Regression with Distance Matrices

TL;DR

This paper introduces a regression framework for data in metric spaces using distance matrices, multidimensional scaling, and scoring/backscoring methods, demonstrated on shape, curve, and correlation data.

Contribution

It proposes a novel approach to regression with non-vector data types via distance matrices and scoring/backscoring, enabling standard analysis methods.

Findings

Effective regression on shape, curve, and correlation data.

Demonstrated methodology on motion capture data.

Provides tools for prediction and interpretation in metric space data.

Abstract

Data types that lie in metric spaces but not in vector spaces are difficult to use within the usual regression setting, either as the response and/or a predictor. We represent the information in these variables using distance matrices which requires only the specification of a distance function. A low-dimensional representation of such distance matrices can be obtained using methods such as multidimensional scaling. Once these variables have been represented as scores, an internal model linking the predictors and the response can be developed using standard methods. We call scoring the transformation from a new observation to a score while backscoring is a method to represent a score as an observation in the data space. Both methods are essential for prediction and explanation. We illustrate the methodology for shape data, unregistered curve data and correlation matrices using motion capture data from an experiment to study the motion of children with cleft lip.