**Workshop: Titles and Abstracts for Talks and Practicals
**

**Andrea Arnold (NC State University) and Franz Hamilton (NC State University)**

**Title: Metropolis-Hastings-based MCMC**

**Abstract:** This practical will discuss various Metropolis-Hastings-based Markov chain Monte Carlo (MCMC) sampling algorithms for use in Bayesian parameter estimation. In particular, an MCMC toolbox for MATLAB (http://helios.fmi.fi/~lainema/mcmc/) will be utilized in exploring parameter distributions for several examples, including a multi-dimensional banana-shaped function and four-parameter SIR model.

**Andrea Arnold (NC State University) and Franz Hamilton (NC State University)**

**Title: Introduction to Bayesian Filtering**

**Abstract:** In this tutorial we will introduce Bayesian filtering methods for sequential state and parameter estimation. We will derive the classic Kalman filter equations from a Bayesian perspective and provide an intuitive understanding of the filtering methodology. We will briefly discuss the extension of these ideas to the case of nonlinear system dynamics. The application of these methods will be illustrated through simple numerical examples.

**Andrea Arnold (NC State University) and Franz Hamilton (NC State University)**

**Title: Kalman Filtering in a Mass-Spring System**

**Abstract:** In this practical we will implement a Kalman filter for state estimation in a damped mass-spring oscillator. In implementing the Kalman filter, we will investigate the influence of several of the filter parameters on achieving optimal performance.

**H. T. Banks (NC State University)**

**Title: A mathematical methodology for determining information content in demographic data for ecotoxicology studies**

**Abstract:** Although pesticides protect crops by killing or repelling pest species, they can also present a risk to non-target species and ecosystems. Thus, ecological risk assessments (ERAs) are used to evaluate the effects of a chemical on the environment. Traditionally, the experiments for ERAs use endpoints at the individual level, even though the protection goals are often defined at the population level [2]. However, some ecotoxicologist are presenting supporting research for the use of demographic data with the population growth rate as the endpoint of interest. In order to determine the population growth rate, demographic data is collected and life tables are produced. Since the collection of this data is both time consuming and costly, an important question is whether or not partial demographic data can be used instead of full demographic data, while still providing an accurate picture of the impact of a toxicant on a population.

This question was recently investigated by Stark and Banks [3], who collected the full demographic data and then used statistical methods to determine the smallest amount of data necessary to still obtain an accurate population growth rate. However, in this process, the full demographic data was still needed. We recently developed an adaptive feedback methodology, which can be used as the stable population data is being collected [1]. This statistical and mathematical based methodology determines when the dataset (which gets larger as the experiment progresses) contains enough information to determine the stable population growth rate with a given confidence level. The ideas in this study have a wide application to the health and social sciences where experimental data are expensive and difficult to obtain.

We are currently continuing this work by determining the underlying assumptions made in the development of life tables, which may not necessarily apply to all datasets and population studies that currently use life tables. Furthermore, we hope to apply our methodology to populations with non-constant rates; for example, considering populations with cyclic growth patterns. This represents joint efforts with J. E. Banks, R. A. Everett, and J. D. Stark.

References

[1] H. T. Banks, J. E. Banks, R. A. Everett, and J. D. Stark, An adaptive feedback methodology for determining information content in population studies, to appear in Mathematical Biosciences and Engineering, (2016).

[2] N. Hanson and J. D. Stark, Utility of population models to reduce uncertainty and increase value relevance in ecological risk assessments of pesticides: an example based on acute mortality data for Daphnids, Integrated Environmental Assessment and Management, 8 (2011), 262-270.

[3] J. D. Stark and J. E. Banks, Developing demographic toxicity data: optimizing effort for predicting population outcomes, PeerJ, to appear, 2016.

**Daniela Calvetti (Case Western Reserve University)**

**Title: Inverse problems in the Bayesian Framework**

**Abstract:** In this tutorial, I will introduce the idea of modeling all unknown parameters as random variables and visit the inverse problem of estimating parameters from noisy observations according to the philosophy of Bayesian inference. As part of the lecture, we will see how to model different levels of a priori belief about the unknowns, as well as how to condition our estimates on the knowledge of the values of a subset of the unknowns. We will conclude the tutorial with an introduction to the Metropolis-Hastings sampling algorithm.

**Daniela Calvetti (Case Western Reserve University)**

**Title: Compensating for lack of data with expert knowledge: how hierarchical models can be used to solve the very underdetermined MEG inverse problem**

**Abstract:** The inverse problem of MEG aims at estimating electromagnetic cerebral activity from a few measurements of the magnetic fields outside the head. The problem is notoriously ill-conditioned and very underdetermined.

After formulating the problem within the Bayesian framework, a hierarchical conditionally Gaussian prior model is introduced, including a physiologically inspired prior model that takes into account the preferred directions of the source currents. The hyperparameter vector consists of prior variances of the dipole moments, assumed to follow a non-conjugate gamma distribution with variable scaling and shape parameters. A point estimate of both dipole moments and their variances can be computed by a very efficient algorithm shown to be globally convergent. The numerical solution is based on computing an approximation of the dipole moments using a Krylov subspace iterative linear solver equipped with statistically inspired preconditioning and a suitable termination rule. The shape parameters of the model are shown to control the focality, and furthermore, using an empirical Bayes argument, it is shown that the scaling parameters can be naturally adjusted to provide a statistically well justified depth sensitivity scaling. The validity of this interpretation is verified through computed numerical examples.

**Marisa Eisenberg (University of Michigan)**

**Title: Introduction to Structural and Practical Identifiability**

**Abstract:** In this tutorial talk, I will introduce concepts and approaches for examining structural and practical identifiability. We will discuss analytical approaches, numerical methods using the Fisher information matrix and profile likelihood, and identifiable combinations.

**Marisa Eisenberg (University of Michigan) and Chris Durden (NC State University)**

**Title: Identifiability Practical**

**Abstract:** The practical will give participants a chance to try out the concepts discussed in the tutorial, including exercises in structural and practical identifiability. Using a cholera transmission model as an example, students will examine parameter uncertainty, identifiable combinations, and different techniques for exploring parameter identifiability.

**Marisa Eisenberg (University of Michigan)**

**Title: Connecting models with data: identifiability and parameter estimation of multiple transmission pathways**

**Abstract:** In this talk, we will examine how parameter estimation and disease forecasting are affected when examining disease transmission via multiple types or pathways of transmission. Using examples taken from the West Africa Ebola epidemic and cholera outbreaks in Angola, Haiti, and Thailand, we illustrate some of the potential difficulties in estimating the relative contributions of different transmission pathways, and show how alternative data collection may help resolve this unidentifiability. We also illustrate how even in the presence of large uncertainties in the data and model parameters, it may still be possible to successfully forecast the disease dynamics.

**Kevin Flores (NC State University)**

**Title: Introduction to parameter estimation in a least squares framework**

**Abstract:** We outline fundamental mathematical and statistical methodologies for inverse or parameter estimation problems. Topics to be discussed include least-squares formulations (ordinary, weighted and generalized least squares), sensitivity equations, asymptotic theory for confidence intervals and uncertainty, use of residual plots to analyze correctness of statistical model formulation.

**Alun Lloyd (NC State University)**

**Title: Sensitivity Analysis and Least Squares Parameter Estimation for an Epidemic Modelâ€¨**

**Abstract:** In this practical we shall derive and implement the sensitivity equations for a simple epidemic model, the SIR model. We shall fit this model to some data from an influenza outbreak at a boarding school and use asymptotic statistical theory to obtain standard errors for the estimated parameters.

**Mette Olufsen (NC State University)**

**Title: Parameter estimation in cardiovascular dynamics modeling**

**Abstract:** This study develops a lumped-parameter compartmental model to predict cardiovascular regulation in response to blood withdrawal. This phenomenon is important to understand to better design clinical strategies for patients who have experienced severe blood loss. This study uses data from Rats that have been exposed to multiple controlled blood withdrawals. Data include time-series measurements of left ventricular blood pressure and volume. Physiological information is used to determine a set of patient specific model parameters, which are adjusted (using optimization) to match measured quantities. Yet only a subset of parameters is identifiable. This study uses local sensitivity and covariance analysis to obtain a subset of parameters that can be estimated to fit baseline data, and asymptotic methods are subsequently used to construct a 95% confidence and prediction intervals around the model solutions. To capture effects of blood withdrawal parameters controlled by the cardiovascular control system are made time-varying and estimated to predict the time-varying dynamics in response to blood withdrawal.

**Ralph Smith (NC State University)**

**Title: Uncertainty Quantification for Biological Models**

**Abstract:** The quantification of uncertainties inherent to biological models, parameters, and experiments is critical to assess the accuracy of predictions. In this presentation, we will discuss issues that must be addressed when propagating input uncertainties through models to quantify the accuracy of responses. We will briefly discuss the use of Bayesian techniques to infer input distributions to motivate the presentations on Saturday. We will detail the manner in which previously discussed identifiability analysis is critical for uncertainty quantification and illustrate how global sensitivity analysis and active subspaces can additionally be used for parameter subset and subspace selection. We will also discuss the construction of surrogate models for complex simulation codes. Open questions and future research directions will be noted throughout the presentation.

**Ralph Smith (NC State University)**

**Title: Numerical Techniques for Active Subspaces, Surrogate Model Construction and Uncertainty Propagation**

**Abstract:** In this practical, we will demonstrate and experiment with numerical techniques that can be used for efficient active subspace construction, surrogate model construction, and uncertainty propagation. The active subspace techniques rely on QR and SVD algorithms whereas we will illustrate the manner in which polynomial surrogates can be constructed to reduce model complexity. All simulations will be performed with MATLAB.

**Hien Tran (NC State University)**

**Title: Introduction to mixed-effects modeling**

**Abstract:** In this tutorial we will introduce mixed-effects modeling, a powerful method for the analysis of repeated measurements. This modeling framework consists of both fixed-effects for population parameters and random effects from the uncertainty associated with inter- and intra- individual variability. We will discuss the theory behind this method and also consider some examples using SimBiology.

**Hulin Wu (University of Texas Health Science Center at Houston)**

**Title: Mixed-Effects Biological Models: Estimation and Inference**

**Abstract:** A biological system can be described or quantified by a dynamic model such as differential equations or difference equations. The experimental data, in particular time course data, collected from a biological system are often sparse for each individual due to the high cost and availability of individual subjects. It is challenging to use the sparse data to estimate the parameters and perform inferences for complicated dynamic models. A natural idea is to use the mixed-effects modeling approach to borrow information across subjects so that the estimation and inference for the dynamic models can be reliable. In this talk, I will review mixed-effects differential equation models, state-space models and high-dimensional dynamic models. The parameter estimation, variable selection and statistical inferences for the complicated dynamic models will be discussed.