Interactive hidden Markov models and their applications

^** Email: wching{at}hkusua.hku.hk In this paper, we propose an Interactive hidden Markov model(IHMM). In a traditional HMM, the observable states are affecteddirectly by the hidden states, but not vice versa. In the proposedIHMM, the transitions of hidden states depend on the observablestates. We also develop an efficient estimation method for themodel parameters. Numerical examples on the sales demand dataand economic data are given to demonstrate the applicabilityof the model.     

Combining a regression model with a multivariate Markov chain in a forecasting problem

This paper proposes a new concept: the usage of Multivariate Markov Chains (MMC) as covariates. Our approach is based on the observation that we can treat possible categorical (or discrete) regressors, whose values are unknown in the forecast period, as an MMC in order to improve the forecast error of a certain dependent variable. Hence, we take advantage of the information about the past state interactions between the MMC categories to forecast the categorical (or discrete) regressors and improve the forecast of the actual dependent variable.     

Sensitivity analysis of the combined travel demand model with applications

The conventional sequential four-step procedure of travel demand forecasting has been widely adopted by practitioners. However, it suffers from inconsistent consideration of travel times and congestion effects in various steps of the procedure. A combined travel demand model overcomes the problems associated with the sequential four-step procedure by integrating travel-destination-mode-route choice together. In this paper, a standard sensitivity analysis for non-linear programming is employed for conducting the sensitivity analysis of the combined travel demand model. Explicit expressions of the derivatives of model variables with respect to perturbations of input variables and parameters of the combined travel demand model are developed. These derivatives could be used to assess changes in solution variables and various system performance measures when the network characteristics are changed slightly. To gain insight into the usefulness of the sensitivity expressions, five applications, such as identification of critical parameters, paradox analysis, access control, destination choice, and error and uncertainty analysis, are presented with numerical results.     

Holdings of financial assets: A Markov chain analysis

This paper introduces the Markov chain model as a simple tool for analyzing the pattern of financial asset holdings over time. The model is based on transition probabilities which give the probability of switching $1 of wealth from one asset to another. An illustrative application is provided.     

On generating multivariate Poisson data in management science applications

Generating multivariate Poisson random variables is essential in many applications, such as multi echelon supply chain systems, multi‐item/multi‐period pricing models, accident monitoring systems, etc. Current simulation methods suffer from limitations ranging from computational complexity to restrictions on the structure of the correlation matrix, and therefore are rarely used in management science. Instead, multivariate Poisson data are commonly approximated by either univariate Poisson or multivariate Normal data. However, these approximations are often not adequate in practice. In this paper, we propose a conceptually appealing correction for NORTA (NORmal To Anything) for generating multivariate Poisson data with a flexible correlation structure and rates. NORTA is based on simulating data from a multivariate Normal distribution and converting it into an arbitrary continuous distribution with a specific correlation matrix. We show that our method is both highly accurate and computationally efficient. We also show the managerial advantages of generating multivariate Poisson data over univariate Poisson or multivariate Normal data. Copyright © 2011 John Wiley & Sons, Ltd.     

An improved customer lifetime value model based on Markov chain

Firms are increasingly looking to provide a satisfactory prediction of customer lifetime value (CLV), a determining metric to target future profitable customers and to optimize marketing resources. One of the major challenges associated with the measurement of CLV is the choice of the appropriate model for predicting customer value because of the large number of models proposed in the literature. Earlier models to forecast CLV are relatively unsuccessful, whereas simple models often provide results which are equivalent or even better than sophisticated ones. To predict CLV, Rust et al. (2011) proposed a framework model that performs better than simple managerial heuristic models, but its implementation excludes cases where customer's profit is negative and does not handle lost‐for‐good situations. In this paper, we propose a modified model that handles both negative and positive profits based on Markov chain model (MCM), hence offering a greater flexibility by covering always‐a‐share and lost‐for‐good situations. The proposed model is compared with the Pareto/Negative Binomial Distribution (Pareto/NBD), the Beta Geometric/Negative Binomial Distribution (BG/NBD), the MCM, and the Rust et al. (2011) models. Based on customer credit card transactions provided by the North African retail bank, an empirical study shows that the proposed model has better forecasting performance than competing models. Copyright © 2014 John Wiley & Sons, Ltd.     

A Markov modulated Poisson model for software reliability

In this paper, we consider a latent Markov process governing the intensity rate of a Poisson process model for software failures. The latent process enables us to infer performance of the debugging operations over time and allows us to deal with the imperfect debugging scenario. We develop the Bayesian inference for the model and also introduce a method to infer the unknown dimension of the Markov process. We illustrate the implementation of our model and the Bayesian approach by using actual software failure data.     

Score and Wald tests for the multivariate growth curve model with missing data

We present the score and Wald test analogues to Srivastava's (1985, Comm. Statist. A—Theory Methods, 14, 775–792) likelihood ratio tests for the multivariate growth curve model with missing data, and illustrate their use with data from an immunotherapy experiment (Fukushima et al. (1982, Int. J. Cancer, 29, 107–112, 113–117)).     

Prediction of multivariate time series by autoregressive model fitting

Suppose the stationary r-dimensional multivariate time series {y_t} is generated by an infinite autoregression. For lead times h≥1, the linear prediction of y_t+h based on y_t, y_t−1,… is considered using an autoregressive model of finite order k fitted to a realization of length T. Assuming that k → ∞ (at some rate) as T → ∞, the consistency and asymptotic normality of the estimated autoregressive coefficients are derived, and an asymptotic approximation to the mean square prediction error based on this autoregressive model fitting approach is obtained. The asymptotic effect of estimating autoregressive parameters is found to inflate the minimum mean square prediction error by a factor of (1 + kr/T).     

A higher order Markov model for analyzing covariate dependence

During the recent past, there has been a renewed interest in Markov chain for its attractive properties for analyzing real life data emerging from time series or longitudinal data in various fields. The models were proposed for fitting first or higher order Markov chains. However, there is a serious lack of realistic methods for linking covariate dependence with transition probabilities in order to analyze the factors associated with such transitions especially for higher order Markov chains. L.R. Muenz and L.V. Rubinstein [Markov models for covariate dependence of binary sequences, Biometrics 41 (1985) 91–101] employed logistic regression models to analyze the transition probabilities for a first order Markov model. The methodology is still far from generalization in terms of formulating a model for higher order Markov chains. In this study, it is aimed to provide a comprehensive covariate-dependent Markov model for higher order. The proposed model generalizes the estimation procedure for Markov models for any order. The proposed models and inference procedures are simple and the covariate dependence of the transition probabilities of any order can be examined without making the underlying model complex. An example from rainfall data is illustrated in this paper that shows the utility of the proposed model for analyzing complex real life problems. The application of the proposed method indicates that the higher order covariate dependent Markov models can be conveniently employed in a very useful manner and the results can provide in-depth insights to both the researchers and policymakers to resolve complex problems of underlying factors attributing to different types of transitions, reverse transitions and repeated transitions. The estimation and test procedures can be employed for any order of Markov model without making the theory and interpretation difficult for the common users.     

层次模型Markov链的观测与统计

对于连续时间的层次模型M arkov链,所有的转移速率都可以由最底层状态的逗留时间和击中时间分布惟一决定,因而整个M arkov链的统计性质由它们的统计所决定.并给出了相应的算法和数例.     

Hitting time in a finite non-homogeneous Markov chain with applications

This paper deals with the computation of the hitting time for a non-homogeneous discrete time Markov chain (NHDTMC or NHMC). We first give the basic definitions of NHMC, then we analyse the hitting time and its survivor function. We also give the sufficient conditions for the existence of the mean hitting time. Finally, a numerical example and an application of this development in reliability evaluation are given in order to illustrate our results. © 1998 John Wiley & Sons, Ltd.     

The asymptotic validity of invariant procedures for the repeated measures model and multivariate linear model

Fairly general sufficient conditions are given to guarantee that invariant tests about means in the multivariate linear model and the repeated measures model have the correct asymptotic size when the normal assumption under which the tests are derived is relaxed. These conditions are the same as Huber's condition which guarantees asymptotic validity of the size of the F-test for the univariate linear model.     

A generalized additive, categorical model in data envelopment analysis

This paper considers efficiency of a Decision Making Unit (DMU) in Data Envelopment Analysis (DEA) with a generalized additive model and a categorical structure. Specifically, it extends the categorical framework in DEA for controllable and noncontrollable situations, and it gives simple, but powerful, tests to determine whether or not a given DMU is efficient.     

多元秩-序模型中回归系数的估计

通过对多元秩.序模型的研究得到了模型的逆回归性质,基于该性质提出了回归系数的估计方法.当自变量满足线性条件时,不用预先设定扰动项的具体分布便可以得到回归系数方向的估计,并且这个估计与回归系数只相差一个正常数因子.证明了估计是√n相目合的.模拟结果表明估计有良好的大样本性质.     

On the Type I multivariate zero-truncated hurdle model with applications in health insurance

In the general insurance modeling literature, there has been a lot of work based on univariate zero-truncated models, but little has been done in the multivariate zero-truncation cases, for instance a line of insurance business with various classes of policies. There are three types of zero-truncation in the multivariate setting: only records with all zeros are missing, zero counts for one or some classes are missing, or zeros are completely missing for all classes. In this paper, we focus on the first case, the so-called Type I zero-truncation, and a new multivariate zero-truncated hurdle model is developed to study it. The key idea of developing such a model is to identify a stochastic representation for the underlying random variables, which enables us to use the EM algorithm to simplify the estimation procedure. This model is used to analyze a health insurance claims dataset that contains claim counts from different categories of claims without common zero observations.     

A newsvendor model with fuzzy price-dependent demand

Price-dependence is an important characteristic for some inventory problems. This paper proposes a newsvendor model with fuzzy price-dependent demand, and discusses the conditions to determine the optimal pricing and inventory decisions jointly so that the expected profit could be maximized. Then an algorithm combining the method of ranking fuzzy numbers is developed to tackle the problem. Furthermore, comparison is made between the fuzzy model and the deterministic model to study the effect of the uncertain price-dependent demand, and the sensitivity properties of the joint optimal decisions are illustrated through numerical examples.     

二阶离散隐马尔科夫模型的严格定义及等价性质

隐马氏模型作为一种具有双重随机过程的统计模型,具有可靠的概率统计理论基础和强有力的数学结构,已被广泛应用于语音识别、生物序列分析、金融数据分析等领域.由于传统的一阶隐马氏模型无法表示更远状态距离间的依赖关系,就可能会忽略很多有用的统计特征,故有人提出二阶隐马氏模型的概念,但此概念并不严格.本文给出二阶离散隐马尔科夫模型的严格定义,并研究了二阶离散隐马尔科夫模型的两个等价性质.