Some Issues Concerning the Iteration Process 132

For the iteration process of the Gauss-Newton method, some major issues will be encountered. They are discussed in detail in what follows.

6.5.1 Choice of Objective Function

In theory either the reflection or transmission coefficient can be chosen to establish the objective function. The one with higher sensitivity to the parameters (this is usually frequency dependent) is more preferable. In practice, the accuracy and ease of the measurements of the coefficients should also be considered.

6.5.2 Selection of Frequency Range

The choice of the frequency range for the parameter identification process is usually based on the sensitivity of the objective function to the parameters. High sensitivity of the objective function to the parameters to be updated can improve the goodness of estimation.

6.5.3 Initial Estimate of Parameters

As mentioned at the end of section 6.3.2.2, the initial estimates are of importance to the convergence of the iteration. Good estimates of the initial parameters can lead to a fast convergence while poor estimates might result in slow convergence, converging to other local minima, or even divergence.

Taking the mass-like discontinuity on a beam for example, there are two joint parameters to be updated. One is the non-dimensional mass, μ and the other is the non- dimensional moment of inertia, ϑ. In practice, an initial estimate of the mass can be expected to be more accurately known than that of the moment of inertia. However, an estimate of ϑ can be obtained based on that of μ. Referring to equation (6.13), when

( )( )

3 3

2 2 0

μξ ϑξ + ϑξ − μξ = (6.28)

there is no reflection. From the ‘measured’ power reflection coefficient against non- dimensional frequency ξ2, the frequency ξ2 at which ρ =0 might be estimated. From equation (6.28), it yields 3 2 2 2 μ ϑ μξ ξ = + . (6.29)

The parameter ϑ can be estimated based on an estimate for μ and the value of ξ at this frequency.

A more generally applicable technique for finding good estimates of the initial parameters is possible if a range for each parameter is assumed within which the global minimum is located. Then their ranges can be divided into a coarse grid (Figure 6.5). The initial estimates of the parameters can be chosen as the pair of μ and ϑ where the objective function is lowest. However, it must be noted that this still does not guarantee

good initial parameter estimates since the global minimum may not lie within the ranges chosen. lower μ μupper lower ϑ upper ϑ Fmin

Figure 6.5 Grid of the range of estimated μ and ϑ for the simple mass-like discontinuity.

6.5.4 Termination of Iteration

A theoretical model is always an approximation of the true system. Owing to the existence of measurement noise and inaccuracy of the joint model, the solution at the

jth iteration X_j need never converge to the true value. It may converge to a value close to the true value depending on the noise level and nonsingularity of S W ST_{j RR j} in equation (6.10). Many criteria for terminating the iteration can be defined. Considering the orders of magnitudes of these two parameters and that μ_j =0 or ϑ_j =0 is possible, a weighted norm of the difference between the parameter vectors of two successive iterations is used, which is

T

j j XX j

δ = ΔX W ΔX . (6.30)

where W_XX is the weighting matrix. When δ_j is small enough, the iteration can be terminated. It should be noted that even if the value of equation (6.30) is very small, it is not sufficient to say that the method has converged. Only when _j

j

δ

→∞

∑

tends to a constant, can it be said that the method is converged.

6.5.5 Evaluating the Goodness of the Estimates

When the iteration process is complete, the identification result is obtained. The result needs to be evaluated before drawing any final conclusion. Two ways, graphical and numerical, may be used to measure the goodness of the result. From the plot of the ‘measured’ and estimated power reflection or transmission coefficient, the result can be viewed easily. A more quantitative way for parametric models is to evaluate the result statistically, among which the sum of squares due to errors (SSE), R-square, adjusted R- square and root mean squared error are quite often used. They are discussed in references [71, 72], here only a brief introduction is given.

1) The sum of squares due to errors (SSE)

This statistic measures the total deviation from the estimate to the response values and is given by

(

)

2 1 n i mi i i SSE w y y = =

∑

− (6.31)

where y_mi is the ith observed or measured response value, y_i is the corresponding response predicted by the model after each iteration, w_i is the ith weighting factor (w_i ≠0) and n is the number of response values. A value closer to zero means a better fit of the model. Here the power reflection coefficient ρ or transmission coefficient τ can be the substitute for y.

2) R-square

R-square is the square of the correlation between the measured and predicted response values. It is defined as the ratio of the sum of squares of the regression (SSR) and the total sum of squares (SST). SSR is defined as

(

)

2 1 n i i m i SSR w y y = =

∑

− (6.32)

where y_m is the mean value of the observed response. SST is also called the sum of squares about the mean, and is defined as

(

)

2 1 n i mi m i SST w y y = =

∑

− (6.33)

It can be proved that SST =SSR+SSE. Therefore, R-square can be expressed as

2 1 SSR SSE R SST SST = = − (6.34)

Note that it is possible to get a negative R-square. In this case, R-square cannot be interpreted as the square of a correlation.

3) Adjusted R-square

If the number of estimated parameters in the model is increased, R-square might increase although the estimation may not improve. To avoid this, the degrees of freedom adjusted R-square statistic is used. The adjusted R-square is defined as

(

)

(

)

2 a 1 1 SSE n R SST n q − = − − (6.35)

where q is the number of parameters to be estimated. Here the power reflection coefficient ρ or transmission coefficient τ can be the substitute for y. The R-square and adjusted R-square can take on any value less than or equal to 1, with a value closer to 1 indicating a better estimate.

4) Root mean squared error

The root mean square error is defined as

SSE RMSE

n q =

− . (6.36)

A RMSE value closer to zero means a better estimate.

The R-square and the adjusted R-square include both information of SSE and SST. Only the R-square will be given in what follows.

In document Joint identification in structural waveguides using wave reflection and transmission coefficients (Page 150-154)