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# 统计代写|多元统计分析代考MULTIVARIATE STATISTICAL ANALYSIS代考|HRD6355 Sample Geometry and Random Sampling

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## 统计代写|多元统计分析代考MULTIVARIATE STATISTICAL ANALYSIS代考|INTRODUCTION

With the vector concepts introduced in the previous chapter, we can now delve deeper into the geometrical interpretations of the descriptive statistics $\overline{\mathbf{x}}, \mathbf{S}_{n}$, and $\mathbf{R}$; we do so in Section 3.2. Many of our explanations use the representation of the columns of $\mathbf{X}$ as $p$ vectors in $n$ dimensions. In Section $3.3$ we introduce the assumption that the observations constitute a random sample. Simply stated, random sampling implies that (1) measurements taken on different items (or trials) are unrelated to one another and (2) the joint distribution of all $p$ variables remains the same for all items. Ultimately, it is this structure of the random sample that justifies a particular choice of distance and dictates the geometry for the $n$-dimensional representation of the data. Furthermore, when data can be treated as a random sample, statistical inferences are based on a solid foundation.

Returning to geometric interpretations in Section 3.4, we introduce a single number, called generalized variance, to describe variability. This generalization of variance is an integral part of the comparison of multivariate means. In later sections we use matrix algebra to provide concise expressions for the matrix products and sums that allow us to calculate $\overline{\mathbf{x}}$ and $\mathbf{S}{n}$ directly from the data matrix $\mathbf{X}$. The connection between $\overline{\mathbf{x}}, \mathbf{S}{n}$, and the means and covariances for linear combinations of variables is also clearly delineated, using the notion of matrix products.

## 统计代写|多元统计分析代考MULTIVARIATE STATISTICAL ANALYSIS代考|THE GEOMETRY OF THE SAMPLE

A single multivariate observation is the collection of measurements on $p$ different variables taken on the same item or trial. As in Chapter 1 , if $n$ observations have been obtained, the entire data set can be placed in an $n \times p$ array (matrix):
$$\underset{(n \times p)}{\mathbf{X}}=\left[\begin{array}{cccc} x_{11} & x_{12} & \cdots & x_{1 p} \ x_{21} & x_{22} & \cdots & x_{2 p} \ \vdots & \vdots & \ddots & \vdots \ x_{n 1} & x_{n 2} & \cdots & x_{n p} \end{array}\right]$$
Each row of $\mathbf{X}$ represents a multivariate observation. Since the entire set of measurements is often one particular realization of what might have been observed, we say that the data are a sample of size $n$ from a $p$-variate “population.” The sample then consists of $n$ measurements, each of which has $p$ components.

As we have seen, the data can be plotted in two different ways. For the $p$-dimensional scatter plot, the rows of $\mathbf{X}$ represent $n$ points in $p$-dimensional space. We can write
$$\underset{(n \times p)}{\mathbf{X}}=\left[\begin{array}{rclc} x_{11} & x_{12} & \cdots & x_{1 p} \ x_{21} & x_{22} & \cdots & x_{2 p} \ \vdots & \vdots & \ddots & \vdots \ x_{n 1} & x_{n 2} & \cdots & x_{n p} \end{array}\right]=\left[\begin{array}{c} \mathbf{x}{1}^{\prime} \ \mathbf{x}{2}^{\prime} \ \vdots \ \mathbf{x}{n}^{\prime} \end{array}\right] \leftarrow n \text {th (multivariate) observation }$$ The row vector $\mathbf{x}{j}^{\prime}$, representing the $j$ th observation, contains the coordinates of a point.

The scatter plot of $n$ points in $p$-dimensional space provides information on the locations and variability of the points. If the points are regarded as solid spheres, the sample mean vector $\overline{\mathbf{x}}$, given by (1-8), is the center of balance. Variability occurs in more than one direction, and it is quantified by the sample variance-covariance matrix $\mathbf{S}_{n}$. A single numerical measure of variability is provided by the determinant of the sample variance-covariance matrix. When $p$ is greater than 3 , this scatter plot representation cannot actually be graphed. Yet the consideration of the data as $n$ points in $p$ dimensions provides insights that are not readily available from algebraic expressions. Moreover, the concepts illustrated for $p=2$ or $p=3$ remain valid for the other cases.

## 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。