Simple_PCA_example.Rmd

---
title: "R Notebook"
output: html_notebook
---
Shift+cmd+enter - run cell
option+command+i 



A Simple PCA

In this example, we will try a PCA using a simple and easy to understand dataset. You will use the mtcars dataset, which is built into R. This dataset consists of data on 32 models of car, taken from an American motoring magazine (1974 Motor Trend magazine). For each car, you have 11 features, expressed in varying units (US units), They are as follows:

```{r}
mtcars
mtcars_refined <- mtcars[,c(1:7,10,11)]
mtcars_refined


```

```{r}
mtcars.pca <- prcomp(mtcars[,c(1:7,10,11)], center = TRUE,scale. = TRUE)
mtcars.pca
summary(mtcars.pca)

```


You obtain 9 principal components, which you call PC1-9. Each of these explains a percentage of the total variation in the dataset. That is to say: PC1 explains 63% of the total variance, which means that nearly two-thirds of the information in the dataset (9 variables) can be encapsulated by just that one Principal Component. PC2 explains 23% of the variance. So, by knowing the position of a sample in relation to just PC1 and PC2, you can get a very accurate view on where it stands in relation to other samples, as just PC1 and PC2 can explain 86% of the variance.



Let's call str() to have a look at your PCA object.

```{r}
str(mtcars.pca)
```

Install next 
```{r}
library(devtools)
#install_github("vqv/ggbiplot", force=TRUE)
library(ggbiplot)
ggbiplot(mtcars.pca)
```

The axes are seen as arrows originating from the center point. Here, you see that the variables hp, cyl, and disp all contribute to PC1, with higher values in those variables moving the samples to the right on this plot. This lets you see how the data points relate to the axes, but it's not very informative without knowing which point corresponds to which sample (car).




You'll provide an argument to ggbiplot: let's give it the rownames of mtcars as labels. This will name each point with the name of the car in question:

```{r}
ggbiplot(mtcars.pca, labels=rownames(mtcars))
```

Now you can see which cars are similar to one another. For example, the Maserati Bora, Ferrari Dino and Ford Pantera L all cluster together at the top. This makes sense, as all of these are sports cars.


How else can you try to better understand your data?



Maybe if you look at the origin of each of the cars. You'll put them into one of three categories (cartegories?), one each for the US, Japanese and European cars. You make a list for this info, then pass it to the groups argument of ggbiplot. You'll also set the ellipse argument to be TRUE, which will draw an ellipse around each group.

```{r}
mtcars.country <- c(rep("Japan", 3), rep("US",4), rep("Europe", 7),rep("US",3), "Europe", rep("Japan", 3), rep("US",4), rep("Europe", 3), "US", rep("Europe", 3))

ggbiplot(mtcars.pca,ellipse=TRUE,  labels=rownames(mtcars), groups=mtcars.country)

```

Now you see something interesting: the American cars form a distinct cluster to the right. Looking at the axes, you see that the American cars are characterized by high values for cyl, disp, and wt. Japanese cars, on the other hand, are characterized by high mpg. European cars are somewhat in the middle and less tightly clustered than either group.

Of course, you have many principal components available, each of which map differently to the original variables. You can also ask ggbiplot to plot these other components, by using the choices argument.


Let's have a look at PC3 and PC4:

```{r}
ggbiplot(mtcars.pca,ellipse=TRUE,choices=c(3,4),   labels=rownames(mtcars), groups=mtcars.country)
```