Models For Discrimination And Simulation Of Positive-valued, Correlated Random Processes

Abstract

ModelsThere are two primary purposes for the use of a random process model. First of all, a model provides a framework for statistical inference, such as estimation or detection. An area of statistical inference which we consider here is discrimination, where a decision is to be made between two or more possible sources for the observed data. Second, a model may needed for simulation, or, specifically, for the generation of synthetic data to be used in computer simulations. Fidelity of the model to the precise statistical nature of the process is more important for simulation than for discrimination. However, for discrimination, tractability of the model probability distributions is usually required.One of the most universally used models is the transformed Gaussian model, where one assumes that if { X ; } is the observed time series, then the series {Zi} where 2; = $-'(XI) is Gaussian. The lognormal process provides a typical example. This approach provides both simplicity (usually resulting in mathematical tractability) and generality. In such a model, one typically measures and matches univariate marginal distributions and autocorrelation. The density function may be evaluated in an especially efficient manner if the Gaussian process is further assumed to be kth-order autoregressive process. If the process {X,} assumes only positive values, then the transformed Gaussian model may not be the most desirable. The linear dependency structure of the Gaussian process becomes distorted by the transformation. An alternative is to assume that the underlying process {Z;} has a distribution other than a Gaussian distribution; in particular, we may assume a x2 distribution for {Zi}. Since the x2 distribution also admits only positive values, the dependency structure is likely to be less distorted by the transformation. Multivariate x2 random variables which are correlated may be generated simply by Zi = U?i + . . . + U:& where each {Uli,z E N} has a multivariate Gaussian distribution. The transformed x2 model yields a dependency structurewhich differs significantly from the transformed Gaussian model, even when the autocorrelation is the same. Although the density involved in the x 2 model is more difficult to evaluate than the transformed Gaussian density, with certain assumptions, the transformed x2 density may also be efficiently computed. Discriminat ionIf a transformed Gaussian model is assumed, where the Gaussian process is a kth-order stationary autoregressive process, then the LRT for any discrimination problem may be evaluated as a sum of k-step memory nonlinearities. That is, we have n Tn = gO(x1,. . . > Z k ) + g(xj,. . x j -k ) . j = k + lwhere Tn is the log-likelihood ratio Tn = log(f,(")/fi")). This result is obtained by using the factorization Q = LLT of the inverse of the covariance matrix, where L is a lower triangular matrix with L,, = 0 for 12 -j l > k. Thus the likelihood ratio (LR) may be efficiently evaluated. The method works equally efficiently for testing multiple hypotheses, as in pattern recogni...