4.4 Surrogate Modeling
53
y i =
m
j=0
β j ϕ
(i)
j + ∈ i (0, σ
2
∈ )
(4.3)
where ϕ j are basis-functions. The design matrix, X now contains additional terms
obtained using non-linear transformation of the input variables for sampled points
using the basis-functions. For model approximation, the coefficients β j (j = 0…, p)
need to be determined. Using sample data, the least-square estimate of β is:
β = X
† Y
( 4 . 4 )
X
†
= (X
T X)
−1 X
T
(4.5)
X
† is known as pseudo-inverse of X. The choice of basis-functions will affect
the accuracy of RSA model, and it becomes important to determine the right basisfunctions ϕ j as well as associated model coefficients β j to obtain the best model
on the sample data. To determine the best model which approximates the data, the
goodness of fit, R
2
ad j is estimated. For a good fit, R
2
ad j must be closer to 1.
4.4.2 Radial Basis Neural Networks
Radial basis neural networks (RBNN) is a two-layered network consisting of a hidden
layer of radial basis neurons and an output layer of linear neurons, characterized by a
set of inputs and a set of outputs [36]. The radial basis functions act as processing units
between the input and output. The hidden layer performs a non-linear transformation
of the input space to an intermediate space using a set of radial basis units.
A Gaussian function of the following form is used as a transfer function for a
radial basis neuron.
φ(x) = exp
−γ x − c i
2
(4.6)
The net input to the transfer function is the vector distance between the neurons
center, c i and the input vector, x multiplied by the parameter, γ . The parameter γ
allows the sensitivity of the neurons to be adjusted. The output layer, then implements
a linear combiner to produce the desired targets. The output function can be expressed
as:
h(x) =
K
i=1
w i exp
−γ x − c i
2
(4.7)
The prediction ability of the network is stored in the weights, w i which can be
obtained from a set of training data [36].
53
y i =
m
j=0
β j ϕ
(i)
j + ∈ i (0, σ
2
∈ )
(4.3)
where ϕ j are basis-functions. The design matrix, X now contains additional terms
obtained using non-linear transformation of the input variables for sampled points
using the basis-functions. For model approximation, the coefficients β j (j = 0…, p)
need to be determined. Using sample data, the least-square estimate of β is:
β = X
† Y
( 4 . 4 )
X
†
= (X
T X)
−1 X
T
(4.5)
X
† is known as pseudo-inverse of X. The choice of basis-functions will affect
the accuracy of RSA model, and it becomes important to determine the right basisfunctions ϕ j as well as associated model coefficients β j to obtain the best model
on the sample data. To determine the best model which approximates the data, the
goodness of fit, R
2
ad j is estimated. For a good fit, R
2
ad j must be closer to 1.
4.4.2 Radial Basis Neural Networks
Radial basis neural networks (RBNN) is a two-layered network consisting of a hidden
layer of radial basis neurons and an output layer of linear neurons, characterized by a
set of inputs and a set of outputs [36]. The radial basis functions act as processing units
between the input and output. The hidden layer performs a non-linear transformation
of the input space to an intermediate space using a set of radial basis units.
A Gaussian function of the following form is used as a transfer function for a
radial basis neuron.
φ(x) = exp
−γ x − c i
2
(4.6)
The net input to the transfer function is the vector distance between the neurons
center, c i and the input vector, x multiplied by the parameter, γ . The parameter γ
allows the sensitivity of the neurons to be adjusted. The output layer, then implements
a linear combiner to produce the desired targets. The output function can be expressed
as:
h(x) =
K
i=1
w i exp
−γ x − c i
2
(4.7)
The prediction ability of the network is stored in the weights, w i which can be
obtained from a set of training data [36].