### 统计代写|贝叶斯统计代写Bayesian statistics代考|Isotropy

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• Foundations of Data Science 数据科学基础

## 统计代写|贝叶斯统计代写beyesian statistics代考|Isotropy

So far the semi-variogram $\gamma(\mathbf{h})$, or the covariance function $C(\mathbf{h})$, has been assumed to depend on the multidimensional $h$. This is very general and too broad, giving the modeler a tremendous amount of flexibility regarding the stochastic process as it varies over the spatial domain, $\mathbb{D}$. However, this flexibility throws upon a lot of burdens arising from the requirement of precise specification of the dependence structure as the process travels from one location to the next. That is, the function $C(\mathbf{h})$ needs to be specified for every possible value of multidimensional h. Not only is this problematic from the purposes of model specification, but also it is hard to estimate all such precise features from data. Hence, the concept of isotropy is introduced to simplify the specification.

A covariance function $C(\mathbf{h})$ is said to be isotropic if it depends only on the length $|\mathbf{h}|$ of the separation vector $\mathbf{h}$. Isotropic covariance functions only depend on the distance but not on the angle or direction of travel. Assuming space to be in two dimensions, an isotropic covariance function guarantees that the covariance between two random variables, one at the center of a circle and the other at any point on the circumference is the same as the covariance between the two random variables one at the center and another at other point on the circumference of the same circle. Thus, the covariance does not depend on where and which direction the random variables are recorded on the circumference of the circle. Hence, such covariance functions are called omni-directional.

Abusing notations an isotropic covariance function, $C(\cdot)$ is denoted by $C(|\mathbf{h}|))$ or simply by $C(d)$ where $d \geq 0$ is a scalar distance between two locations. The notation $C(\cdot)$ has been abused here since earlier we talked about $C(\mathbf{h})$ where $\mathbf{h}$ is a multi-dimensional separation vector, but now the same $C$ is used to denote the one-dimensional covariance function $C(d)$. A covariance function is called anisotropic if it is not isotropic.

In practice it may seem that the isotropic covariance functions are too restrictive as they are rigid in not allowing flexible covariance structure for the underlying stochastic process. For example, a pollution plume can only spread through using the prevailing wind direction, e.g. east-west. Indeed, this is true, and often, the assumption of isotropy is seen as a limitation of the modeling work. However, the overwhelming simplicity still trumps all the disadvantages, and isotropic covariance functions are used for the underlying true error process. Many mathematical constructs and practical tricks are used to build anisotropic covariance functions, see e.g. Chapter 3 of Banerjee et al. $(2015)$.

## 统计代写|贝叶斯统计代写beyesian statistics代考|Matèrn covariance function

In practical modeling work we need to explicitly specify a particular covariance function so that the likelihood function can be written for the purposes of parameter estimation. In this section we discuss the most commonly used covariance function, namely the Matèrn family (Matérn, 1986) of covariance functions as an example of isotropic covariance functions. We discuss its special cases, such as the exponential and Gaussian. To proceed further recall from elementary definitions that covariance is simply variance times correlation if the two random variables (for which covariance is calculated) have the same variance. In spatial and spatio-temporal modeling, we assume equal spatial variance, which we denote by $\sigma^{2}$. Isotropic covariance functions depend on the distance between two locations, which we denote by $d$. Thus, the covariance function we are about to introduce will have the form
$$C(d)=\sigma^{2} \rho(d), d>0$$
where $\rho(d)$ is the correlation function. Note also that when $d=0$, the covariance is the same as the variance and should be equal to $\sigma^{2}$. Indeed, we shall assume that $\rho(d) \rightarrow 1$ as $d \rightarrow 0$. Henceforth we will only discuss covariance functions in the domain when $d>0$.

How should the correlation functions behave as the distance $d$ increases? For most natural and environmental processes, the correlation should decay with increasing $d$. Indeed, the Tobler’s first law of Geography (Tobler, 1970) states that, “everything is related to everything else, but near things are more related than distant things.” Indeed, there are stochastic processes where we may want to assume no correlation at all for any distance $d$ above a threshold value. Although this sounds very attractive and intuitively simple there are technical difficulties in modeling associated with this approach since an arbitrary covariance function may violate the requirement of non-negative definiteness of the variances. More about this requirement is discussed below in Section 2.7. There are mathematically valid ways to specify negligible amounts of correlations for large distances. See for example, the method of tapering discussed by Kaufman et al. (2008).

Usually, the correlation function $\rho(d)$ should monotonically decrease with increasing value of $d$ due to the Tobler’s law stated above. The particular value of $d$, say $d_{0}$, which is a solution of the equation $\rho(d)=0$ is called the range. This implies that the correlation is exactly zero between any two random observations observed at least the range $d_{0}$ distance apart. Note that due to the monotonicity of the correlation function, it cannot climb up once it reaches the value zero for some value of the distance $d$. With the added assumption of Gaussianity for the data, the range $d_{0}$ is the minimum distance beyond which any two random observations are deemed to be independent. With such assumptions we claim that the underlying process does not get affected by the same process, which is taking place at least $d_{0}$ distance away.

## 统计代写|贝叶斯统计代写beyesian statistics代考|Gaussian processes

Often Gaussian processes are assumed as components in spatial and spatiotemporal modeling. These stochastic processes are defined over a continuum, e.g. a spatial study region and specifying the resulting infinite dimensional random variable is often a challenge in practice. Gaussian processes are very convenient to work in these settings since they are fully defined by a mean function, say $\mu(\mathrm{s})$ and a valid covariance function, say $C\left(\left|\mathbf{s}-\mathbf{s}^{}\right|\right)=\operatorname{Cov}\left(Y(\mathbf{s}), Y\left(\mathbf{s}^{}\right)\right)$, which is required to be positive definite. A covariance function is said to be positive definite if the covariance matrix, implied by that covariance function, for a finite number of random variables belonging to that process is positive definite.

Suppose that the stochastic process $Y($ s ), defined over a continuous spatial region $\mathbb{D}$, is assumed to be a GP with mean function $\mu(\mathrm{s})$ and covariance function $C\left(\mathbf{s}, \mathbf{s}^{*}\right)$. Note that since $s$ is any point in $\mathbb{D}$, the process $Y$ (s) defines a non-countably infinite number of random variables. However, in practice

the GP assumption guarantees that for any finite $n$ and any set of $n$ locations $\mathbf{s}{1}, \ldots, \mathbf{s}{n}$ within $\mathbb{D}$ the $n$-variate random variable $\mathbf{Y}=\left(Y\left(\mathbf{s}{1}\right), \ldots, Y\left(\mathbf{s}{n}\right)\right)$ is normally distributed with mean $\boldsymbol{\mu}$ and covariance matrix $\Sigma$ given by:
$$\boldsymbol{\mu}=\left(\begin{array}{c} \mu\left(\mathbf{s}{1}\right) \ \mu\left(\mathbf{s}{2}\right) \ \vdots \ \mu\left(\mathbf{s}{n}\right) \end{array}\right), \quad \Sigma=\left(\begin{array}{cccc} C(0) & C\left(d{12}\right) & \cdots & C\left(d_{1 n}\right) \ C\left(d_{21}\right) & C(0) & \cdots & C\left(d_{2 n}\right) \ \vdots & \vdots & \ddots & \vdots \ C\left(d_{n 1}\right) & C\left(d_{n 2}\right) & \cdots & C(0) \end{array}\right)$$
where $d_{i j}=\left|\mathbf{s}{i}-\mathbf{s}{j}\right|$ is the distance between the two locations $\mathbf{s}{i}$ and $\mathbf{s}{j}$. From the multivariate normal distribution in Section A.1 in Appendix A, we can immediately write down the joint density of $\mathbf{Y}$ for any finite value of $n$. However, the unresolved matter is how do we specify the two functions $\mu\left(\mathbf{s}{i}\right)$ and $C\left(d{i j}\right)$ for any $i$ and $j$. The GP assumption is often made for the error process just as in usual regression modeling the error distribution is assumed to be Gaussian. Hence often a GP assumption comes with $\mu(\mathbf{s})=0$ for all s. The next most common assumption is to assume the Matèrn covariance function $C\left(d_{i j} \mid \psi\right)$ written down in $(2.1)$ for $C\left(d_{i j}\right)$. The Matèrn family provides a valid family of positive definite covariance functions, and it is the only family used in this book.

C(d)=σ2ρ(d),d>0

## 统计代写|贝叶斯统计代写beyesian statistics代考|Gaussian processes

GP 假设保证对于任何有限n和任何一组n地点s1,…,sn之内D这n- 变量随机变量是=(是(s1),…,是(sn))正态分布，均值μ和协方差矩阵Σ给出：
μ=(μ(s1) μ(s2) ⋮ μ(sn)),Σ=(C(0)C(d12)⋯C(d1n) C(d21)C(0)⋯C(d2n) ⋮⋮⋱⋮ C(dn1)C(dn2)⋯C(0))

## 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。