[Paper Review] Extensions of linear regression models based on set arithmetic for interval data

Ángela Blanco, Marta García-Bárzana+2 more|arXiv (Cornell University)|Oct 22, 2012

Fuzzy Systems and Optimization2 references3 citations

TL;DR

This paper extends linear regression models for interval-valued data using set arithmetic and canonical decomposition, enabling flexible modeling of cross-relationships between midpoints and spreads. The proposed models, solved via constrained least-squares estimation with KKT conditions, demonstrate improved flexibility and consistency through simulations and a real-world application.

ABSTRACT

Extensions of previous linear regression models for interval data are presented. A more flexible simple linear model is formalized. The new model may express cross-relationships between mid-points and spreads of the interval data in a unique equation based on the interval arithmetic. Moreover, extensions to the multiple case are addressed. The associated least-squares estimation problem are solved. Empirical results and a real-life application are presented in order to show the applicability and the differences among the proposed models.

Motivation & Objective

• To develop a more flexible simple linear regression model for interval data that captures cross-relationships between midpoints and spreads.
• To extend the proposed model to multiple regression settings using set arithmetic and canonical decomposition.
• To solve least-squares estimation problems under constraints ensuring model coherency and residual existence.
• To validate the models empirically through simulations and a real-life application.
• To provide a foundation for future theoretical studies on estimator properties such as bias, consistency, and asymptotic distributions.

Proposed method

• Formalizes a new simple linear model based on canonical decomposition, allowing joint modeling of midpoints and spreads via interval arithmetic.
• Applies Minkowski addition and scalar multiplication in the (mid, spr) representation to define interval operations.
• Solves constrained least-squares estimation problems using Karush-Kuhn-Tucker (KKT) conditions due to non-negativity constraints on spreads.
• Introduces two distinct multiple regression models: one based on canonical decomposition and another derived from the basic model in [9], both solved via KKT-based algorithms.
• Uses the $L_2$-type metric with a weight parameter $\theta$ to measure interval distances in the estimation framework.
• Employs simulation studies with 10,000 iterations per sample size to evaluate estimator performance and consistency.

Experimental results

Research questions

• RQ1 Can a more flexible simple linear regression model be developed for interval data that captures cross-relationships between midpoints and spreads?
• RQ2 How can the canonical decomposition framework be extended to multiple regression models for interval-valued data?
• RQ3 What constrained optimization techniques are necessary to ensure valid least-squares estimation under non-negativity constraints on spreads?
• RQ4 How do the proposed models compare in performance to existing models in terms of bias, mean squared error, and consistency?
• RQ5 What is the empirical behavior of the LS estimators across varying sample sizes in simulation studies?

Key findings

• The proposed flexible simple model successfully captures cross-relationships between midpoints and spreads through a single equation based on interval arithmetic.
• Least-squares estimators are asymptotically unbiased, with mean values converging to true parameters and mean squared error decreasing as sample size increases.
• Box plots of estimators show reduced interquartile range around true values with larger sample sizes, confirming estimator consistency.
• The KKT conditions provide a viable method for solving constrained least-squares problems in both simple and multiple regression models.
• The multiple basic model (M1) is found to be too restrictive for modeling real physical magnitudes, indicating the need for more flexible alternatives.
• Simulation results demonstrate that the proposed models yield stable and accurate parameter estimates, with performance improving with larger sample sizes.