Weakly Convergent Nonparametric Forecasting of Stationary Time Series

TL;DR

This paper introduces a weakly consistent nonparametric method for forecasting stationary time series that efficiently utilizes finite past data, applicable to infinite alphabets, with potential uses in forecasting, regression, and classification.

Contribution

It develops a new weakly consistent nonparametric approach for estimating conditional distributions of stationary time series, especially for infinite alphabets, improving data efficiency.

Findings

The method achieves weak consistency in estimating conditional distributions.

It is applicable to infinite alphabet processes where strong consistency is impossible.

Applications include online forecasting, regression, and classification.

Abstract

The conditional distribution of the next outcome given the infinite past of a stationary process can be inferred from finite but growing segments of the past. Several schemes are known for constructing pointwise consistent estimates, but they all demand prohibitive amounts of input data. In this paper we consider real-valued time series and construct conditional distribution estimates that make much more efficient use of the input data. The estimates are consistent in a weak sense, and the question whether they are pointwise consistent is still open. For finite-alphabet processes one may rely on a universal data compression scheme like the Lempel-Ziv algorithm to construct conditional probability mass function estimates that are consistent in expected information divergence. Consistency in this strong sense cannot be attained in a universal sense for all stationary processes with values in an infinite alphabet, but weak consistency can. Some applications of the estimates to on-line forecasting, regression and classification are discussed.