Informatica

Second order properties of nearly nonstationary ARMA processes are investigated in the cases when the autoregressive polynomial equation has (i) a real root close to 1; (ii) a real root close to −1; (iii) a pair of complex roots close to the unit circle.

The idea of predicting the case, when the considered long-term ARMA model, fitted to the observed time series tends to become unstable because of deep changes in the structural stability of data, is developed in this paper. The aim is to predict a possible unstable regime of the process {Xt,t∈T}τ-steps in advance before it will express itself by a high level crossing or large variance of an output variable Xt. The problem is solved here for locally stationary AR(p) sequences {Xt,t∈T}, whose estimated parameters can reach critical sets located at the boundary of the stability area. An alarm function and an alarm set are fitted here to predict catastrophic failures in systems output τ units in advance for given τ>0 and a confidence level γ. The probability of false alarm is derived explicitly for AR(1) depending on τ,γ and N – the number of the last observations of {Xt}.

A likelihood approach is considered to the problem of making inferences about the point t = ν in a Gaussian autoregressive sequence {Xt, t = 1 ÷ N} at which the underlying AR(p) parameters undergo a sudden change. The statistics of a loglikelihood function L(n, ν) is investigated over the admissible values n ∈ (p + 1, $\dots$ , N - 1) of a change point ν under validity of hypothesis of a change and no change. The expressions of L(n, ν) implying the loss of plausibility when moving away from the true change point ν are presented, and the probabilities P{$\bar{v}_{N}$ = ν± r}, r = 0,1,2, $\dots$, where $\bar{v}_{N}$ is the MLH estimate of a change point ν from the available realization x1,x2,…,xN of {Xt, t = 1 ÷ N} are considered.