The profound impact of negative power law noise on the estimation of causal behavior

This paper demonstrates that highly correlated negative power law (neg-p) noise - noise with a PSD prop |f|^p when p<0 - and causal behavior contained in data cannot be properly separated from each other by any causal fitting or estimation technique. It then shows: (a) that this leads such techniques to generate anomalous estimates of the true causal behavior and to underreport the true fit error, and (b) that these anomalies cannot be corrected by adding modeling or increasing the data collection interval T. Further consequences of the above are also discussed. These include anomalous behavior in noise whitening, M-corner hat, and cross-correlation techniques and the non-observability of unbiased measures of pure random error when neg-p noise is present. The paper also investigates the relationship between ergodic-like behavior - the approximate equality of time averages over a finite T and ensemble averages - and such anomalous behavior in general statistical processing techniques. It is shown that such ergodic-like behavior is a necessary condition for practical implementations of these techniques to behave like their theoretical counterparts and, furthermore, a necessary condition for ergodic-like behavior is TGttau_c, where tau_c is the correlation time of the noise process involved. Finally, it is shown that this is unachievable for neg-p noise, because tau_c = infin for such noise (unless system highpass filtering makes tau_c finite for the system filtered noise variable).