Random distortion testing with linear measurements

Abstract

This paper addresses the problem of testing whether, after linear transformations and possible dimensionality reductions, a random matrix of interest Θ deviates significantly from some matrix model θ 0 , when Θ is observed in additive independent Gaussian noise with known covariance matrix. In contrast to standard likelihood theory, the probability distribution of Θ is assumed to be unknown. This problem generalizes the Random Distortion Testing (RDT) problem addressed in a former paper. Although the notions of size and power can be extended so as to deal with this generalized problem, no Uniformly Most Powerful (UMP) test exists for it. We can however exhibit a relevant subclass of tests and prove the existence of an optimal test within this class, that is, a test with specified size and maximal constant conditional power. As a consequence, this test is also UMP among invariant tests. The method fits within a wide range of signal processing scenarios. It is here specifically applied to sequential detection, subspace detection and random distortion testing after space-time compressive sensing. It is also used to extend the GLRT optimality properties for testing a waveform amplitude in noise.

Truncated Sequential Non-Parametric Hypothesis Testing Based on Random Distortion Testing

“…Specifically, suppose that we have only N samples from our observation Y so (5). To solve this testing problem, the authors in [28], [34] consider all the FSS tests D N0 (N ) = T(Y ), where T is any (measurable) map T : R N → {0, 1}. All such maps T are hereafter called Ndimensional tests.…”

“…All such maps T are hereafter called Ndimensional tests. In the BlockRDT framework [28], [34], we define the size of a given N -dimensional test T as:…”

On Random Distortion Testing Based Sequential Non-Parametric Hypothesis Testing

“…BlockRDT framework tests the hypotheses defined in (3) for a fixed number of samples, i.e., Y = Ξ + X ∈ M(Ω, R) 1,N . A solution to this problem is proposed in [15] & [19]. Next, we discuss the optimality properties of the test defined in (4) for BlockRDT.…”

“…However, no UMP test with level γ exists for BlockRDT. We therefore show the subclass of BlockRDT-coherent tests, among which a "best" test exists [19]. An N -dimensional test T is for BlockRDT-coherent if it satisfies the following properties:…”

“…The N -dimensional test defined for all y ∈ R N given by (4) is maximal in K γ , i.e., for any T ∈ K γ we have T T N,γ [15], [19].…”