**Tutorial Workshop on Parameter Estimation for Biological Models**

**Talks:** SAS Hall 1102

**Coffee Breaks:** SAS Hall 1st floor lobby (Wednesday)

**Overall plan of themes:**

Day 1 (Wednesday, July 25th): **Least squares and Identifiability**

Day 2 (Thursday, July 26th): **Sensitivity Analysis and Uncertainty Propagation**

Day 3 (Friday, July 27th): **Bayesian Approaches to Parameter Estimation and Model Selection**

Day 4 (Saturday, July 28th; morning only): **Kalman Filtering**

**More Detailed Schedule:**

Wednesday July 28th: (Theme: Identifiability and Least Squares)

8:00 – 8:30 Registration in SAS Hall first floor lobby

8:30 – 8:45 Welcoming remarks (Lloyd and Olufsen)

8:45 – 10:00 Tutorial Lecture: Least squares (Lloyd)

10:00 – 10.20 Coffee Break

10:20 – 12.00 Computer Practical: Least squares (Lloyd) code and slides

12.00 – 1.00 Lunch Break

1:00 – 1:45 Research Talk, H.T. Banks (NC State University): Population Models-The Prohorov Metric Framework and Aggregate Data Inverse Problems slides

1:45 – 3:00 Tutorial Lecture: Identifiability (Eisenberg)

3:00 – 3:30 Coffee Break (set up posters)

3:30 – 5:15 Computer Practical: Identifiability (Eisenberg) Materials

5:30 – 7:30 Poster Session (light refreshments served)

Thursday July 28th: (Theme: Sensitivity Analysis)

8:30 – 10:30 Tutorial Lecture: Tutorial Lecture: Sensitivity Analysis (Alexandrian/ Gremaud) slides

10:30 – 10:50 Coffee Break

10:50 – 12:00 Lecture: Uncertainty Propagation (Smith) Materials

12:00 – 1:15 Lunch Break

1:15 – 2:00 Research Talk (Olufsen)

2:00 – 4:00 Computer practical (Gremaud/Alexandrian/Smith) Slides Codes (.zip file)

4:00 – 4:30 Coffee Break

4:30 – 5:30 Research Talks (Students/Postdocs)

Friday July 29th: (Themes: Bayesian Approximation)

8:30 – 8:45 Announcements (Lloyd and Olufsen)

8:45 – 10:00 Tutorial Lecture: Introduction to Bayesian Analysis (Reich) Slides, Code and Assignment Worksheet

10:00 – 10.30 Coffee Break

10:30 – 12:15 Computer Practical: Introduction to Bayesian Analysis (Reich)

12:15 – 1:15 Lunch Break

1:15 – 3:00 Tutorial Lecture: Statistical emulation and Bayesian optimisation (Husmeier)

Talk slides: Part 0: Introduction: Modern statistical inference in complex biological systems, Part 1: Statistical Emulation, Part 2: A brief introduction to non parametric modeling with Gaussian Processes, Part 3: Bayesian Optimization, Part 4: Uncertainty Quantification

3:00 – 3:30 Coffee Break

3:30 – 5:30 Computer practical: Bayesian Statistics and Gaussian processes (Husmeier + Paun) Slides and material for practical

Saturday July 30th: (Theme: Uncertainty Quantification)

8:30 – 8:45 Announcements (Lloyd and Olufsen)

8:45 – 10:00 Tutorial Lecture: Nonlinear Filtering (Berry) Slides

10:00 – 10.30 Coffee Break

10:30 – 12:30 Computer practical: Filtering (Berry) Practical slides

12:30 – 12:45 Closing Remarks (Lloyd and Olufsen)

**Abstracts**

**Title: Introduction to Least Squares Parameter estimation
Speaker: Alun Lloyd, NC State University**

Abstract: Tutorial: This lecture outlines 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.

Practical: In this computer tutorial 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.

**Title: Population Models – The Prohorov Metric Framework and Aggregate Data Inverse Problems
Speaker: H.T. Banks, NC State University**

Abstract: We consider nonparametric estimation of probability measures for parameters in problems where only aggregate (population level) data are available. We summarize an existing computational method for the estimation problem which has been developed over the past several decades. Theoretical results are presented which establish the existence and consistency of very general (ordinary, generalized and other) least squares estimates and estimators for the measure estimation problem. The material will be presented so as to be accessible to individuals with elementary background at the level of introductory differential equations and probability theory.

**Title: Introduction to Structural and Practical Identifiability
Speaker: Marisa Eisenberg, University of Michigan**

Abstract: Tutorial: In this lecture, 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.

Practical: The computer tutorial 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.

**Title: Global sensitivity analysis for biological models
Speakers: Alen Alexanderian and Pierre Gremaud, NC State University**

Abstract: Models of biological systems often include a number of uncertain input parameters that are typically modelled as random variables. Not all uncertain inputs contribute equally to the uncertainty in predictions. Global sensitivity analysis (GSA) provides a framework to assess the relative importance of various input parameters to variability in model predictions. This guides risk assessment, model calibration, and input parameter dimension reduction. In our presentation, we will discuss variance based and derivative based GSA. Methods for computing the global sensitivity measures, as well as the links between variance-based and derivative-based GSA will be discussed. Model application problems from biochemistry will be used to elucidate the methods. A hands-on computer module will provide the participants an opportunity to experiment with GSA techniques.

**Title: Uncertainty Propagation for Biological Models
Speaker: Ralph C. Smith, NC State University**

Abstract: 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 initially demonstrate uncertainty propagation for linearly parameterized models, where analytic expressions can be derived. For computationally efficient models, we will subsequently demonstrate the propagation of previously inferred distributions using sampling techniques, including the MATLAB Delayed Rejection Adaptive Metropolis (DRAM) code. Finally, we will briefly discuss the construction of surrogate models for complex simulation codes. Open questions and future research directions will be noted throughout the presentation.

**How to best incorporate data in the study of cardiovascular dynamics – the inverse problem
Speaker: M.S. Olufsen, NC State University**

Abstract: Modeling has long been used to study cardiovascular dynamics and results are often compared to data. Her we discuss two model types fluid models set up to understand wave propagation in arterial networks and 0D compartmental models used to understand how the system is controlled. Both the models and the data are inherently uncertain. Fluid dynamics models are derived from well-known theory, they are run in geometry extracted from images, and needs input and output boundary conditions all associated with significant level of uncertainty. Lumped compartment models are even more uncertain as they are build based on approximation of quantities that cannot easily be measured, and typically contain many correlated parameters that cannot easily be specified a priori. This presentation will discuss estimation of nominal parameters, sensitivity analysis, subset selection, and parameter estimation using both local and global approaches.

**Title: Introduction to Bayesian Analysis
Speaker: Brian Reich, NC State University**

Abstract:

Tutorial:

1) Introduction to Bayes, including definition of prior, posterior, etc.

2) Example cases including beta-binomial and simple linear regression

3) Gibbs sample

Practical:

1) Beta-binomial example in R

2) simple regression in Gibbs in R or OpenBUGS

**Title: Statistical emulation and Bayesian optimisation
Speaker: Dirk Husmeier, University of Glasgow, UK**

Abstract:

Part 1: Statistical emulation

Biomechanical modelling of the heart can provide insight into heart (mal)function and help to improve prognostication of heart disease. A grand challenge of applying such models to patient-specific studies, however, is to estimate the material parameters inversely based on in-vivo clinical measurements. This is because the forward modelling is usually computationally expensive, and limited in vivo data make it difficult to find the global minimum of the loss or objective function quantifying the goodness of fit. In the first part of the lecture, I will discuss the concept of statistical emulation for statistical parameter estimation. Emulation methods avoid simulating from the biomechanical model by replacing it with a surrogate model inferred from previous simulations generated before the arrival of a patient at the clinic. These forward simulations can be run by massive parallelisation before a patient arrives. I will compare and contrast two emulation strategies: (a) emulation of the outputs of the computational model and (b) emulation of the loss function quantifying the discrepancy between the observed data and the computational model outputs. The two schemes are assessed in a comparative evaluation study, using both simulated and real MRI data. The lecture will conclude with an outlook on future challenges and developments and the next steps towards personalised medicine.

Part 2: Bayesian optimization

Parameter estimation in biomechanical models is typically computationally expensive, as each parameter adaptation requires a numerical integration of a system of coupled partial differential equations. In the second part of the lecture, I will give an overview of Bayesian optimization for solving such computationally challenging optimization tasks. Different versions of the algorithm vary in the choice of the acquisition function, which recommends the point to query the objective function at next within an iterative optimization procedure. In the lecture, I will provide an overview of the current state of the art, and discuss the pros and cons of three different families of acquisition functions: optimistic measures, improvement-based measures, and information-theoretical measures. The algorithms are empirically assessed based on a broad portfolio of challenging benchmark optimization problems.

**Title: Introduction to Nonlinear Filtering
Speaker: Tyrus Berry, George Mason University**

Abstract: Filtering is the process of identifying a hidden state from noisy observations that may be nonlinearly related to the state and may only contain partial information. Filtering naturally combines ideas of least squares, identifiability, uncertainty quantification, and (often) Bayesian analysis. After introducing the general filter problem, we will briefly describing the three principal approaches, Kalman filtering, Particle filtering and Variational filtering. We will then focus on Kalman filtering, deriving the Kalman equations as a Bayesian combination of Gaussian random variables. Finally, we will show how to generalize the Kalman filter to nonlinear systems using the ensemble transform. In our Computational session we will implement the ensemble Kalman filter (EnKF) and apply it to the chaotic Lorenz-63 dynamical system.