Sequential Analysis and Change-Point Problems

M. Baron. Nonparametric adaptive change-point estimation and on-line detection. Sequential Analysis, 19 (1&2), 1-23, 2000.

Abstract.
Under standard conditions of change-point problems with one or both distributions being unknown, we propose efficient on-line and off-line nonparametric algorithms for detecting and estimating the change-point. They are based on histogram density estimators, which allows applications involving ordinal and categorical data. Also, they are designed to detect any changes in distribution, not necessarily related to the location or scale parameters. Efficiency of the proposed schemes is demonstrated by relevant inequalities for the mean delay and the mean time between false alarms. Asymptotically, they are shown to behave similarly to the most efficient procedures based on the known distributions. The stopping rule achieves an asymptotically linear mean delay and an exponential mean time between false alarms. The guidelines on selecting the threshold and the partition for the histogram density estimation are given, based on the obtained results. Proposed methods are applied to the England temperatures data and the Vostok ice core record to detect the global climate changes.

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M. Baron and A. L. Rukhin. Perpetuities and asymptotic change-point analysis. Statistics and Probability Letters, 55 (1), 29-38, 2001.

Abstract.
The distribution of stochastically discounted sums (perpetuities) is studied. For Bernoulli-type variables a canonical representation of this distribution is obtained, and it is proven to be singular continuous. In the asymptotic setting of the change-point estimation problem the limiting behavior of the posterior distribution is shown to be given by two independent perpetuities.

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M. Baron. Bayes stopping rules in a change-point model with a random hazard rate. Sequential Analysis, 20 (3), 147-163, 2001.

Abstract.
In the Bayes sequential change-point problem, an assumption of a fully known prior distribution of a change-point is usually impracticable. At every moment, one often knows only the discrete hazard function, that is, the probability of a change occurring before the next observation is collected, given that it has not occurred so far. In the randomized model, the observed or predicted values of the hazard function are assumed to form a Markov chain. Under these assumptions, the optimal change-point detection stopping rules are derived for two popular loss functions introduced in Shiryaev (1978) and Ritov (1990). Derivations are based on the theory of optimal stopping of Markov sequences.

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M. Baron. Bayes and asymptotically pointwise optimal stopping rules for the detection of influenza epidemics. C. Gatsonis, R. E. Kass, A. Carriquiry, A. Gelman, D. Higdon, D. K. Pauler and I. Verdinelli, Eds., Case Studies in Bayesian Statistics, vol. 6, pages 153--163, Springer-Verlag, New York, 2002.

Abstract.
Whereas it is customary to announce epidemics when influenza mortality exceeds the epidemic threshold, one can often detect the beginning of epidemics earlier, by solving a suitable change-point problem. We propose a hierarchical Bayesian change-point model for influenza epidemics. Prior probabilities of a change point depend on (random) factors that affect the spread of influenza. Theory of optimal stopping is used to obtain Bayes stopping rules for the detection of epidemic trends under the loss functions penalizing for delays and false alarms. The Bayes solution involves rather complicated computation of the corresponding payoff function. Alternatively, asymptotically pointwise optimal stopping rules can be computed easily and under weaker assumptions. Both methods are applied to the 1996--2001 influenza mortality data published by CDC.

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M. Baron and N. Granott. Consistent estimation of early and frequent change points. In J. Haitovsky, H. R. Lerche, and Y. Ritov, eds., Foundations of Statistical Inference, pages 181--194, Springer, Physica-Verlag, Heidelberg, New York, 2003.

Abstract.
We address two types of processes with change points that often arise in practical situations. These are processes with early change points and processes with frequent change points. Early change points may occur after very few observations and may be followed by additional change points or more complicated patterns. Frequent change points separate different homogeneous phases of the observed process with the possibility of very short phases. Uncertainty of the considered processes during their later phases forces the use of sequential tools, in order to minimize samples from later phases. Change-point detection and post-estimation schemes for these situations are developed. They possess a number of desired properties, not satisfied by procedures proposed in the earlier literature. One of them is distribution-consistency. Unlike the traditional concept of consistency, it implies convergence of small-sample change-point estimators to the corresponding parameters as the magnitude of changes tends to infinity.

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R. Gill and M. Baron. Consistent estimation in generalized broken-line regression. J. Statist. Plann. Inference, 126(2), 441--460, 2004

Abstract.
A change-point model is considered where the canonical parameter of an exponential family drifts from its control value at an unknown time and changes according to a broken-line regression. Necessary conditions and sufficient conditions are obtained for the existence of consistent change-point estimators. When sufficient conditions are met, it is shown that the maximum likelihood estimator of the change point is consistent, unlike the classical abrupt change-point models. Results are extended to the case of nonlinear trends and non-equidistant observations.

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M. Baron. Sequential Methods for Multistate Processes. In N. Mukhopadhyay, S. Datta, S. Chattopadhyay, eds., Applications of Sequential Methodologies, pages 55--73, Marcel Dekker, Inc., New York, 2004.

Abstract.
This paper proposes and justifies the use of sequential methods for non-sequential problems that arise in the analysis of multistate processes. It concentrates on identifying modes and segments of an observed multistate process followed by the parameter estimation and further analysis. Direct applications are shown in energy finance, economics, and meteorology.

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C. Schmegner and M. Baron. Principles of optimal sequential planning, Sequential Analysis, 23(1), 11--32, 2004.

Abstract.
It is often impractical or expensive to collect data according to a classical sequential scheme, that is, one observation at a time. Sequential planning extends and generalizes the ``pure'' sequential procedures by allowing to sample observations in groups. At any moment, all the collected data are used to determine the size of the next group and to decide whether or not sampling should be terminated.

This article discusses optimality of sequential plans in terms of a suitable risk function that balances an observation cost and a group cost. It is shown that only non-randomized sequential plans based on a sufficient statistic have to be considered in order to achieve optimality. Performance of several such plans is evaluated.

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M. Baron and N. Granott. Small sample change-point analysis with applications to problem solving. Submitted.

Abstract.
The proposed scheme detects and post-estimates change points that can occur during early stages of an observed multistage process. The algorithm is designed to analyze change points that are likely to occur after very few observations and to be followed by other change points or more complicated patterns. Such models are justified in problem solving, quality control, and other processes. Special methods are derived in order to: (1) detect a change point even after a very brief period of observation, (2) estimate it with the theoretically highest degree of accuracy, (3) report a no-change case when a significant change has not occurred during the observed period, and (4) use minimum data after the change point to prevent mixing the post-change phase with subsequent phases and patterns. Unlike existing methods, the proposed algorithm produces a distribution consistent estimator of a change point. Details are elaborated for the case of Gamma distributions and demonstrated for a process of problem solving.

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M. Baron. On statistical inference under asymmetric loss functions. Statistics & Decisions, 18 (4), 367-388, 2000.

Abstract.
We introduce a wide class of asymmetric loss functions and show how to obtain decision rules optimal under these losses from the commonly used standard Bayesian procedures. Important properties of minimum risk and minimax estimators are established. In particular, we discuss their sensitivity to the asymmetry of the loss function.

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M. Baron. Bayes stopping rules in a change-point model with a random hazard rate. Sequential Analysis, 20 (3), 147-163, 2001.

Abstract.
In the Bayes sequential change-point problem, an assumption of a fully known prior distribution of a change-point is usually impracticable. At every moment, one often knows only the discrete hazard function, that is, the probability of a change occurring before the next observation is collected, given that it has not occurred so far. In the randomized model, the observed or predicted values of the hazard function are assumed to form a Markov chain. Under these assumptions, the optimal change-point detection stopping rules are derived for two popular loss functions introduced in Shiryaev (1978) and Ritov (1990). Derivations are based on the theory of optimal stopping of Markov sequences.

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M. Baron. Bayes and asymptotically pointwise optimal stopping rules for the detection of influenza epidemics. In A. Carriquiry, C. Gatsonis, A. Gelman, D. Higdon, R. Kass, D. Pauler and I. Verdinell, Eds., Case Studies in Bayesian Statistics, vol. 6, Springer-Verlag, New York, 2002.

Abstract.
Whereas it is customary to announce epidemics when influenza mortality exceeds the epidemic threshold, one can often detect the beginning of epidemics earlier, by solving a suitable change-point problem. We propose a hierarchical Bayesian change-point model for influenza epidemics. Prior probabilities of a change point depend on (random) factors that affect the spread of influenza. Theory of optimal stopping is used to obtain Bayes stopping rules for the detection of epidemic trends under the loss functions penalizing for delays and false alarms. The Bayes solution involves rather complicated computation of the corresponding payoff function. Alternatively, asymptotically pointwise optimal stopping rules can be computed easily and under weaker assumptions. Both methods are applied to the 1996--2001 influenza mortality data published by CDC.

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C. Schmegner and M. Baron. Principles of optimal sequential planning, Sequential Analysis, 23(1), 11--32, 2004.

Abstract.
It is often impractical or expensive to collect data according to a classical sequential scheme, that is, one observation at a time. Sequential planning extends and generalizes the ``pure'' sequential procedures by allowing to sample observations in groups. At any moment, all the collected data are used to determine the size of the next group and to decide whether or not sampling should be terminated.

This article discusses optimality of sequential plans in terms of a suitable risk function that balances an observation cost and a group cost. It is shown that only non-randomized sequential plans based on a sufficient statistic have to be considered in order to achieve optimality. Performance of several such plans is evaluated.

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Applications of Statistics in Energy Finance, Semiconductor Manufacturing, Epidemiology, Developmental Psychology

M. Baron, C. K. Lakshminarayan, Z. Chen. Markov random fields in pattern recognition for semiconductor manufacturing. Technometrics, 43 (1), 66-72, 2001.

Abstract.
Under the most general conditions of a Markov random field, we model the two-dimensional spatial distribution of microchips on a silicon wafer. Its canonical parameters represent the density of failures, main effects and interactions of neighboring chips. Explicit forms of conditional distributions are derived, and maximum pseudo-likelihood estimates of canonical parameters are obtained. This ten-dimensional numerical characteristic summarizes general patterns of clusters of failing chips on a wafer, capturing their size, shape, direction density and thickness. It is used to classify incoming wafers to known root cause categories of failures by matching them to the closest pattern.

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M. Baron, M. Rosenberg, N. Sidorenko. Electricity pricing: modelling and prediction with automatic spike detection. Energy, Power, and Risk Management, 36-39, October 2001.

Abstract.
Power prices are modelled by a Markov chain switching between "regular" and "spike" phases according to the time of the year and other factors. Here we present simple methods of model calibration and optimal prediction.

M. Baron, M. Rosenberg, N. Sidorenko. Divide and conquer: forecasting power via automatic price regime separation. Energy, Power, and Risk Management, 70-73, March 2002.

Abstract.

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M. Rosenberg, J. D. Bryngelson, N. Sidorenko; M. Baron. Price spikes and real options: transmission valuation. In E. I. Ronn, ed., Real Options and Energy Management, pages 323--370, Risk Books, London, 2002.

In the same volume -

M. Rosenberg, J. D. Bryngelson; M. Baron. Probability and stochastic calculus: review of probability concepts. In E. I. Ronn, ed., Real Options and Energy Management, pages 659--697, Risk Books, London, 2002.

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M. Baron. Bayes and asymptotically pointwise optimal stopping rules for the detection of influenza epidemics. C. Gatsonis, R. E. Kass, A. Carriquiry, A. Gelman, D. Higdon, D. K. Pauler and I. Verdinelli, Eds., Case Studies in Bayesian Statistics, vol. 6, pages 153--163, Springer-Verlag, New York, 2002.

Abstract.
Whereas it is customary to announce epidemics when influenza mortality exceeds the epidemic threshold, one can often detect the beginning of epidemics earlier, by solving a suitable change-point problem. We propose a hierarchical Bayesian change-point model for influenza epidemics. Prior probabilities of a change point depend on (random) factors that affect the spread of influenza. Theory of optimal stopping is used to obtain Bayes stopping rules for the detection of epidemic trends under the loss functions penalizing for delays and false alarms. The Bayes solution involves rather complicated computation of the corresponding payoff function. Alternatively, asymptotically pointwise optimal stopping rules can be computed easily and under weaker assumptions. Both methods are applied to the 1996--2001 influenza mortality data published by CDC.

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C. K. Lakshminarayan, M. Baron, Z. Chen. Pattern recognition in IC diagnostics using the linear discriminant classifier and artificial neural networks. Under review.

Abstract.
It is important in IC manufacturing to identify probable root causes, given a signature. The signature is a vector of electrical test parameters measured on process control bars on a wafer. Linear discriminant analysis and artificial neural networks are used to classify a signature of test electrical measurements of a failed chip to one of several pre-assigned root cause categories. An optimal decision rule that assigns a new incoming signature of a failed chip to a particular root cause category is employed such that the probability of misclassification is minimized. The problem of classifying patterns with missing data, outliers, collinearity, and non-normality are also addressed. The selected similarity metric in linear discriminant analysis, and the network topology, used in neural networks, result in a small number of misclassifications. An alternative classification scheme based on the locations of failed chips on a wafer and their spatial dependence is proposed.

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M. Baron and N. Granott. Small sample change-point analysis with applications to problem solving. Submitted.

Abstract.
The proposed scheme detects and post-estimates change points that can occur during early stages of an observed multistage process. The algorithm is designed to analyze change points that are likely to occur after very few observations and to be followed by other change points or more complicated patterns. Such models are justified in problem solving, quality control, and other processes. Special methods are derived in order to: (1) detect a change point even after a very brief period of observation, (2) estimate it with the theoretically highest degree of accuracy, (3) report a no-change case when a significant change has not occurred during the observed period, and (4) use minimum data after the change point to prevent mixing the post-change phase with subsequent phases and patterns. Unlike existing methods, the proposed algorithm produces a distribution consistent estimator of a change point. Details are elaborated for the case of Gamma distributions and demonstrated for a process of problem solving.