Modelling and inference for massive populations of heterogeneous point processes

Increasingly, handling large volumes of very heterogeneous data sets is necessary in most application domains. The past decade has seen considerable computational and theoretical developments as a consequence of this fact, enabling us to understand these new types of large volumes of data. This field of mathematics is known as "high dimensional data analysis" where typically the models we need to understand superficially are as complex as the observations they represent. Theory has been developed for many types of models in this setting. An outstanding challenge is understanding observations that come in the form of the spatial locations of a number of points, or events, which may belong to a number of distinct groups. Such data are referred to as "point processes", where the locations of objects of interest are exactly the points. Point processes are ubiquitous in applications, for example in ecology, seismology, and astronomy, and so new methods to understand such forms of data have a clear pathway to impact.

The challenge in the high dimensional setting for point processes is developing simple and flexible models that can be understood, and characterised, within realistic sampling scenarios. To enable the characterisation of observed data, the project will build new models through considering new forms of structure that the data can possess. To incorporate realistic features, we will build models with forms of scale-based heterogeneity, but also including more complex spatial structure. For many realistic processes this includes strong spatial forms of anisotropy, namely patterns associated with given spatial directions. This project will develop such models, and the methods necessary to characterise the structure from data. Computational feasibility will be a strong constraint, as the number of spatial patterns that we will analyse simultaneously will place a clear computational burden on the analysis.

The project will construct new methods to understand data collected in forest ecology. Here the data are locations of different tree species across time, and we consider a particularly high-dimensional, rich source of data that consists of over 275,000 individual trees, belonging to 312 different species. These data exhibit patterns of spatial aggregation and segregation associated with different spatial scales, but these patterns also show anisotropy associated with explanatory variables, which may be broadly classed as having a biotic or abiotic influence. Biotic factors, such as competition for the same nutrients, typically act independently of direction, whereas abiotic, or environmental factors, can have a rotationally asymmetric influence on plant dispersal. Abiotic features, such as features of the landscape like rivers, elevation and soil type, are normally found at large spatial scales relative to that of direct interaction between individuals. This means that whilst competition may lead to segregation of individuals at small scales, they may occur in the same areas of the landscape, giving an appearance of aggregation at larger scales.

The project will thus determine how to best model strongly heterogeneous multiscale structure in forest ecology and develop the mathematics necessary to quantify their form, which is not possible with current methodology. More broadly, this project will provide a flexible set of tools, and a mathematical framework to understand highly heterogeneous and anisotropic classes of point processes.

The work proposed in this project is fundamental research in statistics, and also directly impacts ecology. Statistics has an additional impact on society via collaborations and users of developed technologies. Statistical methodology underpins all study and interpretation of data, and point processes are ubiquitous in applications. Impact will follow via other statisticians who engage with other application areas.

The work in this project will be particularly relevant to research in ecology. This work is useful to solve problems associated with understanding changing ecosystems and so could benefit policy makers and government agencies responsible for environmentally sensitive ecosystems, such as Defra. In particular we plan to define new spatio-temporal summaries or indicators for multivariate data sets that quantify joint spatial structure in species abundances and potential changes to such. This is important in helping to form a notion of climate change impacts. There is good evidence that species rich communities are more productive, meaning more carbon is sequestered from the atmosphere, therefore understanding how numerous species are maintained, and whether they are vulnerable to environment is a key problem in ecology. We therefore need to understand the richness of species communities, and any potential evolution of such richness.

To ensure impact, we will take the following steps:
1) We will publish in both statistical and ecological journals, as well as on the open-access platforms, ArXiv and UCL Discovery. In addition, any code developed for the project will be formatted and published on our website.

2) Dissemination of our results will also be achieved through presentation at national and international conferences. The investigators will present results at both statistical and ecological meetings such as the RSS International Conference and meetings of the British Ecological Society (see Justification of Resources). At larger meetings, such as the Joint Statistical Meetings and the Ecological Society of America Annual Meeting, we will apply to host sessions on statistical ecology and methods for point process data. Attendance and presentation of our research at larger international meetings will ensure delivery of our methodology to a wide range of principal academic beneficiaries, as well as to end users in ecology, such as forest management bodies.

3) We will use the UCL Public Policy unit to forge new collaborations in the application of these methods, and the new UCL-Defra pilot partnership scheme.

4) The impact of our results will be further boosted by our engagement with the public communications office at UCL; through this office, we will be able to coordinate a high-level 'Lunch Hour Lecture' as well as attendance to the Bloomsbury Scientific Salon, and the University of the Third Age events to communicate our research to the public.