System Identification Technique and Neural Networks for Material Lifetime Assessment Application - Complex System Modelling and Control Through Intelligent Soft Computations - page 772

cost function reduction, I is the total number of initial weights during initialization,
Q is the number of training examples and ʻ is lambda parameter (the parameter of

LM).
Based on the above formulas, the Levenberg-Marquardt algorithm implementing
Bayesian regularization can be stated as follows:

Step 1. The weights w and parameters ʻ , ʱ and ʲ were initialized. For example:

ʻ = 1. The algorithm is not too sensitive to the initial

choice of the parameters. In addition, the choice of

= 0.005, ʱ = 0 and ʲ

ʱ = 0 and ʲ = 1 means

that the NN is starting from the original cost function.
Step 2. One step of the Levenberg-Marquardt algorithm to minimize the objective
function was taken as per Eq. ( 16 ).
Step 3. If E

ð w þ D w Þ \ E ð w Þ , then w new = w + ʔ w was accepted as a new

iteration.
Step 4. The effective number of parameter was computed using Eq. ( 17 ) and the

Hessian formulation of Eq. ( 15 ) was utilized.
Step 5. The parameters

c

ʱ and ʲ were updated using Eq. ( 18 ) for ʱ and Eq. ( 19 ) for ʲ .

Step 6. Steps 2 - 5 were repeated until the stopping criterion was satis ed or con-

vergence was achieved.

It is clear that the NARX-SP architecture is being currently employed and by
sliding over one-step to one-step of stress level, the prediction will be dynamically
covering all the spectrum loadings of the testing sets according to the CLD. As a
result, material lifetime assessment can be fashioned for a wide spectrum of loading
in an ef cient manner based upon solely the training data as the basis of the NARX

regressor, thus developed variable amplitude or spectrum fatigue analysis.
It is important to note that the number of hidden nodes employed was 10 and the
parameter c of the RBFNN also took the same number. For the RBFNN, the rest of
the corresponding parameters were determined accordingly using K-means tech-
nique (Haykin 2009 ).
NN parameters used in the present NARX modeling using the Levenberg-
Marquardt algorithm with Bayesian regularization are described in Table 3 .

Table 3 NN parameters used
in the present NARX
modeling using the
Levenberg-Marquardt
algorithm with Bayesian
regularization

NN parameters Value

Initial lambda, ʻ 0.005

init

Initial weight decay, ʱ 0

init

Initial inverse noise, ʲ − 1 init

Maximum number of iterations 200

10 − 10 Minimum gradient, g min 1 ×

10 10 Maximum lambda, ʻ 1 × max

Performance goal 0

Number of hidden nodes 10