df <- USArrests
dim(df)[1] 50 4head(df) Murder Assault UrbanPop Rape
Alabama 13.2 236 58 21.2
Alaska 10.0 263 48 44.5
Arizona 8.1 294 80 31.0
Arkansas 8.8 190 50 19.5
California 9.0 276 91 40.6
Colorado 7.9 204 78 38.7df <- na.omit(df)
df <- scale(df)scale -> standardize (a data based linear transformation!)
d <- dist(df, method = "euclidean")Euclidean distance -> measure of similarity (closeness)
hc <- hclust(d)
sub_grp <- cutree(hc, k = 4)
library(factoextra)Loading required package: ggplot2Warning: package 'ggplot2' was built under R version 4.2.3Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBafviz_cluster(list(data = df, cluster = sub_grp))
principal components (reduction for visualization purposes) along the two axes
## Example 98
library(rgl)
# Create some dummy data
dat <- replicate(2, 1:3)
# Initialize the scene, no data plotted
plot3d(dat, type = 'n', xlim = c(-1, 8), ylim = c(-1, 8), zlim = c(-10, 20), xlab = '', ylab = '', zlab = '')
# Define the linear plane
planes3d(2, -3, -1, 0, col = 'red', alpha = 0.6)
# Define the origin
points3d(x=0, y=0, z=0)
library(ISLR)
data(Auto)
dim(Auto)[1] 392 9summary(Auto) mpg cylinders displacement horsepower weight
Min. : 9.00 Min. :3.000 Min. : 68.0 Min. : 46.0 Min. :1613
1st Qu.:17.00 1st Qu.:4.000 1st Qu.:105.0 1st Qu.: 75.0 1st Qu.:2225
Median :22.75 Median :4.000 Median :151.0 Median : 93.5 Median :2804
Mean :23.45 Mean :5.472 Mean :194.4 Mean :104.5 Mean :2978
3rd Qu.:29.00 3rd Qu.:8.000 3rd Qu.:275.8 3rd Qu.:126.0 3rd Qu.:3615
Max. :46.60 Max. :8.000 Max. :455.0 Max. :230.0 Max. :5140
acceleration year origin name
Min. : 8.00 Min. :70.00 Min. :1.000 amc matador : 5
1st Qu.:13.78 1st Qu.:73.00 1st Qu.:1.000 ford pinto : 5
Median :15.50 Median :76.00 Median :1.000 toyota corolla : 5
Mean :15.54 Mean :75.98 Mean :1.577 amc gremlin : 4
3rd Qu.:17.02 3rd Qu.:79.00 3rd Qu.:2.000 amc hornet : 4
Max. :24.80 Max. :82.00 Max. :3.000 chevrolet chevette: 4
(Other) :365 auto <- Auto[,1:7]
summary(auto) mpg cylinders displacement horsepower weight
Min. : 9.00 Min. :3.000 Min. : 68.0 Min. : 46.0 Min. :1613
1st Qu.:17.00 1st Qu.:4.000 1st Qu.:105.0 1st Qu.: 75.0 1st Qu.:2225
Median :22.75 Median :4.000 Median :151.0 Median : 93.5 Median :2804
Mean :23.45 Mean :5.472 Mean :194.4 Mean :104.5 Mean :2978
3rd Qu.:29.00 3rd Qu.:8.000 3rd Qu.:275.8 3rd Qu.:126.0 3rd Qu.:3615
Max. :46.60 Max. :8.000 Max. :455.0 Max. :230.0 Max. :5140
acceleration year
Min. : 8.00 Min. :70.00
1st Qu.:13.78 1st Qu.:73.00
Median :15.50 Median :76.00
Mean :15.54 Mean :75.98
3rd Qu.:17.02 3rd Qu.:79.00
Max. :24.80 Max. :82.00 new.data <- Auto[, 1:8]
new.data$origin <- as.character(new.data$origin)
summary(new.data) mpg cylinders displacement horsepower weight
Min. : 9.00 Min. :3.000 Min. : 68.0 Min. : 46.0 Min. :1613
1st Qu.:17.00 1st Qu.:4.000 1st Qu.:105.0 1st Qu.: 75.0 1st Qu.:2225
Median :22.75 Median :4.000 Median :151.0 Median : 93.5 Median :2804
Mean :23.45 Mean :5.472 Mean :194.4 Mean :104.5 Mean :2978
3rd Qu.:29.00 3rd Qu.:8.000 3rd Qu.:275.8 3rd Qu.:126.0 3rd Qu.:3615
Max. :46.60 Max. :8.000 Max. :455.0 Max. :230.0 Max. :5140
acceleration year origin
Min. : 8.00 Min. :70.00 Length:392
1st Qu.:13.78 1st Qu.:73.00 Class :character
Median :15.50 Median :76.00 Mode :character
Mean :15.54 Mean :75.98
3rd Qu.:17.02 3rd Qu.:79.00
Max. :24.80 Max. :82.00 ### Practical Applications
pc <- prcomp(auto, scale=TRUE)
plot(pc)
plot(cumsum(pc$sdev^2/sum(pc$sdev^2)))
## Need this library
library(ggfortify)
autoplot(pc, data=new.data, colour = "origin")
Prove that the set of all \(2 \times2\) symmetric matrices with real entries forms a vector space.
Prove that the set of all polynomials with real coefficients forms a vector space.
