Spectral estimation: An overdetermined rational model equation approach

In seeking rational models of time series, the concept of approximating second-order statistical relationships (i.e., the Yule-Walker equations) is often explicitly or implicitly invoked. The parameters of the hypothesized rational model are typically selected so that these relationships "best represent" a set of autocorrelation lag estimates computed from time series observations. One of the objectives of this paper will be that of establishing this fundamental approach to the generation of rational models. An examination of many popular contemporary spectral estimation methods reveals that the parameters of a hypothesized rational model are estimated upon using a "minimal" set of Yule-Walker equation evaluations. This results in an undesired parameter hypersensitivity and a subsequent decrease in estimation performance. To counteract this parameter hypersensitivity, the concept of using more than the minimal number of Yule-Walker equation evaluations is herein advocated. It is shown that by taking this overdetermined parametric evaluation approach, a reduction in data-induced model parameter hypersensitivity is obtained, and a corresponding improvement in modeling performance results. Moreover, upon adapting a singular value decomposition representation of an extended-order autocorrelation matrix estimate to this procedure, a desired model order determination method is obtained and a further significant improvement in modeling performance is achieved. This approach makes possible the generation of low-order high-quality rational spectral estimates from short data lengths.