A direct prediction of the shape parameter—A purely scattered data approach

Abstract

In this paper we present an approach which predicts directly without search the optimal choice of the shape parameter c contained in the multiquadrics $(-1){}^{\lceil \frac{\beta }{2}\rceil }(c^2+{\parallel x\parallel }^2){}^{\frac{\beta }{2}},\beta >0,$ and the inverse multiquadrics $(c^2+{\parallel x\parallel }^2){}^{\frac{\beta }{2}},\beta <0$. Unlike the simplex scheme where the data points are required to be evenly spaced, as in a recent paper of the author, here we allow them to be arbitrarily scattered in the simplex, making it much more useful. Besides this, we aim at helping non-mathematicians use our approach and hence remove some complicated requirements for the domain. The drawback is that its theoretical ground is not so strong as in the evenly spaced data setting. However, experiments show that it works well. The experimentally optimal value of c coincides with the theoretically predicted one. Since the fill distance, which reflects the amount of data points needed, involved is always of reasonable size, this approach is supposed to be practically useful.

Introduction

In this paper the approximated functions lie in a space called B_σ, as in the following definition.

Definition 1.1

For any σ > 0, the class of band-limited functions f in L²(Rⁿ), the family of square-integrable functions, is

$$B_{\sigma }:=\{f\in L^2\left(R^n\right):\widehat{f}\left(\xi \right)=0if\parallel \xi \parallel >\sigma \},$$

where $\widehat{f}$ denotes the Fourier transform of f.

This function space looks small. In fact, it plays only an intermediate role in the interpolation. Via the B_σ functions, all functions in the Sobolev space can be interpolated by the multiquadrics or inverse multiquadrics, as will be explained further in the paper.

The radial function we adopt is

$$h\left(x\right):=\Gamma (-\frac{\beta }{2})(c^2+{\parallel x\parallel }^2){}^{\frac{\beta }{2}},\beta \in R\setminus 2N_{\ge 0},c>0,$$

where ‖x‖ is the Euclidean norm of x ∈ Rⁿ, Γ is the classical gamma function, and β, c are constants. Note that this definition is slightly different from the one mentioned in the abstract. The definition (1) will simplify its Fourier transform and the presentation of our central theorem. The function h(x) in (1) is conditionally positive definite (c.p.d.) of order $m=\max \{0,\lceil \frac{\beta }{2}\rceil \}$ where $\lceil \frac{\beta }{2}\rceil$ means the smallest integer greater than or equal to $\frac{\beta }{2}$. For further details we refer the reader to Madych and Nelson [1] and Wendland [2].

For any set of data points $(x_j,y_j),j=1,\dots ,N,$ where $X=\{x_1,\dots ,x_N\}$ is a subset of Rⁿ and y_j are real or complex numbers, we can always find an interpolant of the form

$$s\left(x\right)=p\left(x\right)+\sum _{j=1}^N{c}_{j}h(x-x_j),$$

where p(x) is a polynomial in $P_{m-1}^n$ to be determined, $m=\text{max}\{0,\lceil \frac{\beta }{2}\rceil \},$ and c_j are coefficients to be chosen, as long as X is a determining set for $P_{m-1}^n$. Thus $s\left(x_j\right)=y_j$ for $j=1,\dots ,N$. The requirement that X determine $P_{m-1}^n$ means that two polynomials in this polynomial space are exactly the same as long as they are equal at $x_1,\dots ,x_N$. Interested readers can find these in Madych and Nelson [1].

Although we are interested only in scattered data, our criteria of choosing c are developed from a core theorem which involves a simplex scheme with evenly spaced data points, as mentioned in the abstract. Therefore it is necessary to make a brief description of the evenly spaced scheme.

Let T_n denote an n-simplex in Rⁿ. Then T₁ is just a line segment, T₂ is a triangle, and T₃ is a tetrahedron with four vertices. The exact definition can be found in Fleming [3].

Let $v_i,1\le i\le n+1$ be the vertices of T_n. Then any point x ∈ T_n can be written as a convex combination of the vertices:

$$x=\sum _{i=1}^{n+1}{c}_i{v}_i,\sum _{i=1}^{n+1}{c}_i=1,c_i\ge 0.$$

The numbers $c_1,\dots ,c_{n+1}$ are called the barycentric coordinates of x.

For any n-simplex T_n, the evenly spaced points of degree l are those points whose barycentric coordinates are of the form

$$(\frac{k_1}{l},\frac{k_2}{l},\dots ,\frac{k_{n+1}}{l}),k_{i}nonnegativeintegerswith\sum _{i=1}^{n+1}{k}_i=l.$$

Obviously, the number of evenly spaced points of degree l is $N=\left(\begin{array}{c}n+l\\ n\end{array}\right)$. It’s proven in Bos [4] that such points do form a determining set for $P_l^n$.

Before entering our core theorem, some ingredients must be explained. Each function of the form (1) induces a function space ${\mathcal{C}}_{h,m},m=\max \{0,\lceil \frac{\beta }{2}\rceil \},$ called native space. Also, there is a seminorm ‖f‖_h for each $f\in {\mathcal{C}}_{h,m}$. These can be found in Luh [5], [6], [7], Madych and Nelson [1], [8] and Wendland [2]. The constants ρ and Δ₀, which are usually very small positive numbers for low dimensions, in the theorem are determined by n and β. We omit their complicated definitions and refer the reader to Luh [9].

The following theorem is just our core theorem. We omit its complicated proof and take it directly from [9].

Theorem 1.2

Let h be as in (1). For any positive number b₀, let $C=\max \{\frac{2}{3b_0},\frac{8\rho }{c}\}$ and ${\delta }_0=\frac{1}{3C}$. For any n-simplex T_n of diameter r satisfying $\frac{1}{3C}\le r\le \frac{2}{3C}$ (note that $\frac{2}{3C}\le b_0$), if $f\in {\mathcal{C}}_{h,m},$

$$|f\left(x\right)-s\left(x\right)|\le 2^{\frac{n+\beta -7}{4}}{\pi }^{\frac{n-1}{4}}\sqrt{n{\alpha }_n}{c}^{\frac{\beta }{2}}\sqrt{{\Delta }_0}\sqrt{3C}\sqrt{\delta }\left({\lambda }^{\prime }\right){}^{\frac{1}{\delta }}{\parallel f\parallel }_h$$

holds for all x ∈ T_n and 0 < δ ≤ δ₀, where s(x) is defined as in (2) with $x_1,\dots ,x_N$ the evenly spaced points of degree l in T_n satisfying $\frac{1}{3C\delta }\le l\le \frac{2}{3C\delta }$. The constant α_n denotes the volume of the unit ball in Rⁿ, and 0 < λ′ < 1 is given by

$${\lambda }^{\prime }=\left(\frac{2}{3}\right){}^{\frac{1}{3C}}$$

which only in some cases mildly depends on the dimension n.

Remark. Note that as the degree l of the evenly spaced data points increases, the number δ will decrease, making the upper bound in (3) small. Hence δ can be regarded in spirit as the well-known fill distance. It is natural to ask what will happen if one regards δ completely the same as the fill distance. If so, the requirement that the centers $x_1,\dots ,x_N$ be evenly spaced in the simplex can be dropped, making this theorem much more useful. In fact, this is just the central idea of this paper.