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# 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

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## 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

The eigenvalues and eigenvectors, which are often known as variances and component loadings, of the sample variance-covariance matrix, are given in Table 4.1.

Figure 4.2 shows a useful plot, which is often known as a scree plot or screeplot. It is a plot of eigenvalues $\hat{\lambda}i$ versus $i$, that is, the magnitude of an eigenvalue versus its number. To determine the suitable number, $k$, of components, we look for an elbow in the plot, where the component preceding the vertex of the elbow is chosen to be cutoff point $k$. In our example, we will choose $k$ to be 3 or 4 . The eigenvalues of the first four components account for \begin{aligned} & \left(\frac{\hat{\lambda}_1+\hat{\lambda}_2+\hat{\lambda}_3+\hat{\lambda}_4}{\hat{\lambda}_1+\hat{\lambda}_2+\hat{\lambda}_3+\hat{\lambda}_4+\hat{\lambda}_5+\hat{\lambda}_6+\hat{\lambda}_7+\hat{\lambda}_8+\hat{\lambda}_9+\hat{\lambda}{10}}\right) 100 \% \ & =\left(\frac{0.00058+0.0003+0.00018+0.00011}{0.00151}\right) 100 \%=77.5 \% \end{aligned}
of the total sample variance.
The four sample principal components are
\begin{aligned} \hat{Y}1= & \hat{\alpha}_1 \mathbf{Z}=-0.287 Z_1-0.354 Z_2 \ & -0.47 Z_6-0.369 Z_7-0.539 Z_8-391 Z_9-0.324 Z{10} \ \hat{Y}2= & \hat{\alpha}_2 \mathbf{Z}=-0.104 Z_1-0.126 Z_2-0.431 Z_3-0.513 Z_4-0.362 Z_5 \ & -0.329 Z_6-0.167 Z_7+0.289 Z_8+0.175 Z_9+0.379 Z{10} \ \hat{Y}3= & \hat{\alpha}_3 \mathbf{Z}=-0.407 Z_1-0.243 Z_2-0.173 Z_3-0.319 Z_4-0.286 Z_5 \ & +0.596 Z_6+0.316 Z_7-0.191 Z_8-0.251 Z{10} \ \hat{Y}4= & \hat{\alpha}_4 \mathbf{Z}=0.576 Z_1+0.552 Z_2-0.354 Z_3-0.231 Z_4-0.197 Z_5 \ & +0.126 Z_7-0.222 Z_8-0.256 Z{10} \end{aligned}

Now, let us examine the four components more carefully. The first component represents the general market other than communication technology sector. The second component represents the contrast between financial and non-financial sectors. The third component represents the contrast between health and non-health sectors. The fourth component represents the contrast between oil and non-oil industries. Thus, the PCA has provided us with four components that contain a vast amount of information for the 10 stock returns traded on the New York Stock Exchange.

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

Now let us try the PCA using the sample correlation matrix, which is based on the standardized
The eigenvalues and eigenvectors, which are also known as variances and component loadings, of the sample correlation matrix are given in Table 4.2. The screeplot is shown in Figure 4.3 .

The screeplot again indicates $k=4$. The eigenvalues of the first four components account for
$$\left(\frac{\hat{\lambda}1+\hat{\lambda}_2+\hat{\lambda}_3+\hat{\lambda}_4}{m}\right) 100 \%=\left(\frac{3.393+2.21+1.196+0.939}{10}\right) 100 \%=77.38 \%$$ which is almost the same as the one obtained using the covariance matrix. The four sample principal components are now \begin{aligned} \hat{Y}_1= & \hat{\alpha}_1 \hat{\mathbf{U}}=-0.287 U_1-0.354 U_2-0.143 U_4-0.15 U_5 \ & -0.346 U_6-0.364 U_7-0.433 U_8-0.458 U_9-0.31 U{10} \ \hat{Y}2= & \hat{\alpha}_2 \hat{\mathbf{U}}=-0.155 U_1-0.137 U_2-0.491 U_3-0.506 U_4-0.49 U_5 \ & +0.236 U_8+0.231 U_9+0.32 U{10} \ \hat{Y}3= & \hat{\alpha}_3 \hat{\mathbf{U}}=0.654 U_1+0.464 U_2-0.199 U_3-0.418 U_6-0.354 U_7 \ \hat{Y}_4= & \hat{\alpha}_4 \hat{\mathbf{U}}=-0.136 U_1-0.32 U_2+0.233 U_3+0.171 U_4+0.341 U_5 \ & -0.405 U_6-0.429 U_7+0.281 U_8+0.206 U_9+0.458 U{10} \end{aligned}

Now, let us examine the four components. The first component now represents the general stock market. The second component represents the contrast mainly between financial and nonhealth related sectors. The third component represents the contrast between oil and health sectors. The fourth component now represents the contrast between financial/technology and nonfinancial/non-technology industry. For this data set, the PCA results from the covariance matrix and the correlation matrix are very much equivalent.

## 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} \hat{Y}1= & \hat{\alpha}_1 \mathbf{Z}=-0.287 Z_1-0.354 Z_2 \ & -0.47 Z_6-0.369 Z_7-0.539 Z_8-391 Z_9-0.324 Z{10} \ \hat{Y}2= & \hat{\alpha}_2 \mathbf{Z}=-0.104 Z_1-0.126 Z_2-0.431 Z_3-0.513 Z_4-0.362 Z_5 \ & -0.329 Z_6-0.167 Z_7+0.289 Z_8+0.175 Z_9+0.379 Z{10} \ \hat{Y}3= & \hat{\alpha}_3 \mathbf{Z}=-0.407 Z_1-0.243 Z_2-0.173 Z_3-0.319 Z_4-0.286 Z_5 \ & +0.596 Z_6+0.316 Z_7-0.191 Z_8-0.251 Z{10} \ \hat{Y}4= & \hat{\alpha}_4 \mathbf{Z}=0.576 Z_1+0.552 Z_2-0.354 Z_3-0.231 Z_4-0.197 Z_5 \ & +0.126 Z_7-0.222 Z_8-0.256 Z{10} \end{aligned}

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

$$\left(\frac{\hat{\lambda}1+\hat{\lambda}_2+\hat{\lambda}_3+\hat{\lambda}_4}{m}\right) 100 \%=\left(\frac{3.393+2.21+1.196+0.939}{10}\right) 100 \%=77.38 \%$$这与使用协方差矩阵得到的结果几乎相同。四个样本主成分为 \begin{aligned} \hat{Y}_1= & \hat{\alpha}_1 \hat{\mathbf{U}}=-0.287 U_1-0.354 U_2-0.143 U_4-0.15 U_5 \ & -0.346 U_6-0.364 U_7-0.433 U_8-0.458 U_9-0.31 U{10} \ \hat{Y}2= & \hat{\alpha}_2 \hat{\mathbf{U}}=-0.155 U_1-0.137 U_2-0.491 U_3-0.506 U_4-0.49 U_5 \ & +0.236 U_8+0.231 U_9+0.32 U{10} \ \hat{Y}3= & \hat{\alpha}_3 \hat{\mathbf{U}}=0.654 U_1+0.464 U_2-0.199 U_3-0.418 U_6-0.354 U_7 \ \hat{Y}_4= & \hat{\alpha}_4 \hat{\mathbf{U}}=-0.136 U_1-0.32 U_2+0.233 U_3+0.171 U_4+0.341 U_5 \ & -0.405 U_6-0.429 U_7+0.281 U_8+0.206 U_9+0.458 U{10} \end{aligned}

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