Radial Basis Function

The Radial Basis Function (RBF) surrogate interpolates the data by passing through each sample point. In the presence of sufficient sample data points, RBF can represent complex data accurately.

Consider a design study containing $n$ input parameters that runs for $m$ designs. This design study provides $m$ sample points as input to a surrogate fitting calculation for each response.

$y$ denotes the vector of sample response values involving $n$ parameters.

Figure 1. EQUATION_DISPLAY

y = [y_{j}] = {[y_{1}, \dots, y_{m}]}^{t}, j \in [1, m]

(5164)

$X$ denotes the input parameter matrix

Figure 2. EQUATION_DISPLAY

X = [x_{i, j}] = (\begin{matrix} x_{1, 1} & \begin{matrix} {\begin{matrix} x \end{matrix}}_{2, 1} & \dots \end{matrix} & x_{n, 1} \\ \begin{matrix} x_{1, 2} \\ ⋮ \end{matrix} & \begin{matrix} {\begin{matrix} x \end{matrix}}_{2, 2} & \dots \end{matrix} & \begin{matrix} \begin{matrix} {\begin{matrix} x \end{matrix}}_{n, 2} \end{matrix} \\ ⋮ \end{matrix} \\ x_{1, m} & \begin{matrix} \begin{matrix} x_{2, m} \end{matrix} & \dots \end{matrix} & {\begin{matrix} x \end{matrix}}_{n, m} \end{matrix}), i \in [1, n] j \in [1, m]

(5165)

The surrogate calculation indicates the determination of a surrogate function, which outputs a response value for the input parameter

x

in the entire design space. The RBF surrogate function

f (x)

is constructed with a low-order polynomial and a data blending term added to the polynomial:

Figure 3. EQUATION_DISPLAY

y (x) = p (x) + \sum_{i = 1}^{m} α_{i} \cdot ϕ (r_{i} (x))

(5166)

where

$p (x)$ is the low-order polynomial, which is set automatically either to constant or to linear.
$r_{i} (x) = \sqrt{\sum_{i = 1}^{m} {(x - x_{i})}^{2}}$ is the Euclidean distance, also called the $L^{2}$ norm, between $x$ and each data point $x_{i}$ .
$ϕ$ is a radial basis function specified by the fitting function.
$α_{i}$ is the radial basis function coefficient of the $i^{t h}$ data point. During the surrogate function calculation, the coefficients to each data points are determined.

Fitting Functions

The following fitting functions (correlation functions) are available in Design Manager, where

θ

is the shape factor (also known as a correlation parameter). RBF has one generic shape factor applied for all sample points:

Gaussian basis function:

Figure 4. EQUATION_DISPLAY

$ϕ (r_{i}, θ) = \exp [θ {\cdot r_{i}}^{2}]$

(5167)
Multi-Quadratic Basis Function:

Figure 5. EQUATION_DISPLAY

$ϕ (r_{i}, θ) = \sqrt{{r_{i}}^{2} + θ^{2}}$

(5168)
Inverse Multi-Quadratic Basis Function:

Figure 6. EQUATION_DISPLAY

$ϕ (r_{i}, θ) = \frac{1}{\sqrt{{r_{i}}^{2} + θ^{2}}}$

(5169)
Linear Basis Function:

Figure 7. EQUATION_DISPLAY

$ϕ (r_{i}) = r_{i}$

(5170)
Cubic Basis Function:

Figure 8. EQUATION_DISPLAY

$ϕ (r_{i}) = {r_{i}}^{3}$

(5171)
Thin Plate Spline Basis Function:

Figure 9. EQUATION_DISPLAY

$ϕ (r_{i}) = {r_{i}}^{2} \ln (r_{i})$

(5172)

Polynomial Order

In Design Manager, the order of the polynomial term in Eqn. (5166) is set automatically based on the selected fitting function:

Gaussian basis function: no polynomial term
Multi-Quadratic Basis Function: constant
Inverse Multi-Quadratic Basis Function: no polynomial term
Linear Basis Function: constant
Cubic Basis Function: linear
Thin Plate Spline Basis Function: linear

Shape Factor and Tuning

In the fitting functions, the shape factors control how much each sample point affects the surrogate approximation. Small shape factors enhance the influence of the sample point, while large values weaken the influence.

The Gaussian Basis Function, Multi-Quadratic Basis Function, and Inverse Multi-Quadratic Basis Function each require a shape function value. You must find a good fit through manual experiments, where you view the plot and compare cross-validation values. These RBF factors that require tuning are a disadvantage compared to the Kriging method because there is no objective to guide this tuning process.