Multiquadric Function Research Articles

Least squares surface approximation to scattered data using multiquadratic functions

The paper documents an investigation into some methods for fitting surfaces to scattered data. The form of the fitting function is a multiquadratic function with the criteria for the fit being the least mean squared residual for the data points. The principal problem is the selection of knot points (or base points for the multiquadratic basis functions), although the selection of the multiquadric parameter also plays a nontrivial role in the process. We first describe a greedy algorithm for knot selection, and this procedure is used as an initial step in what follows. The minimization including knot locations and the multiquadric parameter is explored, with some unexpected results in terms of “near repeated” knots. This phenomenon is explored, and leads us to consider variable parameter values for the basis functions. Examples and results are given throughout.

On radial basis approximation on periodic grids

A radial basis function approximation in n variables has the formwhere ø:ℝn → ℝ denotes the n-variate, spherically symmetric function associated with a prescribed radial basis function ø+:ℝ+ → ℝ, i.e. ø = ø+(‖ · ‖), the norm being Euclidean. The are real coefficients (often, approximants s above are considered where only finitely many λjs are non-zero), and is a fixed set of points in ℝn (of course, only the xj with non-zero coefficient λj affect s). Thus s is a linear combination of translates of a radially symmetric function which can be of global support, the simplest choice being , where c is a positive parameter. The latter is referred to as the multiquadric function and is usefull in applications.

Univariate multiquadric approximation: Quasi-interpolation to scattered data

The univariate multiquadric function with centerx j ∈R has the form {ϕ j (x)=[(x−x j )2+c 2]1/2, x∈R} wherec is a positive constant. We consider three approximations, namely, ℒ A f, ℒℬ f, and ℒ C f, to a function {f(x),x 0≤x≤x N } from the space that is spanned by the multiquadrics {ϕ j :j=0, 1, ...,N} and by linear polynomials, the centers {x j :j=0, 1,...,N} being given distinct points of the interval [x 0,x N ]. The coefficients of ℒ A f and ℒℬ f depend just on the function values {f(x j ):j=0, 1,...,N}. while ℒ A f, ℒ C f also depends on the extreme derivativesf′(x 0) andf′(x N ). These approximations are defined by quasi-interpolation formulas that are shown to give good accuracy even if the distribution of the centers in [x 0,x N ] is very irregular. Whenf is smooth andc=O(h), whereh is the maximum distance between adjacent centers, we find that the error of each quasi-interpolant isO(h 2|logh|) away from the ends of the rangex 0≤x≤x N. Near the ends of the range, however, the accuracy of ℒ A f and ℒℬ f is onlyO(h), because the polynomial terms of these approximations are zero and a constant, respectively. Thus, some of the known accuracy properties of quasiinterpolation when there is an infinite regular grid of centers {x j =jh:j ∈F} given by Buhmann (1988), are preserved in the case of a finite rangex 0≤x≤x N , and there is no need for the centers {x j :j=0, 1, ...,N} to be equally spaced.

Identification of non-linear processes using reciprocal multiquadric functions

In this paper radial basis function (RBF) networks are used to model general non-linear discrete-time systems. In particular, reciprocal multiquadric functions are used as activation functions for the RBF networks. A stepwise regression algorithm based on orthogonalization and a series of statistical tests is employed for designing and training of the network. The identification method yields non-linear models, which are stable and linear in the model parameters. The advantages of the proposed method compared to other radial basis function methods and backpropagation neural networks are described. Finally, the effectiveness of the identification method is demonstrated by the identification of two non-linear chemical processes, a simulated continuous stirred tank reactor and an experimental pH neutralization process.