Calculating the Cumulative Distributions

2.5. 7 Degree of Damage

2.5.8 Calculating the Cumulative Distributions

One final step remains before the Bayesian probabilistic SHM method defined in this section can be applied: the marginal PDFs and CDFs for the parameters must be found. Given an updated PDF for all the structural model parameters, p(OID) formed using data D, the conditional CDF for the ith structural model parameter, Ci(Oi), is found by integrating the updated PDF over the entire possible space for all parameters except for the ^ith structural model parameter.

This results in (2.89)

where

ei

is the set where all the (j vary across their full range, except

ei,

^which

varies from 0 to

ei.

This integral cannot be evaluated analytically for anything more than a one-DOF problem. Numerical integration is feasible only for problems of very small dimension. Therefore, the value of the integral must be approximated

in some other manner. Two procedures are suggested in this study for ap- proximating the value of the integral. One is easy to calculate, but gives a poor approximation when the uncertainty is large. The other takes more time to compute, but is a better approximation. Both methods are based on a Laplace's method (Bleistein and Handelsman 1986) of asymptotic expansion for evaluating integrals.

Coarse Asymptotic Expansion of the CDF Integral

Because of its form, the updated PDF (2. 72) lends itself well to approximation by a multivariable joint Gaussian distribution function. Let () maximize the updated PDF, and assume that only one maximum exists. Approximate ](O) about

iJ,

by

(2.90) ](O)

~

^](iJ)

⁺ ~(0-

iJfLo(B)(O- B)

where L₀(B) is the Hessian matrix of ](O) with respect to () evaluated at iJ.

The form of the updated PDF (2.72) reveals that using (2.90) in (2.72) will produce a Gaussian approximation for the updated PDF.

Substituting (2.72) and (2.90) into (2.89) leads to (2.91)

Integrals of this type occur often in reliability problems (Madsen, Krenk, and Lind 1986). In terms of the terminology from reliability theory, the boundary of the region defined by 8i defines a failure surface. The failure surface for this problem can be viewed as a hyperplane. In this case, the solution of the integral is very simple to express (Madsen, Krenk, and Lind 1986):

(2.92)

where <I> is the cumulative distribution function for a Normally distributed random variable of zero mean and unit variance, and

(2.93)

TA

·(()·) _ ()i- ei () Pz z - ~=========

V

^{ef Le(e)-1ei}

This process can also give an approximation to the PDF for each ()i by eval- uating the normal Gaussian function of zero mean and unit variance at the calculated values of p.

Since this approach is based on approximating the original PDF about its maximum by a Gaussian distribution function, the results should only be good when the Hessian matrix has large eigenvalues. Increasing eigenvalues in the Hessian matrix correspond to smaller variances, and thus less uncertainty. For reference, this approach is the coarse approximation.

The approximation method just described is a specific example of a general method for asymptotic approximation of integrals described by Papadimitriou, Beck, and Katafygiotis (1995). In order to use this method, the maxima of the updated PDF with respect to the model parameters needs to be found.

Further, the Hessian matrix of J(e) at the maxima must be determined. Ap- pendix B presents the details of these evaluations.

Fine Asymptotic Expansion of the CDF Integral

As noted previously, the form of the updated PDF (2. 72) lends itself well to approximation by a multivariable joint Gaussian distribution function. In the coarse approximation, the PDF was approximated about its maximum with respect to all of the parameters and integrated using a result from reliability theory. If improved accuracy were necessary, a small change could be made to the coarse method, resulting in the fine approximation.

Consider the marginal PDF for the ^ith model parameter.

(2.94)

where Oi = [01, ... , Oi-l, Oi+l, ... , 0N₀]T and 8i is the set where Oi varies across its full possible range. For a fixed

ei,

this integral can be approximated using the previously described asymptotic approximation technique. By approxi- mating the integral at different values of

ei,

an approximation to the marginal PDF for

ei

is derived. This marginal PDF can be numerically integrated to find the CDF for

ei·

Difficulties such as missing the maxima and determining what size step to take can be easily addressed. First, find the maximum of the PDF with respect to all of the structural parameters. This serves as a starting point for the calculation of the marginal PDF for each of the structural parameters, and will ensure that the maxima are not missed. Also, if the PDF is well-behaved, the maxima for the value of

ei

at which the marginal PDF is to be evaluated will be near the maxima for the previous value of

ei.

This improves the efficiency of finding the maximum in each step. An appropriate step size can be determined from the marginal PDF found using the coarse approximation. The standard deviation for the coarse marginal PDF will indicate a characteristic length of the problem, so that a proper initial choice can be made for the necessary step size. The step size can be reduced and the PDF recalculated until the desired level of convergence is achieved. The number of steps can be found by continuing to step until the area added by an additional point is a specified fraction of the total area.

Since the Gaussian distribution function approximation is performed for each point

ei

in the marginal PDF in this method, the results are better than for the coarse approximation. The cost for the improved performance is that forming the CDF for each

ei

requires that the PDF be maximized with respect

to the remaining parameters for each desired point in the marginal PDF. This is a computationally expensive procedure.

For the results presented in this study, the Hessian matrix of the MOF always has large enough eigenvalues that the coarse approximation is suffi- cient for all calculations. Since a study validating the use of the asymptotic expansion in this case has already been completed by Papadimitriou, Beck, and Katafygiotis (1995), these efforts are not duplicated. The fine approxi- mation is not needed for the application presented here, so exploration of its capabilities and an extensive comparison to the coarse approximation is left for future work. The coarse approximation is used without further comment.