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Statistical Learning methods are very useful for building adequate surrogates for the numerical solvers needed in a reliability analysis of a complex structure. This first chapter has the aim of demonstrating the relevance of such an approximation from several points of view. The chapter is also a critical discussion of reliability analysis methods published up to date.
One of the main purposes of applying analytic or synthetic methods in structural reliability is to reduce to a minimum the number of samples, because of their computational cost. This makes relevant the consideration of the Theory of Statistical Learning (SLT) developed in the last decades but specially in the last years1, as it is aimed to solve, upon the basis of a few available samples, the following...
The discussion made in the preceding chapter on the curse of dimensionality points out the need of the dimensionality reduction of the learning problem. Related to this are feature extraction techniques which transform the input data into another set of lower dimensionality that shows better the inner structure of the given population. In general, these techniques can be viewed as preprocessing methods...
As noted in Chap. 1, the reliability problem can be regarded under a classification viewpoint, because the limit-state function determines two well defined classes of samples, namely the safe and failure ones. Despite this clear association, the structural reliability community has paid little attention to solve this problem under such a paradigm. This is perhaps due to the fact that classical pattern...
This chapter is devoted exclusively to the classification approach to structural reliability using Support Vector Machines, which at a difference to Neural Networks were developed apart from biological paradigms and were instead construed entirely in the framework of Statistical Learning Theory. For structural reliability applications they show significant advantages over other classification methods...
After having examined the classification approaches to reliability problems in the previous two chapters, we will be concerned herein with the applicability of function approximation methods, with special regard to Neural Networks and Support Vector Machines. Three kinds of networks are of interest in this respect: Multi-Layer Perceptrons, Radial Basis Function Networks and Networks with Time-Dependent...
In structural safety the computation of the probability of failure obviously requires the knowledge of the joint density function of the basic variables x. In some cases, however, only the marginal distributions and, perhaps, the correlation matrix are known, with the consequence that approximate models for the joint density function should be applied [36]. A more frequent situation is that even the...
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