Browsing by Subject "Stochastic processes."

Now showing 1 - 9 of 9

Open Access
Alternative approaches and noise benefits in hypothesis-testing problems in the presence of partial information
(2011) Bayram, Suat
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.
Open Access
Asymptotic analysis of highly reliable retrial queueing systems
(2000) Kurtuluş, Mümin
The thesis is concerned with the asymptotic analysis of the time of first loss of a customer and the flow of lost customers in some types of Markov retrial queueing systems with flnite buffer. A retrial queueing system is characterized by the following feature: an arriving customer finding all of the servers busy must leave the service area and join a special buffer. After this it may re-apply for service after some random time. If the buffer is full the customer is lost. The analysis of the time of first loss of a customer is based on the method of so-called S — sets and the results about the asymptotic behavior of the first exit time from the fixed subset of states of semi-Markov process of a special structure (so-called monotone structure). Single server retrial queueing systems [M IM IlIm with retrials) as well as multiple server retrial queueing systems {M IM fsfm with retrials) are analyzed in cases of fast service and both fast service and fast retrials. Exponential approximation for the time of first loss and Poisson approximation for the flow of lost customers are proved for all of the considered cases.
Open Access
Feature point classification and matching
(2007) Ay, Avşar Polat
A feature point is a salient point which can be separated from its neighborhood. Widely used definitions assume that feature points are corners. However, some non-feature points also satisfy this assumption. Hence, non-feature points, which are highly undesired, are usually detected as feature points. Texture properties around detected points can be used to eliminate non-feature points by determining the distinctiveness of the detected points within their neighborhoods. There are many texture description methods, such as autoregressive models, Gibbs/Markov random field models, time-frequency transforms, etc. To increase the performance of feature point related applications, two new feature point descriptors are proposed, and used in non-feature point elimination and feature point sorting-matching. To have a computationally feasible descriptor algorithm, a single image resolution scale is selected for analyzing the texture properties around the detected points. To create a scale-space, wavelet decomposition is applied to the given images and neighborhood scale-spaces are formed for every detected point. The analysis scale of a point is selected according to the changes in the kurtosis values of histograms which are extracted from the neighborhood scale-space. By using descriptors, the detected non-feature points are eliminated, feature points are sorted and with inclusion of conventional descriptors feature points are matched. According to the scores obtained in the experiments, the proposed detection-matching scheme performs more reliable than the Harris detector gray-level patch matching scheme. However, SIFT detection-matching scheme performs better than the proposed scheme.
Open Access
Harmonic analysis in finite phase space
(2005) Korkmaz, Sayit
The Wigner distribution and linear canonical transforms are important tools for optics, signal processing, quantum mechanics, and mathematics. In this thesis, we study the discrete versions of Wigner distributions and linear canonical transforms. In the definition of a discrete entity we focus on two aspects: structural analogy and continuum approximation and/or limits. Based on this framework, the tradeoffs are analyzed and a compromise for a discrete Wigner distribution that meets both objectives to a high degree is presented by consolidating sampling theory and the algebraic approach. Such a compromise is necessary since it is impossible to meet the conditions to the highest possible degree. The differences between discrete and continuous time-frequency analysis are also discussed in a group theoretical perspective. In the second part of the thesis, the discrete versions of linear canonical transforms are reviewed and their connections to the continuous theory is established. As a special case the discrete fractional Fourier transform is defined and its properties are derived.
Open Access
Noise benefits in joint detection and estimation systems = Birlikte sezim ve kestirim sistemlerinde gürültünün faydaları
(2014) Akbay, Abdullah Başar
Adding noise to inputs of some suboptimal detectors or estimators can improve their performance under certain conditions. In the literature, noise benefits have been studied for detection and estimation systems separately. In this thesis, noise benefits are investigated for joint detection and estimation systems. The analysis is performed under the Neyman-Pearson (NP) and Bayesian detection frameworks and the Bayesian estimation framework. The maximization of the system performance is formulated as an optimization problem. The optimal additive noise is shown to have a specific form, which is derived under both NP and Bayesian detection frameworks. In addition, the proposed optimization problem is approximated as a linear programming (LP) problem, and conditions under which the performance of the system cannot be improved via additive noise are obtained. With an illustrative numerical example, performance comparison between the noise enhanced system and the original system is presented to support the theoretical analysis.
Open Access
Noise enhanced detection
(2009) Bayram, Suat
Performance of some suboptimal detectors can be improved by adding independent noise to their measurements. Improving the performance of a detector by adding a stochastic signal to the measurement can be considered in the framework of stochastic resonance (SR), which can be regarded as the observation of “noise benefits” related to signal transmission in nonlinear systems. Such noise benefits can be in various forms, such as a decrease in probability of error, or an increase in probability of detection under a false-alarm rate constraint. The main focus of this thesis is to investigate noise benefits in the Bayesian, minimax and Neyman-Pearson frameworks, and characterize optimal additional noise components, and quantify their effects. In the first part of the thesis, a Bayesian framework is considered, and the previous results on optimal additional noise components for simple binary hypothesis-testing problems are extended to M-ary composite hypothesis-testing problems. In addition, a practical detection problem is considered in the Bayesian framework. Namely, binary hypothesis-testing via a sign detector is studied for antipodal signals under symmetric Gaussian mixture noise, and the effects of shifting the measurements (observations) used by the sign detector are investigated. First, a sufficient condition is obtained to specify when the sign detectorbased on the modified measurements (called the “modified” sign detector) can have smaller probability of error than the original sign detector. Also, two suf- ficient conditions under which the original sign detector cannot be improved by measurement modification are derived in terms of desired signal and Gaussian mixture noise parameters. Then, for equal variances of the Gaussian components in the mixture noise, it is shown that the probability of error for the modified detector is a monotone increasing function of the variance parameter, which is not always true for the original detector. In addition, the maximum improvement, specified as the ratio between the probabilities of error for the original and the modified detectors, is specified as 2 for infinitesimally small variances of the Gaussian components in the mixture noise. Finally, numerical examples are presented to support the theoretical results, and some extensions to the case of asymmetric Gaussian mixture noise are explained. In the second part of the thesis, the effects of adding independent noise to measurements are studied for M-ary hypothesis-testing problems according to the minimax criterion. It is shown that the optimal additional noise can be represented by a randomization of at most M signal values. In addition, a convex relaxation approach is proposed to obtain an accurate approximation to the noise probability distribution in polynomial time. Furthermore, sufficient conditions are presented to determine when additional noise can or cannot improve the performance of a given detector. Finally, a numerical example is presented. Finally, the effects of additional independent noise are investigated in the Neyman-Pearson framework, and various sufficient conditions on the improvability and the non-improvability of a suboptimal detector are derived. First, a sufficient condition under which the performance of a suboptimal detector cannot be enhanced by additional independent noise is obtained according to the Neyman-Pearson criterion. Then, sufficient conditions are obtained to specifywhen the detector performance can be improved. In addition to a generic condition, various explicit sufficient conditions are proposed for easy evaluation of improvability. Finally, a numerical example is presented and the practicality of the proposed conditions is discussed.
Open Access
Optimal stochastic approaches for signal detection and estimation under inequality constraints
(2012) Dülek, Berkan
Fundamental to the study of signal detection and estimation is the design of optimal procedures that operate on the noisy observations of some random phenomenon. For detection problems, the aim is to decide among a number of statistical hypotheses, whereas estimating certain parameters of the statistical model is required in estimation problems. In both cases, the solution depends on some goodness criterion by which detection (or estimation) performance is measured. Despite being a well-established field, the advances over the last several decades in hardware and digital signal processing have fostered a renewed interest in designing optimal procedures that take more into account the practical considerations. For example, in the detection of binary-valued scalar signals corrupted with additive noise, an analysis on the convexity properties of the error probability with respect to the transmit signal power has suggested that the error performance cannot be improved via signal power randomization/sharing under an average transmit power constraint when the noise has a unimodal distribution (such as the Gaussian distribution). On the contrary, it is demonstrated that performance enhancement is possible in the case of multimodal noise distributions and even under Gaussian noise for three or higher dimensional signal constellations. Motivated by these results, in this dissertation we adopt a structured approach built on concepts called stochastic signaling and detector randomization, and devise optimal detection procedures for power constrained communications systems operating over channels with arbitrary noise distributions. First, we study the problem of jointly designing the transmitted signals, decision rules, and detector randomization factors for an M-ary communications system with multiple detectors at the receiver. For each detector employed at the receiver, it is assumed that the transmitter can randomize its signal constellation (i.e., transmitter can employ stochastic signaling) according to some probability density function (PDF) under an average transmit power constraint. We show that stochastic signaling without detector randomization cannot achieve a smaller average probability of error than detector randomization with deterministic signaling for the same average power constraint and noise statistics when optimal maximum a-posteriori probability (MAP) detectors are employed in both cases. Next, we prove that a randomization between at most two MAP detectors corresponding to two deterministic signal vectors results in the optimal performance. Sufficient conditions are also provided to conclude ahead of time whether the correct decision performance can or cannot be improved by detector randomization. In the literature, the discussions on the benefits of stochastic signaling and detector randomization are severely limited to the Bayesian criterion. Therefore, we study the convexity/concavity properties for the problem of detecting the presence of a signal emitted from a power constrained transmitter in the presence of additive Gaussian noise under the Neyman-Pearson (NP) framework. First, it is proved that the detection probability corresponding to the α−level likelihood ratio test (LRT) is either concave or has two inflection points such that the function is concave, convex and finally concave with respect to increasing values of the signal power. Based on this result, optimal and near-optimal power sharing/randomization strategies are proposed for average and/or peak power constrained transmitters. Using a similar approach, the convexity/concavity properties of the detection probability are also investigated with respect to the jammer power. The results indicate that a weak Gaussian jammer should employ on-off time sharing to degrade the detection performance. Next, the previous analysis for the NP criterion is generalized to channels with arbitrary noise PDFs. Specifically, we address the problem of jointly designing the signaling scheme and the decision rule so that the detection probability is maximized under constraints on the average false alarm probability and average transmit power. In the case of a single detector at the receiver, it is shown that the optimal solution can be obtained by employing randomization between at most two signal values for the on-signal and using the corresponding NP-type LRT at the receiver. When multiple detectors are available at the receiver, the optimal solution involves a randomization among no more than three NP decision rules corresponding to three deterministic signal vectors. Up to this point, we have focused on signal detection problems. In the following, the trade-offs between parameter estimation accuracy and measurement device cost are investigateed under the influence of noise. First, we seek to determine the most favorable allocation of the total cost to measurement devices so that the average Fisher information of the resulting measurements is maximized for arbitrary observation and measurement statistics. Based on a recently proposed measurement device cost model, we present a generic optimization problem without assuming any specific estimator structure. Closed form expressions are obtained in the case of Gaussian observations and measurement noise. Finally, a more elaborate analysis of the relationship between parameter estimation accuracy and measurement device cost is presented. More specifically, novel convex measurement cost minimization problems are proposed based on various estimation accuracy constraints assuming a linear system subject to additive Gaussian noise for the deterministic parameter estimation problem. Robust allocation of the total cost to measurement devices is also considered by assuming a specific uncertainty model on the system matrix. Closed form solutions are obtained in the case of an invertible system matrix for two estimation accuracy criteria. Through numerical examples, various aspects of the proposed optimization problems are compared. Lastly, the discussion is extended to the Bayesian framework assuming that the estimated parameter is Gaussian distributed.
Open Access
Optimal time sharing strategies for parameter estimation and channel switching problems
(2014) Soğancı, Hamza
Time sharing (randomization) can offer considerable amount of performance improvement in various detection and estimation problems and communication systems. In the first three chapters of this dissertation, time sharing among different signal levels is considered for parametric estimation problems. In the final chapter, time sharing among different channels is investigated for an average power constrained communication system. In the first chapter, the aim is to improve the performance of a single fixed estimator by the optimal stochastic design of signal values corresponding to parameters. It is obtained that the optimal parameter design corresponds to time sharing between at most two different signal values. In the second chapter, the problem in the first chapter is generalized to a scenario where there are multiple parameters and multiple estimators. In this scenario, two different cost functions are considered. The first cost function is the total risk of all the estimators. The optimal solution for this case is time sharing between at most two different signal values. The second cost function is the maximum risk of all the estimators. For this case, it is shown that the optimal parameter design is time sharing among at most three different signal values. In the third chapter, the linear minimum mean squared error (LMMSE) estimator is considered. It is observed that time sharing is not needed for the LMMSE estimator, but still the performance can be improved by modifying the signal level. In the final chapter, the optimal channel switching problem is studied for Gaussian channels, and the optimal channel switching strategy is determined in the presence of average power and average cost constraints. It is shown that the optimal channel switching strategy is to switch among at most three channels.
Open Access
Poisson disorder problem with control on costly observations
(2012) Kadiyala, Bharadwaj
A Poisson process Xt changes its rate at an unknown and unobservable time θ from λ0 to λ1. Detecting the change time as quickly as possible in an optimal way is described in literature as the Poisson disorder problem. We provide a more realistic generalization of the disorder problem for Poisson process by introducing fixed and continuous costs for being able to observe the arrival process. As a result, in addition to finding the optimal alarm time, we also characterize an optimal way of observing the arrival process. We illustrate the structure of the solution spaces with the help of some numerical examples.