Dynamic Objects on Random Fields

这项研究奖学金为我在德国的进一步科学生涯做好了准备，特别是为申请助理教授职位。自2014年7月以来，我一直是凯泽斯劳滕大学统计学教授于尔根·弗兰克教授的博士生。本论文的主题是空间数据的筛子估计--强混合随机场的非参数回归和带小波的密度模型。博士论文的答辩预计将在2017年1月至3月之间进行。我的进一步学术背景如下：我于2006年至2009年在曼海姆大学学习经济学，毕业时获得学士学位。此外，我于2009年至2013年在凯泽斯劳滕大学学习数学，毕业于学士和硕士学位。现在我想加深我在国外的数学和统计知识，同时加强我的科学关系网。这个项目在一定程度上是我论文的延续。这样，我就能很好地运用我所学到的技能。该项目的目的是将高维和无限维数据的统计建模方面的既定方法推广到随机场。随机场是在空间索引集合上定义的随机过程，例如经度和纬度。到目前为止，这些模型大多用于独立的数据或时间序列。空间背景下的统计调查是一项新的研究。它极大地推广了这一理论，这里的依赖结构要复杂得多，该项目允许我们用数学的方式解释空间现象：在许多研究问题中，高维甚至连续测量的数据都是在空间网络上收集的。在本文中，网络被定义为具有节点和边的图。这些数据经常相互作用，或者至少是强的(随机的)依赖关系。值得注意的例子是交通网络中的现象，如交通密度。这里，在单个节点上的观测结果之间存在明显的因果关系。另一个例子是气候事件，如全国各地的温度和降水分布。首先，新开发的方法使我们能够研究收集到的数据的继承模式。这意味着，我们可以对网络中的观测如何准确地依赖于其邻域中的观测以及是否存在可用函数表示的因果关系进行量化陈述。其次，我们可以识别网络中的结构性中断。这意味着我们使用统计测试来确定网络中彼此显著不同的区域。

The research fellowship prepares me for my further scientific career in Germany, in particular for the application for an assistant professor position. I have been a doctoral student of Prof. Dr. Jürgen Franke at the chair of Statistics at the University of Kaiserslautern since July 2014. The topic of my dissertation is Sieve Estimators for Spatial Data – Nonparametric Regression and Density Models with Wavelets for Strong Mixing Random Fields. The defense of the doctoral thesis is expected to take place between January and March 2017. My further academic background is the following: I studied economics at the University of Mannheim from 2006 to 2009 and graduated with a Bachelor’s degree. Furthermore, I studied mathematics at the University of Kaiserslautern from 2009 to 2013 and graduated with a Bachelor’s and a Master’s degree. Now I want to deepen my knowledge in mathematics and statistics abroad and simultaneously strengthen my scientific network.The project is in parts a continuation of my dissertation. In this way, I can well use my acquired skills. The project aims at extending the established methods in the statistical modelling of high- and infinite-dimensional data to random fields. A random field is a stochastic process which is defined on a spatial index set, e.g., longitude and latitude. So far these models have mostly been used for independent data or time series. The statistical investigation in the spatial context is new. It greatly generalizes this theory, here the dependence structures are much more complex.The project allows us to explain spatial phenomena in a mathematical way: in many research questions high-dimensional or even continuously measured data are collected on a spatial network. In this context a network is defined as a graph with nodes and edges. The data often causally interact with each other or are at least strong (stochastically) dependent. Notable examples are phenomena in traffic networks as the traffic density. Here there is an obvious causal relationship between the observations on the single nodes. Further examples are climatologic events such as temperature and precipitation distributions across a country. Firstly, the new developed methods enable us to study inheritance patterns for the collected data. This means, we can make quantitative statements on how exactly observations in a network depend on the observations in their neighborhood and whether there is a causal relationship which can be expressed by a function. Secondly, we can identify structural breaks within networks. This means that we use statistical tests to determine regions within the network that significantly differ from each other.