eda

Data

data <- data.frame(
  A = c(1,2,3,4,5),
  B = c(2,4,6,8,10),
  C = c(1,4,9,16,25),
  D = c(5,4,3,2,1)
)

mean_vec <- colMeans(data) 
mean_vec

##  A  B  C  D 
##  3  6 11  3

X_minus_mu <- sweep(data, 2, mean_vec, "-")
X_minus_mu

##    A  B   C  D
## 1 -2 -4 -10  2
## 2 -1 -2  -7  1
## 3  0  0  -2  0
## 4  1  2   5 -1
## 5  2  4  14 -2

cov_matrix <- cov(data)
cov_matrix

##      A  B     C     D
## A  2.5  5  15.0  -2.5
## B  5.0 10  30.0  -5.0
## C 15.0 30  93.5 -15.0
## D -2.5 -5 -15.0   2.5

det(cov_matrix)

## [1] 0

data_reduced <- as.matrix(data[, c("A", "C")])
data_reduced

##      A  C
## [1,] 1  1
## [2,] 2  4
## [3,] 3  9
## [4,] 4 16
## [5,] 5 25

mean_red <- colMeans(data_reduced)
cov_red <- cov(data_reduced)

mean_red

##  A  C 
##  3 11

cov_red

##      A    C
## A  2.5 15.0
## C 15.0 93.5

det_S <- det(cov_red)
det_S

## [1] 8.75

a <- cov_red[1,1]
b <- cov_red[1,2]
c <- cov_red[2,1]
d <- cov_red[2,2]

adj <- matrix(c(
  d, -b,
  -c, a
), nrow = 2, byrow = TRUE)

adj

##       [,1]  [,2]
## [1,]  93.5 -15.0
## [2,] -15.0   2.5

cat("Inverse Matrix = (1 /", det_S, ") *\n")

## Inverse Matrix = (1 / 8.75 ) *

print(adj)

##       [,1]  [,2]
## [1,]  93.5 -15.0
## [2,] -15.0   2.5

diff <- sweep(data_reduced, 2, mean_red, "-")
diff

##       A   C
## [1,] -2 -10
## [2,] -1  -7
## [3,]  0  -2
## [4,]  1   5
## [5,]  2  14

Sigma_inv <- solve(cov_red)
Sigma_inv

##           A          C
## A 10.685714 -1.7142857
## C -1.714286  0.2857143

manual_md <- diag(diff %*% Sigma_inv %*% t(diff))
manual_md

## [1] 2.7428571 0.6857143 1.1428571 0.6857143 2.7428571

md <- mahalanobis(data_reduced, center = mean_red, cov = cov_red)
md

## [1] 2.7428571 0.6857143 1.1428571 0.6857143 2.7428571

data$Mahalanobis <- md
data

##   A  B  C D Mahalanobis
## 1 1  2  1 5   2.7428571
## 2 2  4  4 4   0.6857143
## 3 3  6  9 3   1.1428571
## 4 4  8 16 2   0.6857143
## 5 5 10 25 1   2.7428571

threshold <- qchisq(0.975, df = 2)
threshold

## [1] 7.377759

data$Outlier <- md > threshold
data

##   A  B  C D Mahalanobis Outlier
## 1 1  2  1 5   2.7428571   FALSE
## 2 2  4  4 4   0.6857143   FALSE
## 3 3  6  9 3   1.1428571   FALSE
## 4 4  8 16 2   0.6857143   FALSE
## 5 5 10 25 1   2.7428571   FALSE