Inner Products

In \(\mathbb{R}^d\), we have

## Example 126
u <- c(1, 5, 0)
v <- c(1, 0, 3)
sum(u * v)

[1] 1

## Example 128
v <- c(1, 2, 3)
sqrt(sum(v * v))

[1] 3.741657

## Example 129
v <- c(1, 2, 3)
u <- c(2, 2, 2)
sqrt(sum((v - u)^2))

[1] 1.414214

Properties:

Practical Application

df <- USArrests
df <- na.omit(df)
df <- scale(df)
# nstart is the number of random starting points
clusters <- kmeans(df, 4, nstart = 10)
library(factoextra)

Loading required package: ggplot2

Warning: package 'ggplot2' was built under R version 4.2.3

Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBa

fviz_cluster(clusters, df)

### Practical Applications
library(animation)
kmeans.ani(df, 4)

fviz_nbclust(df, kmeans, method = "wss")

k-means clustering - Wikipedia

library(ggplot2)
library(ggfortify)
library(gridExtra)
df <- USArrests
df <- na.omit(df)
df <- scale(df)
pc <- prcomp(df, scale=TRUE)
autoplot(pc, data=df)

#pca<- princomp(data.frame(scale(df)))
#pca$loadings

Angles and Orthogonality

## Example 137
library(pracma)
library(MASS)
A <- matrix(c(1, 1, 1, 1, 1, 0, 0, 0, 1), 3, 3, byrow = TRUE)
B <- orth(A)
fractions(B)

     [,1]     [,2]    
[1,] -780/989  209/989
[2,] -571/989 -571/989
[3,] -153/724  571/724

library(MASS)
library(pracma)

A <- matrix(c(1, 1, 1, 1, 1, 0, 0, 0, 1), 3, 3, byrow = TRUE)
#A <-matrix(c(1,1,1,1,1,0,0,0,1),3,3,byrow=TRUE)
Null(t(A))

              [,1]
[1,] -7.071068e-01
[2,]  7.071068e-01
[3,] -1.110223e-16

fractions(Null(t(A)))

     [,1]      
[1,] -5741/8119
[2,]  2378/3363
[3,]          0

rref(A)

     [,1] [,2] [,3]
[1,]    1    1    0
[2,]    0    0    1
[3,]    0    0    0

Practical Applications

library(ISLR)
data(Hitters)
D <- na.omit(Hitters)
dim(Hitters)

[1] 322  20

Hitters[1,]

               AtBat Hits HmRun Runs RBI Walks Years CAtBat CHits CHmRun CRuns
-Andy Allanson   293   66     1   30  29    14     1    293    66      1    30
               CRBI CWalks League Division PutOuts Assists Errors Salary
-Andy Allanson   29     14      A        E     446      33     20     NA
               NewLeague
-Andy Allanson         A

library(pracma)
res <- odregress(as.matrix(D[,1:2]), as.matrix(D$Salary))
res$coeff

[1]  -43.53176  150.65796 1861.91571

Exercises

(see Vector projection - Wikipedia)