In JacksonDowns/Lab1Intro: This package is used to complete the lab 1 for Dr. Stewart's MATH 4793

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

library(Lab1Intro)

Package Making Website and Github Repository

The package making website is http://r-pkgs.had.co.nz/vignettes.html while the link to my own personal git repository for this package is https://github.com/JacksonDowns/Lab1Intro.git

Equations used

In this package, we were asked to code several different equations listed within the book. The first equation asked for was the calculation of the vector mean. This is given in the book as equation 1-1. The equation for calculating the sample mean is $\bar{x_k} = \frac{1}{n} \sum_{j=1}^{n}{x_{jk}}$ for k = 1,2,...,p.

The next equation, equation 1-4, is the equation used to calculate covariance. we were not allowed to use the cov() function, and had to build this ourselves. The equation for calculating the covariance of a group of variable is $s_{ik} = \frac{1}{n} \sum_{j=1}^{n} (x_{ji}-\bar{x_i})(x_{jk}-\bar{x_k})$ where i and k represent different variables.

The final equation we were asked to program was equation 1-5. The equation is used to calculate the correlation coefficient of a group of vectors. Oncce again, we were not allowed to use the cor() function, and were required to calculate the correlation manually. The equation used to calculate correlation is given as:

$$r_{ik} = \frac{s_{ik}}{\sqrt{s_{ii}}\sqrt{s_{kk}}} = \frac{\sum^{n}{j=1}(x{ji}-\bar{x_i})(x_{jk}-\bar{x_k})}{\sqrt{\sum^{n}{j=1}(x{ji}-\bar{x_i})^2}\sqrt{\sum^{n}{j=1}(x{jk}-\bar{x_k})^2}}$$ For multiple variables, the correlations can be put into a correlation matrix to make patterns easy to recognize. This is using equation 3-29, which is $R =D^{\frac{-1}{2}}SD^{\frac{-1}{2}}$, where R is the correlation matrix, D is the sample standard deviation matrix, and S is the variance covariance matrix. This equation can be derived from using equation 3-28, where:

$$\mathbf{D^{\frac{1}{2}}{pxp}} = \left[\begin{array} {ccc} \sqrt{s{11}} & 0 & ... & 0 \ 0 & \sqrt{s_{22}} & ... & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & ... & \sqrt{s_{pp}} \end{array}\right]$$ Then

$$\mathbf{D^{\frac{1}{2}}{pxp}} = \left[\begin{array} {ccc} \frac{1}{\sqrt{s{11}}} & 0 & ... & 0 \ 0 & \frac{1}{\sqrt{s_{22}}} & ... & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & ... & \frac{1}{\sqrt{s_{pp}}} \end{array}\right]$$ Since

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

and

$$\mathbf{R} = \left[\begin{array} {ccc} \frac{s_{11}}{\sqrt{s_{11}\sqrt{s_{11}}}} & \frac{s_{12}}{\sqrt{s_{11}\sqrt{s_{22}}}} & ... & \frac{s_{1p}}{\sqrt{s_{11}\sqrt{s_{pp}}}} \ \vdots & \vdots & \ddots & \vdots \ \frac{s_{1p}}{\sqrt{s_{11}\sqrt{s_{pp}}}} & \frac{s_{2p}}{\sqrt{s_{22}\sqrt{s_{pp}}}} & ... & s_{pp} \end{array}\right] = \left[\begin{array} {ccc} 1 & r_{12} & ... & r_{1p} \ \vdots & \vdots & \ddots & \vdots \ r_{1p} & r_{2p} & ... & 1 \end{array}\right]$$ Which gives us

$$R =D^{\frac{-1}{2}}SD^{\frac{-1}{2}}$$

Examples

#Load in data
density = c(0.801,0.824,0.841,0.816,0.840,0.842,0.820,0.802,0.828,0.819,
            0.826,0.802,0.810,0.802,0.832,0.796,0.759,0.770,0.759,0.722,
            0.806,0.803,0.845,0.822,0.971,0.816,0.836,0.815,0.822,0.822,
            0.843,0.824,0.788,0.782,0.795,0.805,0.836,0.788,0.772,0.776,
            0.758)

machDir = c(121.41,127.70,129.20,131.80,135.10,131.50,126.70,115.10,130.80,124.60,
            118.31,114.20,120.30,115.70,117.51,109.81,109.10,115.10,118.31,112.60,
            116.20,118.00,131.00,125.70,126.10,125.80,125.50,127.80,130.50,127.90,
            123.90,124.10,120.80,107.40,120.70,121.91,122.31,110.60,103.51,110.71,
            113.80)

crossDir = c(70.42,72.47,78.20,74.89,71.21,78.39,69.02,73.10,79.28,76.48,
             70.25,72.88,68.23,68.12,71.62,53.10,50.85,51.68,50.60,53.51,
             56.53,70.70,74.35,68.29,72.10,70.64,76.33,76.75,80.33,75.68,
             78.54,71.91,68.22,54.22,70.41,73.68,74.93,53.52,48.93,53.67,
             52.42)


table = matrix(c(density,machDir,crossDir),41,3,FALSE)

colnames(table) = c("Density","Machine Direction", "Cross Direction")

Sample Means

Here we have a dataset for paper quality.

head(table)

Looping through the columns gives us these means for each variable, respectively.

  dataSetMeans = vector()

  for (i in 1:ncol(table)) {

    dataSetMeans = c(dataSetMeans,VectorMean(table[,i]))

  }

  dataSetMeans

Covariance

To calculate the covariance of each variable, simply take the table and use it as a parameter in the CovarianceMatrix function

CovarianceMatrix(table)

Correlation

Similarly to the covariance function, simply take the table and use it as the parameter for the CorrelationMatrix function.

CorrelationMatrix(table)

Assessments

The assessments for this course depends on whether the student is take 4793 or 5793. For me personally. I am taking 4793, which means my assessment is as follows.

Clicker quiz each class and lab (10%) 4 assignments (20%) Laboratories (10%) 2 exams (10% each, Total 20%) 2 Projects (10%) 1 final (30%)

JacksonDowns/Lab1Intro documentation built on Jan. 26, 2020, 4:08 a.m.