统计代写|随机过程代写stochastic process代考|MTH3016

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

统计代写|随机过程代写stochastic process代考|Hexagonal Lattice, Nearest Neighbors

Here I dive into the details of the processes discussed in Section 1.5.3. I also discuss Figure 2. The source code to produce Figure 2 is discussed in Sections $6.4$ (nearest neighbor graph) and $6.7$ (visualizations). Some elements of graph theory are discussed here, as well as visualization techniques.

Surprisingly, it is possible to produce a point process with a regular hexagonal lattice space using simple operations on a small number $(m=4)$ of square lattices: superimposition, stretching, and shifting. A stretched lattice is a square lattice turned into a rectangular lattice, by applying a multiplication factor to the $\mathrm{X}$ and/or Y coordinates. A shifted lattice is a lattice where the grid points have been shifted via a translation.

Each point of the process almost surely (with probability one) has exactly one nearest neighbor. However, when the scaling factor $s$ is zero, this is no longer true. On the left plot in Figure 2, each point (also called vertex when $s=0$ ) has exactly 3 nearest neighbors. This causes some challenges when plotting the case $s=0$. The case $s>0$ is easier to plot, using arrows pointing from any point to its unique nearest neighbor. I produced the arrows in question with the arrow function in R, see source code in Section $6.7$, and online documentation here. $\mathrm{A}$ bidirectional arrow between points $\mathrm{A}$ and $\mathrm{B}$ means that $\mathrm{B}$ is a nearest neighbor of $\mathrm{A}$, and $\mathrm{A}$ is a nearest neighbor of B. All arrows on the left plot in Figure 2 are bidirectional. Boundary effects are easily noticeable, as some arrows point to nearest neighbors outside the window. Four colors are used for the points, corresponding to the 4 shifted stretched Poisson-binomial processes used to generate the hexagon-based process. The color indicates which of these 4 process, a point is attached to.

The source code in Section $6.4$ handles points with multiple nearest neighbors. It produces a list of all points with their nearest neighbors, using a hash table. A point with 3 nearest neighbors has 3 entries in that list: one for each nearest neighbor. A group of points that are all connected by arrows, is called a connected component [Wiki]. A path from a point of a connected component to another point of the same connected component, following arrows while ignoring their direction, is called a path in graph theory.

In my definition of connected component, the direction of the arrow does not matter: the underlying graph is considered undirected [Wiki]. An interesting problem is to study the size distribution, that is, the number of points per connected component, especially for standard Poisson processes. See Exercise 20. In graph theory, a point is called a vertex or node, and an arrow is called an edge. More about nearest neighbors is discussed in Exercises 18 and 19.

Finally, if you look at Figure 2, the left plot seems to have more points than the right plot. But they actually have roughly the same number of points. The plot on the right seems to be more sparse, because there are large areas with no points. But to compensate, there are areas where several points are in close proximity.

统计代写|随机过程代写stochastic process代考|Modeling Cluster Systems in Two Dimensions

There are various ways to create points scattered around a center. When multiple centers are involved, we get a cluster structure. The point process consisting of the centers is called the parent process, while the point distribution around each center, is called the child process. So we are dealing with a two-layer, or hierarchical structure, referred to as a cluster point process. Besides clustering, many other types of point process operations [Wiki] are possible when combining two processes, such as thinning or superimposition. Typical examples of cluster point processes include Neyma-Scott (see here) and Matérn (see here).

Useful references include Baddeley’s textbook “Spatial Point Processes and their Applications” [4] available online here, Sigman’s course material (Columbia University) on one-dimensional renewal processes for beginners, entitled “Notes on the Poisson Process” [71], available online here, Last and Kenrose’s book “Lectures on the Poisson Process” [52], and Cressie’s comprehensive 900-page book “Statistics for Spatial Data” [16]. Cluster point processes are part of a larger field known as spatial statistics, encompassing other techniques such as geostatistics, kriging and tessellations. For lattice-based processes known as perturbed lattice point processes, more closely related to the theme of this textbook (lattice processes), and also more recent with applications to cellular networks, see the following references:

• “On Comparison of Clustering Properties of Point Processes” [12]. Online PDF here.
• “Clustering and percolation of point processes” [11]. Online version here.
• “Clustering comparison of point processes, applications to random geometric models” [13]. Online version here.
• “Stochastic Geometry-Based Tools for Spatial Modeling and Planning of Future Cellular Networks” [51]. Online version here.
• “Hyperuniform and rigid stable matchings” [54]. Online PDF here. Short presentation available here.
• “Rigidity and tolerance for perturbed lattices” [68]. Online version here.
• “Cluster analysis of spatial point patterns: posterior distribution of parents inferred from offspring” [66].
• “Recovering the lattice from its random perturbations” [79]. Online version here.
• “Geometry and Topology of the Boolean Model on a Stationary Point Processes” [81]. Online version here.
• “On distances between point patterns and their applications” [56]. Online version here.
More general references include two comprehensive volumes on point process theory by Daley and Vere-Jones [20, 21], a chapter by Johnson [45] (available online here or here), books by Møller and Waagepetersen, focusing on statistical inference for spatial processes [60, 61], and “Point Pattern Analysis: Nearest Neighbor Statistics” by Anselin [3] focusing on point inhibition/aggregation metrics, available here. See also [58] by Møller, available online here, and “Limit Theorems for Network Dependent Random Variables” [48], available online here.

随机过程代考

统计代写|随机过程代写stochastic process代考|Modeling Cluster Systems in Two Dimensions

• 《论点过程的聚类特性比较》[12]。此处为在线 PDF。
• “点过程的聚类和渗透”[11]。在线版本在这里。
• “点过程的聚类比较，在随机几何模型中的应用”[13]。在线版本在这里。
• “用于未来蜂窝网络空间建模和规划的基于随机几何的工具”[51]。在线版本在这里。
• “超均匀和刚性稳定匹配”[54]。此处为在线 PDF。此处提供简短演示。
• “扰动格子的刚度和容忍度”[68]。在线版本在这里。
• “空间点模式的聚类分析：从后代推断出父母的后验分布”[66]。
• “从随机扰动中恢复晶格”[79]。在线版本在这里。
• “驻点过程布尔模型的几何和拓扑”[81]。在线版本在这里。
• “关于点模式之间的距离及其应用”[56]。在线版本在这里。
更一般的参考资料包括 Daley 和 Vere-Jones [20, 21] 的两本关于点过程理论的综合性著作，Johnson [45] 的一章（可在此处或此处在线获取），Møller 和 Waagepetersen 的书籍，侧重于空间的统计推断过程 [60、61] 和 Anselin [3] 的“点模式分析：最近邻统计”重点关注点抑制/聚合指标，可在此处获取。另请参见 Møller 的 [58]，可在此处在线获取，以及“网络相关随机变量的极限定理”[48]，可在此处在线获取。

有限元方法代写

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MATLAB代写

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