On the Sequential Probability Ratio Test in Hidden Markov Models

We consider the Sequential Probability Ratio Test applied to Hidden Markov Models. Given two Hidden Markov Models and a sequence of observations generated by one of them, the Sequential Probability Ratio Test attempts to decide which model produced the sequence. We show relationships between the execution time of such an algorithm and Lyapunov exponents of random matrix systems. Further, we give complexity results about the execution time taken by the Sequential Probability Ratio Test

Learning Discrete-Time Markov Chains Under Concept Drift

Learning under concept drift is a novel and promising research area aiming at designing learning algorithms able to deal with nonstationary data-generating processes. In this research field, most of the literature focuses on learning nonstationary probabilistic frameworks, while some extensions about learning graphs and signals under concept drift exist. For the first time in the literature, this paper addresses the problem of learning discrete-time Markov chains (DTMCs) under concept drift. More specifically, following a hybrid active/passive approach, this paper introduces both a family of change-detection mechanisms (CDMs), differing in the required assumptions and performance, for detecting changes in DTMCs and an adaptive learning algorithm able to deal with DTMCs under concept drift. The effectiveness of both the proposed CDMs and the adaptive learning algorithm has been extensively tested on synthetically generated experiments and real data sets

Quickest Change Detection in Autoregressive Models

The problem of quickest change detection (QCD) in autoregressive (AR) models is investigated. A system is being monitored with sequentially observed samples. At some unknown time, a disturbance signal occurs and changes the distribution of the observations. The disturbance signal follows an AR model, which is dependent over time. Before the change, observations only consist of measurement noise, and are independent and identically distributed (i.i.d.). After the change, observations consist of the disturbance signal and the measurement noise, are dependent over time, which essentially follow a continuous-state hidden Markov model (HMM). The goal is to design a stopping time to detect the disturbance signal as quickly as possible subject to false alarm constraints. Existing approaches for general non-i.i.d. settings and discrete-state HMMs cannot be applied due to their high computational complexity and memory consumption, and they usually assume some asymptotic stability condition. In this paper, the asymptotic stability condition is firstly theoretically proved for the AR model by a novel design of forward variable and auxiliary Markov chain. A computationally efficient Ergodic CuSum algorithm that can be updated recursively is then constructed and is further shown to be asymptotically optimal. The data-driven setting where the disturbance signal parameters are unknown is further investigated, and an online and computationally efficient gradient ascent CuSum algorithm is designed. The algorithm is constructed by iteratively updating the estimate of the unknown parameters based on the maximum likelihood principle and the gradient ascent approach. The lower bound on its average running length to false alarm is also derived for practical false alarm control. Simulation results are provided to demonstrate the performance of the proposed algorithms

Robust quickest detection for hidden Markov models

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 65-66).Quickest Detection is the problem of detecting abrupt changes in the statistical behavior of an observed signal in real-time. The literature has focused much attention on the problem for i.i.d. observations. In this thesis, we assess the feasibility of two HMM quickest detection frameworks recently suggested for detecting rare events in a real data set. The first method is a dynamic programming based Bayesian approach, and the second is a non-Bayesian approach based on the cumulative sum algorithm. We discuss implementation considerations for each method and show their performance through simulations for a real data set. In addition, we examine, through simulations, the robustness of the non-Bayesian method when the disruption model is not exactly known but belongs to a known class of models.by Aliaa Atwi.S.M

Unattended acoustic sensor systems for noise monitoring in national parks

2017 Spring.Includes bibliographical references.Detection and classification of transient acoustic signals is a difficult problem. The problem is often complicated by factors such as the variety of sources that may be encountered, the presence of strong interference and substantial variations in the acoustic environment. Furthermore, for most applications of transient detection and classification, such as speech recognition and environmental monitoring, online detection and classification of these transient events is required. This is even more crucial for applications such as environmental monitoring as it is often done at remote locations where it is unfeasible to set up a large, general-purpose processing system. Instead, some type of custom-designed system is needed which is power efficient yet able to run the necessary signal processing algorithms in near real-time. In this thesis, we describe a custom-designed environmental monitoring system (EMS) which was specifically designed for monitoring air traffic and other sources of interest in national parks. More specifically, this thesis focuses on the capabilities of the EMS and how transient detection, classification and tracking are implemented on it. The Sparse Coefficient State Tracking (SCST) transient detection and classification algorithm was implemented on the EMS board in order to detect and classify transient events. This algorithm was chosen because it was designed for this particular application and was shown to have superior performance compared to other algorithms commonly used for transient detection and classification. The SCST algorithm was implemented on an Artix 7 FPGA with parts of the algorithm running as dedicated custom logic and other parts running sequentially on a soft-core processor. In this thesis, the partitioning and pipelining of this algorithm is explained. Each of the partitions was tested independently to very their functionality with respect to the overall system. Furthermore, the entire SCST algorithm was tested in the field on actual acoustic data and the performance of this implementation was evaluated using receiver operator characteristic (ROC) curves and confusion matrices. In this test the FPGA implementation of SCST was able to achieve acceptable source detection and classification results despite a difficult data set and limited training data. The tracking of acoustic sources is done through successive direction of arrival (DOA) angle estimation using a wideband extension of the Capon beamforming algorithm. This algorithm was also implemented on the EMS in order to provide real-time DOA estimates for the detected sources. This algorithm was partitioned into several stages with some stages implemented in custom logic while others were implemented as software running on the soft-core processor. Just as with SCST, each partition of this beamforming algorithm was verified independently and then a full system test was conducted to evaluate whether it would be able to track an airborne source. For the full system test, a model airplane was flown at various trajectories relative to the EMS and the trajectories estimated by the system were compared to the ground truth. Although in this test the accuracy of the DOA estimates could not be evaluated, it was show that the algorithm was able to approximately form the general trajectory of a moving source which is sufficient for our application as only a general heading of the acoustic sources is desired

Doctor of Philosophy

dissertationChirp signals arise in many applications of digital signal processing. In this dissertation, we address the problem of detection of chirp signals that are encountered in a bistatic radar which we are developing for remote sensing of cosmic ray induced air showers. The received echoes from the air showers are characterized by their large Doppler shift (several tens of MHz), and very short sweep period (~ 10 ^s). This makes our astrophysical problem a challenging one, since a very short sweep period is equivalent to a very low energy chirp signal. Furthermore, the related parameters of the received echoes are nondeterministic since they are tied to the physical parameters of the air showers that are stochastic in nature. In addition, our problem is characterized by the rarity of the expected chirp-echoes to be received, few events per week, and thus, background noise reception is the case most of the time. The primary focus of this research is to address these challenges and find an optimized detection approach under the existing receiver environment which contains non-Gaussian noise and is characterized by low signal-to-noise ratio (SNR). Matched filters are commonly used in radar systems when the chirp signal is known. In our first method, we revisit this context and use a matched filter as a basis of building a rake-like receiver that consists of a set of filters matched to quantized chirp rates, logarithmically distributed within the chirp-rate interval of interest. We examine the detection capability of the proposed structure through extensive theoretical and numerical analysis. Theoretical analysis and simulation results prove that the proposed detector has high detection capability for a range of chirp slopes in a low SNR environment. A major source of false-alarms was found to be due to sudden noise spikes that cover wide frequency bands. These transient signals have high amplitudes and occur at random time instants. This leads to erroneous detection decision. We study the influence of amplitude limiting the noisy signal on reducing the received false-alarms and enhancing the detection performance of the proposed rake-like receiver. In our second method, we use Hough transform (HT), which is widely used in the area of image processing for the purpose of finding parameterized patterns, as a basis of building a robust detection technique. We examine the detection capability of the proposed structure through theoretical and numerical analysis. Our results prove that the proposed detector has high detection capability for a range of chirp slopes in a low SNR environment. The introduced detection algorithms are implemented over a Virtex-5 FPGA. National Instruments modules are used as a high-performance custom hardware. Due to rarity of received echoes, we emulate the expected radar echoes to evaluate the system performance. The detection performance of the emulated echoes is examined using the implemented receiver at the field. Also, we compare the performance of both detectors

Uncertainty quantification and its properties for hidden Markov models with application to condition based maintenance

Condition-based maintenance (CBM) can be viewed as a transformation of data gathered from a piece of equipment into information about its condition, and further into decisions on what to do with the equipment. Hidden Markov model (HMM) is a useful framework to probabilistically model the condition of complex engineering systems with partial observability of the underlying states. Condition monitoring and prediction of such type of system requires accurate knowledge of HMM that describes the degradation of such a system with data collected from the sensors mounted on it, as well as understanding of the uncertainty of the HMMs identified from the available data. To that end, this thesis proposes a novel HMM estimation scheme based on the principles of Bayes theorem. The newly proposed Bayesian estimation approach for estimating HMM parameters naturally yields information about model parametric uncertainties via posterior distributions of HMM parameters emanating from the estimation process. In addition, a novel condition monitoring scheme based on uncertain HMMs of the degradation process is proposed and demonstrated on a large dataset obtained from a semiconductor manufacturing facility. Portion of the data was used to build operating mode specific HMMs of machine degradation via the newly proposed Bayesian estimation process, while the remainder of the data was used for monitoring of machine condition using the uncertain degradation HMMs yielded by Bayesian estimation. Comparison with a traditional signature-based statistical monitoring method showed that the newly proposed approach effectively utilizes the fact that its parameters are uncertain themselves, leading to orders of magnitude fewer false alarms. This methodology is further extended to address the practical issue that maintenance interventions are usually imperfect. We propose both a novel non-ergodic and non-homogeneous HMM that assumes imperfect maintenances and a novel process monitoring method capable of monitoring the hidden states considering model uncertainty. Significant improvement in both the log-likelihood of estimated HMM parameters and monitoring performance were observed, compared to those obtained using degradation HMMs that always assumed perfect maintenance. Finally, behavior of the posterior distribution of parameters of unidirectional non- ergodic HMMs modeling in this thesis for degradation was theoretically analyzed in terms of their evolution as more data become available in the estimation process. The convergence problem is formulated as a Bernstein-von Mises theorem (BvMT), and under certain regularity conditions, the sequence of posterior distributions is proven to converge to a Gaussian distribution with variance matrix being the inverse of the Fisher information matrix. An example of a unidirectional HMM is presented for which the regularity conditions are verified, and illustrations of expected theoretical results are given using simulation. The understanding of such convergence of posterior distributions enables one to determine when Bayesian estimation of degradation HMMs is justified and converges toward true model parameters, as well as how much data one then needs to achieve desired accuracy of the resulting model. Understanding of these issues is of utmost important if HMMs are to be used for degradation modeling and monitoring.Operations Research and Industrial Engineerin

Data Discovery and Anomaly Detection using Atypicality.

Ph.D. Thesis. University of Hawaiʻi at Mānoa 2017

Continuous hidden Markov models and the sequential probability ratio test

We consider Hidden Markov Models that emit sequences of observations which are drawn from continuous distributions. For example, such a model may emit a sequence of numbers, each of which is drawn from a uniform distribution, but the support of the uniform distribution depends on the state of the Hidden Markov Model. Such models generalise the more common version where each observation is drawn from a finite alphabet. We consider a distance measure on Hidden Markov Models called the total variation distance. When this distance is 0 we say the models are equivalent. When this distance is maximal we say the models are distinguishable. We prove that for two Hidden Markov Models with continuous observations one can decide in polynomial time whether they are equivalent and also whether they are distinguishable. We also consider the Sequential Probability Ratio Test applied to Hidden Markov Models with finite observations. Given two distinguishable Hidden Markov Models and a sequence of observations generated by one of them, the Sequential Probability Ratio Test attempts to decide which model produced the sequence. We show relationships between the execution time of such an algorithm and Lyapunov exponents of random matrix products. Further, we give complexity results about the execution time taken by the Sequential Probability Ratio Test