## Student Perspectives: Gaussian Process Emulation

A post by Conor Crilly, PhD student on the Compass programme.

# Introduction

This project investigates uncertainty quantification methods for expensive computer experiments. It is supervised by Oliver Johnson of the University of Bristol, and is partially funded by AWE.

## Outline

Physical systems and experiments are commonly represented, albeit approximately, using mathematical models implemented via computer code. This code, referred to as a simulator, often cannot be expressed in closed form, and is treated as a ‘black-box’. Such simulators arise in a range of application domains, for example engineering, climate science and medicine. Ultimately, we are interested in using simulators to aid some decision making process. However, for decisions made using the simulator to be credible, it is necessary to understand and quantify different sources of uncertainty induced by using the simulator. Running the simulator for a range of input combinations is what we call a computer experiment [1]. As the simulators of interest are expensive, the available data is usually scarce. Emulation is the process of using a statistical model (an emulator) to approximate our computer code and provide an estimate of the associated uncertainty.

Intuitively, an emulator must possess two fundamental properties

• It must be cheap, relative to the code
• It must provide an estimate of the uncertainty in its output

A common choice of emulator is the Gaussian process emulator, which is discussed extensively in [2] and described in the next section.

## Types of Uncertainty

There are many types of uncertainty associated with the use of simulators including input, model and observational uncertainty. One type of uncertainty induced by using an expensive simulator is code uncertainty, described by Kennedy and O’Hagan in their seminal paper on calibration [3]. To paraphrase Kennedy and O’Hagan: In principle the simulator encodes a relationship between a set of inputs and a set of outputs, which we could evaluate for any given combination of inputs. However, in practice, it is not feasible to run the simulator for every combination, so acknowledging the uncertainty in the code output is required. (more…)

## Student Perspectives: The Importance of Stability in Dynamic Network Analysis

A post by Ed Davis, PhD student on the Compass programme.

## Introduction

Today is a great day to be a data scientist. More than ever, our ability to both collect and analyse data allow us to solve larger, more interesting, and more diverse problems. My research focuses on analysing networks, which cover a mind-boggling range of applications from modelling vast computer networks [1], to detecting schizophrenia in brain networks [2]. In this blog, I want to share some of the cool research I have been a part of since joining the COMPASS CDT, which has to do with the analysis of dynamic networks.

## Network Basics

A network can be defined as an ordered pair, $(V, E)$, where $V$ is a node (or vertex) set and $E$ is an edge set. From this definition, we can represent any $n$ node network in terms of an adjacency matrix, $A \in \mathbb{R}^{n \times n}$, where for nodes $i, j \in V$,

$A_{ij} = \Bigg\{ \begin{array}{ll} 1 & (i,j) \in E \\ 0, & (i,j) \not\in E \end{array}$.

When we model networks, we can assume that there are some unobservable weightings which mean that certain nodes have a higher connection probability than others. We then observe these in the adjacency matrix with some added noise (like an image that has been blurred). Under this assumption, there must exist some unobservable noise-free version of the adjacency matrix (i.e. the image) that we call the probability matrix, $\mathbf{P} \in \mathbb{R}^{n \times n}$. Mathematically, we represent this by saying

$A_{ij} \overset{\text{ind}}{\sim} \text{Bernoulli} \left(P_{ij} \right)$,

where we have chosen a Bernoulli distribution as it will return either a 1 or a 0. As the connection probabilities are not uniform across the network (inhomogeneous) and the adjacency is sampled from some probability matrix (random), we say that $\mathbf{A}$ is an inhomogeneous random graph.

Going a step further, we can model each node as having a latent position, which can be used to generate its connection probabilities and, hence define its behaviour. Using this, we can define node communities; a group of nodes that have the same underlying connection probabilities, meaning they have the same latent positions. We call this kind of model a latent position model. For example, in a network of social interactions at a school, we expect that pupils are more likely to interact with other pupils in their class. In this case, pupils in the same class are said to have similar latent positions and are part of a node community. Mathematically, we say there is a latent position $\mathbf{Z}_i \in \mathbb{R}^{k}$ assigned to each node, and then our probability matrix will be the gram matrix of some kernel, $f: \mathbb{R}^k \times \mathbb{R}^k \rightarrow [0,1]$. From this, we generate our adjacency matrix as

$A_{ij} \overset{\text{ind}}{\sim} \text{Bernoulli}\left( f \left\{ \mathbf{Z}_i, \mathbf{Z}_j \right\} \right)$.

Under this model, our goal is then to estimate the latent positions by analysing $\mathbf{A}$.

## Compass Science Focus Lab – November 2021

Using OMIC data to predict breast cancer outcomes

Our Compass students took part in a data challenge in partnership with the MRC Integrative Epidemiology Unit based at the University of Bristol.

Academic leaders Matthew Suderman, Paul Yousefi and Josine Min challenged the students to use data collected on cancer rates, risk factors, complexity of individual cancers and potential treatments to build outcome prediction models from multi-omic data derived from hundreds of breast tumours.

In smaller teams, over a 2 week period, the students used machine learning techniques to feedback to academic staff a number of different and creative approaches to this challenge.

## Virtual Applicant Open Day: 13 December 2021

### Applicant Open Day: Q&A 13 December 2021

13th December 2021
13:00 - 14:00

## ILRI sponsors Compass PhD project

We are excited to announce a new partnership between Compass – the EPSRC Centre for Doctoral Training in Computational Statistics and Data Science – and the International Livestock Research Institute (ILRI).

The first step in this new partnership is a co-funded and co-created PhD research project entitled A spatially explicit assessment of agro-pastoral sustainability in Kenya and Ethiopia. The aim of the PhD project is to develop a framework for the assessment of sustainability dynamics in ecologically important areas used by agro-pastoral and pastoral households. Mountainous areas are important water towers and reserves of biodiversity in East Africa, and conservation of such areas is important to stop degradation of the surrounding arid lowlands. However, population pressure and food demands continue to rise, so a sustainable balance between land use and land stewardship must be struck. The PhD project will build upon methods of agricultural sustainability assessment, and make use of spatial statistics to bring together data from household surveys, soil and water measurements, and remote sensing. The resulting analysis will contribute to the understanding of current human-environment interactions in the two study locations, and form the basis for developing scenarios considering the pros and cons of potential future changes. The PhD contributes to the ESSA project, and will operate in Yabelo, South-East Ethiopia, and the Taita Hills, South East Kenya.

“Coming from a geography background, the Compass-ILRI partnership is a fantastic opportunity for me to elevate my skill-set and apply cutting edge statistical techniques to the challenge of sustainable food security. ILRI are a world leader in agricultural research and I am really looking forward to learning from them and contributing to their important goal.” Dan Milner, Compass PhD student.

The International Livestock Research Institute (ILRI) works for better lives through livestock in developing countries. ILRI is co-hosted by Kenya and Ethiopia, has 14 offices across Asia and Africa, employs some 700 staff.

## Student perspectives: Discovering the true positions of objects from pairwise similarities

A post by Annie Gray, PhD student on the Compass programme.

# Introduction

Initially, my Compass mini-project aimed to explore what we can discover about objects given a matrix of similarities between them. More specifically, how to appropriately measure the distance between objects if we represent each as a point in $\mathbb{R}^p$ (the embedding), and what this can tell us about the objects themselves. This led to discovering that the geodesic distances in the embedding relate to the Euclidean distance between the original positions of the objects, meaning we can recover the original positions of the objects. This work has applications in fields that work with relational data for example: genetics, Natural Language Processing and cyber-security.

This work resulted in a paper [3] written with my supervisors (Nick Whiteley and Patrick Rubin-Delanchy), which has been accepted at NeurIPS this year. The following gives an overview of the paper and how the ideas can be used in practice.

## Welcome Cohort 3

A huge welcome to Compass to our 3rd Cohort of CDT students.  Look out for their updates during this year about their research and experiences.

Edward Milsom, Ben Griffiths, Emerald Dilworth, Hannah Sansford, Daniel Milner, Harry Tata, Dominic Broadbent, Tennessee Hickling, Josh Givens, Ettore Fincato

## Compass Away Day

At the end of July 2021, Compass students and staff travelled together to the Brecon Beacons National Park in Wales for a day full of adventure, which was carefully planned by Call of the Wild.

Activities on the day started with some fun team tasks called the ‘Mini Olympics’. Some of the tasks tested logical thinking, the ability to do a task under time pressure, or simply work as a team to complete a certain objective but ultimately to have fun and a laugh.

The tasks were a great opportunity to work together and get to know each other better. Some of them have been more difficult to complete than our students and staff initially expected, but very enjoyable.

After lunch Compass students and staff started a 3-hour Canyoning adventure, guided by the very well trained Call of the Wild team.

The best way of describing this canyoning activity is white water rafting but without the raft. With qualified guides, our students and staff descended a stunning steep sided gorge by various ways and means. This involved sliding down rapids, swimming down rapids, floating down fast flowing chutes and waves, walking behind some breathtaking waterfalls and of course jumping off some jaw dropping waterfalls.

After this thrilling adventure, Compass students and staff travelled to the Vale Resort where they enjoyed dinner together and leisure time until the day after when it was time to come back to Bristol.

It was wonderful to be able to spend time together after the long months of working from home.

## Student Perspectives: Contemporary Ideas in Statistical Philosophy

A post by Alessio Zakaria, PhD student on the Compass programme.

## Introduction

Probability theory is a branch of mathematics centred around the abstract manipulation and quantification of uncertainty and variability. It forms a basic unit of the theory and practice of statistics, enabling us to tame the complex nature of observable phenomena into meaningful information. It is through this reliance that the debate over the true (or more correct) underlying nature of probability theory has profound effects on how statisticians do their work. The current opposing sides of the debate in question are the Frequentists and the Bayesians. Frequentists believe that probability is intrinsically linked to the numeric regularity with which events occur, i.e. their frequency. Bayesians, however, believe that probability is an expression of someones degree of belief or confidence in a certain claim. In everyday parlance we use both of these concepts interchangeably: I estimate one in five of people have Covid; I was 50% confident that the football was coming home. It should be noted that the latter of the two is not a repeatable event per se. We cannot roll back time to check what the repeatable sequence would result in.