In the previous sections we learned about PCA. We worked out an example from scratch to emphasis the mechanics behind PCA. In general, however, we rely on one of the implementations built-in in the R *stats* package to conduct a PCA. In R there are two main implementations for PCA; `prcomp()`

and `princomp()`

. Both functions implement PCA, however the `princomp()`

function uses the spectral decomposition approach, whereas the the `prcomp()`

function uses singular value decomposition (SVD). According to the R help, SVD has slightly better numerical accuracy.

In this section we revisit the *food-texture* data set and briefly showcase PCA by applying the R machinery and in particular the `prcomp()`

function.

Consider the *food-texture* data set (download here).

```
food <- read.csv("https://userpage.fu-berlin.de/soga/300/30100_data_sets/food-texture.csv")
rownames(food) <- food$X
food <- food[, 2:ncol(food)] # exclude first column from data set
str(food)
```

```
## 'data.frame': 50 obs. of 5 variables:
## $ Oil : num 16.5 17.7 16.2 16.7 16.3 19.1 18.4 17.5 15.7 16.4 ...
## $ Density : int 2955 2660 2870 2920 2975 2790 2750 2770 2955 2945 ...
## $ Crispy : int 10 14 12 10 11 13 13 10 11 11 ...
## $ Fracture: int 23 9 17 31 26 16 17 26 23 24 ...
## $ Hardness: int 97 139 143 95 143 189 114 63 123 132 ...
```

The data set consists of 50 rows (observations) and 5 columns (features). By calculating the mean and the standard deviation of the features we realize that the variables are not on the same scale. Hence, we center and scale the variables to have a mean of zero and standard deviation of one.

`apply(food, 2, mean)`

```
## Oil Density Crispy Fracture Hardness
## 17.202 2857.600 11.520 20.860 128.180
```

`apply(food, 2, sd)`

```
## Oil Density Crispy Fracture Hardness
## 1.592007 124.499980 1.775571 5.466073 31.127578
```

`prcomp`

objectIn order to center and standardize the variables we add `center = TRUE`

and `scale = TRUE`

to the function call `prcomp()`

.

`food.pca = prcomp(food, center = TRUE, scale = TRUE)`

The output from `prcomp()`

, a `prcomp`

object, contains a number of useful quantities.

`names(food.pca)`

`## [1] "sdev" "rotation" "center" "scale" "x"`

The `prcomp`

object stores the eigenvectors as a matrix in the `rotation`

attribute. The name relates to the term *rotation matrix* and emphasis that a matrix-multiplication of the data-matrix with the rotation matrix returns the coordinates of the data in the rotated coordinate system, and thus the principal component scores.

`food.pca$rotation`

```
## PC1 PC2 PC3 PC4 PC5
## Oil 0.4575334 -0.3704389 0.65903020 -0.4679449 0.01204121
## Density -0.4787455 0.3567500 0.01623973 -0.7184632 0.35648161
## Crispy 0.5323877 0.1976610 -0.17888443 0.1325269 0.79242064
## Fracture -0.5044769 -0.2212399 0.54227938 0.4569317 0.44011646
## Hardness 0.1534026 0.8046661 0.48923298 0.1961843 -0.22614798
```

The `sdev`

attribute of the `prcomp`

object contains the standard deviation of each principal component as a measure of variance.

`food.pca$sdev`

`## [1] 1.7410380 1.1382907 0.5568207 0.4918537 0.3480110`

In order to get the variance explained by each principal component we square the standard deviation.

```
food.pca.var = food.pca$sdev^2
food.pca.var
```

`## [1] 3.0312132 1.2957058 0.3100493 0.2419201 0.1211116`

To compute the proportion of variance explained by each principal component, we simply divide the variance explained by each principal component by the total variance explained by all principal components:

```
food.pca.ve <- food.pca.var/sum(food.pca.var)
food.pca.ve
```

`## [1] 0.60624263 0.25914115 0.06200987 0.04838402 0.02422233`

We see that the first principal component explains 61% of the variance in the data, the second principal component explains 26% of the variance, and so forth.

Finally, we are interested in the principal component scores, which correspond to the projection of the original data on the directions of the principal component. The principal component scores are stored in `x`

attribute of the `prcomp`

object. We take a look at the first 10 entries of `food.pca$x`

by applying the `head()`

function.

```
# Principal component scores
head(food.pca$x, 10)
```

```
## PC1 PC2 PC3 PC4 PC5
## B110 -1.3832112 -0.6194061 -0.4025115 -0.48680785 -0.005948156
## B136 2.7944769 0.3537246 -1.0760269 0.25580128 -0.488773246
## B171 0.2375564 0.8614582 -0.6115486 0.02952179 -0.176323309
## B192 -1.9393406 -1.1417366 0.4379471 0.31253089 0.554021980
## B225 -1.2679371 0.6634646 0.4371652 0.07190367 0.403450293
## B237 1.9974564 1.2983543 1.1015333 -0.08025960 -0.351879906
## B261 1.4881878 -0.6326525 -0.2730264 -0.03277424 0.153700961
## B264 -0.8288636 -2.3825485 -0.2494352 0.32334864 -0.039526092
## B353 -1.1851533 0.3501810 -0.4257859 -0.01315433 0.245396437
## B360 -0.9934626 0.3508258 0.1033445 -0.02088279 0.237188926
```

The columns of the matrix correspond to principal component score vectors. That is, the k^{th} column is the k^{th} principal component score vector.

The biplot is a very popular way for visualization of results from PCA, as it combines both, the principal component scores and the loading vectors in a single biplot display. In R we simply call the `biplot()`

function. The `scale = 0`

argument to `biplot()`

ensures that the arrows are scaled to represent the loadings.

`biplot(food.pca, scale = 0, cex = 0.6)`

In the biplot the observations are labeled by the observation number (e.g. the row name in data frame). The position in the plot represents the scores for the first two principal components. The original variables are shown as vectors (arrows). They begin at the origin \([0,0]\) and extend to coordinates given by the first two principal component loading vectors. For example, the loading for Oil on the first component is 0.46, and its loading on the second principal component -0.48 (the label “Oil” is centered at the point (0.46, -0.48)).