SACOBRA (version 1.2)

trainMQRBF: Fit multiquadric RBF model to training data for d>1.

Description

The model for a point $z=(z_1,...,z_d)$ is fitted using n sample points $x_1,...,x_n$

$$s(z)={\lambda }_1\ast \mathrm{\Phi }(||z-x_1||)+...+{\lambda }_n\ast \mathrm{\Phi }(||z-x_n||)+c_0+c_1\ast z_1+...+c_d\ast z_d$$

where $\mathrm{\Phi }(r)=\sqrt{(1+(r/\sigma {)}^2)}$ denotes the multiquadrics radial basis function with width $\sigma$. The coefficients ${\lambda }_1,...,{\lambda }_n,c_0,c_1,...,c_d$ are determined by this training procedure. This is for the default case squares==FALSE. In case squares==TRUE there are d additional pure square terms and the model is

$$s_{sq}(z)=s(z)+c_{d+1}\ast z_1^2+...+c_{d+d}\ast z_d^2$$ In case ptail==FALSE the polynomial tail (all coefficients $c_i$) is omitted completely.

Usage

trainMQRBF(
  xp,
  U,
  ptail = TRUE,
  squares = FALSE,
  width,
  RULE = "One",
  widthFactor = 1,
  rho = 0,
  DEBUG2 = F
)

Arguments

xp

n points $x_i$ of dimension d are arranged in (n x d) matrix xp

U

vector of length n, containing samples $u(x_i)$ of the scalar function $u$ to be fitted - or - (n x m) matrix, where each column 1,...,m contains one vector of samples $u_j(x_i)$ for the m'th model, j=1,...,m

ptail

[TRUE] flag, see description

squares

[FALSE] flag, see 'Description'

width

[-1] either a positive real value which is the constant width $\sigma$ for all Gaussians in all iterations, or -1. If -1, the appropriate width $\sigma$ is calculated anew in each iteration with one of the rules RULE, based on the distribution of data points xp.

RULE

["One"] one out of ["One" | "Two" | "Three"], different rules for automatic estimation of width $\sigma$. Only relevant if width = -1,

widthFactor

[1.0] additional constant factor applied to each width $\sigma$

rho

[0.0] experimental: 0.0: interpolating, >0.0, approximating (spline-like) Gaussian RBFs

DEBUG2

[FALSE] if TRUE, save M and rhs on return value

Value

rbf.model, an object of class RBFinter, which is basically a list with elements:

coef

(n+d+1 x m) matrix holding in column m the coefficients for the m'th model: ${\lambda }_1,...,{\lambda }_n,c_0,c_1,...,c_d$. In case squares==TRUE it is an (n+2d+1 x m) matrix holding additionally the coefficients $c_{d+1},...,c_{d+d}$.

xp

matrix xp

d

size of the polynomial tail. If length(d)==0 it means no polynomial tail will be used for the model. In case of ptail==T && squares==F d will be dimension+1 and in case of ptail==T && squares==T d will be 2*dimension+1

npts

number n of points $x_i$

ptail

TRUE or FALSE (see description)

squares

TRUE or FALSE (see description)

width

the calculated width $\sigma$

type

"MQ"

Details

The linear equation system is solved via SVD inversion. Near-zero elements in the diagonal matrix $D$ are set to zero in $D^{-1}$. This makes rank-deficient systems numerically stable.

See Also

trainCubicRBF, predict.RBFinter, interpRBF