Posted on Categories:Time Series, 数据科学代写, 时间序列, 统计代写, 统计代考

统计代写|时间序列分析代写Time-Series Analysis代考|The PCA based on the sample covariance matrix

avatest™

avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

统计代写|时间序列分析代写Time-Series Analysis代考|The orthogonal factor model

Given a weakly stationary $m$-dimensional random vector at time $t, \mathbf{Z}t=\left[Z{1, t}, Z_{2, t}, \ldots, Z_{m, t}\right]^{\prime}$ with mean $\boldsymbol{\mu}=\left(\mu_1, \mu_2, \ldots, \mu_m\right)^{\prime}$, and covariance matrix $\boldsymbol{\Gamma}$, the factor model assumes that $\mathbf{Z}t$ is dependent on a small number of $k$ unobservable factors, $F{j, t}, j=1,2, \ldots, k$, known as common factors, and $m$ additional noises $\varepsilon_{i, t}, i=1,2, \ldots, m$, also known as specific factors, that is
\begin{aligned} & Z_{1, t}-\mu_1=\ell_{1,1} F_{1, t}+\ell_{1,2} F_{2, t}+\cdots+\ell_{1, k} F_{k, t}+\varepsilon_{1, t}, \ & Z_{2, t}-\mu_2=\ell_{2,1} F_{1, t}+\ell_{2,2} F_{2, t}+\cdots+\ell_{2, k} F_{k, t}+\varepsilon_{2, t}, \ & \vdots \ & Z_{m, t}-\mu_m=\ell_{m, 1} F_{1, t}+\ell_{m, 2} F_{2, t}+\cdots+\ell_{m, k} F_{k, t}+\varepsilon_{m, t} . \end{aligned}

More compactly, we can write the system in following matrix form,
$$\underset{m \times 1}{\dot{\mathbf{Z}}t}=\underset{(m \times k)(k \times 1)}{\mathbf{L}}+\underset{(m \times 1)}{\mathbf{F}_t},$$ where $\dot{\mathbf{Z}}_t=\left(\mathbf{Z}_t-\boldsymbol{\mu}\right), \mathbf{F}_t=\left(F{1, t}, F_{2, t}, \ldots, F_{k, t}\right)^{\prime}$ is a $(k \times 1)$ vector of factors at time $t, \mathbf{L}=\left[\ell_{i, j}\right]$ is a $(m \times k)$ loading matrix, with $\ell_{i, j}$ is the loading of the $i$ th variable on the $j$ th factor, $i=1,2, \ldots, m$, $j=1,2, \ldots, k$, and $\varepsilon_t=\left(\varepsilon_{1, t}, \varepsilon_{2, t}, \ldots, \varepsilon_{m, t}\right)^{\prime}$ is a $(m \times 1)$ vector of noises, with $E\left(\boldsymbol{\varepsilon}_t\right)=\mathbf{0}$, and $\operatorname{Cov}\left(\boldsymbol{\varepsilon}_t\right)=\operatorname{diag}\left{\sigma_1^2, \sigma_2^2, \ldots, \sigma_m^2\right}$.

The factor model in Eq. (5.2) is an orthogonal factor model if it satisfies the following assumptions:

1. $E\left(\mathbf{F}_t\right)=\mathbf{0}$, and $\operatorname{Cov}\left(\mathbf{F}_t\right)=\mathbf{I}_k$, the $(k \times k)$ identity matrix,
2. $E\left(\boldsymbol{\varepsilon}_t\right)=\mathbf{0}$, and $\operatorname{Cov}\left(\varepsilon_t\right)=\mathbf{\Sigma}=\operatorname{diag}\left{\sigma_1^2, \sigma_2^2, \ldots, \sigma_m^2\right}$, a $(m \times m)$ diagonal matrix, and
3. $\mathbf{F}_t$ and $\boldsymbol{\varepsilon}_t$ are independent and so $\operatorname{Cov}\left(\mathbf{F}_t, \boldsymbol{\varepsilon}_t\right)=E\left(\mathbf{F}_t \boldsymbol{\varepsilon}_t^{\prime}\right)=\mathbf{0}$, a $(k \times m)$ zero matrix.

统计代写|时间序列分析代写Time-Series Analysis代考|The principal component method

Given observations $\mathbf{Z}t=\left(Z{1, t}, Z_{2, t}, \ldots, Z_{m, t}\right)^{\prime}$, for $t=1,2, \ldots, n$, and its $m \times m$ sample covariance matrix $\hat{\mathbf{\Gamma}}=\left[\hat{\gamma}_{i, j}\right]$, a natural method of estimation is simply to use the principle component analysis introduced in Chapter 4 and choose $k$, which is much less than $m$, common factors from the first $k$ largest eigenvalue-eigenvector pairs in $\left(\hat{\lambda}_1, \hat{\boldsymbol{\alpha}}_1\right),\left(\hat{\lambda}_2, \hat{\boldsymbol{\alpha}}_2\right), \ldots,\left(\hat{\lambda}_m, \hat{\boldsymbol{\alpha}}_m\right)$, with $\hat{\lambda}_1 \geq \hat{\lambda}_2 \geq, \ldots, \geq \hat{\lambda}_m$. Let $\hat{\mathbf{L}}$ be the estimate of $\mathbf{L}$. Then,
$$\underset{m \times k}{\hat{\mathbf{L}}}=\left[\sqrt{\hat{\lambda}_1} \hat{\boldsymbol{\alpha}}_1 \sqrt{\hat{\lambda}_2} \hat{\boldsymbol{\alpha}}_2 \ldots \sqrt{\hat{\lambda}_k} \hat{\boldsymbol{\alpha}}_k\right],$$
and the estimated specific variances are obtained by
$$\hat{\boldsymbol{\Sigma}}=\left[\begin{array}{cccccc} \hat{\sigma}_1^2 & 0 & . & \cdots & . & 0 \ 0 & \hat{\sigma}_2^2 & 0 & \cdots & . & 0 \ . & 0 & . & \cdots & . & . \ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \ 0 & . & . & \cdots & . & 0 \ 0 & . & . & \cdots & 0 & \hat{\sigma}_m^2 \end{array}\right],$$

with the $i$ th specific variance estimate being
$$\hat{\sigma}i^2=\hat{\gamma}{i, i}-\left(\hat{\ell}{i, 1}^2+\hat{\ell}{i, 2}^2+\cdots+\hat{\ell}{i, k}^2\right),$$ where the sum of squares is the estimate of the $i$ th communality $$\hat{c}_i^2=\left(\hat{\ell}{i, 1}^2+\hat{\ell}{i, 2}^2+\cdots+\hat{\ell}{i, k}^2\right) .$$

统计代写|时间序列分析代写Time-Series Analysis代考|The PCA based on the sample covariance matrix

\begin{aligned} & Z_{1, t}-\mu_1=\ell_{1,1} F_{1, t}+\ell_{1,2} F_{2, t}+\cdots+\ell_{1, k} F_{k, t}+\varepsilon_{1, t}, \ & Z_{2, t}-\mu_2=\ell_{2,1} F_{1, t}+\ell_{2,2} F_{2, t}+\cdots+\ell_{2, k} F_{k, t}+\varepsilon_{2, t}, \ & \vdots \ & Z_{m, t}-\mu_m=\ell_{m, 1} F_{1, t}+\ell_{m, 2} F_{2, t}+\cdots+\ell_{m, k} F_{k, t}+\varepsilon_{m, t} . \end{aligned}

$$\underset{m \times 1}{\dot{\mathbf{Z}}t}=\underset{(m \times k)(k \times 1)}{\mathbf{L}}+\underset{(m \times 1)}{\mathbf{F}t},$$其中$\dot{\mathbf{Z}}_t=\left(\mathbf{Z}_t-\boldsymbol{\mu}\right), \mathbf{F}_t=\left(F{1, t}, F{2, t}, \ldots, F_{k, t}\right)^{\prime}$是$(k \times 1)$的因子向量，$t, \mathbf{L}=\left[\ell_{i, j}\right]$是$(m \times k)$的加载矩阵，$\ell_{i, j}$是$i$变量对$j$因子的加载，$i=1,2, \ldots, m$, $j=1,2, \ldots, k$, $\varepsilon_t=\left(\varepsilon_{1, t}, \varepsilon_{2, t}, \ldots, \varepsilon_{m, t}\right)^{\prime}$是$(m \times 1)$的噪声向量，有$E\left(\boldsymbol{\varepsilon}_t\right)=\mathbf{0}$，和$\operatorname{Cov}\left(\boldsymbol{\varepsilon}_t\right)=\operatorname{diag}\left{\sigma_1^2, \sigma_2^2, \ldots, \sigma_m^2\right}$。

Eq.(5.2)中的因子模型是一个正交因子模型，它满足以下假设:

$E\left(\mathbf{F}_t\right)=\mathbf{0}$， $\operatorname{Cov}\left(\mathbf{F}_t\right)=\mathbf{I}_k$, $(k \times k)$单位矩阵，

$E\left(\boldsymbol{\varepsilon}_t\right)=\mathbf{0}$， $\operatorname{Cov}\left(\varepsilon_t\right)=\mathbf{\Sigma}=\operatorname{diag}\left{\sigma_1^2, \sigma_2^2, \ldots, \sigma_m^2\right}$, $(m \times m)$对角矩阵，和

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。