مرکز منطقه ای اطلاع رساني علوم و فناوري - Quantization complexity and training sample size in detection

Abstract :

For the $k$ -hypothesis detection problem, it is shown that, among the $k$ -classes of probability density functions with $m$ fixed quantiles, histograms achieve the least favorable performance as measured by the probability of correct detection and Chernoff distance. It is assumed that the $m$ cell probabilities are estimated using $n$ training samples per class. With the aid of the estimated cell probabilities, new observations are processed. A distribution-free upper bound to the probability of $\\epsilon$ -deviation between the actual probability of correct detection and the theoretical (known quantiles) probability is derived as a function of $(m,n,\\epsilon,k,u_{o})$ , where $u_{o}$ is a uniform upper bound to the true class densities. The bound converges exponentially to zero as $n \\rightarrow \\infty$ . Exponential convergence is obtained by choosing $m = n^{\\alpha }, 0 < \\alpha < 1$ . Hence, the rule $m = n^{\\alpha }$ answers the long standing question of how to relate $m$ and $n$ in a distribution-free manner. The question of the optimal choice of a is also discussed.

Keywords :

Quantization (signal); Signal detection; Signal quantization; Circuit noise; Circuit testing; Circuits and systems; Detectors; Probability density function; Quantization; Robustness; Signal detection; Statistics; Upper bound;