On Sequential Random Distortion Testing of Non-Stationary Processes

Abstract

In this work, we propose a non-parametric sequential hypothesis test based on random distortion testing (RDT). RDT addresses the problem of testing whether or not a random signal, Ξ, observed in independent and identically distributed (i.i.d) additive noise deviates by more than a specified tolerance, τ , from a fixed model, ξ0. The test is non-parametric in the sense that the underlying signal distributions under each hypothesis are assumed to be unknown. The need to control the probabilities of false alarm (PFA) and missed detection (PMD), while reducing the number of samples required to make a decision, leads to a novel sequential algorithm, SeqRDT. We show that under mild assumptions on the signal, SeqRDT follows the properties desired by a sequential test. We introduce the concept of a buffer and derive bounds on PFA and PMD, from which we choose the buffer size. Simulations show that SeqRDT leads to faster decisionmaking on an average compared to its fixed-sample-size (FSS) counterpart, BlockRDT. These simulations also show that the proposed algorithm is robust to model mismatches compared to the sequential probability ratio test (SPRT).

On Random Distortion Testing Based Sequential Non-Parametric Hypothesis Testing

“…First is BlockRDT proposed in [15], which solves this problem for a fixed-sample-size test, i.e., for a fixed N . And the second is SeqRDT proposed in [16], [17], which solves the problem using a sequential approach.…”

“…In this section, we briefly review SeqRDT proposed in [16], [17]. SeqRDT tests the hypothesis defined in (3) in a sequential manner.…”

“…Remark 2: In [16], [17], SeqRDT used this assumption to design a buffer, M , which made it possible to control the false alarm and missed detection probabilities below α and β, respectively. The assumption implies that the empirical mean of the signal under the two hypotheses is bounded away from τ .…”

“…In contrast, SeqRDT [16], [17] required additional assumptions on the signal (see Assumption 4.1). Now, we show that unlike the likelihood based approaches, T-SeqRDT can be designed with only the knowledge of this τ + and τ instead of knowing the complete distribution of the signal.…”

“…They generalized the RDT formulation by replacing Ξ with its empirical mean over time. Later, in [16], [17] the authors further extended RDT framework for sequential testing where SeqRDT was proposed to guarantee both false alarm and missed detection probabilities below certain levels, while making a decision faster on average compared to BlockRDT. In this work, we first briefly review BlockRDT and SeqRDT and then propose the truncated version of SeqRDT: T-SeqRDT.…”