Flexible spatiotemporal models for environmental processes

环境统计方法涉及对自然现象的各种全球计量。在这一领域可能感兴趣的问题的一些例子是：污染物的平均浓度的建模，空气污染和肺部疾病之间的关系的研究，和时空最高温度的趋势在一个区域。环境过程通常涉及在不同时刻在固定地点进行的观测。我的研究计划侧重于地质统计过程，即在感兴趣的区域连续变化的位置观察到的单变量和多变量过程。地质统计学的主要兴趣在于预测未来时间（时间预测）和空间中未观察到的位置（空间插值）的过程，同时考虑复杂的相关结构。 在环境研究中，大多数时空过程的分析，观测呈现偏态分布。通常，使用数据的单个变换来近似正态性，并且将平稳且各向同性（在围绕原点的平移和旋转下不变）的高斯过程（GP）拟合到变换的数据。然而，它可以表明，常用的转换（如对数和平方根）诱导非平稳性的数据时，考虑在原始规模。因此，应避免使用转换，因为原始过程会产生某种非平稳性。 我的研究计划在未来五年的一个重点将是开发单变量和多变量时空过程的模型，不需要转换的数据，而是可以应用到他们的原始规模的数据。 我将研究混合分布，以模拟偏度和峰度大于偏正态分布的过程。所提出的模型，如所得的协方差结构，峰度和偏度的理论属性，将被导出。这些模型将涉及高维过程;因此，我将把降维技术结合到大型数据集的拟议模型中。将在贝叶斯范式下进行推断，并将使用马尔可夫链蒙特卡罗（MCMC）方法从后验分布中获取样本。将提供R语言的软件包，以协助向广大受众传播和实施拟议的模型。

Environmental statistics methodology is concerned with diverse global measurements of natural phenomena. Some examples of problems which can be of interest in this field are: the modeling of mean concentrations of pollutants, the study of relationships between airborne pollution and lung diseases, and spatiotemporal maximum temperature trends over a region. Environmental processes commonly involve observations made at fixed locations across different instants in time. My research programme focuses on geostatistical processes, i.e. uni- and multivariate processes observed at locations that vary continuously across a region of interest. The main interest in geostatistics lies in predicting the process of interest at future times (temporal prediction) and at unobserved locations in space (spatial interpolation), while accounting for complex correlation structures. In the analysis of most spatiotemporal processes in environmental studies, observations present skewed distributions. Typically a single transformation of the data is used to approximate normality, and a stationary and isotropic (invariant under translation and rotation about the origin) Gaussian process (GP) is fitted to the transformed data. However, it can be shown that commonly-used transformations (e.g. log and square-root) induce non-stationarity in the data when considered on the original scale. Therefore, the use of transformations should be avoided as some sort of non-stationarity is induced to the original process. A key focus of my research programme over the next five years will be to develop models for uni- and multivariate spatiotemporal processes that do not require transformation of the data, but rather can be applied to the data on their original scale. I will investigate mixtures of distributions to model processes that show skewness and kurtosis greater than those of the skew normal distribution. Theoretical properties of the proposed models, such as the resultant covariance structure, kurtosis, and skewness, will be derived. These models will involve high-dimensional processes; I will therefore incorporate dimension-reduction techniques to the proposed models for large datasets. Inference will be performed under the Bayesian paradigm, and Markov chain Monte Carlo (MCMC) methods will be used to obtain samples from the posterior distribution. Packages in R will be made available to assist in the dissemination and implementation of the proposed models to a wide audience.