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# 统计代写|时间序列分析代写Time Series Analysis代考|Correlation and partial correlation matrix functions

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## 统计代写|时间序列分析代写Time Series Analysis代考|Correlation and partial correlation matrix functions

Let $\boldsymbol{Z}t=\left[Z{1, t}, Z_{2, t}, \ldots, Z_{m, t}\right]^{\prime}, t=0, \pm 1, \pm 2, \ldots$ be a $m$-dimensional stationary real-valued vector process so that $E\left(Z_{i, t}\right)=\mu_i$ is constant for each $i=1,2, \ldots, m$ and the cross-covariance between $Z_{i, t}$ and $Z_{j, s}$, for all $i=1,2, \ldots, m$ and $j=1,2, \ldots, m$, are functions only of the time difference $(s-t)$. Hence, we have the mean vector
$$E\left(\mathbf{Z}t\right)=\boldsymbol{\mu}=\left[\begin{array}{c} \mu_1 \ \mu_2 \ \vdots \ \mu_m \end{array}\right]$$ and the lag $k$ covariance matrix \begin{aligned} \boldsymbol{\Gamma}(k) & =\operatorname{Cov}\left{\mathbf{Z}_t, \mathbf{Z}{t+k}\right}=E\left[\left(\mathbf{Z}t-\boldsymbol{\mu}\right)\left(\mathbf{Z}{t+k}-\boldsymbol{\mu}\right)^{\prime}\right] \ & =E\left[\begin{array}{c} Z_{1, t}-\mu_1 \ Z_{2, t}-\mu_2 \ \vdots \ Z_{m, t}-\mu_m \end{array}\right]\left[Z_{1, t+k}-\mu_1, Z_{2, t+k}-\mu_2, \ldots, Z_{m, t+k}-\mu_{\mathrm{m}}\right] \ & =\left[\begin{array}{cccc} \gamma_{1,1}(k) & \gamma_{1,2}(k) & \cdots & \gamma_{1, m}(k) \ \gamma_{2,1}(k) & \gamma_{2,2}(k) & \cdots & \gamma_{2, m}(k) \ \vdots & \vdots & \vdots & \vdots \ \gamma_{m, 1}(k) & \gamma_{m, 2}(k) & \cdots & \gamma_{m, m}(k) \end{array}\right], \end{aligned}
where
$$\gamma_{i, j}(k)=E\left(Z_{i, t}-\mu_i\right)\left(Z_{j, t+k}-\mu_j\right)$$
for $k=0, \pm 1, \pm 2, \ldots, i=1,2, \ldots, m$, and $j=1,2, \ldots, m$. As a function of $k, \boldsymbol{\Gamma}(k)$ is called the covariance matrix function for the vector process $\mathbf{Z}t$. Also, $i=j, \gamma{i, i}(k)$ is the autocovariance function for the $i$ th component process, $Z_{i, t}$, and $i \neq j, \gamma_{i, j}(k)$ is the cross-covariance function between component series $Z_{i, t}$ and $Z_{j, t}$. The matrix $\Gamma(0)$ can be easily seen to be the contemporaneous variance-covariance matrix of the process.

## 统计代写|时间序列分析代写Time Series Analysis代考|Vector white noise process

The $m$-dimensional vector process, $\mathbf{a}t$, is said to be a vector white noise process with mean vector $\mathbf{0}$ and covariance matrix function $\mathbf{\Sigma}$ if $$E\left[\mathbf{a}_t \mathbf{a}{t+k}^{\prime}\right]=\left{\begin{array}{l} \mathbf{\Sigma}, \text { if } k=0, \ \mathbf{0}, \text { if } k \neq 0, \end{array}\right.$$
where $\mathbf{\Sigma}$ is a $m \times m$ symmetric positive definite matrix. Note that although the components of the white noise process are uncorrelated at different times, they may be contemporaneously correlated. It is a Gaussian white noise process if $\mathbf{a}_t$ also follows a multivariate normal distribution. Unless mentioned otherwise, $\mathbf{a}_t$ will be used to denote a Gaussian vector white noise process with mean vector $\mathbf{0}$ and covariance matrix function $\mathbf{\Sigma}, \mathbf{V W N}(\mathbf{0}, \mathbf{\Sigma})$, in this book.

## 统计代写|时间序列分析代写Time Series Analysis代考|Correlation and partial correlation matrix functions

$$E\left(\mathbf{Z}t\right)=\boldsymbol{\mu}=\left[\begin{array}{c} \mu_1 \ \mu_2 \ \vdots \ \mu_m \end{array}\right]$$和滞后$k$协方差矩阵\begin{aligned} \boldsymbol{\Gamma}(k) & =\operatorname{Cov}\left{\mathbf{Z}t, \mathbf{Z}{t+k}\right}=E\left[\left(\mathbf{Z}t-\boldsymbol{\mu}\right)\left(\mathbf{Z}{t+k}-\boldsymbol{\mu}\right)^{\prime}\right] \ & =E\left[\begin{array}{c} Z{1, t}-\mu_1 \ Z_{2, t}-\mu_2 \ \vdots \ Z_{m, t}-\mu_m \end{array}\right]\left[Z_{1, t+k}-\mu_1, Z_{2, t+k}-\mu_2, \ldots, Z_{m, t+k}-\mu_{\mathrm{m}}\right] \ & =\left[\begin{array}{cccc} \gamma_{1,1}(k) & \gamma_{1,2}(k) & \cdots & \gamma_{1, m}(k) \ \gamma_{2,1}(k) & \gamma_{2,2}(k) & \cdots & \gamma_{2, m}(k) \ \vdots & \vdots & \vdots & \vdots \ \gamma_{m, 1}(k) & \gamma_{m, 2}(k) & \cdots & \gamma_{m, m}(k) \end{array}\right], \end{aligned}

$$\gamma_{i, j}(k)=E\left(Z_{i, t}-\mu_i\right)\left(Z_{j, t+k}-\mu_j\right)$$

## 统计代写|时间序列分析代写Time Series Analysis代考|Vector white noise process

$m$维向量过程$\mathbf{a}t$被认为是一个向量白噪声过程，其平均向量$\mathbf{0}$和协方差矩阵函数$\mathbf{\Sigma}$ if $$E\left[\mathbf{a}_t \mathbf{a}{t+k}^{\prime}\right]=\left{\begin{array}{l} \mathbf{\Sigma}, \text { if } k=0, \ \mathbf{0}, \text { if } k \neq 0, \end{array}\right.$$

## 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。