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# 统计代写|多元统计分析代考MULTIVARIATE STATISTICAL ANALYSIS代考|STA675 MATRIX INEQ\.!ALITIES AND MAXIMIZATION

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## 统计代写|多元统计分析代考MULTIVARIATE STATISTICAL ANALYSIS代考|MATRIX INEQ.!ALITIES AND MAXIMIZATION

Maximization principles play an important role in several multivariate techniques. Linear discriminant analysis, for example, is concerned with allocating observations to predetermined groups. The allocation rule is often a linear function of measurements that maximizes the separation between groups relative to their withingroup variability. As another example, principal components are linear combinations of measurements with maximum variability.

The matrix inequalities presented in this section will easily allow us to derive certain maximization results, which will be referenced in later chapters.

Cauchy-Schwarz Inequality. Let $\mathbf{b}$ and $\mathbf{d}$ be any two $p \times 1$ vectors. Then
$$\left(\mathbf{b}^{\prime} \mathbf{d}\right)^{2} \leqslant\left(\mathbf{b}^{\prime} \mathbf{b}\right)\left(\mathbf{d}^{\prime} \mathbf{d}\right)$$
with equality if and only if $\mathbf{b}=c \mathbf{d}($ or $\mathbf{d}=c \mathbf{b}$ ) for some constant $c$.
Proof. The inequality is obvious if either $\mathbf{b}=\mathbf{0}$ or $\mathbf{d}=\mathbf{0}$. Excluding this possibility, consider the vector $\mathbf{b}-x \mathbf{d}$, where $x$ is an arbitrary scalar. Since the length of $\mathbf{b}-x \mathbf{d}$ is positive for $\mathbf{b}-x \mathbf{d} \neq \mathbf{0}$, in this case
\begin{aligned} 0<(\mathbf{b}-x \mathbf{d})^{\prime}(\mathbf{b}-x \mathbf{d}) &=\mathbf{b}^{\prime} \mathbf{b}-x \mathbf{d}^{\prime} \mathbf{b}-\mathbf{b}^{\prime}(x \mathbf{d})+x^{2} \mathbf{d}^{\prime} \mathbf{d} \ &=\mathbf{b}^{\prime} \mathbf{b}-2 x\left(\mathbf{b}^{\prime} \mathbf{d}\right)+x^{2}\left(\mathbf{d}^{\prime} \mathbf{d}\right) \end{aligned}
The last expression is quadratic in $x$. If we complete the square by adding and subtracting the scalar $\left(\mathbf{b}^{\prime} \mathbf{d}\right)^{2} / \mathbf{d}^{\prime} \mathbf{d}$, we get
\begin{aligned} 0 &<\mathbf{b}^{\prime} \mathbf{b}-\frac{\left(\mathbf{b}^{\prime} \mathbf{d}\right)^{2}}{\mathbf{d}^{\prime} \mathbf{d}}+\frac{\left(\mathbf{b}^{\prime} \mathbf{d}\right)^{2}}{\mathbf{d}^{\prime} \mathbf{d}}-2 x\left(\mathbf{b}^{\prime} \mathbf{d}\right)+x^{2}\left(\mathbf{d}^{\prime} \mathbf{d}\right) \ &=\mathbf{b}^{\prime} \mathbf{b}-\frac{\left(\mathbf{b}^{\prime} \mathbf{d}\right)^{2}}{\mathbf{d}^{\prime} \mathbf{d}}+\left(\mathbf{d}^{\prime} \mathbf{d}\right)\left(x-\frac{\mathbf{b}^{\prime} \mathbf{d}}{\mathbf{d}^{\prime} \mathbf{d}}\right)^{2} \end{aligned}

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

Definition 2A.13. An $m \times k$ matrix, generally denoted by a boldface uppercase letter such as $\mathbf{A}, \mathbf{R}, \mathbf{\Sigma}$, and so forth, is a rectangular array of elements having $m$ rows and $k$ columns.
Examples of matrices are
$$\begin{gathered} \mathbf{A}=\left[\begin{array}{rr} -7 & 2 \ 0 & 1 \ 3 & 4 \end{array}\right], \quad \mathbf{B}=\left[\begin{array}{rrr} x & 3 & 0 \ 4 & -2 & 1 / x \end{array}\right], \quad \mathbf{I}=\left[\begin{array}{lll} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\right] \ \mathbf{\Sigma}=\left[\begin{array}{rrr} 1 & .7 & -.3 \ .7 & 2 & 1 \ -.3 & 1 & 8 \end{array}\right], \quad \mathbf{E}=\left[e_{1}\right] \end{gathered}$$
In our work, the matrix elements will be real numbers or functions taking on values in the real numbers.

Definition 2A.14. The dimension (abbreviated dim) of an $m \times k$ matrix is the ordered pair $(m, k) ; m$ is the row dimension and $k$ is the column dimension. The dimension of a matrix is frequently indicated in parentheses below the letter representing the matrix. Thus, the $m \times k$ matrix $\mathbf{A}$ is denoted by $\mathbf{A}$. In the preceding examples, the dimension of the matrix $\mathbf{\Sigma}$ is $3 \times 3$, and this information can be conveyed by writing $\mathbf{\Sigma}$.
$(3 \times 3)$
An $m \times k$ matrix, say, $\mathbf{A}$, of arbitrary constants can be written
$$\underset{(m \times k)}{\mathbf{A}}=\left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 k} \ a_{21} & a_{22} & \cdots & a_{2 k} \ \vdots & \vdots & \ddots & \vdots \ a_{m 1} & a_{m 2} & \cdots & a_{m k} \end{array}\right]$$
or more compactly as $\underset{(m \times k)}{\mathbf{A}}=\left{a_{i j}\right}$, where the index $i$ refers to the row and the index $j$ refers to the column.

## 统计代写|多元统计分析代考MULTIVARIATE STATISTIC ALANALYSIS代考|RANDOM VECTORS AND MATRICES

〈left 的分隔符缺失或无法识别 豆 $n \times p$ 随机矩阵。那么期望值 $\mathbf{X}$ ，表示

$E\left(X_{i j}\right)=\left{\int_{-\infty}^{\infty} x_{i j} f_{i j}\left(x_{i j}\right) d x_{i j} \quad\right.$ if $X_{i j}$ is a continuous random variable with probability density function $f_{i j}\left(x_{i j}\right) \sum_{\text {all } x_{i j}} x_{i j} p_{i j}\left(x_{i j}\right) \quad$ if $X$

## 统计代写|多元统计分析代考MULTIVARIATE STATISTICAL ANALYSIS代考|MEAN VECTORS AND COVARIANCE MATRICES

$\mu_{i}=\left{\int_{-\infty}^{\infty} x_{i} f_{i}\left(x_{i}\right) d x_{i} \quad\right.$ if $X_{i}$ is a continuous random variable with probability density function $f_{i}\left(x_{i}\right) \sum_{\text {all } x_{i}} x_{i} p_{i}\left(x_{i}\right) \quad$ if $X_{i}$ is a discrete
$\sigma_{i}^{2}=\left{\int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)^{2} f_{i}\left(x_{i}\right) d x_{i} \quad\right.$ if $X_{i}$ is a continuous random variable with probability density function $f_{i}\left(x_{i}\right) \sum_{\text {all } x_{i}}\left(x_{i}-\mu_{i}\right)^{2} p_{i}\left(x_{i}\right) \quad$ i

$\sigma_{i k}=E\left(X_{i}-\mu_{i}\right)\left(X_{k}-\mu_{k}\right)$
$=\left{\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)\left(x_{k}-\mu_{k}\right) f_{i k}\left(x_{i}, x_{k}\right) d x_{i} d x_{k} \quad\right.$ if $X_{i}, X_{k}$ are continuous random variables with the joint density $\sum_{a l l} x_{i}$ all $x_{k}\left(x_{i}-\mu_{i}\right)$

## 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。