Zonotopic representation of multi-variable regression with interval dependent variables

This paper extends existing work on imprecise regression by elucidating the set-valued representation of the computational problem of training regression models with interval-valued data. We frame the problem in a zonotopic sense to clearly expose the computational cost and limitations of training imprecise regression models with intervals. We focus on the regression problem where the independent predictor variables are precisely measured while the dependent response variable has uncertainty. Unlike classical regression, the dependent variable here is an intervalvalued random variable. We demonstrate that ordinary least-square regression can be reduced to a multi-dimensional Minkowski sum of line segments, each corresponding to an individual data interval. This means the problem is closed to zonotope representation and the uncertainty in the form of intervals can be propagated precisely. Recall that intervals are a crude but efficient method for computing with imprecise probability. The zonotopic representation enables the deconstruction of the feasible set of regression coefficients using affine transformations of the unit hypercube. This avoids the exponential enumeration of vertices characteristic of convex-hull propagation in multivariable regression. Consequently, the proposed zonotopic regression provides an exact and computationally efficient alternative to the expensive vertex-based computation of tight bounds and a preferable option to numeric-grade linear programming and overly conservative naive interval bounds. Additional advantages include the compact storage of uncertainty structures in generator matrices and a clear geometric interpretation of parameter uncertainty when working with interval-valued data for the dependent variable. Since multilinear regression is closed under affine transformations, this representation can be extended to nonlinear and neural-network regression models providing a framework for zonotopic uncertainty propagation. The method's efficiency is demonstrated on a variety of data affected by imprecision.

Keywords: Zonotopes, Imprecise Regression, Uncertainty Propagation, Interval-Valued Data.