---
title: "PCA: Houses"
author: "Gustavo Cepparo and Milica Cudina"
output:
  pdf_document: default
---

First, we read in our house data. 

```{r}
data=read.csv("housepriceall.csv",header=TRUE)
attach(data)
View(data)
```

Let me do a bit of exploratory data analysis.

```{r}
plot(data, pch=20, col="lightblue")
cor(data)
```

Obviously, the scale of the price is different from the scale of the square footage. Also, the assessed price and the price are artificially on a different scale. Let's look at the means and variances. 

```{r}
apply(data, 2, mean)
apply(data, 2, sd)
```
Let's take a look at what the principal component analysis is telling us. 

```{r}
pr.out<-prcomp(data, scale=TRUE)
```

Here are the outputs of `prcomp`. 

```{r}
pr.out$center
pr.out$scale
pr.out$rotation
dim(pr.out$x)
```
Of course, it's much better to look at the `biplot`. 

```{r}
biplot(pr.out, scale=0, cex=0.5, xlabs=rep("*", length(price)),
       col=c("lightblue", "red"))
```

How many principal components are "sufficient"?

```{r}
#transforming standard deviations to variances
pr.var=pr.out$sdev^2
#pr.var
#proportion of variance explained
pve=pr.var/sum(pr.var)
#pve
#plots
par(mfrow = c(1, 2))
plot(pve,xlab="Principal Component", 
     ylab="Proportion of Variance Explained", col="peru", 
     pch=20, ylim=c(0,1),type='b')
plot(cumsum(pve),xlab="Principal Component", 
     ylab="Cumulative Proportion of Variance Explained", 
     col="peru", pch=20, ylim=c(0,1),type='b')
```