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Which is the relationship between the Mahalanobis distance and the mean?
In the case of a multivariate normal distribution we can take the squared Mahalanobis distance between a point of the multivariate normal distribution and its mean as such a random variable. Then the p-quantile computation will answer the following question: Which value is required so that a random point fulfills ?
In a Quantile-Quantile Plot one can see that quantiles of the Mahalanobis distance of a sample drawn from a Gaussian distribution is very similar to the corresponding quantiles computed on the Chi-Square distribution. The following R-script shows this:
How to find the squared Mahalanobis distance of a random vector?
The squared Mahalanobis Distance follows a Chi-Square Distribution: More formal Derivation. Thus, the squared Mahalanobis distance of a random vector matr X and the center vec mu of a multivariate Gaussian distribution is defined as: where is a covariance matrix and is the mean vector.
How are Gaussian distributions used in anomaly detection?
In practice, sometimes (multivariate) Gaussian distributions are used for anomaly detection tasks (assuming that the considered data is approx. normally distributed): the parameters of the Gaussian can be estimated using maximum likelihood estimation (MLE) where the maximum likelihood estimate is the sample mean and sample covariance matrix.
The squared Mahalanobis distance can be expressed as: D = ℓ ∑ k = 1Y2 k. where Yk ∼ N(0, 1). Now the Chi-square distribution with ℓ degrees of freedom is exactly defined as being the distribution of a variable which is the sum of the squares of ℓ random variables being standard normally distributed.