Alternative approaches and noise benefits in hypothesis-testing problems in the presence of partial information
Author
Bayram, Suat
Advisor
Gezici, Sinan
Date
2011Publisher
Bilkent University
Language
English
Type
ThesisItem Usage Stats
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Abstract
Performance of some suboptimal detectors can be enhanced by adding independent
noise to their observations. In the first part of the dissertation, the effects
of additive noise are studied according to the restricted Bayes criterion, which
provides a generalization of the Bayes and minimax criteria. Based on a generic
M-ary composite hypothesis-testing formulation, the optimal probability distribution
of additive noise is investigated. Also, sufficient conditions under which
the performance of a detector can or cannot be improved via additive noise are
derived. In addition, simple hypothesis-testing problems are studied in more
detail, and additional improvability conditions that are specific to simple hypotheses
are obtained. Furthermore, the optimal probability distribution of the
additive noise is shown to include at most M mass points in a simple M-ary
hypothesis-testing problem under certain conditions. Then, global optimization,
analytical and convex relaxation approaches are considered to obtain the optimal
noise distribution. Finally, detection examples are presented to investigate the
theoretical results.
In the second part of the dissertation, the effects of additive noise are studied
for M-ary composite hypothesis-testing problems in the presence of partial
prior information. Optimal additive noise is obtained according to two criteria,
which assume a uniform distribution (Criterion 1) or the least-favorable distribution
(Criterion 2) for the unknown priors. The statistical characterization of
the optimal noise is obtained for each criterion. Specifically, it is shown that the
optimal noise can be represented by a constant signal level or by a randomization
of a finite number of signal levels according to Criterion 1 and Criterion 2,
respectively. In addition, the cases of unknown parameter distributions under
some composite hypotheses are considered, and upper bounds on the risks are
obtained. Finally, a detection example is provided to illustrate the theoretical
results.
In the third part of the dissertation, the effects of additive noise are studied
for binary composite hypothesis-testing problems. A Neyman-Pearson (NP)
framework is considered, and the maximization of detection performance under a
constraint on the maximum probability of false-alarm is studied. The detection
performance is quantified in terms of the sum, the minimum and the maximum of
the detection probabilities corresponding to possible parameter values under the
alternative hypothesis. Sufficient conditions under which detection performance
can or cannot be improved are derived for each case. Also, statistical characterization
of optimal additive noise is provided, and the resulting false-alarm
probabilities and bounds on detection performance are investigated. In addition,
optimization theoretic approaches for obtaining the probability distribution of
optimal additive noise are discussed. Finally, a detection example is presented
to investigate the theoretical results.
Finally, the restricted NP approach is studied for composite hypothesistesting
problems in the presence of uncertainty in the prior probability distribution
under the alternative hypothesis. A restricted NP decision rule aims to
maximize the average detection probability under the constraints on the worstcase
detection and false-alarm probabilities, and adjusts the constraint on the
worst-case detection probability according to the amount of uncertainty in the
prior probability distribution. Optimal decision rules according to the restricted
NP criterion are investigated, and an algorithm is provided to calculate the optimal
restricted NP decision rule. In addition, it is observed that the average
detection probability is a strictly decreasing and concave function of the constraint
on the minimum detection probability. Finally, a detection example is
presented, and extensions to more generic scenarios are discussed.
Keywords
Hypothesis-testingNoise Enhanced Detection
Restricted Bayes
Stochastic Resonance
Composite Hypotheses
Bayes Risk
Neyman-Pearson
Maxmin
Least-favorable Prior