Zero Mean Noise Processes that Do Not Appear to be Zero Mean
Victor S. Reinhardt · 2009
This paper shows that correlated but random zero mean noise processes can cause estimation filters (such as Kalman and least squares filters) to generate anoma lous results that are similar to those generated by unmo deled non-zero mean causal behavior. The paper discusses two types of such noise processes: stationary Poisson a nd Gauss-Markov processes and non-stationary negative power law (neg-p) processes. The paper shows that s uch anomalous results are due to two factors. First, it is shown that single ensemble members of such correlated noi se processes exhibit non-ergodic-like behavior over a finite data collection interval T, if τ c the correlation time of the noise process is an appreciable fraction of T, even if the process is strictly ergodic as T →∞. This these causes practical realizations of such filters, which rely on ergodic-like behavior in a single ensemble member f or their proper operation, to generate results that de viate from theoretically predicted behavior. Second, it i s shown that the signatures over T of such individual ensem ble members can mimic those of the desired signal being estimated. This causes the filter to treat such noi se as part of the desired signal and leads to the underestimat ion of the true error generated by the noise. It is then s hown that setting T >> τc allows one to properly separate the noise from the desired signal and to properly estimate th e true error. It is further shown that τc → ∞ for neg-p noise, and thus one can never separate the correlated part of such noise from the desired signal, regardless of the va lue of T. This is shown to lead to infinite true errors for n eg-p noise unless one introduces periodic calibration to bound the divergent effects of such noise. The paper conclude s with a description of (causal) environmentally induced e rrors that mask their causal nature and exhibit behavior similar to correlated ZM noise processes.