statistics or robust statistics for identifying multivariate outliers

Question

For the single variate data sets, we can use some straightforward methods, such as box plot or [5%, 95%] quantile to identify outliers. For multivariate data sets, are there any statistics that can be used to identify outliers?

Answer 1

Multivariate outlier detection can be quite tricky and even 2D data can be difficult to visually decipher at times. You are spot-on in looking for robust statistical treatments analogous to 95% quantiles.

Where as normally distributed data naturally aligns with the chi square distribution, the gold standard for robust statistics in n dimensions would be to use Mahalanobis distances and then eliminate data beyond 95% or 99% quantiles in Mahalanobis space.

Plug and play capabilities are available in scikit-learn and in R.

Here is an excellent theoretical and practical treatment of the methodology:

And here is a big picture viewpoint with some heuristics.

Additionally there is a very sophisticated treatments called PCOUT for outlier detection that instead rely on principal component decomposition. There is a corresponding R package, but the theoretical treatment is behind a paywall:

P. Filzmoser, R. Maronna, M. Werner. Outlier identification in high dimensions, Computational Statistics and Data Analysis, 52, 1694-1711, 2008

Hope this helps!