#include <AMA.h>

Functions
long int	AMA_UnvLstsqr (AMA_OPTIONS options, long int n, double x, double z, double wht, long int degree, long int mlamda, double lamda, double theta, AMA_SPLINE *spline)
	Least Squares Approximation of Univariate Data More...

Function Documentation

long int AMA_UnvLstsqr	(	AMA_OPTIONS *	options,
		long int	n,
		double *	x,
		double *	z,
		double *	wht,
		long int	degree,
		long int	mlamda,
		double *	lamda,
		double	theta,
		AMA_SPLINE **	spline
	)

Least Squares Approximation of Univariate Data

For a given set of data $(x_\ell,z_\ell)$ and weights $w_\ell$ , for $\ell=1,\ldots,N$ , this function employs cnspla to compute the spline

$s(x) = \sum_{j=1}^{m}\alpha_jB_{d,j}(x|{\bf\Lambda})$

that minimizes

$(1.0-\theta)\sum_\ell^N w_\ell ( s(x_\ell) - z_\ell )^2 + \theta \int_{\lambda_1}^{\lambda_{m+d+1}}\left( s^{(2)}(x) \right)^2 dx$

for $0.0\le\theta< 1.0$ . In the above definition of $s(x)$ the $\alpha_j$ , for $j=1,\ldots,m$ , are the coefficients of the $m$ univariate B-splines $B_{d,j}(x|{\bf\Lambda})$ of degree $1 \le d \le 5$ which are based on the knot vector ${\bf\Lambda}\in{\rm R}^M$ given as

${\bf\Lambda} = ( \lambda_1,\cdots,\lambda_{d+1}, \lambda_{d+2}, \cdots, \lambda_m, \lambda_{m+1},\cdots,\lambda_{m+d+1} )^T$

where the knots $\lambda_i$ must satisfy the conditions

$\lambda_1 = \ldots = \lambda_{d+1}$ ;
$\lambda_i \le \lambda_{i+1}$ , for $1\le i\le m+d$ ;
$\lambda_i < \lambda_{i+d+1}$ , for $1\le i\le m$ ;
$\lambda_{m+1} = \ldots = \lambda_{m+d+1}$ .

If the knot vector ${\bf\Lambda}$ is represented by its distinct knot vector

${\bf\rm K}^{{\bf\Lambda}} = ( \kappa_1, \kappa_2, \cdots, \kappa_{m^d-1}, \kappa_{m^d} )^T$

and its knot multiplicity vector

${\bf\rm H}^{{\bf\Lambda}} = ( \eta_1, \eta_2, \cdots, \eta_{m^d-1}, \eta_{m^d} )^T$

then the aforementioned conditions are

$\kappa_i < \kappa_{i+1}$ , for $1\le i\le m^d - 1$ ;
$\eta_i \le d + 1$ , for $2\le i\le m^d-1$ ;
$\eta_1 = \eta_{m^d} = d+1$ .

Also the first and last distinct knots must satisfy the conditions $\kappa_1\le x_1$ and $\kappa_{m^d}\ge x_N$ , respectively. In this case the number of knots is $M = m + d + 1$ .

For convenience this function does not require the definition of the $d+1$ order knots at the left and right hand endpoints and also accepts the knot vector ${\bf\Lambda}^o\in{\rm R}^M$

${\bf\Lambda}^o = ( \lambda_{d+1}, \lambda_{d+2}, \cdots, \lambda_m, \lambda_{m+1} )^T$

where the knots $\lambda_i$ satisfy the conditions

$\lambda_i \le \lambda_{i+1}$ , for $d+2\le i\le m-1$ ;
$\lambda_i < \lambda_{i+d+1}$ , for $d+1\le i\le m-d$ ;
$\lambda_{d+1} < \lambda_{d+2}$ and $\lambda_m < \lambda_{m+1}$ .

If the knot vector ${\bf\Lambda}^o$ is represented by its distinct knot vector

${\bf\rm K}^{{\bf\Lambda}^o} = ( \kappa_1, \kappa_2, \cdots, \kappa_{m^d-1}, \kappa_{m^d} )^T$

and its knot multiplicity vector

${\bf\rm H}^{{\bf\Lambda}^o} = ( \eta_1, \eta_2, \cdots, \eta_{m^d-1}, \eta_{m^d} )^T$

then the aforementioned conditions are

$\kappa_i < \kappa_{i+1}$ , for $1\le i\le m^d - 1$ ;
$\eta_i \le d + 1$ , for $2\le i\le m^d-1$ ;
$\eta_1 = \eta_{m^d} = 1$ .

As before, the first and last distinct knots must satisfy the conditions $\kappa_1\le x_1$ and $\kappa_{m^d}\ge x_N$ , respectively. In this case the number of knots is $M = m - d + 1$ .

Additionally, the spline $s(x)$ is subject to the bounds

$\alpha_l \le s(x) \le \alpha_u$

where $\alpha_l=-\alpha_\infty$ , $\alpha_u=\alpha_\infty$ and $\alpha_\infty$ equals AMA_SplineInfbnd(). By default the spline is unbounded but finite bounds can be set with AMA_OptionsSetBounds().

Depending on the number and distribution of the knots relative to the independent variable data it is possible for the least squares approximation problem to have a non-unique solution. Hence, this function provides the capability of performing a minimum norm optimization subsequent to computing the least squares approximation. The minimum norm optimization computes an alternate spline $\bar s({\bf X})$ by minimizing

$\int_{\lambda_1}^{\lambda_{m+d+1}}\left(\bar s^{(2)}(x)\right)^2dx$

subject to the approximation constraints

$z_l - \epsilon_\ell \le \bar s(x_\ell) \le z_l + \epsilon_\ell,$

for $\ell=1,\ldots,N$ , where $\epsilon_l = s(x_\ell)-z_\ell$ and $s(x_\ell)$ is the least squares approximation. These constraints insure the condition

$\sum_\ell^N w_\ell ( \bar s(x_\ell) - z_\ell )^2 \le \sum_\ell^N w_\ell ( s(x_\ell) - z_\ell )^2,$

that is, the minimum norm optimization does not increase the least squares error.

This function does the following:

Checks input parameters for validity, see AMA_UnvInputCheck().
Defines the spline based on the knot vector ${\bf\Lambda}$ or ${\bf\Lambda}^o$ .
Invokes cnspla to compute a least squares spline approximation.
Invokes cnspla to perform a minimum norm optimization.
Stores approximation into spline.

Note: By default the spline coefficients are initialized to zero but a different initial value for the spline coefficients can be set with AMA_OptionsSetCoefficients().; By default the spline is unbounded but finite bounds can be set with AMA_OptionsSetBounds().; By default the minimum norm optimization is not performed but it can be enabled with AMA_OptionsSetMinimumNorm().; The penalty term and minimum norm optimization options are mutually exclusive; that is, the minimum norm optimization is not performed if $\theta > 0.0$ .

Parameters

options	[in] Pointer to AMA_OPTIONS. Must be initialized with AMA_Options() prior to calling AMA_UnvLstsqr().
n	[in] The number of data points . Must satisfy n $\ge 2$ .
x	[in] Array of size n containing the independent variable data $x_\ell$ , for $\ell=1,\ldots,N$ . Must be in ascending order.
z	[in] Array of size n containing the dependent variable data $z_\ell$ , for $\ell=1,\ldots,N$ .
wht	[in] Array of size n containing the weights $w_\ell$ , for $\ell=1,\ldots,N$ . Must satisfy $w_\ell\ge 0.0$ for all $\ell=1,\ldots,N$ . If wht = NULL, then this function uses $w_\ell = 1.0$ for all $\ell=1,\ldots,N$ .
degree	[in] The degree . Must satisfy $1\le$ degree $\le 5$ .
mlamda	[in] The number of knots . Must satisfy mlamda $\ge 2$ .
lamda	[in] Array of size mlamda containing the knot vector ${\bf\Lambda}$ or ${\bf\Lambda}^o$ .
theta	[in] The penalty term weight $\theta$ . Must satisfy $0.0\le$ theta .
spline	[out] Pointer to AMA_SPLINE pointer containing the least squares spline approximation. Must satisfy spline $\ne$ NULL.

Returns: Success/Error Code.

User Callable Function - Documented 101715

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