A New Analysis of an Adaptive Convex Mixture: A Deterministic Approach

TL;DR

This paper provides a deterministic analysis of an adaptive convex mixture method, guaranteeing performance bounds without relying on statistical assumptions, suitable for highly nonstationary and chaotic signals.

Contribution

It introduces a non-stochastic, guaranteed-performance analysis of the adaptive mixture, applicable to arbitrary signals without statistical modeling.

Findings

Achieves the performance of the best constituent filter in steady-state.

Outperforms individual filters in some cases.

Provides guarantees for nonstationary, chaotic signals.

Abstract

We introduce a new analysis of an adaptive mixture method that combines outputs of two constituent filters running in parallel to model an unknown desired signal. This adaptive mixture is shown to achieve the mean square error (MSE) performance of the best constituent filter, and in some cases outperforms both, in the steady-state. However, the MSE analysis of this mixture in the steady-state and during the transient regions uses approximations and relies on statistical models on the underlying signals and systems. Hence, such an analysis may not be useful or valid for signals generated by various real life systems that show high degrees of nonstationarity, limit cycles and, in many cases, that are even chaotic. To this end, we perform the transient and the steady-state analysis of this adaptive mixture in a "strong" deterministic sense without any approximations in the derivations or statistical assumptions on the underlying signals such that our results are guaranteed to hold. In particular, we relate the time-accumulated squared estimation error of this adaptive mixture at any time to the time-accumulated squared estimation error of the optimal convex mixture of the constituent filters directly tuned to the underlying signal in an individual sequence manner.