfit);

Gets the number of parameters of a nonlinear fitter.

The number of parameters determines the size of covariance matrix obtained by gwy_math_nlfit_get_covar() and valid arguments for gwy_math_nlfit_get_correlations().

Parameters

nlfit

A Marquardt-Levenberg nonlinear fitter.

Returns

The number of parameters for the last fit. If the last fit was not successful zero is returned.

Since: 2.47

gwy_math_nlfit_get_sigma ()

gdouble
gwy_math_nlfit_get_sigma (GwyNLFitter *nlfit,
                          gint par);

Returns the standard deviation of parameter number par .

This function makes sense only after a successful fit and for a free parameter.

Parameters

nlfit	A Marquardt-Levenberg nonlinear fitter.
par	Parameter index.

Returns

The SD of par -th parameter.

gwy_math_nlfit_set_callbacks ()

void
gwy_math_nlfit_set_callbacks (GwyNLFitter *nlfit,
                              GwySetFractionFunc set_fraction,
                              GwySetMessageFunc set_message);

Sets callbacks reporting a non-linear least squares fitter progress.

Parameters

nlfit	A Marquardt-Levenberg nonlinear fitter.
set_fraction	Function that sets fraction to output (or `NULL`).
set_message	Function that sets message to output (or `NULL`).

Since: 2.46

gwy_math_nlfit_diff ()

void
gwy_math_nlfit_diff (gdouble x,
                     gint nparam,
                     const gdouble *param,
                     const gboolean *fixed_param,
                     GwyNLFitFunc func,
                     gpointer user_data,
                     gdouble *der,
                     gboolean *success);

Numerically computes the partial derivatives of a function.

Parameters

x	The value to compute the derivative at.
nparam	The nuber of parameters.
param	Array of parameters (of size `nparam` ).
fixed_param	Which parameters should be treated as fixed (corresponding entries are set to `TRUE`).
func	The fitted function.
user_data	User data to be passed to `func` .
der	Array where the put the result to.
success	Set to `TRUE` if succeeds, `FALSE` on failure.

Since: 2.46

gwy_math_nlfit_derive ()

void
gwy_math_nlfit_derive (gdouble x,
                       gint nparam,
                       const gdouble *param,
                       const gboolean *fixed_param,
                       GwyNLFitFunc func,
                       gpointer user_data,
                       gdouble *der,
                       gboolean *success);

Numerically computes the partial derivatives of a function.

This is a legacy name for function gwy_math_nlfit_diff().

Parameters

x	The value to compute the derivative at.
nparam	The nuber of parameters.
param	Array of parameters (of size `nparam` ).
fixed_param	Which parameters should be treated as fixed (corresponding entries are set to `TRUE`).
func	The fitted function.
user_data	User data to be passed to `func` .
der	Array where the put the result to.
success	Set to `TRUE` if succeeds, `FALSE` on failure.

gwy_math_nlfit_diff_idx ()

void
gwy_math_nlfit_diff_idx (guint i,
                         gint nparam,
                         const gdouble *param,
                         const gboolean *fixed_param,
                         GwyNLFitIdxFunc func,
                         gpointer user_data,
                         gdouble *der,
                         gboolean *success);

Numerically computes the partial derivatives of an opaque function.

This function cannot be passed as derivative calculation function to gwy_math_nlfit_new_idx() because it is not of the GwyNLFitIdxDiffFunc type. Just pass NULL to gwy_math_nlfit_new_idx() if you want automatic numerical derivatives.

You can employ this function in your own GwyNLFitIdxDiffFunc function if you need to modify the derivatives somehow, for instance to apply weighting.

Parameters

i	Data index from the set {0, 1, 2, ..., ndata-1}.
nparam	The nuber of parameters.
param	Array of parameters (of size `nparam` ).
fixed_param	Which parameters should be treated as fixed (corresponding entries are set to `TRUE`).
func	The fitted function.
user_data	User data to be passed to `func` .
der	Array where the put the result to.
success	Set to `TRUE` if succeeds, `FALSE` on failure.

Since: 2.46

gwy_math_nlfit_map_free_params ()

gint
gwy_math_nlfit_map_free_params (const gboolean *fixed_param,
                                gint n,
                                gint *id_map);

Creates a map from free parameter ids to all-parameter ids.

This is a helper function for reduction of full parameter vectors and covariance matrices to only free parameters.

The array id_map must at least as many elements as there are free parameters. However, it is usually better to just pass an array of size n because the number of free parameters is what this function calculates.

Parameters

fixed_param	Which parameters are fixed.
n	Length of `fixed_param` array.
id_map	Parameter id map to fill.

Returns

The number of free parameters. This is the number of ids filled in id_map .

Since: 2.50

Types and Values

struct GwyNLFitter

struct GwyNLFitter {
    GwyNLFitFunc fmarq;  /* fitting function */
    GwyNLFitDerFunc dmarq;  /* fitting function derivations */
    gint maxiter;  /* max number of iterations */
    gboolean eval;  /* success? */
    gdouble *covar; /* covariance matrix  */
    gdouble dispersion; /* dispersion */
    gdouble mfi;    /* fitting parameters --
                       fi, decrease, increase of lambda, minimum lambda */
    gdouble mdec;
    gdouble minc;
    gdouble mtol;
};

Non-linear least-squares fitter.

The fields should be considered private.

Use gwy_math_nlfit_get_eval(), gwy_math_nlfit_get_covar() and gwy_math_nlfit_get_dispersion() and gwy_math_nlfit_succeeded() to examine the fitter state.

Use gwy_math_nlfit_set_max_iterations() to set the maximum iteration.

In rare occasions you might want modify mdec or minc . The rest is better left untouched.

Members

GwyNLFitFunc `fmarq`;	Evaluates the fitted function.
GwyNLFitDerFunc `dmarq`;	Evaluates derivatives of the fitted function.
gint `maxiter`;	Maximum number of iteration.
gboolean `eval`;	`TRUE` if last evaluation succeeded.
gdouble *`covar`;	Covariance matrix (set upon successful fit).
gdouble `dispersion`;	Mean residual sum of squares per point, set to -1 on failure.
gdouble `mfi`;	Lambda parameter is multiplied by it. Probably keep at 1.
gdouble `mdec`;	Decrease of lambda parameter after an unsuccessful step.
gdouble `minc`;	Increase of lambda parameter after a successful step.
gdouble `mtol`;	If lambda parameter becomes zero it is set to this value.