Ronald WESONGA (PhD)
13 September 2022
| Objective | Description |
|---|---|
| Data reduction | index of measurement |
| Sorting and grouping | develop clusters |
| screen procedures | |
| seperation rules | |
| investigate dependence | identify responsible factors |
| examine relationships | |
| predict | predict outcomes |
| identify risks | |
| test hypotheses | test for differences |
Through the VAM system:
\[ X_{np} = \begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1p} \\ \vdots & \vdots & \ddots & \vdots \\ x_{j1} & x_{j2} & \cdots & x_{jp} \\ \vdots & \vdots & \ddots & \vdots \\ x_{n1} & x_{n2} & \cdots & x_{np} \end{bmatrix} \]
\[ \begin{bmatrix} Sales(OMR) &| & 32 & 42 & 52 & 48 & 58 \\ \hline Books(No.) &| & 2 & 4 & 6 & 3 & 6 \\ \end{bmatrix} \]
\[ X_{5X2} = \begin{bmatrix} 32 & 2 \\ 42 & 4 \\ 52 & 6 \\ 48 & 3 \\ 58 & 6 \end{bmatrix} \]
\[ \begin{bmatrix} Age(Yrs.) &| & 23 & 20 & 22 & 20 \\ \hline Mark(No.) &| & 82 & 4 & 6 & 3 \\ \hline Height(M) &| & 1.53 & 1.58 & 1.60 & 1.65 \\ \end{bmatrix} \]
Yes we can: Let the measurements on the first variable be \(x_{11}, x_{21} \cdots,x_{n1}\) and for the \(i^{th}\) variable will be \(x_{1i}, x_{2i} \cdots,x_{ji},\cdots, x_{ni}\). Thus,
\[ X_{n1} = \begin{bmatrix} x_{11} \\ \vdots \\ x_{j1} \\ \vdots \\ x_{n1} \end{bmatrix} X_{ni} = \begin{bmatrix} x_{1i} \\ \vdots \\ x_{ji} \\ \vdots \\ x_{ni} \end{bmatrix}\]
\[\bar{X}_k = \frac{1}{n}\sum_{j=1}^n X_{jk}\]
\[S_{kk}=S_k^2=\frac{1}{n}\sum_{j=1}^n \left(X_{jk}-\bar{X}_k\right)^2\]
\[S_{ik}=\frac{1}{n}\sum_{j=1}^n \left(X_{ji}-\bar{X}_i\right)\left(X_{jk}-\bar{X}_k\right)\] \[r_{ik}=\frac{\sum_{j=1}^n \left(X_{ji}-\bar{X}_i\right)\left(X_{jk}-\bar{X}_k\right)}{\sqrt{\sum_{j=1}^n \left(X_{ji}-\bar{X}_i\right)^2}\sqrt{\sum_{j=1}^n \left(X_{jk}-\bar{X}_k\right)^2}}\]
\[ \bar{X} = \begin{bmatrix} \bar{X}_{1} \\ \vdots \\ \bar{X}_{i} \\ \vdots \\ \bar{X}_{p} \end{bmatrix}\]
\[ S_{n} = \begin{bmatrix} S_{11} & S_{12} & \cdots & S_{1p} \\ \vdots & \vdots & \ddots & \vdots \\ S_{i1} & S_{i2} & \cdots & S_{ip} \\ \vdots & \vdots & \ddots & \vdots \\ S_{p1} & S_{p2} & \cdots & S_{pp} \end{bmatrix} R = \begin{bmatrix} 1 & r_{12} & \cdots & r_{1p} \\ \vdots & \vdots & \ddots & \vdots \\ r_{i1} & r_{i2} & \cdots & r_{ip} \\ \vdots & \vdots & \ddots & \vdots \\ r_{p1} & r_{p2} & \cdots & 1 \end{bmatrix}\]
\[ X_{4X2} = \begin{bmatrix} 42 & 4 \\ 52 & 6 \\ 48 & 3 \\ 58 & 6 \end{bmatrix} \] \[\bar{X}_1=\sum_{j=1}^4X_{j1}=50;~~~~ \bar{X}_2=\sum_{j=1}^4X_{j2}=4\]
\[ S_{4} = \begin{bmatrix} 34 & -1.5 \\ -1.5 & 0.5 \\ \end{bmatrix} R = \begin{bmatrix} 1 & -0.36 \\ -0.36 & 1 \\ \end{bmatrix} \]
If \(P(x_1,x_2, \cdots,x_p)\) and \(O(0,0,\cdots,0)\) are two points, then
\[d(O,P)=\sqrt{x_1^2+x_2^2+\cdots+x_p^2}\] elseif \(P(x_1,x_2, \cdots,x_p)\) and \(Q(y_1,y_2, \cdots,y_p)\), then
\[d(O,P)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2+\cdots+(x_p-y_p)^2}\]
If \(P(x_1,x_2, \cdots,x_p)\) and \(O(0,0,\cdots,0)\) are two points, then
\[d(O,P)=\sqrt{\left(\frac{x_1}{\sqrt{s_{11}}}\right)^2+\cdots+\left(\frac{x_p}{\sqrt{s_{pp}}}\right)^2}\] elseif \(P(x_1,x_2, \cdots,x_p)\) and \(Q(y_1,y_2, \cdots,y_p)\), then
\[d(O,P)=\sqrt{\left(\frac{x_1-y_1}{\sqrt{s_{11}}}\right)^2+\cdots+\left(\frac{x_p-y_p}{\sqrt{s_{pp}}}\right)^2}\]
Notice that when \(S_{11}=S_{22}\), then use the euclidean distance
Example 1.14 assumes two points, \(P(x_1,x_2),O(0,0)\), and \[d^2(O,P)=\frac{x_1^2}{4}+\frac{x_2^2}{1}=1\longrightarrow \{(0,1),(0,-1),(2,0),(1,\frac{\sqrt{3}}{2})\}\]
Distance measurement after the coordinate system is rotated, What is the formula for the distance?
For \(P(\tilde{x}_1, \tilde{x}_2)\) and O(0,0), then
\[d(O,P)=\sqrt{\frac{\tilde{x}_1^2}{\tilde{s}_{11}}+\frac{\tilde{x}_2^2}{\tilde{s}_{22}}}\] where \[\tilde{x}_1=x_1cos\theta+x_2sin\theta\] \[\tilde{x}_2=-x_1sin\theta+x_2cos\theta\] implying that \[d(O,P)=\sqrt{a_{11}x_1^2+2a_{12}x_1x_2+a_{22}x_2^2}\] and \[d(O,P)=\sqrt{a_{11}(x_1-y_1)^2+2a_{12}(x_1-y_1)(x_2-y_2)+a_{22}(x_2-y_2)^2}\]
where: \[{\scriptstyle a_{11}=\frac{cos^2\theta}{cos^2\theta s_{11}+2sin\theta cos\theta s_{12}+sin^2\theta s_{22}} + \frac{sin^2\theta}{cos^2\theta s_{22}-2sin\theta cos\theta s_{12}+sin^2\theta s_{11}}} \] \[{\scriptstyle a_{22}=\frac{sin^2\theta}{cos^2\theta s_{11}+2sin\theta cos\theta s_{12}+sin^2\theta s_{22}} + \frac{cos^2\theta}{cos^2\theta s_{22}-2sin\theta cos\theta s_{12}+sin^2\theta s_{11}}}\] \[{\scriptstyle a_{12}=\frac{cos\theta sin\theta}{cos^2\theta s_{11}+2sin\theta cos\theta s_{12}+sin^2\theta s_{22}} + \frac{sin\theta cos\theta}{cos^2\theta s_{22}-2sin\theta cos\theta s_{12}+sin^2\theta s_{11}}}\]