hw03.Rmd

---
title: "GEOG606 - HW03"
author: "Anika Cartas"
date: "Tuesday, March 24, 2015"
output: word_document
---

### Assignment Description

In 1898, Bumpus did a series study of English sparrows. Attached is his measurement of characteristics of 49 female sparrows:
  
  X1=total length
  X2=alar length,
  X3=length of beak and head,
  X4=length of humerus,
  X5=length of keel and sternum.

Please use principle component analysis to find out the linear combination of 5 variables that best discriminates different sparrows. First, use the PCA routine in R to do it. 

Second, try to reproduce the result of PCA routine by following the key steps of PCA:
  
  - generate covariance matrix
  - solve the eigen value equation to calculate the eigen values
  - derive eigen vectors
  - calculate loading vectors

###1. Read in the data

```{r}
data <- read.table("hw03.csv",header=TRUE, col.names=c("Total","Alar","Beak.Head","Humerus","Keel.Sternum"))
head(data)
```

###2. Use PCA routine in R

```{r}
prcomp.results <- prcomp(data)

#Rotation Matrix
prcomp.results

#Loading Scores
head(prcomp.results$x)
```

###3. Reproduce results manually


```{r}
#3.1 Zero data by subtracting mean 
data.adjusted <- apply(data,2,function(x) { x - mean(x) }) 

#3.2 Generate covariance matrix
cov.matrix <- cov(data.adjusted)
cov.matrix

#3.3 Solve the eigen value equation to calculate the eigen values
cov.eigen <- eigen(cov.matrix)
cov.eigen$values

#3.4 Derive eigen vectors
cov.eigen$vectors

#3.5 Calculate loading scores
manual.results <- t(cov.eigen$vectors) %*% t(as.matrix(data.adjusted))
manual.results <- t(manual.results)
head(manual.results)

#3.6. Compare Results
head(prcomp.results$x)

#Calculate differences between each value and see if any are not nearly identical.
table(abs(manual.results) - abs(prcomp.results$x) < 0.001)
```