In jeffniv/Lab1Intro: Lab1

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(Lab1Intro)

Links

The github repo can be found here. The documentation for making a packing can be found here.

Definitions

The sample mean can be computed from the $n$ measurements on each of the $p$ variables, so that, in general, there will be $p$ sample means: $$\bar{x}k = \frac{1}{n} \sum^n{j=1} x_{jk}, \ \ k=1,2,\ldots,p.$$

The sample covariance measures the association between the ith and kth variables: $$s_{ik} = \frac{1}{n} \sum^n_{j=1} (x_{ji} - \bar{x}i)(x{jk} - \bar{x}_k), \ \ i=1,2,\ldots,p, \ k=1,2,\ldots,p.$$

The sample correlation coefficient for the ith and kth variables is defined as: $$r_{ik} = \frac{s_{ik}}{\sqrt{s_{ii}}\sqrt{s_{kk}}} = \frac{\sum^n_{j=1} (x_{ji} - \bar{x}i)(x{jk} - \bar{x}k)}{\sqrt{\sum^n{j=1} (x_{ji} - \bar{x}i)^2} \sqrt{\sum^n{j=1} (x_{jk} - \bar{x}_k)^2}}$$.

Let bolded items represent vectors and matrices. $\mathbf{1}$ represent an $n\times 1$ vector with entries $1$. The sample mean vector is defined as: $$\bar{\mathbf{x}} = \frac{1}{n}\mathbf{X}'\mathbf{1}$$

The sample covariance matrix is: $$\mathbf{S} = \frac{1}{n-1} \mathbf{X}' \left(\mathbf{I} - \frac{1}{n}\mathbf{1}\mathbf{1}'\right)\mathbf{X} $$.

Once $\mathbf{S}$ is computed then the sample standard deviation matrix is defined as $$\mathbf{D}^{1/2}{(p \times p)} = \left[ \begin{array}{cccc} \sqrt{s{11}} & 0 & \cdots & 0 \ 0 & \sqrt{s_{22}} & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & \sqrt{s_{pp}} \end{array} \right]$$

Then $$\mathbf{D}^{-1/2}{(p \times p)} = \left[ \begin{array}{cccc} \frac{1}{\sqrt{s{11}}} & 0 & \cdots & 0 \ 0 & \frac{1}{\sqrt{s_{22}}} & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & \frac{1}{\sqrt{s_{pp}}} \end{array} \right]$$

Since $$\mathbf{S} = \left[ \begin{array}{cccc} s_{11} & s_{12} & \cdots & s_{1p} \ \vdots & \vdots & \ddots & \vdots \ s_{1p} & s_{2p} & \cdots & s_{pp} \end{array} \right]$$

and $$\mathbf{R} = \left[ \begin{array}{cccc} \frac{s_{11}}{\sqrt{s_{11}}\sqrt{s_{11}}} & \frac{s_{12}}{\sqrt{s_{11}}\sqrt{s_{22}}} & \cdots & \frac{s_{1p}}{\sqrt{s_{11}}\sqrt{s_{pp}}} \ \vdots & \vdots & \ddots & \vdots \ \frac{s_{1p}}{\sqrt{s_{11}}\sqrt{s_{pp}}} & \frac{s_{2p}}{\sqrt{s_{22}}\sqrt{s_{pp}}} & \cdots & \frac{s_{pp}}{\sqrt{s_{pp}}\sqrt{s_{pp}}} \end{array} \right] = \left[ \begin{array}{cccc} 1 & r_{12} & \cdots & r_{1p} \ \vdots & \vdots & \ddots & \vdots \ r_{1p} & r_{2p} & \cdots & 1 \end{array} \right]$$

We have $$\mathbf{R} = \mathbf{D}^{-1/2}\mathbf{S}\mathbf{D}^{-1/2}$$.