多變量分析 HW3
劉昱維, 吳冠瑋, 廖永賦, 謝靖惟, 黃奎鈞
Q1: 9.12
\(\mathbf{S_n} = \frac{23}{24} \mathbf{S} = \begin{pmatrix} 0.01061 & 0.00768 & 0.00782 \\ 0.00768 & 0.00615 & 0.00575 \\ 0.00782 & 0.00575 & 0.00649 \\ \end{pmatrix}\)
(a) Specific Variances
\(\mathbf{S_n} \approx \hat{\mathbf{L}} \hat{\mathbf{L}}^T + \hat{\mathbf{\Psi}}\), where \(diag(\mathbf{S_n}) = diag(\hat{\mathbf{L}} \hat{\mathbf{L}}^T) + diag(\hat{\mathbf{\Psi}})\)
Hence, \(\hat{\mathbf{\Psi}} = diag(\hat{\mathbf{\Psi}}) = diag(\mathbf{S_n} - \hat{\mathbf{L}} \hat{\mathbf{L}}^T)\)
\(\hat{\mathbf{\Psi}} = \begin{pmatrix} 0.000166 & 0.000000 & 0.000000 \\ 0.000000 & 0.000495 & 0.000000 \\ 0.000000 & 0.000000 & 0.000639 \\ \end{pmatrix}\)
(b) Communalities
\(\sigma_{ii} = \ell_{i1}^2 + \ell_{i2}^2 + ~...~ + \ell_{im}^2 + \psi_i\)
\(h_i^2 = \ell_{i1}^2 + \ell_{i2}^2 + ~...~ + \ell_{im}^2 = \ell_{i1}^2\)
\(h_1^2 = 0.0104, ~ h_2^2 = 0.0057, ~ h_3^2 = 0.0059\)
(c) Proportion of variance explained by the factor
\(\frac{s_{11} + s_{22} + ~...~ + s_{pp}}{h_i^2} = 0.9441\)
(d) Residual Matrix
\(\mathbf{S_n} - \mathbf{\hat{L}}\mathbf{\hat{L}}^T - \mathbf{\hat{\Psi}} = \begin{pmatrix} 0.000000 & -0.000166 & -0.000164 \\ -0.000495 & 0.000000 & -0.000493 \\ -0.000637 & -0.000637 & 0.000000 \\ \end{pmatrix}\)
Q2: 9.32
(a) S PC
| RC1 | RC2 | RC3 | |
|---|---|---|---|
| YrHgt | 0.566 | 0.734 | 0.87 |
| FtFrBody | 31.179 | 83.935 | 24.022 |
| PrctFFB | 0.601 | 1.488 | 2.742 |
| Frame | 0.296 | 0.39 | 0.415 |
| BkFat | 0.014 | -0.006 | -0.055 |
| SaleHt | 1.021 | 0.934 | 0.82 |
| SaleWt | 122.551 | 39.068 | -17.48 |
(b) S ML
| ML1 | ML2 | ML3 | |
|---|---|---|---|
| YrHgt | -0.001 | 0.581 | 1.629 |
| FtFrBody | 21.802 | 84.458 | 31.383 |
| PrctFFB | -0.253 | 2.155 | 1.056 |
| Frame | 0.019 | 0.307 | 0.817 |
| BkFat | 0.033 | -0.015 | -0.027 |
| SaleHt | 0.498 | 0.841 | 1.532 |
| SaleWt | 119.136 | 33.925 | 38.8 |
(c) R PC
| RC1 | RC3 | RC2 | |
|---|---|---|---|
| YrHgt | 0.941 | 0.27 | -0.082 |
| FtFrBody | 0.447 | 0.794 | 0.205 |
| PrctFFB | 0.262 | 0.859 | -0.295 |
| Frame | 0.938 | 0.219 | -0.028 |
| BkFat | -0.231 | -0.339 | 0.812 |
| SaleHt | 0.833 | 0.419 | 0.109 |
| SaleWt | 0.352 | 0.43 | 0.722 |
(d) R ML
| ML1 | ML2 | ML3 | |
|---|---|---|---|
| YrHgt | 0.941 | 0.286 | 0.164 |
| FtFrBody | 0.414 | 0.505 | 0.553 |
| PrctFFB | 0.231 | 0.947 | 0.212 |
| Frame | 0.891 | 0.251 | 0.18 |
| BkFat | -0.256 | -0.514 | 0.273 |
| SaleHt | 0.755 | 0.269 | 0.434 |
| SaleWt | 0.253 | -0.05 | 0.879 |
(e) Compare S & R
The results obtained using covariance matrices are hard to interpret here. Since there are three kinds of units used: pound, inch, and self-defined scales(1-8), the factor loadings on some variables are very large and others small.
Interpretation of the factors is straightforward using results obtained from the covariance matrix.
By (c) and (d), 1. factor 1 has larger loadings on YrHgt, Frame, and SaleHt, which are all related to the heights of the bulls. 2. Factor 2 has large loadings on FtFrBody and PrctFFB and is negatively related to BkFat in both methods(PC & ML) of factor analysis. It might be called a lean factor. 3. In both PC & ML method of factor analysis, the loadings of factor 3 is large on SaleWt. The loading is also large on BkFat from the PC method, and the loading is medium on FtFrBody from the ML method. Factor 3 might be related to the weights of the bulls.
(f) scatter plots of factor2 vs factor1 in (a) and (c)
The point, 51, in the scatter plot on the left and the point, 16, in the scatter plot on the right seem to be outliers.
Q3
(a) Appropriate number of factors
By the unity criterion we use 4 factors1.
| RC4 | RC1 | RC2 | RC3 | |
|---|---|---|---|---|
| X1 | 0.131 | -0.215 | 0.905 | -0.044 |
| X2 | 0.953 | 0.212 | 0.05 | -0.023 |
| X3 | 0.293 | 0.899 | -0.165 | -0.043 |
| X4 | 0.027 | -0.091 | 0.955 | 0.054 |
| X5 | 0.933 | 0.261 | 0.111 | -0.076 |
| X6 | -0.083 | -0.043 | 0.041 | 0.994 |
| X7 | 0.228 | 0.918 | -0.23 | -0.028 |
| X8 | 0.057 | -0.224 | 0.935 | 0.051 |
| X9 | 0.935 | 0.276 | 0.073 | -0.048 |
| X10 | 0.273 | 0.896 | -0.199 | -0.016 |
(b) Label the Factors
RC4: 有利成分
此因素在第 2, 5, 9 題有較高的因素負荷量。這些題目似乎皆與健康有關。飲料能提供營養或避免添加不健康的物質，似與此因素有關。RC1: 有效解渴 此因素在第 3, 7, 10 題有較高的因素負荷量，這些題目皆與飲料是否解渴有關。
RC2: 獨特風味
此因素在第 1, 4, 8 題有較高的因素負荷量，這些題目皆強調飲料的口感。RC3: 綠能包裝
此因素僅在第 6 題有較高的因素負荷量。第 6 題是一獨特的題目，所有題目中僅其強調環境因素。因此一個因素僅在第 6 題有較高的因素負荷量顯得相當合理。
(c) Average factor scores of each brand
Average factor scores
| 有利成分 | 有效解渴 | 獨特風味 | 綠能包裝 | |
|---|---|---|---|---|
| Pepsi | -0.629 | -1.008 | -0.107 | 0.354 |
| Coke | -1.215 | -0.403 | 0.194 | -0.177 |
| Gatorade | 0.420 | 0.928 | -0.980 | -0.077 |
| Allsport | -0.177 | 0.666 | -1.074 | -0.181 |
| Lipton tea | 0.885 | 0.085 | 0.613 | 0.031 |
| Nestea | 0.615 | 0.122 | 0.821 | -0.044 |
(d) Interpret the positions of the six brands
在不考慮綠能包裝下，可以從各散布圖中看出 3 種飲料類型：
- Pepsi, Coke: 碳酸飲料
- Gatorade, Allsport: 運動飲料
- Lipton tea, Nestea: 茶飲
相同類型的飲料在散布圖中，通常位置會較為接近。例如，茶飲較為健康且有獨特風味，因此出現在出現在右上圖的右上方；碳酸飲料含糖量高，既不健康又不解渴，因此在左上圖中，出現在左下方；運動飲料則在有效解渴上，出現在類似的位置。
Although factor 4 is less than 1, it’s eigenvalue is very close to factor 3, hence we retain factor 4.↩