Interval Principal Component Analysis of Non-probabilistic Convex Model

Abstract

The non-probabilistic convex model describes uncertainty of parameters in the form of a bounded set rather than the probability distribution and is generally applied to interval uncertainty analysis based on the fluctuation ranges of structural parameters. When performing the structural response bounds analysis or uncertain optimization design, too many interval parameters is not beneficial in most circumstances. It may lead to problems such as inefficient computation and non-convergence of the optimization process. This paper proposes an interval principal component analysis method for the non-probabilistic multidimensional ellipsoid convex model, which aims to reduce the dimensionality of the uncertainty domain of correlated intervals, while retaining as much as possible of the variation present in the bounded uncertainty. Firstly, the mathematical mapping between the coordinate axes and the marginal intervals of the multidimensional ellipsoid convex model on these axes is established for an arbitrary coordinate system. Secondly, the difference index is introduced to describe the differentiation degree of the group of marginal intervals, the maximizing of which yields the principal axes of the multidimensional ellipsoid convex model. With the magnitudes of the marginal intervals, the most critical axes and corresponding interval components in uncertainty quantification can then be identified. By neglecting those axes with small marginal intervals, the dimensionality can be thus reduced. An error index is also suggested to indicate the accuracy of uncertainty quantification under the dimensionality reduction representation. Finally, two numerical examples are presented to show the dimensionality reduction representation procedure by the proposed interval principal component analysis.