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

Perfect sampling for nonhomogeneous Markov chains and hidden Markov models

We obtain a perfect sampling characterization of weak ergodicity for backward products of finite stochastic matrices, and equivalently, simultaneous tail triviality of the corresponding nonhomogeneous Markov chains. Applying these ideas to hidden Markov models, we show how to sample exactly from the finite-dimensional conditional distributions of the signal process given infinitely many observations, using an algorithm which requires only an almost surely finite number of observations to actually be accessed. A notion of "successful" coupling is introduced and its occurrence is characterized in terms of conditional ergodicity properties of the hidden Markov model and related to the stability of nonlinear filters

Identifiability of parameters in latent structure models with many observed variables

While hidden class models of various types arise in many statistical applications, it is often difficult to establish the identifiability of their parameters. Focusing on models in which there is some structure of independence of some of the observed variables conditioned on hidden ones, we demonstrate a general approach for establishing identifiability utilizing algebraic arguments. A theorem of J. Kruskal for a simple latent-class model with finite state space lies at the core of our results, though we apply it to a diverse set of models. These include mixtures of both finite and nonparametric product distributions, hidden Markov models and random graph mixture models, and lead to a number of new results and improvements to old ones. In the parametric setting, this approach indicates that for such models, the classical definition of identifiability is typically too strong. Instead generic identifiability holds, which implies that the set of nonidentifiable parameters has measure zero, so that parameter inference is still meaningful. In particular, this sheds light on the properties of finite mixtures of Bernoulli products, which have been used for decades despite being known to have nonidentifiable parameters. In the nonparametric setting, we again obtain identifiability only when certain restrictions are placed on the distributions that are mixed, but we explicitly describe the conditions.Comment: Published in at http://dx.doi.org/10.1214/09-AOS689 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Deep Learning Models for Planetary Seismicity Detection

Research in planetary seismology is fundamentally constrained by a lack of data. Seismo-logical science products of future missions can typically only be informed by theoretical signal/noise characteristics of the environment or likely Earth-analogues. Although objectives can be re-assessed after some initial data-collection upon lander arrival, transfer of high-resolution data back to Earth is costly on lander power usage. Over the last several years, development of GPU computing techniques and open-source high-level APIs have led to rapid advances in deep learning within the fields of computer vision, natural language processing, and collaborative filtering. These techniques are actively being adapted in seismology for a variety of tasks, including: earthquake detection, seismic phase discrimination, and ground-motion prediction. Until the recent detection of mars quakes during the Mars InSight mission, the only other measurements of seismicity recorded outside of Earth was on the Moon during the Apollo missions between 1969 to 1977. These unique data sets have been periodically revisited using new seismological methods, including ambient noise interferometry and Hidden Markov Models. Our objective is to develop a deep learning seismic detector and use it to catalog moonquakes from the Apollo 17 Lunar Seismic Profiling Experiment (LSPE) and compare the results with those obtained by other methods. Additionally, we will assess the accuracy tradeoff between using a training set of lunar data and one composed of Earth seismicity. In this document, we present preliminary results using a prototype classifier trained on a small set of earthquakes that was able to obtain detections for LSPE moonquakes with a greater accuracy than a recent study using Hidden Markov Models

Matrix products for the synthesis of stationary time series with a priori prescribed joint distributions

Inspired from non-equilibrium statistical physics models, a general framework enabling the definition and synthesis of stationary time series with a priori prescribed and controlled joint distributions is constructed. Its central feature consists of preserving for the joint distribution the simple product struc- ture it has under independence while enabling to input con- trolled and prescribed dependencies amongst samples. To that end, it is based on products of d-dimensional matrices, whose entries consist of valid distributions. The statistical properties of the thus defined time series are studied in details. Having been able to recast this framework into that of Hidden Markov Models enabled us to obtain an efficient synthesis procedure. Pedagogical well-chosen examples (time series with the same marginal distribution, same covariance function, but different joint distributions) aim at illustrating the power and potential of the approach and at showing how targeted statistical prop- erties can be actually prescribed.Comment: 4 pages, 2 figures, conference publication published in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 201

Stochastic modeling of retail mortgage loans based on past due, prepaid, and default states

Stochastic models were developed that provide important measures related to retail mortgages and credit cards for the management of a bank. Based on Markov theory, two models were developed that predict mortgage portfolio size and expected duration of stay in each of the states, which are defined according to the criteria of Basel Accord II and the Federal Reserve Bank. Also, to facilitate comparisons among different types of credit products and different time periods, a model was developed to generate a health index for a retail mortgage. This model could be easily extended, using multivariate regression or multivariate time series techniques, to analyze the interaction between a mortgage and local macroeconomic factors. Furthermore, the models in this dissertation address decision making on the part of the management of a bank concerning business strategy such as collection policies and loan officer compensation policies. Extending the basic assumption of the Markov property to a higher-order Markov model and a multivariate Markov model, this work also analyzed the correlation between the payment pattern for retail mortgages and credit cards. To complete this correlation analysis, a comparison among 3 models (higher-order, multivariate, and a higher-order multivariate Markov model (HMMM)) has also been provided. Finally, an interaction analysis between the payment behavior of a retail mortgage and local macroeconomic variables has been performed using an Interactive Hidden Markov Model (IHMM). For IHMM and HMMM models, the number of unknown parameters increases exponentially with the increase of the order of the models. Hence, to deal with this situation, a linear programming algorithm has been used to obtain solutions for the HMMM and IHMM. The models provided in this study are of practical importance to the bank management. Not only do they give quantitative measures about loan stand-alone characteristics, but also they provide cross-section comparisons among different credit products and multi-period loan performance tracking as well. These models, used to analyze retail mortgages and credit cards, could be easily applied to other credit products issued by a commercial bank. The data used in this study have been obtained from an Ohio local commercial bank. It includes monthly paid 20-year retail mortgages and personal credit cards. A contract has also been signed to guarantee that the data would be used only for academic research

Asymptotic operating characteristics of an optimal change point detection in hidden Markov models

Let \xi_0,\xi_1,...,\xi_{\omega-1} be observations from the hidden Markov model with probability distribution P^{\theta_0}, and let \xi_{\omega},\xi_{\omega+1},... be observations from the hidden Markov model with probability distribution P^{\theta_1}. The parameters \theta_0 and \theta_1 are given, while the change point \omega is unknown. The problem is to raise an alarm as soon as possible after the distribution changes from P^{\theta_0} to P^{\theta_1}, but to avoid false alarms. Specifically, we seek a stopping rule N which allows us to observe the \xi's sequentially, such that E_{\infty}N is large, and subject to this constraint, sup_kE_k(N-k|N\geq k) is as small as possible. Here E_k denotes expectation under the change point k, and E_{\infty} denotes expectation under the hypothesis of no change whatever. In this paper we investigate the performance of the Shiryayev-Roberts-Pollak (SRP) rule for change point detection in the dynamic system of hidden Markov models. By making use of Markov chain representation for the likelihood function, the structure of asymptotically minimax policy and of the Bayes rule, and sequential hypothesis testing theory for Markov random walks, we show that the SRP procedure is asymptotically minimax in the sense of Pollak [Ann. Statist. 13 (1985) 206-227]. Next, we present a second-order asymptotic approximation for the expected stopping time of such a stopping scheme when \omega=1. Motivated by the sequential analysis in hidden Markov models, a nonlinear renewal theory for Markov random walks is also given.Comment: Published at http://dx.doi.org/10.1214/009053604000000580 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org