Incorporating parameter uncertainty into quantitative microbial risk assessment (QMRA)

Modern statistical models and computational methods can now incorporate uncertainty of the parameters used in Quantitative Microbial Risk Assessments (QMRA). Many QMRAs use Monte Carlo methods, but work from fixed estimates for means, variances and other parameters. We illustrate the ease of estimating all parameters contemporaneously with the risk assessment, incorporating all the parameter uncertainty arising from the experiments from which these parameters are estimated. A Bayesian approach is adopted, using Markov Chain Monte Carlo Gibbs sampling (MCMC) via the freely available software, WinBUGS. The method and its ease of implementation are illustrated by a case study that involves incorporating three disparate datasets into an MCMC framework. The probabilities of infection when the uncertainty associated with parameter estimation is incorporated into a QMRA are shown to be considerably more variable over various dose ranges than the analogous probabilities obtained when constants from the literature are simply ‘plugged’ in as is done in most QMRAs. Neglecting these sources of uncertainty may lead to erroneous decisions for public health and risk management.

Effect of soil parameter uncertainty on seismic response of buried segmented pipeline

Pipelines are important lifeline facilities spread over a large area and they generally encounter a range of seismic hazards and different soil conditions. The seismic response of a buried segmented pipe depends on various parameters such as the type of buried pipe material and joints, end restraint conditions, soil characteristics, burial depths, and earthquake ground motion, etc. This study highlights the effect of the variation of geotechnical properties of the surrounding soil on seismic response of a buried pipeline. The variations of the properties of the surrounding soil along the pipe are described by sampling them from predefined probability distribution. The soil-pipe interaction model is developed in OpenSEES. Nonlinear earthquake time-history analysis is performed to study the effect of soil parameters variability on the response of pipeline. Based on the results, it is found that uncertainty in soil parameters may result in significant response variability of the pipeline.

Analysis of parameter uncertainty and sensitivity in PCPF-1 modeling for predicting concentrations of rice herbicides

This paper demonstrates the procedures for probabilistic assessment of a pesticide fate and transport model, PCPF-1, to elucidate the modeling uncertainty using the Monte Carlo technique. Sensitivity analyses are performed to investigate the influence of herbicide characteristics and related soil properties on model outputs using four popular rice herbicides: mefenacet, pretilachlor, bensulfuron-methyl and imazosulfuron. Uncertainty quantification showed that the simulated concentrations in paddy water varied more than those of paddy soil. This tendency decreased as the simulation proceeded to a later period but remained important for herbicides having either high solubility or a high 1st-order dissolution rate. The sensitivity analysis indicated that PCPF-1 parameters requiring careful determination are primarily those involve with herbicide adsorption (the organic carbon content, the bulk density and the volumetric saturated water content), secondary parameters related with herbicide mass distribution between paddy water and soil (1st-order desorption and dissolution rates) and lastly, those involving herbicide degradations. © Pesticide Science Society of Japan.

Sequential Monte Carlo for Bayesian sequentially designed experiments for discrete data

In this paper we present a sequential Monte Carlo algorithm for Bayesian sequential experimental design applied to generalised non-linear models for discrete data. The approach is computationally convenient in that the information of newly observed data can be incorporated through a simple re-weighting step. We also consider a flexible parametric model for the stimulus-response relationship together with a newly developed hybrid design utility that can produce more robust estimates of the target stimulus in the presence of substantial model and parameter uncertainty. The algorithm is applied to hypothetical clinical trial or bioassay scenarios. In the discussion, potential generalisations of the algorithm are suggested to possibly extend its applicability to a wide variety of scenarios

Design of fixed order robust power oscillation damper for TCSC

This paper illustrates robust fixed order power oscillation damper design for mitigating power systems oscillations. From implementation and tuning point of view, such low and fixed structure is common practice for most practical applications, including power systems. However, conventional techniques of optimal and robust control theory cannot handle the constraint of fixed-order as it is, in general, impossible to ensure a target closed-loop transfer function by a controller of any given order. This paper deals with the problem of synthesizing or designing a feedback controller of dynamic order for a linear time-invariant plant for a fixed plant, as well as for an uncertain family of plants containing parameter uncertainty, so that stability, robust stability and robust performance are attained. The desired closed-loop specifications considered here are given in terms of a target performance vector representing a desired closed-loop design. The performance of the designed controller is validated through non-linear simulations for a range of contingencies.

Visual predictive control of spiral motion

This paper deals with constrained image-based visual servoing of circular and conical spiral motion about an unknown object approximating a single image point feature. Effective visual control of such trajectories has many applications for small unmanned aerial vehicles, including surveillance and inspection, forced landing (homing), and collision avoidance. A spherical camera model is used to derive a novel visual-predictive controller (VPC) using stability-based design methods for general nonlinear model-predictive control. In particular, a quasi-infinite horizon visual-predictive control scheme is derived. A terminal region, which is used as a constraint in the controller structure, can be used to guide appropriate reference image features for spiral tracking with respect to nominal stability and feasibility. Robustness properties are also discussed with respect to parameter uncertainty and additive noise. A comparison with competing visual-predictive control schemes is made, and some experimental results using a small quad rotor platform are given.

Quantifying uncertainty in parameter estimates for stochastic models of collective cell spreading using approximate Bayesian computation

Wound healing and tumour growth involve collective cell spreading, which is driven by individual motility and proliferation events within a population of cells. Mathematical models are often used to interpret experimental data and to estimate the parameters so that predictions can be made. Existing methods for parameter estimation typically assume that these parameters are constants and often ignore any uncertainty in the estimated values. We use approximate Bayesian computation (ABC) to estimate the cell diffusivity, D, and the cell proliferation rate, λ, from a discrete model of collective cell spreading, and we quantify the uncertainty associated with these estimates using Bayesian inference. We use a detailed experimental data set describing the collective cell spreading of 3T3 fibroblast cells. The ABC analysis is conducted for different combinations of initial cell densities and experimental times in two separate scenarios: (i) where collective cell spreading is driven by cell motility alone, and (ii) where collective cell spreading is driven by combined cell motility and cell proliferation. We find that D can be estimated precisely, with a small coefficient of variation (CV) of 2–6%. Our results indicate that D appears to depend on the experimental time, which is a feature that has been previously overlooked. Assuming that the values of D are the same in both experimental scenarios, we use the information about D from the first experimental scenario to obtain reasonably precise estimates of λ, with a CV between 4 and 12%. Our estimates of D and λ are consistent with previously reported values; however, our method is based on a straightforward measurement of the position of the leading edge whereas previous approaches have involved expensive cell counting techniques. Additional insights gained using a fully Bayesian approach justify the computational cost, especially since it allows us to accommodate information from different experiments in a principled way.

Bounded parameter Markov Decision Processes with average reward criterion

Bounded parameter Markov Decision Processes (BMDPs) address the issue of dealing with uncertainty in the parameters of a Markov Decision Process (MDP). Unlike the case of an MDP, the notion of an optimal policy for a BMDP is not entirely straightforward. We consider two notions of optimality based on optimistic and pessimistic criteria. These have been analyzed for discounted BMDPs. Here we provide results for average reward BMDPs. We establish a fundamental relationship between the discounted and the average reward problems, prove the existence of Blackwell optimal policies and, for both notions of optimality, derive algorithms that converge to the optimal value function.

Using Bayesian methods for the estimation of uncertainty in complex statistical models

The research objectives of this thesis were to contribute to Bayesian statistical methodology by contributing to risk assessment statistical methodology, and to spatial and spatio-temporal methodology, by modelling error structures using complex hierarchical models. Specifically, I hoped to consider two applied areas, and use these applications as a springboard for developing new statistical methods as well as undertaking analyses which might give answers to particular applied questions. Thus, this thesis considers a series of models, firstly in the context of risk assessments for recycled water, and secondly in the context of water usage by crops. The research objective was to model error structures using hierarchical models in two problems, namely risk assessment analyses for wastewater, and secondly, in a four dimensional dataset, assessing differences between cropping systems over time and over three spatial dimensions. The aim was to use the simplicity and insight afforded by Bayesian networks to develop appropriate models for risk scenarios, and again to use Bayesian hierarchical models to explore the necessarily complex modelling of four dimensional agricultural data. The specific objectives of the research were to develop a method for the calculation of credible intervals for the point estimates of Bayesian networks; to develop a model structure to incorporate all the experimental uncertainty associated with various constants thereby allowing the calculation of more credible credible intervals for a risk assessment; to model a single day’s data from the agricultural dataset which satisfactorily captured the complexities of the data; to build a model for several days’ data, in order to consider how the full data might be modelled; and finally to build a model for the full four dimensional dataset and to consider the timevarying nature of the contrast of interest, having satisfactorily accounted for possible spatial and temporal autocorrelations. This work forms five papers, two of which have been published, with two submitted, and the final paper still in draft. The first two objectives were met by recasting the risk assessments as directed, acyclic graphs (DAGs). In the first case, we elicited uncertainty for the conditional probabilities needed by the Bayesian net, incorporated these into a corresponding DAG, and used Markov chain Monte Carlo (MCMC) to find credible intervals, for all the scenarios and outcomes of interest. In the second case, we incorporated the experimental data underlying the risk assessment constants into the DAG, and also treated some of that data as needing to be modelled as an ‘errors-invariables’ problem [Fuller, 1987]. This illustrated a simple method for the incorporation of experimental error into risk assessments. In considering one day of the three-dimensional agricultural data, it became clear that geostatistical models or conditional autoregressive (CAR) models over the three dimensions were not the best way to approach the data. Instead CAR models are used with neighbours only in the same depth layer. This gave flexibility to the model, allowing both the spatially structured and non-structured variances to differ at all depths. We call this model the CAR layered model. Given the experimental design, the fixed part of the model could have been modelled as a set of means by treatment and by depth, but doing so allows little insight into how the treatment effects vary with depth. Hence, a number of essentially non-parametric approaches were taken to see the effects of depth on treatment, with the model of choice incorporating an errors-in-variables approach for depth in addition to a non-parametric smooth. The statistical contribution here was the introduction of the CAR layered model, the applied contribution the analysis of moisture over depth and estimation of the contrast of interest together with its credible intervals. These models were fitted using WinBUGS [Lunn et al., 2000]. The work in the fifth paper deals with the fact that with large datasets, the use of WinBUGS becomes more problematic because of its highly correlated term by term updating. In this work, we introduce a Gibbs sampler with block updating for the CAR layered model. The Gibbs sampler was implemented by Chris Strickland using pyMCMC [Strickland, 2010]. This framework is then used to consider five days data, and we show that moisture in the soil for all the various treatments reaches levels particular to each treatment at a depth of 200 cm and thereafter stays constant, albeit with increasing variances with depth. In an analysis across three spatial dimensions and across time, there are many interactions of time and the spatial dimensions to be considered. Hence, we chose to use a daily model and to repeat the analysis at all time points, effectively creating an interaction model of time by the daily model. Such an approach allows great flexibility. However, this approach does not allow insight into the way in which the parameter of interest varies over time. Hence, a two-stage approach was also used, with estimates from the first-stage being analysed as a set of time series. We see this spatio-temporal interaction model as being a useful approach to data measured across three spatial dimensions and time, since it does not assume additivity of the random spatial or temporal effects.

A sequential Monte Carlo algorithm to incorporate model uncertainty in Bayesian sequential design

Here we present a sequential Monte Carlo (SMC) algorithm that can be used for any one-at-a-time Bayesian sequential design problem in the presence of model uncertainty where discrete data are encountered. Our focus is on adaptive design for model discrimination but the methodology is applicable if one has a different design objective such as parameter estimation or prediction. An SMC algorithm is run in parallel for each model and the algorithm relies on a convenient estimator of the evidence of each model which is essentially a function of importance sampling weights. Other methods for this task such as quadrature, often used in design, suffer from the curse of dimensionality. Approximating posterior model probabilities in this way allows us to use model discrimination utility functions derived from information theory that were previously difficult to compute except for conjugate models. A major benefit of the algorithm is that it requires very little problem specific tuning. We demonstrate the methodology on three applications, including discriminating between models for decline in motor neuron numbers in patients suffering from neurological diseases such as Motor Neuron disease.

Inherent errors in pollutant build-up estimation in considering urban land use as a lumped parameter

Stormwater quality modelling results is subject to uncertainty. The variability of input parameters is an important source of overall model error. An in-depth understanding of the variability associated with input parameters can provide knowledge on the uncertainty associated with these parameters and consequently assist in uncertainty analysis of stormwater quality models and the decision making based on modelling outcomes. This paper discusses the outcomes of a research study undertaken to analyse the variability related to pollutant build-up parameters in stormwater quality modelling. The study was based on the analysis of pollutant build-up samples collected from 12 road surfaces in residential, commercial and industrial land uses. It was found that build-up characteristics vary appreciably even within the same land use. Therefore, using land use as a lumped parameter would contribute significant uncertainties in stormwater quality modelling. Additionally, it was also found that the variability in pollutant build-up can also be significant depending on the pollutant type. This underlines the importance of taking into account specific land use characteristics and targeted pollutant species when undertaking uncertainty analysis of stormwater quality models or in interpreting the modelling outcomes.

Developments of the total entropy utility function for the dual purpose of model discrimination and parameter estimation in Bayesian design

The total entropy utility function is considered for the dual purpose of Bayesian design for model discrimination and parameter estimation. A sequential design setting is proposed where it is shown how to efficiently estimate the total entropy utility for a wide variety of data types. Utility estimation relies on forming particle approximations to a number of intractable integrals which is afforded by the use of the sequential Monte Carlo algorithm for Bayesian inference. A number of motivating examples are considered for demonstrating the performance of total entropy in comparison to utilities for model discrimination and parameter estimation. The results suggest that the total entropy utility selects designs which are efficient under both experimental goals with little compromise in achieving either goal. As such, the total entropy utility is advocated as a general utility for Bayesian design in the presence of model uncertainty.

Populations of models, experimental designs and coverage of parameter space by Latin Hypercube and Orthogonal Sampling

In this paper we have used simulations to make a conjecture about the coverage of a t-dimensional subspace of a d-dimensional parameter space of size n when performing k trials of Latin Hypercube sampling. This takes the form P(k,n,d,t) = 1 - e^(-k/n^(t-1)). We suggest that this coverage formula is independent of d and this allows us to make connections between building Populations of Models and Experimental Designs. We also show that Orthogonal sampling is superior to Latin Hypercube sampling in terms of allowing a more uniform coverage of the t-dimensional subspace at the sub-block size level. These ideas have particular relevance when attempting to perform uncertainty quantification and sensitivity analyses.