## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Theory of Estimation

statistics-lab™ 为您的留学生涯保驾护航 在代写多元统计分析Multivariate Statistical Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写多元统计分析Multivariate Statistical Analysis代写方面经验极为丰富，各种代写多元统计分析Multivariate Statistical Analysis相关的作业也就用不着说。

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Theory of Estimation

We know from our basic knowledge of statistics that one of the objectives in statistics is to better understand and model the underlying process which generates the data. This is known as statistical inference: we infer from information contained in a sample properties of the population from which the observations are taken. In multivariate statistical inference, we do exactly the same. The basic ideas were introduced in Section 4.5 on sampling theory: we observed the values of a multivariate random variable $X$ and obtained a sample $\mathcal{X}=\left{x_i\right}_{i=1}^n$. Under random sampling, these observations are considered to be realizations of a sequence of i.i.d. random variables $X_1, \ldots, X_n$ where each $X_i$ is a $p$-variate random variable which replicates the parent or population random variable $X$. In this chapter, for notational convenience, we will no longer differentiate between a random variable $X_i$ and an observation of it, $x_i$, in our notation. We will simply write $x_i$ and it should be clear from the context whether a random variable or an observed value is meant.

Statistical inference infers from the i.i.d. random sample $\mathcal{X}$ the properties of the population: typically, some unknown characteristic $\theta$ of its distribution. In parametric statistics, $\theta$ is a $k$-variate vector $\theta \in \mathbb{R}^k$ characterizing the unknown properties of the population pdf $f(x ; \theta)$ : this could be the mean, the covariance matrix, kurtosis, etc.

The aim will be to estimate $\theta$ from the sample $\mathcal{X}$ through estimators $\hat{\theta}$ which are functions of the sample: $\widehat{\theta}=\widehat{\theta}(\mathcal{X})$. When an estimator $\widehat{\theta}$ is proposed, we must derive its sampling distribution to analyze its properties (is it related to the unknown quantity $\theta$ it is supposed to estimate?).

In this chapter the basic theoretical tools are developed which are needed to derive estimators and to determine their properties in general situations. We will basically rely on the maximum likelihood theory in our presentation. In many situations, the maximum likelihood estimators indeed share asymptotic optimal properties which make their use easy and appealing.

We will illustrate the multivariate normal population and also the linear regression model where the applications are numerous and the derivations are easy to do. In multivariate setups, the maximum likelihood estimator is at times too complicated to be derived analytically. In such cases, the estimators are obtained using numerical methods (nonlinear optimization). The general theory and the asymptotic properties of these estimators remain simple and valid. The following chapter, Chapter 7 , concentrates on hypothesis testing and confidence interval issues.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Likelihood Function

Suppose that $\left{x_i\right}_{i=1}^n$ is an i.i.d. sample from a population with pdf $f(x ; \theta)$. The aim is to estimate $\theta \in \mathbb{R}^k$ which is a vector of unknown parameters. The likelihood function is defined

as the joint density $L(\mathcal{X} ; \theta)$ of the observations $x_i$ considered as a function of $\theta$ :
$$L(\mathcal{X} ; \theta)=\prod_{i=1}^n f\left(x_i ; \theta\right),$$
where $\mathcal{X}$ denotes the sample of the data matrix with the observations $x_1^{\top}, \ldots, x_n^{\top}$ in each row. The maximum likelihood estimator (MLE) of $\theta$ is defined as
$$\widehat{\theta}=\arg \max \theta L(\mathcal{X} ; \theta)$$ Often it is easier to maximize the log-likelihood function $$\ell(\mathcal{X} ; \theta)=\log L(\mathcal{X} ; \theta)$$ which is equivalent since the logarithm is a monotone one-to-one function. Hence $$\widehat{\theta}=\arg \max \theta L(\mathcal{X} ; \theta)=\arg \max _\theta \ell(\mathcal{X} ; \theta)$$
The following examples illustrate cases where the maximization process can be performed analytically, i.e., we will obtain an explicit analytical expression for $\widehat{\theta}$. Unfortunately, in other situations, the maximization process can be more intricate, involving nonlinear optimization techniques. In the latter case, given a sample $\mathcal{X}$ and the likelihood function, numerical methods will be used to determine the value of $\theta$ maximizing $L(\mathcal{X} ; \theta)$ or $\ell(\mathcal{X} ; \theta)$. These numerical methods are typically based on Newton-Raphson iterative techniques.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Cramer-Rao Lower Bound

As pointed out above, an important question in estimation theory is whether an estimator $\widehat{\theta}$ has certain desired properties, in particular, if it converges to the unknown parameter $\theta$ it is supposed to estimate. One typical property we want for an estimator is unbiasedness, meaning that on the average, the estimator hits its target: $E(\widehat{\theta})=\theta$. We have seen for instance (see Example 6.2) that $\bar{x}$ is an unbiased estimator of $\mu$ and $\mathcal{S}$ is a biased estimator of $\Sigma$ in finite samples. If we restrict ourselves to unbiased estimation then the natural question is whether the estimator shares some optimality properties in terms of its sampling variance. Since we focus on unbiasedness, we look for an estimator with the smallest possible variance.

In this context, the Cramer-Rao lower bound will give the minimal achievable variance for any unbiased estimator. This result is valid under very general regularity conditions (discussed below). One of the most important applications of the Cramer-Rao lower bound is that it provides the asymptotic optimality property of maximum likelihood estimators. The Cramer-Rao theorem involves the score function and its properties which will be derived first.
The score function $s(\mathcal{X} ; \theta)$ is the derivative of the log likelihood function w.r.t. $\theta \in \mathbb{R}^k$
$$s(\mathcal{X} ; \theta)=\frac{\partial}{\partial \theta} \ell(\mathcal{X} ; \theta)=\frac{1}{L(\mathcal{X} ; \theta)} \frac{\partial}{\partial \theta} L(\mathcal{X} ; \theta) .$$
The covariance matrix $\mathcal{F}_n=\operatorname{Var}{s(\mathcal{X} ; \theta)}$ is called the Fisher information matrix. In what follows, we will give some interesting properties of score functions.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Likelihood Function

$$Y \sim N_p\left(\mathcal{A} \mu+c, \mathcal{A} \Sigma \mathcal{A}^{\top}\right)$$

$$Y=\Sigma^{-1 / 2}(X-\mu) \sim N_p\left(0, \mathcal{I}_p\right)$$

$$Y^{\top} Y=(X-\mu)^{\top} \Sigma^{-1}(X-\mu) \sim \chi_p^2$$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Conditional Approximations

$$X_2=E\left(X_2 \mid X_1\right)+U=\mu_2+\Sigma_{21} \Sigma_{11}^{-1}\left(X_1-\mu_1\right)+U$$

$$X_2=\beta_0+\mathcal{B} X_1+U$$

$$X_2=\beta_0+\beta^{\top} X_1+U$$

$$\sigma_{22}=\beta^{\top} \Sigma_{11} \beta+\sigma_{22.1}=\sigma_{21} \Sigma_{11}^{-1} \sigma_{12}+\sigma_{22.1} .$$

$$\rho_{2.1 \ldots r}^2=\frac{\sigma_{21} \Sigma_{11}^{-1} \sigma_{12}}{\sigma_{22}}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Heavy-Tailed Distributions

statistics-lab™ 为您的留学生涯保驾护航 在代写多元统计分析Multivariate Statistical Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写多元统计分析Multivariate Statistical Analysis代写方面经验极为丰富，各种代写多元统计分析Multivariate Statistical Analysis相关的作业也就用不着说。

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Heavy-Tailed Distributions

Heavy-tailed distributions were first introduced by the Italian-born Swiss economist Pareto and extensively studied by Paul Lévy. Although in the beginning these distributions were mainly studied theoretically, nowadays they have found many applications in areas as diverse as finance, medicine, seismology, structural engineering. More concretely, they have been used to model returns of assets in financial markets, stream flow in hydrology, precipitation and hurricane damage in meteorology, earthquake prediction in seismology, pollution, material strength, teletraffic and many others.

A distribution is called heavy-tailed if it has higher probability density in its tail area compared with a normal distribution with same mean $\mu$ and variance $\sigma^2$. Figure 4.6 demonstrates the differences of the pdf curves of a standard Gaussian distribution and a Cauchy distribution with location parameter $\mu=0$ and scale parameter $\sigma=1$. The graphic shows that the probability density of the Cauchy distribution is much higher than that of the Gaussian in the tail part, while in the area around the center, the probability density of the Cauchy distribution is much lower.

In terms of kurtosis, a heavy-tailed distribution has kurtosis greater than 3 (See Chapter 4 , formula (4.40)), which is called leptokurtic, in contrast to mesokurtic distribution (kurtosis $=3$ ) and platykurtic distribution (kurtosis $<3$ ). Since univariate heavy-tailed distributions serve as basics for their multivariate counterparts and their density properties have been proved useful even in multivariate cases, we will start from introducing some univariate heavy-tailed distributions. Then we will move on to analyze their multivariate counterparts, and their tail behavior.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Student’s t-distribution

The $t$-distribution was first analyzed by Gosset (1908). He published his results under his pseudonym “Student” by request of his employer. Let $X$ be a normally distributed random variable with mean $\mu$ and variance $\sigma^2$, and $Y$ be the random variable such that $Y^2 / \sigma^2$ has a chi-square distribution with $n$ degrees of freedom. Assume that $X$ and $Y$ are independent, then
$$t \stackrel{\text { def }}{=} \frac{X \sqrt{n}}{Y}$$

is distributed as Student’s $t$ with $n$ degrees of freedom. The $t$-distribution has the following density function
$$f_t(x ; n)=\frac{\Gamma\left(\frac{n+1}{2}\right)}{\sqrt{n \pi} \Gamma\left(\frac{n}{2}\right)}\left(1+\frac{x^2}{n}\right)^{-\frac{n+1}{2}}$$
where $n$ is the number of degrees of freedom, $-\infty4)$ are:
\begin{aligned} \mu & =0 \ \sigma^2 & =\frac{n}{n-2} \ \text { Skewness } & =0 \ \text { Kurtosis } & =3+\frac{6}{n-4} \end{aligned}
The $t$-distribution is symmetric around 0 , which is consistent with the fact that its mean is 0 and skewness is also 0 .
Student’s $t$-distribution approaches the normal distribution as $n$ increases, since
$$\lim _{n \rightarrow \infty} f_t(x ; n)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{x^2}{2}}$$
In practice the $t$-distribution is widely used, but its flexibility of modeling is restricted because of the integer-valued tail index.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Student’s t-distribution

$t$ -分布最早由Gosset(1908)分析。应雇主的要求，他用“学生”这个笔名发表了他的研究结果。设$X$为均值$\mu$、方差$\sigma^2$的正态分布随机变量，$Y$为使$Y^2 / \sigma^2$具有卡方分布、自由度$n$的随机变量。假设$X$和$Y$是独立的
$$t \stackrel{\text { def }}{=} \frac{X \sqrt{n}}{Y}$$

$$f_t(x ; n)=\frac{\Gamma\left(\frac{n+1}{2}\right)}{\sqrt{n \pi} \Gamma\left(\frac{n}{2}\right)}\left(1+\frac{x^2}{n}\right)^{-\frac{n+1}{2}}$$

\begin{aligned} \mu & =0 \ \sigma^2 & =\frac{n}{n-2} \ \text { Skewness } & =0 \ \text { Kurtosis } & =3+\frac{6}{n-4} \end{aligned}
$t$ -分布是围绕0对称的，这与它的均值为0，偏度也为0的事实是一致的。

$$\lim _{n \rightarrow \infty} f_t(x ; n)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{x^2}{2}}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Conditional Expectations

statistics-lab™ 为您的留学生涯保驾护航 在代写多元统计分析Multivariate Statistical Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写多元统计分析Multivariate Statistical Analysis代写方面经验极为丰富，各种代写多元统计分析Multivariate Statistical Analysis相关的作业也就用不着说。

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Conditional Expectations

The conditional expectations are
$$E\left(X_2 \mid x_1\right)=\int x_2 f\left(x_2 \mid x_1\right) d x_2 \quad \text { and } \quad E\left(X_1 \mid x_2\right)=\int x_1 f\left(x_1 \mid x_2\right) d x_1$$
$E\left(X_2 \mid x_1\right)$ represents the location parameter of the conditional pdf of $X_2$ given that $X_1=x_1$. In the same way, we can define $\operatorname{Var}\left(X_2 \mid X_1=x_1\right)$ as a measure of the dispersion of $X_2$ given that $X_1=x_1$. We have from $(4.20)$ that
$$\operatorname{Var}\left(X_2 \mid X_1=x_1\right)=E\left(X_2 X_2^{\top} \mid X_1=x_1\right)-E\left(X_2 \mid X_1=x_1\right) E\left(X_2^{\top} \mid X_1=x_1\right) .$$
Using the conditional covariance matrix, the conditional correlations may be defined as:
$$\rho_{X_2 X_3 \mid X_1=x_1}=\frac{\operatorname{Cov}\left(X_2, X_3 \mid X_1=x_1\right)}{\sqrt{\operatorname{Var}\left(X_2 \mid X_1=x_1\right) \operatorname{Var}\left(X_3 \mid X_1=x_1\right)}} .$$
These conditional correlations are known as partial correlations between $X_2$ and $X_3$, conditioned on $X_1$ being equal to $x_1$.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Characteristic Functions

The characteristic function (cf) of a random vector $X \in \mathbb{R}^p$ (respectively its density $f(x)$ ) is defined as
$$\varphi_X(t)=E\left(e^{\mathbf{i} t^{\top} X}\right)=\int e^{\mathbf{i} t^{\top} x} f(x) d x, \quad t \in \mathbb{R}^p,$$
where $\mathbf{i}$ is the complex unit: $\mathbf{i}^2=-1$. The cf has the following properties:
$$\varphi_X(0)=1 \text { and }\left|\varphi_X(t)\right| \leq 1$$

If $\varphi$ is absolutely integrable, i.e., the integral $\int_{-\infty}^{\infty}|\varphi(x)| d x$ exists and is finite, then
$$f(x)=\frac{1}{(2 \pi)^p} \int_{-\infty}^{\infty} e^{-\mathbf{i} t^{\top} x} \varphi_X(t) d t .$$
If $X=\left(X_1, X_2, \ldots, X_p\right)^{\top}$, then for $t=\left(t_1, t_2, \ldots, t_p\right)^{\top}$
$$\varphi_{X_1}\left(t_1\right)=\varphi_X\left(t_1, 0, \ldots, 0\right), \quad \ldots \quad, \varphi_{X_p}\left(t_p\right)=\varphi_X\left(0, \ldots, 0, t_p\right)$$
If $X_1, \ldots, X_p$ are independent random variables, then for $t=\left(t_1, t_2, \ldots, t_p\right)^{\top}$
$$\varphi_X(t)=\varphi_{X_1}\left(t_1\right) \cdot \ldots \cdot \varphi_{X_p}\left(t_p\right)$$
If $X_1, \ldots, X_p$ are independent random variables, then for $t \in \mathbb{R}$
$$\varphi_{X_1+\ldots+X_p}(t)=\varphi_{X_1}(t) \cdot \ldots \cdot \varphi_{X_p}(t)$$
The characteristic function can recover all the cross-product moments of any order: $\forall j_k \geq$ $0, k=1, \ldots, p$ and for $t=\left(t_1, \ldots, t_p\right)^{\top}$ we have
$$E\left(X_1^{j_1} \cdot \ldots \cdot X_p^{j_p}\right)=\frac{1}{\mathbf{i}^{j_1+\ldots+j_p}}\left[\frac{\partial \varphi_X(t)}{\partial t_1^{j_1} \ldots \partial t_p^{j_p}}\right]_{t=0} .$$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Conditional Expectations

$$E\left(X_2 \mid x_1\right)=\int x_2 f\left(x_2 \mid x_1\right) d x_2 \quad \text { and } \quad E\left(X_1 \mid x_2\right)=\int x_1 f\left(x_1 \mid x_2\right) d x_1$$
$E\left(X_2 \mid x_1\right)$表示$X_2$的条件PDF的位置参数，假设$X_1=x_1$。同样，我们可以将$\operatorname{Var}\left(X_2 \mid X_1=x_1\right)$定义为$X_2$离散度的度量，假设$X_1=x_1$。我们从$(4.20)$得到这个
$$\operatorname{Var}\left(X_2 \mid X_1=x_1\right)=E\left(X_2 X_2^{\top} \mid X_1=x_1\right)-E\left(X_2 \mid X_1=x_1\right) E\left(X_2^{\top} \mid X_1=x_1\right) .$$

$$\rho_{X_2 X_3 \mid X_1=x_1}=\frac{\operatorname{Cov}\left(X_2, X_3 \mid X_1=x_1\right)}{\sqrt{\operatorname{Var}\left(X_2 \mid X_1=x_1\right) \operatorname{Var}\left(X_3 \mid X_1=x_1\right)}} .$$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Characteristic Functions

$$\varphi_X(t)=E\left(e^{\mathbf{i} t^{\top} X}\right)=\int e^{\mathbf{i} t^{\top} x} f(x) d x, \quad t \in \mathbb{R}^p,$$

$$\varphi_X(0)=1 \text { and }\left|\varphi_X(t)\right| \leq 1$$

$$f(x)=\frac{1}{(2 \pi)^p} \int_{-\infty}^{\infty} e^{-\mathbf{i} t^{\top} x} \varphi_X(t) d t .$$

$$\varphi_{X_1}\left(t_1\right)=\varphi_X\left(t_1, 0, \ldots, 0\right), \quad \ldots \quad, \varphi_{X_p}\left(t_p\right)=\varphi_X\left(0, \ldots, 0, t_p\right)$$

$$\varphi_X(t)=\varphi_{X_1}\left(t_1\right) \cdot \ldots \cdot \varphi_{X_p}\left(t_p\right)$$

$$\varphi_{X_1+\ldots+X_p}(t)=\varphi_{X_1}(t) \cdot \ldots \cdot \varphi_{X_p}(t)$$

pcp的一个特点是，许多线被画在彼此的顶部。用成对的散点图来描述变量可以减少这个问题。在一个大的散点图矩阵中包含所有14个变量是可能的，但很难从图中看到任何东西。因此，为了便于说明，我们将只分析图1.25中变量子集中的一个这样的矩阵。在PCP和散点图矩阵的基础上，我们想解释13个变量中的每一个以及它们与第14个变量的最终关系。图中包括$X_1-X_5$和$X_{14}$的图像，下面将详细讨论每个变量。下面对散点图的所有引用参见图1.25。

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Derivatives

$$\begin{gathered} \frac{\partial a^{\top} x}{\partial x}=\frac{\partial x^{\top} a}{\partial x}=a, \ \frac{\partial x^{\top} \mathcal{A} x}{\partial x}=2 \mathcal{A} x . \end{gathered}$$

$$\frac{\partial^2 x^{\top} \mathcal{A} x}{\partial x \partial x^{\top}}=2 \mathcal{A} .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Boston Housing

statistics-lab™ 为您的留学生涯保驾护航 在代写多元统计分析Multivariate Statistical Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写多元统计分析Multivariate Statistical Analysis代写方面经验极为丰富，各种代写多元统计分析Multivariate Statistical Analysis相关的作业也就用不着说。

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Aim of the Analysis

The Boston Housing data set was analyzed by Harrison and Rubinfeld (1978) who wanted to find out whether “clean air” had an influence on house prices. We will use this data set in this chapter and in most of the following chapters to illustrate the presented methodology. The data are described in Appendix B.1.
What Can Be Seen from the PCPs
In order to highlight the relations of $X_{14}$ to the remaining 13 variables we color all of the observations with $X_{14}>$ median $\left(X_{14}\right)$ as red lines in Figure 1.24. Some of the variables seem to be strongly related. The most obvious relation is the negative dependence between $X_{13}$ and $X_{14}$. It can also be argued that there exists a strong dependence between $X_{12}$ and $X_{14}$ since no red lines are drawn in the lower part of $X_{12}$. The opposite can be said about $X_{11}$ : there are only red lines plotted in the lower part of this variable. Low values of $X_{11}$ induce high values of $X_{14}$.
For the PCP, the variables have been rescaled over the interval $[0,1]$ for better graphical representations. The $\mathrm{PCP}$ shows that the variables are not distributed in a symmetric manner. It can be clearly seen that the values of $X_1$ and $X_9$ are much more concentrated around 0 . Therefore it makes sense to consider transformations of the original data.
The Scatterplot Matrix
One characteristic of the PCPs is that many lines are drawn on top of each other. This problem is reduced by depicting the variables in pairs of scatterplots. Including all 14 variables in one large scatterplot matrix is possible, but makes it hard to see anything from the plots. Therefore, for illustratory purposes we will analyze only one such matrix from a subset of the variables in Figure 1.25. On the basis of the PCP and the scatterplot matrix we would like to interpret each of the thirteen variables and their eventual relation to the 14th variable. Included in the figure are images for $X_1-X_5$ and $X_{14}$, although each variable is discussed in detail below. All references made to scatterplots in the following refer to Figure 1.25.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Per-capita crime rate X1

Taking the logarithm makes the variable’s distribution more symmetric. This can be seen in the boxplot of $\widetilde{X}1$ in Figure 1.27 which shows that the median and the mean have moved closer to each other than they were for the original $X_1$. Plotting the kernel density estimate (KDE) of $\widetilde{X}_1=\log \left(X_1\right)$ would reveal that two subgroups might exist with different mean values. However, taking a look at the scatterplots in Figure 1.26 of the logarithms which include $X_1$ does not clearly reveal such groups. Given that the scatterplot of $\log \left(X_1\right)$ vs. $\log \left(X{14}\right)$ shows a relatively strong negative relation, it might be the case that the two subgroups of $X_1$ correspond to houses with two different price levels. This is confirmed by the two boxplots shown to the right of the $X_1$ vs. $X_2$ scatterplot (in Figure 1.25): the right boxplot’s shape differs a lot from the black one’s, having a much higher median and mean.
Proportion of residential area zoned for large lots $X_2$
It strikes the eye in Figure 1.25 that there is a large cluster of observations for which $X_2$ is equal to 0 . It also strikes the eye that-as the scatterplot of $X_1$ vs. $X_2$ shows-there is a strong, though non-linear, negative relation between $X_1$ and $X_2$ : Almost all observations for which $X_2$ is high have an $X_1$-value close to zero, and vice versa, many observations for which $X_2$ is zero have quite a high per-capita crime rate $X_1$. This could be due to the location of the areas, e.g., downtown districts might have a higher crime rate and at the same time it is unlikely that any residential land would be zoned in a generous manner.
As far as the house prices are concerned it can be said that there seems to be no clear (linear) relation between $X_2$ and $X_{14}$, but it is obvious that the more expensive houses are situated in areas where $X_2$ is large (this can be seen from the two boxplots on the second position of the diagonal, where the red one has a clearly higher mean/median than the black one).

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Aim of the Analysis

Harrison和Rubinfeld(1978)分析了Boston Housing的数据集，他们想要找出“清洁空气”是否对房价有影响。我们将在本章和接下来的大部分章节中使用这个数据集来说明所提出的方法。数据见附录B.1。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。