Extensions of Morse-Smale Regression with Application to Actuarial Science

TL;DR

This paper enhances Morse-Smale regression by integrating various machine learning algorithms, improving prediction accuracy and interpretability in modeling complex subgroups, especially in actuarial science applications.

Contribution

It introduces the use of machine learning methods within Morse-Smale regression, expanding its applicability and effectiveness for nonlinear and interaction effects.

Findings

Machine learning approaches improve Morse-Smale regression performance.

Methods perform well on simulated Tweedie regression problems.

Algorithms provide insights into predictor relationships within data partitions.

Abstract

The problem of subgroups is ubiquitous in scientific research (ex. disease heterogeneity, spatial distributions in ecology...), and piecewise regression is one way to deal with this phenomenon. Morse-Smale regression offers a way to partition the regression function based on level sets of a defined function and that function's basins of attraction. This topologically-based piecewise regression algorithm has shown promise in its initial applications, but the current implementation in the literature has been limited to elastic net and generalized linear regression. It is possible that nonparametric methods, such as random forest or conditional inference trees, may provide better prediction and insight through modeling interaction terms and other nonlinear relationships between predictors and a given outcome. This study explores the use of several machine learning algorithms within a Morse-Smale piecewise regression framework, including boosted regression with linear baselearners, homotopy-based LASSO, conditional inference trees, random forest, and a wide neural network framework called extreme learning machines. Simulations on Tweedie regression problems with varying Tweedie parameter and dispersion suggest that many machine learning approaches to Morse-Smale piecewise regression improve the original algorithm's performance, particularly for outcomes with lower dispersion and linear or a mix of linear and nonlinear predictor relationships. On a real actuarial problem, several of these new algorithms perform as good as or better than the original Morse-Smale regression algorithm, and most provide information on the nature of predictor relationships within each partition to provide insight into differences between dataset partitions.