Complex Stochastic Models Research Articles

Lumping and reversed processes in cooperating automata

Performance evaluation of computer software or hardware architectures may rely on the analysis of a complex stochastic model whose specification is usually given in terms of a high level formalism such as queueing networks, stochastic Petri nets, stochastic automata or Markovian process algebras. Compositionality is a key-feature of many of these formalisms and allows the modeller to combine several (simple) components to form a complex architecture. However, although these formalisms lead to relative compact specifications of possibly complex models, the derivation of the performance indices may be computationally very time and space consuming since the set of possible states of the model tends to grow exponentially (or even faster) with the number of components. In this paper we focus on models with underlying continuous time Markov chains (CTMCs) and we introduce a notion of typed lumpability, which gives sufficient conditions under which a lumping of the process can be derived, allowing the exact computation of marginal stationary probabilities of the cooperating components. The peculiarity of our method relies on the fact that lumping is applied at the component-level rather than to the CTMC underlying the joint process, thus reducing both the memory requirements and the computational cost of the subsequent solution of the model. Moreover, we investigate the properties of the lumping of reversed automata and we prove that, if these are reversible, a conditional product-form solution of their cooperation with other non-blocking automata can be derived. Although conditional product-forms have been previously investigated by other authors with the aim of approximating non-product-form models, the contribution of this paper consists in giving sufficient conditions for this approach to yield exact results and providing examples to support the modeller’s intuition.

Generalized method of wavelet moments for inertial navigation filter design

The integration of observations issued from a satellite-based system (GNSS) with an inertial navigation system (INS) is usually performed through a Bayesian filter such as the extended Kalman filter (EKF). The task of designing the navigation EKF is strongly related to the inertial sensor error modeling problem. Accelerometers and gyroscopes may be corrupted by random errors of complex spectral structure. Consequently, identifying correct error-state parameters in the INS/GNSS EKF becomes difficult when several stochastic processes are superposed. In such situations, classical approaches like the Allan variance (AV) or power spectral density (PSD) analysis fail due to the difficulty of separating the error processes in the spectral domain. For this purpose, we propose applying a recently developed estimator based on the generalized method of wavelet moments (GMWM), which was proven to be consistent and asymptotically normally distributed. The GMWM estimator matches theoretical and sample-based wavelet variances (WVs), and can be computed using the method of indirect inference. This article mainly focuses on the implementation aspects related to the GMWM, and its integration within a general navigation filter alibration procedure. Regarding this, we apply the GMWM on error signals issued from MEMS-based inertial sensors by building and estimating composite stochastic processes for which classical methods cannot be used. In a first stage, we validate the resulting models using AV and PSD analyses and then, in a second stage, we study the impact of the resulting stochastic models design in terms of positioning accuracy using an emulated scenario with statically observed error signatures. We demonstrate that the GMWM-based calibration framework enables to estimate complex stochastic models in terms of the resulting navigation accuracy that are relevant for the observed structure of errors.

Using ensembles of decision trees to predict transport mode choice decisions: Effects on predictive success and uncertainty estimates

The application of activity-based models of travel demand to planning practice has triggered interest in issues that potentially improve the accuracy and/or usefulness of model forecasts. The limited knowledge of uncertainty propagation in complex stochastic model systems has put uncertainty analysis high on the research agenda to differentiate between simulation error and policy effects. Focusing on transport mode choice, this paper draws attention to the use of model ensembles, which has hardly been explored in travel demand forecasting. Prior studies predicting transport mode choice has typically relied on a single equation, relating observed transport mode choices to a set of personal and contextual variables. The estimated single model is then assumed to apply to all individuals. This paper explores the idea of replacing a single equation/representation with an ensemble of model predictions, using the decision tree formalism. Potentially, ensembles better capture the notion that travellers may use different heuristics in their transport mode decisions. The aim of the study is to investigate whether the use of a model ensemble of different decision heuristics will reduce the error/uncertainty in predicting transport mode decisions. Results of the study, conducted in the Rotterdam region, The Netherlands, suggest that the accuracy of predicting transport mode choice is improved, albeit non-monotonically, with increasing ensemble size. Simultaneously, the uncertainty related to these predictions is decreasing. Finally, it is shown that the importance of the selected explanatory variables co-varies with ensemble size. Estimation results tend to become stable in this study with an ensemble size of approximately 20 decision trees.

Computer simulation and stochastic programming to reduce patient wait times in an outpatient infusion center.

181 Background: As cancer treatment demand outpaces the capacity of outpatient facilities, providers are challenged to provide timely, cost-effective, safe, and patient-centered care. Reduction in patient wait times can help address these challenges through more efficient care delivery. Operations research techniques such as simulation and stochastic programming can inform more efficient patient scheduling. Methods: After 60 hours of observation by two students in one NCI-designated comprehensive cancer center's outpatient infusion center, 9 months of data from the electronic medical record and scheduling systems were used to calculate patient wait times, defined as the difference between scheduled time to begin infusion and actual time and actual administration time. After mapping the patient flow through the center, a computer simulation model was created to examine the effects on patient wait times and total hours of operation under different scheduling paradigms. To create different appointment schedules, both simple heuristic approaches and a more complex stochastic programming model were developed. These methods take into account the variability in infusion times. We generated 35 appointment schedules and evaluated their performance by running 100 replications for each one. Results: The baseline average patient wait time was 130.48 minutes (95% CI 122.38-138.68). Computer simulation results show that schedules generated by the stochastic programming model perform better than baseline and simple heuristics, reaching a 70% reduction in waiting times. Schedules that allocated patients with longer infusion times to earlier appointments resulted in both reduced average patient wait times and total hours of operation. Conclusions: Operations research techniques can improve care delivery in an outpatient infusion center. Based on our results from the computer simulation model, scheduling patients with longer infusion times earlier in the day results in shorter patient waiting times and total length of day of operations. Next steps include guideline development for appointment scheduling staff that attains maximal efficiency.

Conflict Diagnostics in Directed Acyclic Graphs, with Applications in Bayesian Evidence Synthesis

Complex stochastic models represented by directed acyclic graphs (DAGs) are increasingly employed to synthesise multiple, imperfect and disparate sources of evidence, to estimate quantities that are difficult to measure directly. The various data sources are dependent on shared parameters and hence have the potential to conflict with each other, as well as with the model. In a Bayesian framework, the model consists of three components: the prior distribution, the assumed form of the likelihood and structural assumptions. Any of these components may be incompatible with the observed data. The detection and quantification of such conflict and of data sources that are inconsistent with each other is therefore a crucial component of the model criticism process. We first review Bayesian model criticism, with a focus on conflict detection, before describing a general diagnostic for detecting and quantifying conflict between the evidence in different partitions of a DAG. The diagnostic is a p-value based on splitting the information contributing to inference about a "separator" node or group of nodes into two independent groups and testing whether the two groups result in the same inference about the separator node(s). We illustrate the method with three comprehensive examples: an evidence synthesis to estimate HIV prevalence; an evidence synthesis to estimate influenza case-severity; and a hierarchical growth model for rat weights.

Bayesian computation via empirical likelihood.

Approximate Bayesian computation has become an essential tool for the analysis of complex stochastic models when the likelihood function is numerically unavailable. However, the well-established statistical method of empirical likelihood provides another route to such settings that bypasses simulations from the model and the choices of the approximate Bayesian computation parameters (summary statistics, distance, tolerance), while being convergent in the number of observations. Furthermore, bypassing model simulations may lead to significant time savings in complex models, for instance those found in population genetics. The Bayesian computation with empirical likelihood algorithm we develop in this paper also provides an evaluation of its own performance through an associated effective sample size. The method is illustrated using several examples, including estimation of standard distributions, time series, and population genetics models.

Bayesian Approach to MSD-Based Analysis of Particle Motion in Live Cells

Quantitative tracking of particle motion using live-cell imaging is a powerful approach to understanding the mechanism of transport of biological molecules, organelles, and cells. However, inferring complex stochastic motion models from single-particle trajectories in an objective manner is nontrivial due to noise from sampling limitations and biological heterogeneity. Here, we present a systematic Bayesian approach to multiple-hypothesis testing of a general set of competing motion models based on particle mean-square displacements that automatically classifies particle motion, properly accounting for sampling limitations and correlated noise while appropriately penalizing model complexity according to Occam's Razor to avoid over-fitting. We test the procedure rigorously using simulated trajectories for which the underlying physical process is known, demonstrating that it chooses the simplest physical model that explains the observed data. Further, we show that computed model probabilities provide a reliability test for the downstream biological interpretation of associated parameter values. We subsequently illustrate the broad utility of the approach by applying it to disparate biological systems including experimental particle trajectories from chromosomes, kinetochores, and membrane receptors undergoing a variety of complex motions. This automated and objective Bayesian framework easily scales to large numbers of particle trajectories, making it ideal for classifying the complex motion of large numbers of single molecules and cells from high-throughput screens, as well as single-cell-, tissue-, and organism-level studies.

Constructing Summary Statistics for Approximate Bayesian Computation: Semi-Automatic Approximate Bayesian Computation

SummaryMany modern statistical applications involve inference for complex stochastic models, where it is easy to simulate from the models, but impossible to calculate likelihoods. Approximate Bayesian computation (ABC) is a method of inference for such models. It replaces calculation of the likelihood by a step which involves simulating artificial data for different parameter values, and comparing summary statistics of the simulated data with summary statistics of the observed data. Here we show how to construct appropriate summary statistics for ABC in a semi-automatic manner. We aim for summary statistics which will enable inference about certain parameters of interest to be as accurate as possible. Theoretical results show that optimal summary statistics are the posterior means of the parameters. Although these cannot be calculated analytically, we use an extra stage of simulation to estimate how the posterior means vary as a function of the data; and we then use these estimates of our summary statistics within ABC. Empirical results show that our approach is a robust method for choosing summary statistics that can result in substantially more accurate ABC analyses than the ad hoc choices of summary statistics that have been proposed in the literature. We also demonstrate advantages over two alternative methods of simulation-based inference.

In This Issue

In This Issue

Lack of confidence in approximate Bayesian computation model choice

Approximate Bayesian computation (ABC) have become an essential tool for the analysis of complex stochastic models. Grelaud et al. [(2009) Bayesian Anal 3:427-442] advocated the use of ABC for model choice in the specific case of Gibbs random fields, relying on an intermodel sufficiency property to show that the approximation was legitimate. We implemented ABC model choice in a wide range of phylogenetic models in the Do It Yourself-ABC (DIY-ABC) software [Cornuet et al. (2008) Bioinformatics 24:2713-2719]. We now present arguments as to why the theoretical arguments for ABC model choice are missing, because the algorithm involves an unknown loss of information induced by the use of insufficient summary statistics. The approximation error of the posterior probabilities of the models under comparison may thus be unrelated with the computational effort spent in running an ABC algorithm. We then conclude that additional empirical verifications of the performances of the ABC procedure as those available in DIY-ABC are necessary to conduct model choice.

Assessing Wastewater Micropollutant Loads with Approximate Bayesian Computations

Wastewater production, like many other engineered and environmental processes, is inherent stochastic in nature and requires the use of complex stochastic models, for example, to predict realistic patterns of down-the-drain chemicals or pharmaceuticals and personal care products. Up until now, a formal method of statistical inference has been lacking for many of those models, where explicit likelihood functions were intractable. In this Article, we investigate Approximate Bayesian Computation (ABC) methods to infer important parameters of stochastic environmental models. ABC methods have been recently suggested to perform model-based inference in a Bayesian setting when model likelihoods are analytically or computationally intractable and have not been applied to environmental systems analysis or water quality modeling before. In a case study, we investigate the performance of three different algorithms to infer the number of wastewater pulses contained in three high-resolution data series of benzotriazole and total nitrogen loads in sewers. We find that all algorithms perform well and that the uncertainty in the inferred number of corresponding wastewater pulses varies between 6% and 28%. In our case, the results are more sensitive to substance characteristics than to catchment properties. Although the application of ABC methods requires careful tuning and attention to detail, they have a great general potential to update stochastic model parameters with monitoring data and improve their predictive capabilities.

Sequential Monte Carlo Methods for Option Pricing

In this article, we provide a review and development of sequential Monte Carlo (SMC) methods for option pricing. SMC are a class of Monte Carlo-based algorithms, that are designed to approximate expectations w.r.t a sequence of related probability measures. These approaches have been used successfully for a wide class of applications in engineering, statistics, physics, and operations research. SMC methods are highly suited to many option pricing problems and sensitivity/Greek calculations due to the nature of the sequential simulation. However, it is seldom the case that such ideas are explicitly used in the option pricing literature. This article provides an up-to-date review of SMC methods, which are appropriate for option pricing. In addition, it is illustrated how a number of existing approaches for option pricing can be enhanced via SMC. Specifically, when pricing the arithmetic Asian option w.r.t a complex stochastic volatility model, it is shown that SMC methods provide additional strategies to improve estimation.

MCDB-R

Enterprises often need to assess and manage the risk arising from uncertainty in their data. Such uncertainty is typically modeled as a probability distribution over the uncertain data values, specified by means of a complex (often predictive) stochastic model. The probability distribution over data values leads to a probability distribution over database query results, and risk assessment amounts to exploration of the upper or lower tail of a query-result distribution. In this paper, we extend the Monte Carlo Database System to efficiently obtain a set of samples from the tail of a query-result distribution by adapting recent "Gibbs cloning" ideas from the simulation literature to a database setting.

A Bayesian Approach to the Evolution of Metabolic Networks on a Phylogeny

The availability of genomes of many closely related bacteria with diverse metabolic capabilities offers the possibility of tracing metabolic evolution on a phylogeny relating the genomes to understand the evolutionary processes and constraints that affect the evolution of metabolic networks. Using simple (independent loss/gain of reactions) or complex (incorporating dependencies among reactions) stochastic models of metabolic evolution, it is possible to study how metabolic networks evolve over time. Here, we describe a model that takes the reaction neighborhood into account when modeling metabolic evolution. The model also allows estimation of the strength of the neighborhood effect during the course of evolution. We present Gibbs samplers for sampling networks at the internal node of a phylogeny and for estimating the parameters of evolution over a phylogeny without exploring the whole search space by iteratively sampling from the conditional distributions of the internal networks and parameters. The samplers are used to estimate the parameters of evolution of metabolic networks of bacteria in the genus Pseudomonas and to infer the metabolic networks of the ancestral pseudomonads. The results suggest that pathway maps that are conserved across the Pseudomonas phylogeny have a stronger neighborhood structure than those which have a variable distribution of reactions across the phylogeny, and that some Pseudomonas lineages are going through genome reduction resulting in the loss of a number of reactions from their metabolic networks.

A multi-objective multi-period stochastic programming model for public debt management

While raising debt on behalf of the government, public debt managers need to consider several possibly conflicting objectives and have to find an appropriate combination for government debt taking into account the uncertainty with regard to the future state of the economy. In this paper, we explicitly consider the underlying uncertainties with a complex multi-period stochastic programming model that captures the trade-offs between the objectives. The model is designed to aid the decision makers in formulating the debt issuance strategy. We apply an interactive procedure that guides the issuer to identify good strategies and demonstrate this approach for the public debt management problem of Turkey.

Combining analysis tools and mathematical modeling to enhance and harmonize food safety and food defense regulatory requirements

The past decade has seen rapid developments in risk assessment techniques and tools related to food safety and food defense. These advances are beginning to have a significant impact on the decision making processes used by risk managers in both government and industry. Furthermore, the highly complex issues currently facing the management of food safety risks on an international basis are likely to require even more sophisticated assessments. One of the key challenges that must be met to advance the adoption of risk analysis principles is learning how risk managers can best utilize the results of risk assessments as they must take the results of complex stochastic models and adapt the findings to the binary decision criteria that are the basis for legal decisions. This will require a better articulation of key principles of food safety risk management, such as the recently adopted Codex Alimentarius “Principles and Guidelines for the Conduct of Microbial Risk Management.”

Theory, methods and tools for determining environmental flows for riparian vegetation: riparian vegetation‐flow response guilds

Summary1. Riparian vegetation composition, structure and abundance are governed to a large degree by river flow regime and flow‐mediated fluvial processes. Streamflow regime exerts selective pressures on riparian vegetation, resulting in adaptations (trait syndromes) to specific flow attributes. Widespread modification of flow regimes by humans has resulted in extensive alteration of riparian vegetation communities. Some of the negative effects of altered flow regimes on vegetation may be reversed by restoring components of the natural flow regime.2. Models have been developed that quantitatively relate components of the flow regime to attributes of riparian vegetation at the individual, population and community levels. Predictive models range from simple statistical relationships, to more complex stochastic matrix population models and dynamic simulation models. Of the dozens of predictive models reviewed here, most treat one or a few species, have many simplifying assumptions such as stable channel form, and do not specify the time‐scale of response. In many cases, these models are very effective in developing alternative streamflow management plans for specific river reaches or segments but are not directly transferable to other rivers or other regions.3. A primary goal in riparian ecology is to develop general frameworks for prediction of vegetation response to changing environmental conditions. The development of riparian vegetation‐flow response guilds offers a framework for transferring information from rivers where flow standards have been developed to maintain desirable vegetation attributes, to rivers with little or no existing information.4. We propose to organise riparian plants into non‐phylogenetic groupings of species with shared traits that are related to components of hydrologic regime: life history, reproductive strategy, morphology, adaptations to fluvial disturbance and adaptations to water availability. Plants from any river or region may be grouped into these guilds and related to hydrologic attributes of a specific class of river using probabilistic response curves.5. Probabilistic models based on riparian response guilds enable prediction of the likelihood of change in each of the response guilds given projected changes in flow, and facilitate examination of trade‐offs and risks associated with various flow management strategies. Riparian response guilds can be decomposed to the species level for individual projects or used to develop flow management guidelines for regional water management plans.

Model criticism based on likelihood-free inference, with an application to protein network evolution

Mathematical models are an important tool to explain and comprehend complex phenomena, and unparalleled computational advances enable us to easily explore them without any or little understanding of their global properties. In fact, the likelihood of the data under complex stochastic models is often analytically or numerically intractable in many areas of sciences. This makes it even more important to simultaneously investigate the adequacy of these models-in absolute terms, against the data, rather than relative to the performance of other models-but no such procedure has been formally discussed when the likelihood is intractable. We provide a statistical interpretation to current developments in likelihood-free Bayesian inference that explicitly accounts for discrepancies between the model and the data, termed Approximate Bayesian Computation under model uncertainty (ABCmicro). We augment the likelihood of the data with unknown error terms that correspond to freely chosen checking functions, and provide Monte Carlo strategies for sampling from the associated joint posterior distribution without the need of evaluating the likelihood. We discuss the benefit of incorporating model diagnostics within an ABC framework, and demonstrate how this method diagnoses model mismatch and guides model refinement by contrasting three qualitative models of protein network evolution to the protein interaction datasets of Helicobacter pylori and Treponema pallidum. Our results make a number of model deficiencies explicit, and suggest that the T. pallidum network topology is inconsistent with evolution dominated by link turnover or lateral gene transfer alone.

Automated Trace Analysis of Discrete-Event System Models

In this paper, we describe a novel technique that helps a modeler gain insight into the dynamic behavior of a complex stochastic discrete event simulation model based on trace analysis. We propose algorithms to distinguish progressive from repetitive behavior in a trace and to extract a minimal progressive fragment of a trace. The implied combinatorial optimization problem for trace reduction is solved in linear time with dynamic programming. We present and compare several approximate and one exact solution method. Information on the reduction operation as well as the reduced trace itself helps a modeler to recognize the presence of certain errors and to identify their cause. We track down a subtle modeling error in a dependability model of a multi-class server system to illustrate the effectiveness of our approach in revealing the cause of an observed effect. The proposed technique has been implemented and integrated in Traviando, a trace analyzer to debug stochastic simulation models.

Use of heterogeneous operation-specific contact parameters changes predictions for foot-and-mouth disease outbreaks in complex simulation models

The role of contact parameters in a complex spatial simulation model of foot-and-mouth disease spread was determined by comparing predictions of number of infected premises, epidemic duration, and relative infection risk for different production sectors between a model that included the Full, heterogeneous (differing by production type) type-specific information about animal, vehicle and personnel movement between premises, and models that used partial and homogeneous (similar across production types) weighted-mean or proxy parameter sets for contacts between premises of all types. The model was run using a dataset of known premises locations in a three-county area in the Central Valley of California and categorized into 13 premises types and six production sectors. Results from models run with homogeneous contact parameters were always different from those obtained from the Full model, demonstrating that model predictions are affected by heterogeneity in contact parameters. Models simplified by using weighted-mean parameters predicted fewer infected premises. Models that were simplified by using medium dairy farm or large swine operation proxy parameters predicted longer epidemics with more infected premises, while those using small beef operation proxy parameters predicted shorter epidemics with fewer infected premises. Simplified-parameter models underestimated the impact on the economically important dairy sector, while overestimating the impact on beef and backyard operations. Results establish a need for heterogeneous, operation-specific contact parameters in complex stochastic simulation models that must be weighed against the cost of obtaining and coding premises type-specific contact information.