Multivariate Error Overbounding with a Gaussian–Pareto Model

Abstract

This paper extends the overbounding technique for univariate error monitoring to the multivariate error domain while using modeling techniques based on the extreme value theory. It derives an overbounding condition called norm overbounding by using hyperspherical coordinates to define errors in the multivariate domain as a vector norm, which in turn can be compared to a scalar threshold probability (such as an alert limit). This new approach provides an overbound model that bounds the probabilities of any norm exceeding any value. In particular, this method employs a constant lower bound for the marginal angular distribution and a Gaussian–Pareto hybrid distribution to bound the conditional radial distribution. In this case, a half-Gaussian distribution is used to overbound the region of the distribution near zero, and a Pareto distribution is used for the tail. This is motivated by results from extreme value theory for modeling heavy-tailed distributions. Furthermore, this approach uses a hypercone partitioning approach to estimate the norm overbounds from sampled data. The entire technique is demonstrated through a two-dimensional example.