Bootstrapped mahalanobis distance about correlation HOT 3 CLOSED

stats::mahalanobis(data, center = colMeans(dat), cov = stats::cov(dat))

stats::mahalanobis(dat, center = colMeans(dat), cov = stats::cov(dat))

set.seed(123)
mean <- c(4,6)
cov <- matrix(c(1, 0.5, 0.5, 1),2,2)
d <- MASS::mvrnorm(100, mean, cov)

#Adding Outliers
d[3,1] <- 12
d[5,2] <- -8

.distance_mahalanobis <- function(data, indices = 1:nrow(data), ...) {
  dat <- data[indices, ] # allows boot to select sample
  row.names(dat) <- NULL
  stats::mahalanobis(data, center = colMeans(dat), cov = stats::cov(dat))
}

rez <- boot::boot(data = d, statistic = .distance_mahalanobis, R = 1000)
bayestestR::point_estimate(as.data.frame(rez$t), centrality="all")

set.seed(123)
mean <- c(4,6)
cov <- matrix(c(1, 0.5, 0.5, 1),2,2)

d <- MASS::mvrnorm(100, mean, cov)

#Correlation with no outliers
cor(d[,1],d[,2])
#> [1] 0.455973

#Adding Outliers
d[3,1] <- 12
d[5,2] <- -8

#Correlation with outliers
cor(d[,1],d[,2])
#> [1] 0.2037985

j <- 1000
n <- nrow(d)
Ms <- matrix(data=NA,nrow=j,ncol=n)

# Bootstrap the distances
for(i in 1:j)
{
  #Draw random numbers from 1:n with replacement
  x <- sample(1:n,n,replace = TRUE)
  #Resampled data
  dat <- d[x,]
  #Calculating the Mahalanobis distance for each actual observation using resampled data
  m <- stats::mahalanobis(d, center = colMeans(dat), cov = stats::cov(dat))
  Ms[i,] <- m
}

# Average across all bootstraps to get the bootstrapped Mahalanobis distance
boot_m = colMeans(Ms)

#Determine the outliers
outlier_indicies <- which(boot_m >=6)

#Remove Outliers
new_d <- d[-outlier_indicies,]

# Shepherd's pi correlation
cor(new_d[,1],new_d[,2])
#> [1] 0.4930791
cor(new_d[,1],new_d[,2],method = "spearman")
#> [1] 0.4525431