Error exponents for Neyman-Pearson detection of a continuous-time Gaussian Markov process from noisy irregular samples

TL;DR

This paper characterizes the exponential decay of the Type II error probability in detecting a Gaussian Markov process from noisy irregular samples, linking it to Kalman filter behavior and providing insights for sensor network applications.

Contribution

It introduces a complete characterization of error exponents for Neyman-Pearson detection of Gaussian Markov processes with irregular sampling, connecting detection performance to Kalman filter asymptotics.

Findings

Error exponents decrease exponentially with sample size.

Error exponents are linked to Kalman filter asymptotics.

Numerical results demonstrate detection performance in sensor networks.

Abstract

This paper addresses the detection of a stochastic process in noise from irregular samples. We consider two hypotheses. The \emph{noise only} hypothesis amounts to model the observations as a sample of a i.i.d. Gaussian random variables (noise only). The \emph{signal plus noise} hypothesis models the observations as the samples of a continuous time stationary Gaussian process (the signal) taken at known but random time-instants corrupted with an additive noise. Two binary tests are considered, depending on which assumptions is retained as the null hypothesis. Assuming that the signal is a linear combination of the solution of a multidimensional stochastic differential equation (SDE), it is shown that the minimum Type II error probability decreases exponentially in the number of samples when the False Alarm probability is fixed. This behavior is described by \emph{error exponents} that are completely characterized. It turns out that they are related with the asymptotic behavior of the Kalman Filter in random stationary environment, which is studied in this paper. Finally, numerical illustrations of our claims are provided in the context of sensor networks.