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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 4.3.
The two sample principal components are
\begin{aligned} & \hat{Y}_1=0.516 Z_1+0.003 Z_2+0.227 Z_3+0.461 Z_4+0.686 Z_5 \ & \hat{Y}_2=0.081 Z_1-0.057 Z_2-0.331 Z_3-0.755 Z_4+0.557 Z_5 \end{aligned}
The first component explains $95.76 \%$ of the total sample variance, and the first two explain $99.04 \%$. Thus, sample variation is very much summarized by the first principle component or the first two principle components. Figure 4.5 shows the useful screeplot where the vertex of the elbow can be easily seen to be $k=1$.

Now, let us examine component 1 more carefully. In this component, the loadings are all positive. The component can be regarded as the CPI growth component that grew over the time period that we observed. The five variables are combined into a composite score, which is plotted in Figure 4.6, and it follows a combination of patterns observed mainly for gasoline and energy in Figure 4.4.

Thus, the PCA has provided us with a single component that contains the vast majority of information for the five individual variables. From this, we can conclude that gasoline and energy were the true drivers of the overall economy for the Greater New York City area during the period between 1986 and 2014 .

## 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. The eigenvalues and eigenvectors, which are also known as variances and component loadings, of the sample correlation matrix are given in Table 4.4.

The two sample principal components are
\begin{aligned} & \hat{Y}_1=0.503 Z_1+0.044 Z_2+0.501 Z_3+0.499 Z_4+0.495 Z_5 \ & \hat{Y}_2=0.100 Z_1-0.987 Z_2-0.107 Z_3+0.032 Z_4+0.061 Z_5 \end{aligned}
The first component explains $76.74 \%$ of the total sample variance, and the first two explain $97.1 \%$. Thus, sample variation of the five industries is primarily summarized by the first two principle components. Figure 4.7 shows the screeplot, which clearly indicates $k=2$.
From Table 4.4, we see that the loadings in component 1 are all positive, almost equal for energy, commodities, housing, and gas, and have strong positive correlations among them. It represents the CPI growth over the time period that we observed. The loadings in component 2 are relatively positive small numbers for energy, housing, and gas, and negative for apparel and commodities. It represents the market contrast between consumer goods and utility housing. Since the loading for apparel is especially dominating, component 2 can also be simply regarded as representing the apparel sector.

The five variables are combined into two composite scores, which are plotted in Figure 4.8.

## 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=0.516 Z_1+0.003 Z_2+0.227 Z_3+0.461 Z_4+0.686 Z_5 \ & \hat{Y}_2=0.081 Z_1-0.057 Z_2-0.331 Z_3-0.755 Z_4+0.557 Z_5 \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

\begin{aligned} & \hat{Y}_1=0.503 Z_1+0.044 Z_2+0.501 Z_3+0.499 Z_4+0.495 Z_5 \ & \hat{Y}_2=0.100 Z_1-0.987 Z_2-0.107 Z_3+0.032 Z_4+0.061 Z_5 \end{aligned}

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