Several signal processing problems have been considered in the nonstochastic (or fraction-of-time probability) approach in [J5], [J8], [J9], [J11], [J14], [J15], [J16]. 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.
In [J14], the concept of the quantile in fraction-of-time probability framework is introduced. Moreover, 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. Furthermore, applications to the design of constant false-alarm rate radar processors and the analysis of real financial data are presented. The analysis of economic time-series in the nonstochastic approach is considered in [BC2], where the problem of predicting the Value-at-Risk is treated.
In [J24], the mathematical foundation of the functional (or nonstochastic) approach for signal analysis is established. The considered approach is alternative to the classical one that models signals as realizations of stochastic processes. The work follows the fraction-of-time probability approach introduced by Gardner. By applying the concept of relative measure used by Bochner, Bohr, Haviland, Jessen, Wiener, and Wintner and by Kac and Steinhaus, a probabilistic –but nonstochastic– model is built starting from a single function of time (the signal at hand). Therefore, signals are modelled without resorting to an underlying ensemble of realizations, i.e., the stochastic process model. Several existing results are put in a common, rigorous, measure-theory based setup. It is shown that by using the relative measure concept, a distribution function, the expectation operator, and all the familiar probabilistic parameters can be constructed starting from a single function of time. The new concept of joint relative measurability of two or more functions is introduced in this paper which is shown to be necessary for the joint characterization of signals. Moreover, by using such a concept, the independence of signals is defined. The joint relative measurability property is then used to prove the nonstochastic counterparts of several useful theorems for signal analysis. It is shown that the convergence of parameter estimators requires (analytical) assumptions on the single function of time that are much easier to verify than the classical ergodicity assumptions on stochastic processes. As an example of application, non relatively measurable functions are shown to be useful in the design of secure information transmission systems.
In [J33], the central limit theorem is proved within the framework of the functional approach for signal analysis. It is shown that if a sequence of independent signals fulfills some mild regularity assumptions, then the asymptotic distribution of the appropriately scaled average of such signals has a limiting normal distribution. The obtained results also allow one to rigorously justify stochastic models for signals and channels that up to now have been derived starting from a deterministic description of phenomena and for which the inferred stochastic model is built invoking a not proved ergodicity property. An application to the statistical characterization of the output signal of a multipath Doppler channel is presented.
In [J41], the problem of time average estimation is addressed in the fraction-of-time probability framework. Under mild regularity assumptions on temporal cumulants of the signal, a central limit theorem (CLT) is proved for the normalized error of the time average estimate, where the normalizing factor is the square root of the observation interval length. This rate of convergence is the same as that obtained in the classical stochastic approach, but is derived here without resorting to mixing assumptions. That is, no ergodicity hypothesis i needed. Examples of functions of interest in communications are provided whose time average estimate satisfies the CLT. For the class of the almost-periodic functions, a non normal limit distribution for the normalized error of the time average estimate can be obtained when the normalizing factor equals the observation interval length. The limit distribution, nevertheless, depends on the sequence adopted to perform the limit, a result that cannot be obtained in the classical stochastic approach. Numerical examples illustrate the theoretical results and an application to test the presence of a nonzero-mean cyclostationary signal is presented.
For a description of the work made in [J9], [J15], and [J16], see the Nonstationary Signal Analysis Section .