Optimal Noise Benefits in Neyman-pearson and Inequality-constrained Statistical Signal Detection

We present theorems and an algorithm to find optimal or near-optimal “stochastic resonance” (SR) noise benefits for Neyman-Pearson hypothesis testing and for more general inequality-constrained signal detection problems. The optimal SR noise distribution is just the randomization of two noise realizations when the optimal noise exists for a single inequality constraint on the average cost. The theorems give necessary and sufficient conditions for the existence of such optimal SR noise in inequality-constrained signal detectors. There exists a sequence of noise variables whose detection performance limit is optimal when such noise does not exist. Another theorem gives sufficient conditions for SR noise benefits in Neyman-Pearson and other signal detection problems with inequality cost constraints. An upper bound limits the number of iterations that the algorithm requires to find near-optimal noise. The appendix presents the proofs of the main results. I. NOISE BENEFITS IN SIGNAL DETECTION Stochastic resonance (SR) occurs when noise benefits a nonlinear system [1]–[14]. SR noise benefits occur in a wide range of applications in physics, biology, and medicine [15]– [26]. The noise benefit can take the form of an increase in an entropy-based bit count [27]–[29] , a signal-to-noise ratio [30]–[32], a cross-correlation [3], [30], or a detection probability for a preset level of false-alarm probability [33], [34], or a decrease in the error probability [35] or in the average sample number of sequential detection problems [36]. An SR noise benefit requires some form of nonlinear signal detection [10]. Its signature often takes the form of an invertedU curve or a nonmonotonic plot of a bit count or SNR against the variance or dispersion of the noise process. We focus first on the special case of SR in signal detection that uses Neyman-Pearson (N-P) hypothesis testing [37] to decide between two simple alternatives. We define the noise as N-P SR noise if adding such noise to the received signal before making a decision increases the signal detection probability PD while the false-alarm probability PFA stays at or below a preset level α for a given detection strategy. Figure 1 shows this type of noise benefit for a suboptimal receiver and does not involve the typical inverted-U curve of SR (but would if it used uniform noise and we plotted the detection probability against the noise variance). An SR noise benefit does not occur in an optimal receiver if the noise is independent of the concurrent received signal and the hypotheses. This follows from the so-called irrelevance theorem of optimal detection [38], [39]. But Section V shows that SR noise benefits can occur even if the receiver is optimal when the noise depends on the received signal. Figure 3 Ashok Patel and Bart Kosko are with the Signal and Image Processing Institute, Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089-2564, USA. Email: [email protected] shows such an SR noise benefit in optimal anti-podal signal detection when the average signal power constrains the signal transmission. Sections II and III present three SR results for NeymanPearson signal detection. The first SR result gives necessary and sufficient conditions for the existence of optimal N-P SR noise. The existence of some N-P SR noise does not itself imply the existence of optimal noise. But there exists a sequence of noise variables whose detection performance limit is optimal when the optimal N-P SR noise does not exist. The second SR result is a sufficient condition for SR noise benefits in N-P signal detection. The third SR result is an algorithm that finds near-optimal N-P SR noise from a finite set Ñ of noise realizations. This noise is nearly optimal if the detection and false alarm probabilities in Ñ and in the actual noise space N ⊃ Ñ are sufficiently close. An upper bound limits the number of iterations that the algorithm needs to find near-optimal noise. Section IV extends these results to more general statistical decision problems that have one inequality constraint on the average cost. These SR results extend and correct prior work in “detector randomization” or adding noise for improving the performance of N-P signal detection. Tsitsiklis [40] explored the mechanism of detection-strategy randomization for a finite set of detection strategies (operating points) in decentralized detection. He showed that there exists a randomized detection strategy that uses a convex or random combination of at most two existing detection strategies and that gives the optimal N-P detection performance. Such optimal detection strategies lie on the upper boundary of the convex hull of the receiver operating characteristic (ROC) curve points. Scott [41] later used the same optimization principle in classification systems while Appadwedula [42] used it for energy-efficient detection in sensor networks. Then Chen et al. [33] used a fixed detector structure: they injected independent noise in the received signal to obtain a proper random combination of operating points on the ROC curve for a given suboptimal detector. They showed that the optimal N-P SR noise for suboptimal detectors randomizes no more than two noise realizations. But Chen et al. [33] assumed that the convex hull V of the set of ROC curve operating points U ⊆ R always contains its boundary ∂V and thus that the convex hull V is closed. This is not true in general. The topological problem is that the convex hull V need not be closed if U is not compact [43]: the convex hull of U is open if U itself is open [44]. Chen et al. argued correctly along the lines of the proof of Theorem 3 in [33] when they concluded that the “optimum pair can only exist on the boundary.” But their later claim that “each z on