#include <AMA.h>

Functions
long int	AMA_MltvApprox (AMA_OPTIONS options, long int nind, long int n, double x, double z, double epsilon, long int degree, AMA_SPLINE **spline)
	Approximation of Multivariate Data More...

Function Documentation

long int AMA_MltvApprox	(	AMA_OPTIONS *	options,
		long int	nind,
		long int	n,
		double **	x,
		double *	z,
		double *	epsilon,
		long int *	degree,
		AMA_SPLINE **	spline
	)

Approximation of Multivariate Data

This function employs cnspla to compute a spline approximation of independent variable data ${\bf X}\in{\rm R}^n$ and dependent variable data ${\bf Z}\in{\rm R}^1$ . For a given set of independent variable data ${\bf X}_\ell=(x_{1,\ell},\cdots,x_{n,\ell})$ , dependent variable data $z_\ell$ and approximation tolerances $\epsilon_\ell$ , for $\ell=1,\ldots,N$ , this function computes the spline

$s({\bf X}) = \sum_{j_n=1}^{m_n}\cdots\sum_{j_1=1}^{m_1}\alpha_{j_1,\ldots,j_n}B_{d_1,j_1}(x_1|{\bf\Lambda}_1)\cdots B_{d_n,j_n}(x_n|{\bf\Lambda_n})$

that minimizes

$F^{(2)}(s({\bf X})) = \sum_{k=1}^n\left[\int_R\left(\partial^2 s({\bf X})\over\partial^2 x_k\right)^2dR + 2.0\sum_{l=k+1}^n\int_R\left(\partial^2 s({\bf X})\over\partial x_k \partial x_l\right)^2dR\right]$

subject to the approximation constraints

$z_\ell - \epsilon_\ell \le s({\bf X}_\ell) \le z_\ell + \epsilon_\ell$

for $\ell=1,\ldots,N$ . The integration region $R = [x^L_1,x^U_1] \times [x^L_2,x^U_2] \times\cdots\times [x^L_n,x^U_n]$ where $x^L_k=\min_\ell\lbrace{x_{k,\ell}\rbrace}$ and $x^U_k=\max_\ell\lbrace{x_{k,\ell}\rbrace}$ , for $1\le k\le n$ .

In the above definition of $s({\bf X})$ the $\alpha_{j_1,\ldots,j_n}$ , for $j_k=1,\ldots,m_k$ and $1\le k\le n$ , are the coefficients of the tensor product B-splines

$\Phi_{j_1,\ldots,j_n}({\bf X}) = B_{d_1,j_1}(x_1|{\bf\Lambda}_1)\cdots B_{d_n,j_n}(x_n|{\bf\Lambda}_n)$

where $B_{d_k,j_k}(x_k|{\bf\Lambda}_k)$ are the $m_k$ univariate B-splines of degree $1 \le d_k\le 5$ defined by the knot vector ${\bf\Lambda}_k\in{\rm R}^{M_k}$ . The knot vectors ${\bf\Lambda}_k$ are

${\bf\Lambda}_k = ( \lambda_{k,1},\cdots,\lambda_{k,d_k+1}, \lambda_{k,d_k+2}, \cdots, \lambda_{k,m_k}, \lambda_{k,m_k+1},\cdots,\lambda_{k,m_k+d_k+1} )^T$

and they depend on the independent variable data and the degree $d_k$ . They are based on the rectilinear grid $(x^g_{1,1},x^g_{1,2},\cdots,x^g_{1,N^g_1-1},x^g_{1,N^g_1})\times\cdots\times(x^g_{n,1},x^g_{n,2},\cdots,x^g_{n,N^g_n-1},x^g_{n,N^g_n})$ upon which the independent variable data lies and they are defined by AMA_LamdaInterp(). That is, the knot vector ${\bf\Lambda}_k$ is defined by AMA_LamdaInterp() based on the points $(x^g_{k,1},x^g_{k,2},\cdots,x^g_{k,N^g_k-1},x^g_{k,N^g_k})$ .

Additionally, the spline $s({\bf X})$ is subject to the bounds

$\alpha_l \le s({\bf X}) \le \alpha_u$

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(). If the bounds specified with AMA_OptionsSetBounds() can not be satisfied in conjunction with the aforementioned approximation constraints, then AMA_MltvApprox() imposes the bounds

$\tilde\alpha_l \le s({\bf X}) \le \tilde\alpha_u$

where $\tilde\alpha_l = \alpha_l - \alpha^s_l$ and $\tilde\alpha_u = \alpha_u + \alpha^s_u$ . The values of $\alpha^s_l$ and $\alpha^s_u$ are defined by minimizing

$f(\alpha^s_l,\alpha^s_u) = ( \alpha^s_l )^2 + ( \alpha^s_u )^2$

subject to the constraints

$\begin{array}{ccc} \alpha^s_l &\ge &0.0, \cr \alpha^s_u &\ge &0.0, \cr \alpha_l - \alpha^s_l &\le &s({\bf X}), \cr \alpha_u + \alpha^s_u &\ge &s({\bf X}) \cr \end{array}$

and the aforementioned approximation constraints. If either $\alpha^s_l > 0.0$ or $\alpha^s_u > 0.0$ , then AMA_MltvApprox() reports AMA_WARNING_ERROR messages which specify the bounds $\tilde\alpha_l$ and $\tilde\alpha_u$ imposed on the spline. Additionally, it sets AMA_OPTIONS::lwrbnd $=\alpha^s_l$ and AMA_OPTIONS::uprbnd $=\alpha^s_u$ .

This function does the following:

Checks the input parameters for validity, see AMA_MltvInputCheck().
Defines the spline $s({\bf X})$ based on the knot vectors ${\bf\Lambda}_k$ , for $1\le k\le n$ .
Defines CNSPLA_CONPNT constraints to represent the approximation constraints, see AMA_MltvConpnt().
Defines CNSPLA_PNLTRM condition to represent the function $F^{(2)}(s({\bf X}))$ , see AMA_MltvPnltrm().
Invokes cnspla to compute $\alpha^s_l$ and/or $\alpha^s_u$ if lower and/or upper bounds are specified with AMA_OptionsSetBounds().
Invokes cnspla to compute a spline which minimizes $F^{(2)}(s({\bf X}))$ subject to the bounds and approximation constraints.
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 cross partial terms are included in the penalty term but they can be excluded with AMA_OptionsSetPenaltyTerm().; Because this function defines knot vectors based on the rectilinear grid upon which the independent variable data lies, it should only be used when the independent variable data either defines a gridded data with holes distribution, defines a gridded data distribution or consists of a small randomly distributed data set. See Multivariate Data Functions for a description of these data distributions.

Parameter Note: In the parameter definitions given below the limits on $k$ are $1\le k\le n$ and k = $k-1$ .

Parameters

options	[in] Pointer to AMA_OPTIONS. Must be initialized with AMA_Options() prior to calling AMA_MltvApprox().
nind	[in] The number of independent variables . Must satisfy $2\le$ nind $\le$ AMA_MXNIND.
n	[in] The number of data points . Must satisfy n $\ge 2$ .
x	[in] Array of size nind containing arrays of size n where x[k] contains the independent variable data $x_{k,\ell}$ , for $\ell=1,\ldots,N$ .
z	[in] Array of size n containing the dependent variable data $z_\ell$ , for $\ell=1,\ldots,N$ .
epsilon	[in] Array of size n containing the approximation tolerances $\epsilon_\ell$ , for $\ell=1,\ldots,N$ . Must satisfy $\epsilon_\ell\ge 0.0$ for all $\ell=1,\ldots,N$ . If epsilon = NULL, then this function uses $\epsilon_\ell = 0.0$ for all $\ell=1,\ldots,N$ .
degree	[in] Array of size nind containing the degree where degree[k] . Must satisfy $1\le$ degree[k] $\le 5$ .
spline	[out] Pointer to AMA_SPLINE pointer containing the spline approximation. Must satisfy spline $\ne$ NULL.

Returns: Success/Error Code.

User Callable Function - Documented 022416

Functions

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