Detecting multivariate outliers

Fraud Detection in R

Tim Verdonck

Professor Data Science at KU Leuven

Animals data

Animals dataset in package MASS contains average brain and body weights for 28 animals

library(MASS)
data("Animals")

                    body brain
Mountain beaver     1.35   8.1
Cow               465.00 423.0
Grey wolf          36.33 119.5
Goat               27.66 115.0

Apply logarithmic transformation on both body and brain weight

X <- data.frame(log_body = log(Animals$body), log_brain = log(Animals$brain))

Animals data: univariate outlier detection

Boxplot of logarithms of body weight and brain weight

animalsbp

Animals data: scatterplot

scatteranimals_ggplot

Mahalanobis distance

Mahalanobis (or generalized) distance for observation is the distance from this observation to the center, taking into account the covariance matrix

mahalanobiseuclidean_ggplot

Mahalanobis distance to detect multivariate outliers

Classical Mahalanobis distances : sample mean as estimate for location and sample covariance matrix as estimate for scatter
To detect multivariate outliers the mahalanobis distance is compared with a cut-off value, which is derived from the chisquare distribution
In two dimensions we can construct corresponding $97.5\%$ tolerance ellipsoid, which is defined by those observations whose Mahalanobis distance does not exceed the cut-off value

Tolerance ellipsoid based on Mahalanobis distance

animals.clcenter <- colMeans(X)
animals.clcov <- cov(X)
rad <- sqrt(qchisq(0.975, df = ncol(X)))

library(car)
ellipse.cl <- data.frame(ellipse(center = animals.clcenter, 
shape = animals.clcov,radius = rad, segments = 100, draw = FALSE))
colnames(ellipse.cl) <- colnames(X)

fig <- fig +
  geom_polygon(data=ellipse.cl, color = "dodgerblue", 
               fill = "dodgerblue", alpha = 0.2) +
  geom_point(aes(x = animals.clcenter[1], y = animals.clcenter[2]), 
                 color = "blue", size = 6)

cltolanimals_ggplot

Robust estimates of location and scatter

Minimum Covariance Determinant (MCD) estimator of Rousseeuw is a popular robust estimator of multivariate location and scatter

MCD looks for those $h$ observations whose classical covariance matrix has the lowest possible determinant
MCD estimate of location is then mean of these $h$ observations
MCD estimate of scatter is then sample covariance matrix of these $h$ points (multiplied by consistency factor)
Reweighting step is applied to improve efficiency at normal data
Computation of MCD is difficult, but several fast algorithms are proposed

Robust distance

Robust estimates of location and scatter using MCD

library(robustbase)
animals.mcd <- covMcd(X)

# Robust estimate of location
animals.mcd$center 

# Robust estimate of scatter
animals.mcd$cov

By plugging in these robust estimates of location and scatter in the definition of the Mahalanobis distances, we obtain robust distances and can create a robust tolerance ellipsoid.

Animals: robust tolerance ellipsoid

library(robustbase)
animals.mcd <- covMcd(X)
ellipse.mcd <- data.frame(ellipse(center = animals.mcd$center, 
                                   shape = animals.mcd$cov,
                                   radius = rad, segments = 100, draw = FALSE))
colnames(ellipse.mcd) <- colnames(X)

fig2 <- fig +
  geom_polygon(data = ellipse.mcd, color = "red", fill = "red", alpha = 0.3) +
  geom_point(aes(x = animals.mcd$center[1], y = animals.mcd$center[2]), 
                 color = "red", size = 6)

Distance-distance plot

When $p>3$ it is not possible to visualize the tolerance ellipsoid.
The distance-distance plot shows the robust distance of each observation versus its classical Mahalanobis distance obtained immediately from MCD object
```
plot(animals.mcd, which = "dd")
```

Animals: check outliers

animalsdetection

Let's practice!

Fraud Detection in R