A Few Results about the Geometry of Model Averages

TL;DR

This paper investigates the geometric properties of model averaging, comparing its performance to individual models in estimating a natural process, and provides conditions and bounds for when averaging outperforms single models.

Contribution

It introduces geometric conditions that determine when a model average outperforms individual models and provides sharp bounds on average performance within a fixed observation interval.

Findings

A condition that predicts when a single model outperforms the average.

Necessary conditions for the average to outperform any individual model.

Sharp bounds on the performance of the average over a fixed interval.

Abstract

Given a collection of computational models that all estimate values of the same natural process, we compare the performance of the average of the collection to the individual member whose estimates are nearest a given set of observations. Performance is the ability of a model, or average, to reproduce a sequence of observations of the process. We identify a condition that determines if a single model performs better than the average. That result also yields a necessary condition for when the average performs better than any individual model. We also give sharp bounds for the performance of the average on a given interval. Since the observation interval is fixed, performance is evaluated in a vector space, and we can add intuition to our results by explaining them geometrically. We conclude with some comments on directions statistical tests of performance might take.