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We develop two models for Bayesian estimation and selection in high-order, discrete-state Markov chains. Both are based on the mixture transition distribution, which constructs a transition probability tensor with additive mixing of probabilities from first-order transition matrices. We demonstrate two uses for the proposed models: parsimonious approximation of high-order dynamics by mixing lower-order transition models, and order/lag selection through over-specification and shrinkage via priors for sparse probability vectors. The priors further shrink all models to an identifiable and interpretable parameterization, useful for data analysis. We discuss properties of the models and demonstrate their utility with simulation studies. We further apply the methodology to a data analysis from the high-order Markov chain literature and to a time series of pink salmon abundance in a creek in Alaska, U.S.A.

Mixture transition distribution time series models build high-order dependence through a weighted combination of first-order transition densities for each one of a specified number of lags. We present a framework to construct stationary transition mixture distribution models that extend beyond linear, Gaussian dynamics. We study conditions for first-order strict stationarity which allow for different constructions with either continuous or discrete families for the first-order transition densities given a pre-specified family for the marginal density, and with general forms for the resulting conditional expectations. Inference and prediction are developed under the Bayesian framework with particular emphasis on flexible, structured priors for the mixture weights. Model properties are investigated both analytically and through synthetic data examples. Finally, Poisson and Lomax examples are illustrated through real data applications.

We propose a flexible approach to modeling for renewal processes. The model is built from a structured mixture of Erlang densities for the renewal process inter-arrival density. The Erlang mixture ...

We develop a Bayesian nonparametric mixture modeling framework for replicated count responses in dose-response settings. We explore this methodology for modeling and risk assessment in developmental toxicity studies, where the primary objective is to determine the relationship between the level of exposure to a toxic chemical and the probability of a physiological or biochemical response, or death. Data from these experiments typically involve features that cannot be captured by standard parametric approaches. To provide flexibility in the functional form of both the response distribution and the probability of positive response, the proposed mixture model is built from a dependent Dirichlet process prior, with the dependence of the mixing distributions governed by the dose level. The methodology is tested with a simulation study, which involves also comparison with semiparametric Bayesian approaches to highlight the practical utility of the dependent Dirichlet process nonparametric mixture model. Further...

Computer simulators are used in science and technology to model physical processes or the behavior of real-world systems. Sensitivity analysis provides a useful tool for quantifying the impact of uncertainty in the computer simulator inputs on the computed output. We focus on global sensitivity analysis, which quantifies output uncertainty as all the inputs vary continuously over the input space. The influence of each input and how uncertainty in the output is apportioned among the inputs are determined by calculating the main effects and sensitivity indices of the computer simulator inputs. Typically, these quantities are computed using Monte Carlo methods, which require a large number of computer simulator runs, making the calculations infeasible if the simulator is computationally expensive. Bayesian methods have been used to tackle sensitivity analysis of computationally expensive simulators through building a statistical emulator for the computer simulator output, typically, based on a Gaussian proce...

The CRASH computer model simulates the effect of a vehicle colliding against different barrier types. If it accurately represents real vehicle crashworthiness, the computer model can be of great value in various aspects of vehicle design, such as the setting of timing of air bag releases. The goal of this study is to address the problem of validating the computer model for such design goals, based on utilizing computer model runs and experimental data from real crashes. This task is complicated by the fact that (i) the output of this model consists of smooth functional data, and (ii) certain types of collision have very limited data. We address problem (i) by extending existing Gaussian process-based methodology developed for models that produce real-valued output, and resort to Bayesian hierarchical modeling to attack problem (ii). Additionally, we show how to formally test if the computer model reproduces reality. Supplemental materials for the article are available online.

Customary modeling for continuous point-referenced data assumes a Gaussian process that is often taken to be stationary. When such models are fitted within a Bayesian framework, the unknown parameters of the process are assumed to be random, so a random Gaussian process results. Here we propose a novel spatial Dirichlet process mixture model to produce a random spatial process that is neither Gaussian nor stationary. We first develop a spatial Dirichlet process model for spatial data and discuss its properties. Because of familiar limitations associated with direct use of Dirichlet process models, we introduce mixing by convolving this process with a pure error process. We then examine properties of models created through such Dirichlet process mixing. In the Bayesian framework, we implement posterior inference using Gibbs sampling. Spatial prediction raises interesting questions, but these can be handled. Finally, we illustrate the approach using simulated data, as well as a dataset involving precipitati...

This article proposes a probability model for k-dimensional ordinal outcomes, that is, it considers inference for data recorded in k-dimensional contingency tables with ordinal factors. The proposed approach is based on full posterior inference, assuming a flexible underlying prior probability model for the contingency table cell probabilities. We use a variation of the traditional multivariate probit model, with latent scores that determine the observed data. In our model, a mixture of normals prior replaces the usual single multivariate normal model for the latent variables. By augmenting the prior model to a mixture of normals we generalize inference in two important ways. First, we allow for varying local dependence structure across the contingency table. Second, inference in ordinal multivariate probit models is plagued by problems related to the choice and resampling of cutoffs defined for these latent variables. We show how the proposed mixture model approach entirely removes these problems. We ill...

Temperature and oxygen profiles evolution in windrows were monitored in windrows located at a commercial composting plant. Controlled turning of the windrows was executed using a turning machine. Measurements were made from the onset of turning, when temperature and oxygen were uniform throughout, onward, to evaluate the development of the relevant profiles. Temperatures rose and oxygen declined soon after turning. In fresh active windrows, a temperature maxima at a depth of 50-70 cm was detected, with temperatures as high as 65°-70°. The outward layers were cooler due to the effect of the ambient air temperature. Inward cooler core region seemed to be due to limited oxygen availability and the resulted restricted metabolic activity. No temperature maxima were observed as windrows approached stability. At this stage, the temperature of the core was as high as that of mid-layers. Oxygen, assumed to be in equilibrium with the atmosphere at the instance of turning, dropped down rapidly and reached steady levels within a period of about 4 hours. Oxygen concentration was lowered with depth, demonstrating a linear gradient down to about 70 cm depth. These data iterated the notion that oxygen is an important limiting factor in windrow composting. For any given point, high temperature may be a limiting factor. However, it seems that in properly managed windrows, high temperature is not limiting overall, due to an intrinsic correction feed back mechanism. If temperature in the hottest mid-layers gets to a level where microbial activity is hampered, less oxygen is consumed as it moves toward the core. Thus, more oxygen can penetrate into the core and the overall activity is not affected. It should be stressed that oxygen and temperature are not the only factors affecting composting rates. Other factors such as moisture, nutrients, substrate and porosity do have an important role.

Widely used parametric generalized linear models are, unfortunately, a somewhat limited class of specifications. Nonparametric aspects are often introduced to enrich this class, resulting in semiparametric models. Focusing on single or k-sample problems, many classical nonparametric approaches are limited to hypothesis testing. Those that allow estimation are limited to certain functionals of the underlying distributions. Moreover, the associated inference often relies upon asymptotics when nonparametric specifications are often most appealing for smaller sample sizes. Bayesian nonparametric approaches avoid asymptotics but have, to date, been limited in the range of inference. Working with Dirichlet process priors, we overcome the limitations of existing simulation-based model fitting approaches which yield inference that is confined to posterior moments of linear functionals of the population distribution. This article provides a computational approach to obtain the entire posterior distribution for mor...

Median regression models become an attractive alternative to mean regression models when employing flexible families of distributions for the errors. Classical approaches are typically algorithmic with desirable properties emerging asymptotically. However, nonparametric error models may be most attractive in the case of smaller sample sizes where parametric specifications are difficult to justify. Hence, a Bayesian approach, enabling exact inference given the observed data, may be appealing. In this context there is little Bayesian work. We develop two fully Bayesian modeling approaches, employing mixture models, for the errors in a median regression model. The associated families of error distributions allow for increased variability, skewness, and flexible tail behavior. The first family is semiparametric with extra variability captured nonparametrically through mixing and skewness handled parametrically. The second family, a fully nonparametric one, includes all unimodal densities on the real line with...

We consider the situation in which the manufacturer of a single-period product first sets the unit wholesale price and then the retailer responds with an order size. We present mostly analytical results on the effects of the problem's environmental parameters (such as shortage cost and demand uncertainty) on the optimal decisions (ie, the unit wholesale price and retailer's order size) and on the expected profits of the manufacturer and of the retailer. Some of these effects are counter-intuitive and/or contradict related results published recently for similar models. The most important finding is that demand uncertainty is always harmful to the manufacturer but is very often beneficial to the retailer. This means that when the manufacturer can set the wholesale price, the manufacturer should be much more supportive (or even aggressive) than previously advised towards activities such as market surveys and ‘Quick Response’ that reduce the retailer's market uncertainty; in contrast, the retailer need not be as enthusiastic about these activities.

This paper presents a method for evaluating the often neglected component, the expected incompletion cost, of the operating cost of a paced line with stochastic task times. This incompletion cost is developed through a three stage process. First presented is a procedure for generating all possible combinations of incompleted tasks. Next is an evaluation of the probabilities and costs of their occurrence. Included are not only a mechanism for generating only those incompletion combinations with a significant probability of occurring, but also various techniques for reducing computer storage requirements and computation time. Finally, these make possible an evaluation of the operating cost of paced lines.

This paper presents some techniques for the realistic treatment of stochastic lead-time-demands in inventory models. First, we suggest the use of the first four moments to describe the diversified distribution forms of the lead times and the daily demands; and formulas for deriving a lead-time-demand's first four moments are presented. Next, we demonstrate the use of a lead-time-demand's first four moments in conjunction with the Johnson et al.'s tables to obtain various probability estimates. Thirdly, we discuss the use of the versatile four-parameter Schmeiser-Deutsch curves to fit a lead-time-demand distribution. The computational advantages of using the fitted Schmeiser-Deutsch curves in solving inventory models are then illustrated.

This paper presents a practical approach for studying the profit probability distribution defined in stochastic breakeven analysis. After presenting a procedure for calculating the mean and the second to fourth central moments of the profit distribution, the several important uses of these four moments are discussed. A major application of the four moments is in the fitting of a Pearson's curve. Using a numerical example in conjunction with a set of published tables, we then demonstrate the simplicity of our proposed approach as well as the resultant high accuracy in estimating probabilities of various profit levels.

This paper presents a heuristic procedure for designing paced production lines with stochastic task times. The procedure adopts explicitly the objective of minimizing the sum of incompletion cost and labour cost. Basically, the first stage of the procedure uses a probabilistic procedure to generate many promising line designs: this probabilistic procedure is conceptually related to the procedures of Arues (1960) and Tongc (1965). The procedures second stage uses the Kottas-Lau (1976) cost model to evaluate the operating costs of the generated line designs and to identify the economical ones. Extensive tests with the computerized procedure using randomly generated-test problems have indicated that the procedure can produce very good line designs with modest computer time

This paper extends an earlier paper in this journal by Murphy (1975) by presenting practical approaches for determining the expected lost sales for a reorder level system of inventory control. First, a set of formulas are derived for computing the first four moments (f.f.m.) of the lead time demand, these formulas are applicable to any distribution forms of daily demand and lead time. By fitting the lead time demand's f.f.m. to either a beta or a Schmeiser-Deutsch distribution, this paper then demonstrates how the expected lost sales can be easily computed. The approaches presented provide accurate modelling for diversified situations and the computations can be easily performed manually.

This paper presents a new heuristic for balancing paced assembly lines involving tasks which exhibit significant time variations. Labor and incompletion costs are explicitly considered in grouping tasks into work assignments to achieve a near minimum cost line design. The procedure is computationally attractive because its calculations are comparably efficient to those of heuristics which consider task times deterministically.