An infinite-dimensional Christoffel function and detection of abnormal trajectories

TL;DR

This paper develops an infinite-dimensional Christoffel function in a Hilbert space to identify the support of measures and detect outlier trajectories, extending finite-dimensional methods to functional data analysis.

Contribution

It introduces a novel infinite-dimensional Christoffel function and demonstrates its effectiveness in detecting abnormal trajectories in functional data.

Findings

The reciprocal of the Christoffel function grows exponentially outside the support.

The function can be computed similarly to the finite-dimensional case using moments.

Numerical examples confirm its utility in outlier detection for trajectories.

Abstract

We introduce an infinite-dimensional version of the Christoffel function, where now (i) its argument lies in a Hilbert space of functions, and (ii) its associated underlying measure is supported on a compact subset of the Hilbert space. We show that it possesses the same crucial property as its finite-dimensional version to identify the support of the measure (and so to detect outliers). Indeed, the growth of its reciprocal with respect to its degree is at least exponential outside the support of the measure and at most polynomial inside. Moreover, for a fixed degree, its computation mimics that of the finite-dimensional case, but now the entries of the moment matrix associated with the measure are moments of moments. To illustrate the potential of this new tool, we consider the following application. Given a data base of registered reference trajectories, we consider the problem of detecting whether a newly acquired trajectory is abnormal (or out of distribution) with respect to the data base. As in the finite-dimensional case, we use the infinite-dimensional Christoffel function as a score function to detect outliers and abnormal trajectories. A few numerical examples are provided to illustrate the theory.