Memory-universal prediction of stationary random processes

• Published in: IEEE transactions on information theory Vol. 44; no. 1; pp. 117 - 133
• Main Authors: Modha, D.S., Masry, E.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.01.1998; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Convergence; Information processing; Least squares approximation; Markov analysis; Markov processes; Mathematical models; Neural networks; Parametric statistics; Polynomials; Predictive models; Pricing; Random processes; Signal processing
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/18.650998

We consider the problem of one-step-ahead prediction of a real-valued, stationary, strongly mixing random process (Xi)/sub i=-/spl infin///sup /spl infin//. The best mean-square predictor of X/sub 0/ is its conditional mean given the entire infinite past (X/sub i/)/sub i=-/spl infin///sup -1/. Given a sequence of observations X/sub 1/, X/sub 2/, X/sub N/, we propose estimators for the conditional mean based on sequences of parametric models of increasing memory and of increasing dimension, for example, neural networks and Legendre polynomials. The proposed estimators select both the model memory and the model dimension, in a data-driven fashion, by minimizing certain complexity regularized least squares criteria. When the underlying predictor function has a finite memory, we establish that the proposed estimators are memory-universal: the proposed estimators, which do not know the true memory, deliver the same statistical performance (rates of integrated mean-squared error) as that delivered by estimators that know the true memory. Furthermore, when the underlying predictor function does not have a finite memory, we establish that the estimator based on Legendre polynomials is consistent.