010. Overview

MVA

Overview of mva (not ended)

Find relationships b/w $x_p,y_q$, e.g.,

• Response variables (variable directed)
  • PCA
  • Factor Analysis
  • mv Regression
  • Cannonical Correlation Analysis
• Experiment units (individual directed)
  • Discriminant Analysis
  • Cluster Analysis
  • MANOVA

Multivariate techniques tend to be exploratory.

• i.e. not hypothesis testing type

Experimental units must be independent. Time series data are not appropriate for this course.

Notation

Variable $y_1,\cdots ,y_p$

One observation $y^{\prime }=(y_1,\cdots ,y_p)$

Data Matrix $Y_{n\times p}=\left[\begin{array}{c}{y}^{\prime }=(y_1,\cdots ,y_p)\\ ⋮\\ y^{\prime }=(y_1,\cdots ,y_p)\end{array}\right]$

Random Samples: Suppose we intend to collect n sets of measurements on p variables, but not been observed yet. If $x_1^{\prime },\cdots ,x_n^{\prime }$ are independent observation from pdf $f(x)=f(x_1,\cdots ,x_p)$, then $x_1^{\prime },\cdots ,x_n^{\prime }$ are said to be rs from $f(x)$.

rvec $X=X_{p\times n}=\left[\begin{array}{c}{x}_1\\ ⋮\\ x_p\end{array}\right]$

mean vector $\mu =E(x)=\left[\begin{array}{c}{\mu }_1\\ ⋮\\ {\mu }_p\end{array}\right]$

Covariance Matrix $\Sigma$

Correlation Matrix $\rho$, ${\rho }_{ij}=\frac{{\sigma }_{ij}}{{\sigma }_{ii}{\sigma }_{jj}}$

• Correlation measures linear association.
• Correlation is 0 if symmetric non-linear association exists.

Summary Statistics

1. Sample Mean Vector $\bar{X}=$ estimate of $\mu$
2. Sample Covariance Matrix
3. Sample Correlation Matrix

Statistical Inference on Correlation

$$H_0:\rho =0$$

test stat $T=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}\sim t_{n-2}$, where $r=Corr(x,y)$ and $x\sim N_2,y\sim N_2$ Notes:

1. Correlation measures a linear relationships
2. it is still difficult to get a CI for $\rho$.

CI for $\rho$

1. Fisher’s Method:

$100(1-\alpha )\%$ CI for $=\tanh \left(\text{inv}\tanh (r)\pm \frac{z_{\alpha /2}}{\sqrt{n-3}}\right)$

2. Ruben’s Method

let $u=z_{\alpha /2}$, $r^{\ast }=\frac{r}{\sqrt{1-r^2}}$

$$\begin{array}{rl}a& =2n-3-u^2\\ b& =r^{\ast }\ast \sqrt{(2n-3)(2n-5)}\\ c& =(2n-5-u^2)\ast (r^{\ast }{)}^2-2u^2\end{array}$$

set $a{y}^2-2by+c=0$, then root of this equation will be $y_1,y_2=\frac{b}{a}\pm \frac{\sqrt{b^2-ac}}{a}$.

이때 $100(1-\alpha )$ CI for $\rho$ =\left[ \dfrac{y_1}{\sqrt{1+y_1^2}, \dfrac{y_2}{\sqrt{1+y_2^2}, \right]

• 이때, input은 $n,\alpha ,r$, output은 $\rho$의 CI.

Standardization

Missing Value Treatment