Signal extraction from nonstationary time series
The paper deals with the extraction problem \(z_ t=s_ t+n_ t\), where \(z_ t\) is an observable time series, \(s_ t\) is an unobservable signal which is to be estimated and \(n_ t\) is an unobservable noise. The considered extraction problem is nonstationary so that either \(s_ t\) or \(n_ t\) or both are nonstationary. Certain assumptions on the generation of the time series \(z_ t\), \(s_ t\) and \(n_ t\) and their starting values are considered in the paper. E.g. it is assumed that processes \(s_ t\) and \(n_ t\) can be modelled by means of \(\delta_ s(B)s_ t=u_ t\) and \(\delta_ n(B)=v_ t\), where \(u_ t\) and \(v_ t\) are independent stationary time series and \(\delta_ s(B)\) and \(\delta_ n(B)\) are polynomials whose zeros lie on or inside the unit circle (it implies the form of the model for \(z_ t).\) Two sets of alternative assumptions regarding the generation of \(z_ t\), \(s_ t\) and \(n_ t\) including starting values are considered in the paper. These assumptions play an important role in the extraction and should not be in any case ignored. The paper gives the solution of the extraction problem in such a way that the exact formulae for the conditional expectation \(E(s_ t| \{z_ t\})\) and for the conditional variance \(var(s_ t| \{z_ t\})\) are derived. Extensions to signal extraction with a finite number of observations, to the non Gaussian case and to the multivariate case are discussed. No practical experiences and applications are reported in the paper.
scientific article; zbMATH DE number 3862287