in built Mahalanobis distance function gives different result

Question

i want to calculate mahalanobis distance, from the formula and also theoretical explanation given here

https://www.machinelearningplus.com/statistics/mahalanobis-distance/

i tried to implement it with help of r, here is my simulated data, their covariance matrix and applied formula : l

ibrary(matlib)
calculus_score <- rnorm(20,12,3)
r_score  <-rnorm(20,15,7)
df <-data.frame(calculus_score,r_score)
data_vector =as.matrix(df[1,])
mean_vector =t(as.matrix(colMeans((df))))
x <- data_vector -mean_vector
cov = as.matrix(cov(df))
inv_r =inv(cov)
print(inv_r)
first_part =(x %*% inv_r)
last_part =(first_part %*% t(x))
print(sqrt(last_part))
print(mahalanobis(df, colMeans(df), cov(df)))

result is given :

> library(matlib)
> calculus_score <- rnorm(20,12,3)
> r_score  <-rnorm(20,15,7)
> df <-data.frame(calculus_score,r_score)
> data_vector =as.matrix(df[1,])
> mean_vector =t(as.matrix(colMeans((df))))
> x <- data_vector -mean_vector
> cov = as.matrix(cov(df))
> inv_r =inv(cov)
> print(inv_r)
                          
[1,] 0.08996774 0.02129345
[2,] 0.02129345 0.05771103
> first_part =(x %*% inv_r)
> last_part =(first_part %*% t(x))
> print(sqrt(last_part))
         1
1 1.131093
> print(mahalanobis(df, colMeans(df), cov(df)))
 [1] 1.2793708 3.8746218 0.6191061 0.2273994 7.8007252 0.6624168 1.0335179
 [8] 4.4952383 0.5713539 3.0517258 1.5961463 0.7668096 1.1018290 1.2884556
[15] 1.5785714 1.5660312 2.3136476 1.7095509 0.8067573 1.6567251

problem is that calculated distance for first row 1.131093 is difference from the value calculated based on built in function, am i making some errors? or there is different algorithms for calculatung mahalanobis distance?

Answer 1

?mahalanobis states that it returns the squared Mahalanobis distance and you are computing the Mahalanobis distance.

Doing

sqrt(1.2793708)

I get your manually computed value:

r$> sqrt(1.2793708)                                                             
[1] 1.131093