Convex-area-wise Linear Regression and Algorithms for Data Analysis

TL;DR

This paper presents Convex-Area-Wise Linear Regression (CALR), a novel interpretable regression method that partitions data into convex regions and fits linear models, capable of handling non-linear and discontinuous relationships.

Contribution

The paper introduces CALR, proposes three algorithms for solving it efficiently, and establishes its equivalence to a mixed integer programming formulation for approximate solutions.

Findings

Algorithms solve CALR in sub-quadratic time under certain conditions.

CALR can interpolate complex datasets with non-linear and discontinuous patterns.

The mixed integer programming formulation enables practical approximate solutions.

Abstract

This paper introduces a new type of regression methodology named as Convex-Area-Wise Linear Regression(CALR), which separates given datasets by disjoint convex areas and fits different linear regression models for different areas. This regression model is highly interpretable, and it is able to interpolate any given datasets, even when the underlying relationship between explanatory and response variables are non-linear and discontinuous. In order to solve CALR problem, 3 accurate algorithms are proposed under different assumptions. The analysis of correctness and time complexity of the algorithms are given, indicating that the problem can be solved in $o(n^2)$ time accurately when the input datasets have some special features. Besides, this paper introduces an equivalent mixed integer programming problem of CALR which can be approximately solved using existing optimization solvers.