Modelling and inference for massive populations of heterogeneous point processes

在大多数应用程序领域中，处理大量异构数据集越来越有必要。由于这一事实，过去十年在计算和理论方面取得了相当大的发展，使我们能够理解这些新型的大量数据。这个数学领域被称为“高维数据分析”，在这个领域中，我们需要从表面上理解的模型与它们所代表的观察结果一样复杂。在这种情况下，许多类型的模型的理论都得到了发展。一个突出的挑战是理解以一些点或事件的空间位置形式出现的观察结果，这些点或事件可能属于许多不同的群体。这样的数据被称为“点过程”，其中感兴趣的对象的位置正是点。点过程在应用中无处不在，例如在生态学、地震学和天文学中，因此理解这种数据形式的新方法具有明确的影响途径。在点过程的高维设置中的挑战是开发简单而灵活的模型，这些模型可以在现实的采样场景中被理解和表征。为了能够描述观测到的数据，该项目将通过考虑数据可能具有的新结构形式来建立新的模型。为了结合现实特征，我们将构建具有基于尺度的异质性形式的模型，但也包括更复杂的空间结构。对于许多现实过程，这包括强各向异性的空间形式，即与给定空间方向相关的模式。该项目将开发这样的模型，以及从数据中描述结构的必要方法。计算可行性将是一个强大的约束，因为我们将同时分析的空间模式的数量将给分析带来明显的计算负担。该项目将构建新的方法来理解在森林生态中收集的数据。这里的数据是不同树种在不同时间的位置，我们考虑了一个特别高维，丰富的数据源，由超过275,000棵树组成，属于312个不同的物种。这些数据显示了与不同空间尺度相关的空间聚集和分离模式，但这些模式也显示了与解释变量相关的各向异性，这些解释变量可大致归类为具有生物或非生物影响。生物因素，例如对相同营养物质的竞争，通常独立于方向起作用，而非生物或环境因素可能对植物扩散具有旋转不对称的影响。非生物特征，如河流、海拔和土壤类型等景观特征，相对于个体之间的直接相互作用，通常在更大的空间尺度上发现。这意味着，虽然竞争可能导致个体在小范围内的隔离，但它们可能发生在景观的同一区域，从而在更大范围内呈现聚集的外观。因此，该项目将确定如何最好地模拟森林生态中的强异质性多尺度结构，并发展量化其形式所需的数学，这是目前的方法无法做到的。更广泛地说，这个项目将提供一套灵活的工具和一个数学框架来理解高度异构和各向异性的点过程。

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.