State-space Chain Research Articles

Markov (Set) chains application to predict mortality rates using extended Milevsky–Promislov generalized mortality models

The mortality rates () measure the frequency of deaths in a fixed: population and time interval. The ability to model and forecast allows determining, among others, fundamental characteristics of life expectancy tables, e.g. used to determine the amount of premium in life insurance, adequate to the risk of death. The article proposes a new method of modelling and forecasting , using the class of stochastic Milevsky–Promislov switch models with excitations. The excitations are modelled by second, fourth and sixth order polynomials of outputs from the non-Gaussian Linear Scalar Filter (nGLSF) model and taking into account the Markov (Set) chain. The Markov (Set) chain state space is defined based on even orders of the nGLSF polynomial. The model order determines the theoretical values of the death rates. The obtained results usually provide a more precise forecast of the mortality rates than the commonly used Lee–Carter model.

Computation of exact probabilities associated with overlapping pattern occurrences

AbstractSearching for patterns in data is important because it can lead to the discovery of sequence segments that play a functional role. The complexity of pattern statistics that are used in data analysis and the need of the sampling distribution of those statistics for inference renders efficient computation methods as paramount. This article gives an overview of the main methods used to compute distributions of statistics of overlapping pattern occurrences, specifically, generating functions, correlation functions, the Goulden‐Jackson cluster method, recursive equations, and Markov chain embedding. The underlying data sequence will be assumed to be higher‐order Markovian, which includes sparse Markov models and variable length Markov chains as special cases. Also considered will be recent developments for extending the computational capabilities of the Markov chain‐based method through an algorithm for minimizing the size of the chain's state space, as well as improved data modeling capabilities through sparse Markov models. An application to compute a distribution used as a test statistic in sequence alignment will serve to illustrate the usefulness of the methodology.This article is categorized under: Statistical Learning and Exploratory Methods of the Data Sciences > Pattern Recognition Data: Types and Structure > Categorical Data Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms

AN IMPROVED MARKOV CHAIN APPROXIMATION METHODOLOGY: DERIVATIVES PRICING AND MODEL CALIBRATION

This paper presents an improved continuous-time Markov chain approximation (MCA) methodology for pricing derivatives and for calibrating model parameters. We propose a generalized nonequidistant grid model for a general stochastic differential equation, and extend the proposed model to accommodate a jump component. Because the prices of derivatives generated by the MCA models are sensitive to the setting of the chain's state space, we suggest a heuristic determination of the grid spacing such that the Kolmogorov–Smirnov distance between the underlying distribution and the MCA distribution is minimized. The continuous time setting allows us to introduce semi-analytical formulas for pricing European and American style options. The numerical examples demonstrate that the proposed model with a nonequidistant grid setting provides superior results over the equidistant grid setting. Finally, we present the MCA maximum likelihood estimator for a jump-diffusion process. The encouraging results from the simulation and empirical studies provide insight into calibration problems in finance where the density function of a jump-diffusion model is unknown.

Exact Smoothing in Hidden Conditionally Markov Switching Linear Models

We consider the problem of the exact calculation of smoothing in hidden switching state-space systems. There is a hidden state-space chain X, the switching chain R, and the observed chain Y. In the classical, widely used conditionally Gaussian state-space linear model (CGSSLM) the exact calculation with complexity linear in time is not feasible and different approximations have to be made. Different alternative models, in which the exact calculations are feasible, have been proposed recently. The key difference between these models and the classical ones is that R is Markovian conditionally on Y in the recent models, while it is not in the classical ones. Moreover, these different models have been extended to models in which X is no longer necessarily Markovian conditionally on (R, Y). Here, we propose a further new extension of the latter models and we derive exact computation of posterior expectation as well as posterior variance-covariance matrix with complexity polynomial in time.

Markovian inventory policy with application to the paper industry

This paper concerns problem formulation and solution procedure for inventory planning with Markov decision process models. Using data collected from a large paper manufacturer, we develop inventory policies for the finished products. To incorporate both variability and regularity of the system into mathematical formulation, we analyze probabilistic distribution of the demand, explore its connection with the corresponding Markov chains, and integrate these into our decision making. In particular, we formulate the Markov decision model by identifying the chain's state space and the transition probabilities, specify the cost structure and evaluate its individual component; and then use the policy-improvement algorithm to obtain the optimal policy. Application examples are provided for illustration.

Bayesian Methods for Hidden Markov Models

Markov chain Monte Carlo (MCMC) sampling strategies can be used to simulate hidden Markov model (HMM) parameters from their posterior distribution given observed data. Some MCMC methods used in practice (for computing likelihood, conditional probabilities of hidden states, and the most likely sequence of states) can be improved by incorporating established recursive algorithms. The most important of these is a set of forward-backward recursions calculating conditional distributions of the hidden states given observed data and model parameters. I show how to use the recursive algorithms in an MCMC context and demonstrate mathematical and empirical results showing a Gibbs sampler using the forward-backward recursions mixes more rapidly than another sampler often used for HMMs. Iintroduce an augmented variables technique for obtaining unique state labels in HMMs and finite mixture models. I show how recursive computing allows the statistically efficient use of MCMC output when estimating the hidden states. I directly calculate the posterior distribution of the hidden chain's state-space size by MCMC, circumventing asymptotic arguments underlying the Bayesian information criterion, which is shown to be inappropriate for a frequently analyzed dataset in the HMM literature. The use of log-likelihood for assessing MCMC convergence is illustrated, and posterior predictive checks are used to investigate application specific questions of model adequacy.