Quantile prediction for time series in the fraction-of-time probability framework

Abstract

The aim of this work is to introduce the concept of the quantile and propose its prediction algorithms using the fraction-of-time probability approach. In such an approach, unlike the classical one based on stochastic processes, statistical functions and probability concepts are defined starting from a single observed time series instead of an ensemble of realizations of a stochastic process. Two prediction algorithms based on a single observed time series and without any distributional assumption are proposed. The former is devoted to deal with statistics not depending on time (stationary case) whereas the latter considers statistics that depend on time (nonstationary case). Convergence and estimation accuracy issues are considered in the paper without checking the usual mixing assumptions, used in the classical stochastic approach. Moreover, applications to the design of constant false-alarm rate radar processors and the analysis of real financial data are presented.

Introduction

An alternative signal analysis framework that does not resort to random processes is the fraction-of-time (FOT) probability approach [8], [9], [11]. In such an approach, statistical parameters are defined through infinite-time averages of signals (i.e., single functions of time) rather than ensemble averages of random processes.

The adoption of the FOT probability approach is motivated by the fact that in several applications, e.g. time series in finance, signal processing and climatology, multiple realizations of a stochastic process do not exist or are not at a disposal of the experimenter. Rather, there is only one realization that can be assumed to be observed for an increasing-length time interval. Furthermore, the adoption of the FOT probability approach simplifies convergence issues. For example, in stochastic approach, convergence of empirical quantiles is based on convergence of empirical distribution functions but the latter convergence requires in most cases some asymptotic independence assumptions. The FOT probability approach does not have such difficulty provided that some very general analytic conditions on the observed time series are fulfilled.

The differences between the FOT and stochastic approach have a variety of implications in the study of properties of inferential techniques involved. For example, in the FOT probability framework a natural way to define estimators is through considering finite-time averages of the same quantities involved in the infinite-time averages. Thus, the kind of convergence of the estimators to be considered as the time of the observation approaches infinity is the convergence of the sequence of the finite-time averages, that is, pointwise, in the temporal mean-square sense [23], or in the sense of generalized functions (distributions) [19]. On the contrary, in the stochastic process framework, the convergence must be demonstrated in the mean-square, almost everywhere or weak convergence sense.

The FOT probability approach was first introduced in [23] with reference to time-invariant statistics of ordinary functions of time. Later, it was developed in [5], [13], and then extended to the case of distributions (generalized functions) in [19]. Furthermore, in [24] an isometric isomorphism (Wold isomorphism) between a stationary stochastic process and the Hilbert space generated by a single sample path was singled out and a rigorous link between FOT probability and the stochastic process frameworks in the stationary case was established. The case of time-variant FOT statistics of almost-cyclostationary time-series was widely treated in [7], [8], [10] with reference to the second-order statistics and in [12], [21] for the higher-order statistics. Moreover, the Wold isomorphism was extended to the case of cyclostationary time-series in [14]. A further development of the FOT probability theory for nonstationary signals was presented in [15]. In that paper, a more general class of nonstationary time-series called the generalized almost-cyclostationary time-series has been introduced and characterized.

In the present paper, quantile prediction algorithms in the FOT probability framework are proposed. Such algorithms are based on the observation of the only time series at disposal of the experimenter and do not make any assumption on the possible stochastic process of which it is a sample path. Therefore, they operate under less restrictive assumptions than those made in the stochastic approach (see, e.g. [3], [4] for the analysis of financial data). Two cases are considered in the paper. The former deals with statistics that do not depend on time and will be referred to as the stationary case. The latter, instead, deals with statistics that are time varying and will be referred to as the nonstationary case. For the stationary case the convergence of the predicted quantile is proved when the observation interval increases. For the nonstationary case it is shown that a trade-off exists between the necessity of obtaining a convergent predicted value and the necessity of obtaining time varying statistics to account for the existence of time varying mechanisms in the generation of the observed time series.

The paper is organized as follows. In Section 2, general concepts and definitions of the FOT probability framework are reviewed. Section 3 is devoted to the definition of the quantile in the FOT probability framework. Moreover, the problem of predicting quantiles (without the independence and same distribution assumption) is also presented. Numerical results for the case of the stepwise function, a radar application, and the analysis of financial data are described in Section 4. Finally, conclusions are drawn in Section 5.