Uncertainty Quantification in Power Spectrum Estimation of Stochastic Processes Subject to Missing Data

Abstract

An analytical expression for the probability density function (PDF) of a random process power spectrum value at a given frequency is derived. The significance of the derived PDF relates to cases where incomplete process realizations are available for power spectrum estimation applications. Specifically, standard power spectrum estimation techniques subject to missing data such as zero-padding and least squares regression normally provide with a deterministic estimate for the power spectrum value at a given frequency. In this regard, no information is provided concerning the uncertainty in the estimates. In this paper, relying on the assumptions of stationarity and ergodicity, and on minimum assumptions about the missing data, an explicit closed-form analytical expression for the PDF of the power spectrum value at each and every frequency is derived. In this manner, the uncertainty propagation from the measured incomplete process realization in the time/space domain to its power spectrum estimate in the frequency domain is efficiently quantified. Further, the derived results can be potentially used for assessing the performance of alternative power spectrum estimation techniques subject to missing data which provide with a deterministic estimate for the power spectrum value at a specific frequency.