## 统计代写|随机过程代写stochastic process代考|STAT6540

2022年12月29日

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## 统计代写|随机过程代写stochastic process代考|Fractal Supervised Classification

The rightmost video in Figure 9 shows supervised clustering in action, from the first frame representing the training set with 4 groups, to the last one showing the cluster assignment of any future observation (an arbitrary point location in the state space). Based on image filtering techniques acting as a neural network, the video illustrates how machine learning algorithms are performed in GPU (graphics processing unit). GPU-based clustering [Wiki] is very fast, not only because it uses graphics processors and memory, but the algorithm itself has a computational complexity that beats (by a long shot) any traditional classifier. It does not require the computation of nearest neighbor distances.

The video medium also explains how the clustering is done, in better ways than any text description could do. You can view the video (also called data animation) on YouTube, here. The source code and instructions to help you create your own videos or replicate this one, is in Section 6.7.2. See Section 3.4.3 for a description of the underlying supervised clustering methodology.

I use the word “fractal” because the shape of the clusters, and their boundaries in particular, is arbitrary. The boundary may be as fractal-like as a shoreline. It also illustrates the concept of fuzzy clustering [Wiki]: towards the middle of the video, when the entire state space is eventually classified, constant cluster re-assignments are taking place along the cluster boundaries. A point, close to the fuzzy border between clusters $\mathrm{A}$ and $\mathrm{B}$, is sometimes assigned to $\mathrm{A}$ in a given video frame, and may be assigned to $\mathrm{B}$ in the next one. By averaging cluster assignments over many frames, it is possible to compute the probability that the point belongs to A or B. Another question is whether the algorithm (the successive frames) converge or not. It depends on the parameters, and in this case, stochastic convergence is observed. In other words, despite boundaries changing all the time, their average location is almost constant, and the changes are small. Small portions of a cluster, embedded in another cluster, don’t disappear over time.

## 统计代写|随机过程代写stochastic process代考|Statistical Inference, Machine Learning

This section covers a lot of material, extending far beyond Poisson-binomial processes. The main type of processes investigated here is the $m$-interlacing defined in Section 1.5.3, as opposed to the radial cluster processes studied in Section 2.1. An $m$-process is a superimposition of $m$ shifted Poisson-binomial processes, well suited to model cluster structures. In Section 3.4.3, I discuss supervised and unsupervised clustering algorithms applied to simulated data generated by $m$-processes. The technique, similar to neural networks, relies on image filtering performed in the GPU (graphics processing unit). It leads to fractal supervised clustering, illustrated with data animations. I discuss how to automatically detect the number of clusters in Section 3.4.4.

Before getting there, I describe different methods to estimate the core parameters of these processes. First in one dimension in Section 3.2, then in two dimensions in Section 3.4.2. The methodology features a new test of independence (Section 3.1.3), model fitting via the empirical distribution, and dual confidence region in the context of minimum contrast estimation (Section 3.1.1). I show that the point count expectations are almost stationary but exhibit small periodic oscillations (Section 3.1.2) and that the increments (point counts across non-overlapping, adjacent intervals) are almost independent.

In many instances, Poisson-binomial processes exhibit patterns that are invisible to the naked eye. In Section 3.3, I show examples of such patterns. Then, I discuss model identifiability, and the need for statistical or machine learning techniques to unearth the invisible patterns. Boundary effects, their impact, and how to fix this problem, is discussed mainly in Section $3.5$.

In 1979, Bradley Efron published his seminal article “Bootstrap Methods: Another Look at the Jackknife” [24], available online here. It marked the beginning of a new era in statistical science: the development of model-free, data driven techniques. Several chapters in my book “Statistics: New Foundations, Toolbox, and Machine Learning Recipes” [37] published in 2019 (available online here) deal with extensions and modern versions of this methodology. I follow the same footsteps here, first discussing the general principles, and then showing how it applies to estimating the intensity $\lambda$ and scaling factor $s$ of a Poisson-binomial process. As in Jesper Møller [58], my methodology is based on minimum contrast estimation: see slides 114-116 here or here. See also [18] for other examples of this method in the context of point process inference.

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|Statistical Inference, Machine Learning

1979 年，Bradley Efron 发表了他的开创性文章“Bootstrap Methods: Another Look at the Jackknife”[24]，可在此处在线获取。它标志着统计科学新时代的开始：无模型、数据驱动技术的发展。我 2019 年出版的《统计：新基础、工具箱和机器学习秘诀》[37] 一书（可在此处在线获取）中的几章涉及该方法的扩展和现代版本。我在这里遵循相同的步骤，首先讨论一般原则，然后展示它如何应用于估计强度λ和比例因子s泊松二项式过程。与 Jesper Møller [58] 一样，我的方法基于最小对比度估计：请在此处或此处查看幻灯片 114-116。有关此方法在点过程推理上下文中的其他示例，另请参见 [18]。

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。