General Markov Model Research Articles

Are information criteria good enough to choose the right the number of regimes in hidden Markov models?

Selecting the number of regimes in Hidden Markov models is an important problem. Many criteria are used to do so, such as Akaike information criterion (AIC), Bayesian information criterion (BIC), integrated completed likelihood (ICL), deviance information criterion (DIC), and Watanabe-Akaike information criterion (WAIC), to name a few. In this article, we introduced goodness-of-fit tests for general Hidden Markov models with covariates, where the distribution of the observations is arbitrary, i.e. continuous, discrete, or a mixture of both. A selection procedure is proposed based on this goodness-of-fit test. The main aim of this article is to compare the classical information criterion with the new criterion when the outcome is either continuous, discrete or zero-inflated. Numerical experiments assess the finite sample performance of the goodness-of-fit tests, and comparisons between the different criteria are made.

Speed and duration of drawdown under general Markov models

We propose an efficient computational method based on continuous-time Markov chain (CTMC) approximation to compute the distributions of the speed and duration of drawdown for general one-dimensional (1D) time-homogeneous Markov processes. We derive linear systems for the Laplace transforms of drawdown quantities and show how to solve them efficiently by recursion. In addition, we prove the convergence of our method and obtain a sharp estimate of the convergence rate under some assumptions. As applications, we consider pricing two financial options that provide protection against drawdown risk. Finally, we demonstrate the accuracy and efficiency of our method through extensive numerical experiments.

Sine-skewed von Mises- and Lindley/Gumbel models as candidates for direction and distance in modelling animal movement

Movement of animals is often characterised by direction (measured on the circle) and distance (measured on the real line); but traditionally employed models often do not account for potential asymmetric directional movement, or departures from the usual von Mises assumption for direction and gamma/Weibull assumptions for distance. This paper focuses on the modelling of circular data in this animal movement context relying on a previously unconsidered circular distribution (the sine-skewed von Mises) which provides a platform for departures from symmetry. In addition, alternative models to usual distance assumptions are considered, namely the power Lindley (as a mixture of gamma andWeibull distributions) as well as a Gumbel candidate. Computational aspects and investigations of this joint modelling (presented as a consensus model) are highlighted, accompanied by an extensive bootstrap study. A general hidden state Markov model is used to incorporate these essential components when estimating via the use of the EM algorithm, and goodness of fit measures verify the validity and viable future consideration of the newly proposed theoretical models within this practical and computational animal movement environment.

Designing Weights for Quartet-Based Methods When Data are Heterogeneous Across Lineages

Homogeneity across lineages is a general assumption in phylogenetics according to which nucleotide substitution rates are common to all lineages. Many phylogenetic methods relax this hypothesis but keep a simple enough model to make the process of sequence evolution more tractable. On the other hand, dealing successfully with the general case (heterogeneity of rates across lineages) is one of the key features of phylogenetic reconstruction methods based on algebraic tools. The goal of this paper is twofold. First, we present a new weighting system for quartets (ASAQ) based on algebraic and semi-algebraic tools, thus especially indicated to deal with data evolving under heterogeneous rates. This method combines the weights of two previous methods by means of a test based on the positivity of the branch lengths estimated with the paralinear distance. ASAQ is statistically consistent when applied to data generated under the general Markov model, considers rate and base composition heterogeneity among lineages and does not assume stationarity nor time-reversibility. Second, we test and compare the performance of several quartet-based methods for phylogenetic tree reconstruction (namely QFM, wQFM, quartet puzzling, weight optimization and Willson’s method) in combination with several systems of weights, including ASAQ weights and other weights based on algebraic and semi-algebraic methods or on the paralinear distance. These tests are applied to both simulated and real data and support weight optimization with ASAQ weights as a reliable and successful reconstruction method that improves upon the accuracy of global methods (such as neighbor-joining or maximum likelihood) in the presence of long branches or on mixtures of distributions on trees.

A general method for analysis and valuation of drawdown risk

Drawdown risk is a major concern in financial markets. We develop a novel method to solve the first passage problem of the drawdown process for general one-dimensional Markov processes (including time-inhomogeneous ones) as well as regime-switching and stochastic volatility models. We compute its Laplace transform based on continuous-time Markov chain (CTMC) approximation and invert the Laplace transform numerically to obtain the first passage probabilities and the distribution of the maximum drawdown. We prove convergence of our method for general Markov models and provide sharp estimate of the convergence rate for a general class of jump-diffusion models. We apply our method to price and hedge maximum drawdown options and demonstrate its accuracy and efficiency through various numerical experiments. In addition, we can apply our method to calculate the Calmar ratio for investment analysis, and quantify the contributions of assets to the drawdown risk of a portfolio when the assets follow multivariate exponential Lévy models.

Asynchronous Boundary Stabilization of Stochastic Markov Jump Reaction-Diffusion Systems

Dissipativity-based asynchronous boundary stabilization problem is addressed for stochastic Markov jump reaction-diffusion systems (SMJRDSs). In practical engineering, nonsynchronous behavior between system modes and controller modes is inevitable, and the incomplete matrix information makes the problem analysis difficult, so this work considers the asynchronous stabilization. Different from the distributed control, we apply a simple boundary control strategy, which greatly reduces the cost of the control design. Note that three issues need to be addressed: 1) how to model the asynchronous behavior? 2) how to design the asynchronous boundary controller? and 3) how to process the incomplete matrix information? We deal with these problems one by one. Based on a general hidden Markov model (HMM), an asynchronous boundary feedback controller is considered. Via the Wirtinger-type inequality, Schur complement technique, and transition matrix properties, sufficient conditions ensuring exponentially mean square stability and strictly <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(W, P, R)-\alpha $ </tex-math></inline-formula> dissipativity are established, which covers several special cases. Finally, a numerical example is presented to illustrate the proposed control strategies.

Data-driven modeling of the temporal evolution of breakers’ states in the French electrical transmission grid

In electrical transmission grids, it is common to observe the states of circuit breakers. While they are known at irregular times, system modeling and grid state estimation are of the highest importance to ensure secure operations. This paper proposes a richer method to estimate the grid state over its reference configurations based on the temporal evolution of its breakers’ states. The first contribution consists in developing a general multi-observation continuous-time finite-state Hidden Markov Model with filter-based parameter estimation to infer the hidden state (e.g., the grid reference configuration) handling multiple observed processes with irregular “jump” times (e.g., the breakers’ states). As a second contribution, we build a numerical scheme with no discretization error adapted to all state jumps generated by the observed processes. Finally, we apply our model to simulated and real data to illustrate the approach’s performance. The available data consists of historical records of breakers’ states during the electrical transmission grid operated normally. For this real-data-driven application, we also present a clustering approach to identify the set of grid reference configurations.

Parametric estimation of hidden Markov models by least squares type estimation and deconvolution

This paper develops a simple and computationally efficient parametric approach to the estimation of general hidden Markov models (HMMs). For non-Gaussian HMMs, the computation of the maximum likelihood estimator (MLE) involves a high-dimensional integral that has no analytical solution and can be difficult to approach accurately. We develop a new alternative method based on the theory of estimating functions and a deconvolution strategy. Our procedure requires the same assumptions as the MLE and deconvolution estimators. We provide theoretical guarantees about the performance of the resulting estimator; its consistency and asymptotic normality are established. This leads to the construction of confidence intervals. Monte Carlo experiments are investigated and compared with the MLE. Finally, we illustrate our approach using real data for ex-ante interest rate forecasts.

Speed and Duration of Drawdown under General Markov Models

Speed and Duration of Drawdown under General Markov Models

Recognizing human activities in Industry 4.0 scenarios through an analysis-modeling- recognition algorithm and context labels

Activity recognition technologies only present a good performance in controlled conditions, where a limited number of actions are allowed. On the contrary, industrial applications are scenarios with real and uncontrolled conditions where thousands of different activities (such as transporting or manufacturing craft products), with an incredible variability, may be developed. In this context, new and enhanced human activity recognition technologies are needed. Therefore, in this paper, a new activity recognition technology, focused on Industry 4.0 scenarios, is proposed. The proposed mechanism consists of different steps, including a first analysis phase where physical signals are processed using moving averages, filters and signal processing techniques, and an atomic recognition step where Dynamic Time Warping technologies and k-nearest neighbors solutions are integrated; a second phase where activities are modeled using generalized Markov models and context labels are recognized using a multi-layer perceptron; and a third step where activities are recognized using the previously created Markov models and context information, formatted as labels. The proposed solution achieves the best recognition rate of 87% which demonstrates the efficacy of the described method. Compared to the state-of-the-art solutions, an improvement up to 10% is reported.

SAQ: Semi-Algebraic Quartet Reconstruction.

We present the phylogenetic quartet reconstruction method SAQ (Semi-Algebraic Quartet reconstruction). SAQ is consistent with the most general Markov model of nucleotide substitution and, in particular, it allows for rate heterogeneity across lineages. Based on the algebraic and semi-algebraic description of distributions that arise from the general Markov model on a quartet, the method outputs normalized weights for the three trivalent quartets (which can be used as input of quartet-based methods). We show that SAQ is a highly competitive method that outperforms most of the well known reconstruction methods on data simulated under the general Markov model on 4-taxon trees. Moreover, it also achieves a high performance on data that violates the underlying assumptions.

A Modeling Framework for Decision Support in Periprosthetic Joint Infection Treatment.

In this paper, we present a framework, which aims at facilitating the choice of the best strategy related to the treatment of periprosthetic joint infection (PJI). The framework includes two models: a detailed non-Markovian model based on the decision tree approach, and a general Markov model, which captures the most essential states of a patient under treatment. The application of the framework is demonstrated on the dataset provided by Russian Scientific Research Institute of Traumatology and Orthopedics "R.R. Vreden", which contains records of patients with PJI occurred after total hip arthroplasty. The methods of cost-effectiveness analysis of treatment strategies and forecasting of individual treatment outcomes depending on the selected strategy are discussed.

An IBD-based mixed model approach for QTL mapping in multiparental populations

Key messageThe identity-by-descent (IBD)-based mixed model approach introduced in this study can detect quantitative trait loci (QTLs) referring to the parental origin and simultaneously account for multilevel relatedness of individuals within and across families. This unified approach is proved to be a powerful approach for all kinds of multiparental population (MPP) designs.Multiparental populations (MPPs) have become popular for quantitative trait loci (QTL) detection. Tools for QTL mapping in MPPs are mostly developed for specific MPPs and do not generalize well to other MPPs. We present an IBD-based mixed model approach for QTL mapping in all kinds of MPP designs, e.g., diallel, Nested Association Mapping (NAM), and Multiparental Advanced Generation Intercross (MAGIC) designs. The first step is to compute identity-by-descent (IBD) probabilities using a general Hidden Markov model framework, called reconstructing ancestry blocks bit by bit (RABBIT). Next, functions of IBD information are used as design matrices, or genetic predictors, in a mixed model approach to estimate variance components for multiallelic genetic effects associated with parents. Family-specific residual genetic effects are added, and a polygenic effect is structured by kinship relations between individuals. Case studies of simulated diallel, NAM, and MAGIC designs proved that the advanced IBD-based multi-QTL mixed model approach incorporating both kinship relations and family-specific residual variances (IBD.MQMkin_F) is robust across a variety of MPP designs and allele segregation patterns in comparison to a widely used benchmark association mapping method, and in most cases, outperformed or behaved at least as well as other tools developed for specific MPP designs in terms of mapping power and resolution. Successful analyses of real data cases confirmed the wide applicability of our IBD-based mixed model methodology.

Pricing American drawdown options under Markov models

The drawdown in the price of an asset shows how much the price falls relative to its historical maximum. This paper considers the pricing problem of perpetual American style drawdown call options, which allow the holder to optimally choose the time to receive a call payoff written on the drawdown. Our pricing framework includes classical Russian options and American lookback puts as special cases after a suitable equivalent measure change. We approximate the original asset price model by a continuous time Markov chain and develop two types of algorithms to solve the optimal stopping problem for the drawdown process. The first one is a transform based algorithm which is applicable to general exponential Lévy models. The second approach solves the linear complementarity problem (LCP) associated with the variational inequalities for the value function and it applies to general Markov models. We propose an efficient Block-LCP (BLCP) method that reduces an LCP with big size to a sequence of sub-LCPs with mild size which can be solved by a variety of LCP solvers and we identify the best solver through numerical experiments. Convergence of Markov chain approximation is proved and various numerical examples are given to demonstrate their computational efficiency and convergence properties. An extension of the BLCP method to the finite maturity case is also provided.

A General Method for Analysis and Valuation of Drawdown Risk under Markov Models

Drawdown risk is a major concern in financial markets. We develop a novel algorithm to solve the first passage problem of the drawdown process of general one-dimensional time-homogeneous Markov processes. We compute its Laplace transform based on continuous time Markov chain (CTMC) approximation and numerically invert the Laplace transform to obtain the first passage probabilities and the distribution of the maximum drawdown. We prove convergence of our algorithm for general Markovian models and provide sharp estimate of the convergence rate for a general class of jump-diffusion models. We apply the algorithm to calculate the Calmar ratio for investment analysis, price maximum drawdown derivatives and hedge the risk of selling such derivatives with a highly volatile asset as the underlying. Various numerical experiments document the computational efficiency of our method. We also develop extensions to solve the drawdown problem in models with time dependence or stochastic volatility or regime switching and for portfolio drawdown analysis.

Do generalized draw-down times lead to better dividends? A Pontryagin principle-based answer

Abstract In the context of maximizing cumulative dividends under barrier policies, generalized Azéma–Yor (draw-down) stopping times receive increasing attention during these past years. Based on Pontryagin’s maximality principle, we illustrate the necessity of such generalizations under the framework of spectrally negative Markov processes. Roughly speaking, starting from the explicit expression of the optimal value of discounted dividends in terms of the scale functions, we write down the optimality conditions (via Pontryagin’s principle). The use of generalized draw-downs is then quantified through a structure term (linked to the existence of non bang-bang optimal controls). We thoroughly study several classes of Lévy processes (Bertoin, Lévy Processes, vol. 121. Cambridge University Press, 1998; Kyprianou, Fluctuations of Lévy Processes with Applications: Introductory Lectures. Springer Science &amp; Business Media, 2014) constituting the usual models of insurance claims and a particular piece-wise deterministic Markov model (extending the premium rate to reserve-dependent settings). In all these models, we disprove the consistency of the aforementioned structure equation, thus denying the necessity of such generalizations. We end the paper with some heuristics on possible non-trivial cases for general Markov models.

Asymptotic analysis of model selection criteria for general hidden Markov models

The paper obtains analytical results for the asymptotic properties of Model Selection Criteria – widely used in practice – for a general family of hidden Markov models (HMMs), thereby substantially extending the related theory beyond typical ‘i.i.d.-like’ model structures and filling in an important gap in the relevant literature. In particular, we look at the Bayesian and Akaike Information Criteria (BIC and AIC) and the model evidence. In the setting of nested classes of models, we prove that BIC and the evidence are strongly consistent for HMMs (under regularity conditions), whereas AIC is not weakly consistent. Numerical experiments support our theoretical results.

Coupled conditional backward sampling particle filter

The conditional particle filter (CPF) is a promising algorithm for general hidden Markov model smoothing. Empirical evidence suggests that the variant of CPF with backward sampling (CBPF) performs well even with long time series. Previous theoretical results have not been able to demonstrate the improvement brought by backward sampling, whereas we provide rates showing that CBPF can remain effective with a fixed number of particles independent of the time horizon. Our result is based on analysis of a new coupling of two CBPFs, the coupled conditional backward sampling particle filter (CCBPF). We show that CCBPF has good stability properties in the sense that with fixed number of particles, the coupling time in terms of iterations increases only linearly with respect to the time horizon under a general (strong mixing) condition. The CCBPF is useful not only as a theoretical tool, but also as a practical method that allows for unbiased estimation of smoothing expectations, following the recent developments by Jacob, Lindsten and Schön (2020). Unbiased estimation has many advantages, such as enabling the construction of asymptotically exact confidence intervals and straightforward parallelisation.

Tailored optimal posttreatment surveillance for cancer recurrence.

A substantial rise in the number of cancer survivors has led to urgent management questions regarding effective posttreatment imaging-based surveillance strategies for cancer recurrence. Current surveillance guidelines provided by a number of professional societies all warn against overly aggressive surveillance, especially for low-risk patients, but all fail to provide more specific directions to accommodate underlying heterogeneity of cancer recurrence. Therefore it is imperative to develop data-driven strategies that can tailor the surveillance schedules to recurrence risk in this era of stricter insurance regulations, provider shortages, and rising costs of health care. Due to a lack of statistical methods for optimizing surveillance scheduling in presence of competing risks, we propose a general approach that uses an intuitive loss function for optimization of early detection of recurrence before death. The proposed strategies can tailor to patient risks of recurrence, in terms of both intensity and amount of surveillance. Using general three-state Markov models, our method is flexible and includes earlier works as special cases. We illustrate our method in both simulation studies and an application to breast cancersurveillance.

Pricing American Drawdown Options under Markov Models

The drawdown in the price of an asset shows how much the price falls relative to its historical maximum. This paper considers the pricing problem of American style drawdown call options, which allow the holder to optimally choose the time to receive a call payoff written on the drawdown. Our pricing framework includes classical Russian options and American lookback puts as special cases after a suitable equivalent measure change. We approximate the original asset price model by a continuous time Markov chain and develop two types of algorithms to solve the optimal stopping problem for the drawdown process. The first one is a transform based algorithm which is applicable to general exponential Levy models. The second approach solves the linear complementarity problem (LCP) associated with the variational inequalities for the value function and it applies to general Markov models. We propose an efficient Block-LCP method that reduces an LCP with big size to a sequence of sub-LCPs with mild size which can be solved by a variety of LCP solvers and we identify the best solver through numerical experiments. Convergence of Markov chain approximation is proved and various numerical examples are given to demonstrate their computational efficiency and convergence properties.