Journal of Applied Mathematics and Stochastic Analysis
Volume 10 (1997), Issue 4, Pages 333-353

The empirical TES methodology: modeling empirical time series

Benjamin Melamed

Faculty of Management, Department of MSIS and RUTCOR, Rutger University Center for Operations Research, New Brunswick 08903, NJ, USA

Received 1 September 1996; Revised 1 February 1997

Copyright © 1997 Benjamin Melamed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


TES (Transform-Expand-Sample) is a versatile class of stochastic sequences defined via an autoregressive scheme with modulo-1 reduction and additional transformations. The scope of TES encompasses a wide variety of sample path behaviors, which in turn give rise to autocorrelation functions with diverse functional forms - monotone, oscillatory, alternating, and others. TES sequences are readily generated on a computer, and their autocorrelation functions can be numerically computed from accurate analytical formulas at a modest computational cost.

This paper presents the empirical TES modeling methodology which uses TES process theory to model empirical records. The novel feature of the TES methodology is that it expressly aims to simultaneously capture the empirical marginal distribution (histogram) and autocorrelation function. We draw attention to the non-parametric nature of TES modeling in that it always guarantees an exact match to the empirical marginal distribution. However, fitting the corresponding autocorrelation function calls for a heuristic search for a TES model over a large parametric space. Consequently, practical TES modeling of empirical records must currently rely on software assistance. A visual interactive software environment, called TEStool, has been designed and implemented to support TES modeling. The paper describes the empirical TES modeling methodology as implemented in TEStool and provides numerically-computable formulas for TES autocorrelations. Two examples illustrate the efficacy of the TES modeling approach. These examples serve to highlight the ability of TES models to capture first-order and second-order properties of empirical sample paths and to mimic their qualitative appearance.