In their original publication, the subroutine of computing distances and updating the estimates of μ, Σ, and |Σ| is called a “C-step”. For each of the k samples, μ, Σ, and |Σ| are estimated, the distances are calculated and sorted in increasing order, and the h smallest distances are used to update the estimates. The algorithm starts with k random samples with (p+1) points. Their method is based on the assumption that at least h out of the n samples are “normal” ( h is a hyperparameter). Rousseeuw and Van Driessen developed a computationally efficient algorithm that can yield robust covariance estimates. Consequently, |Σ| (the determinant) will also be larger, which would theoretically decrease by removing extreme events. Robust covariance methods are based on the fact that outliers lead to an increase of the values (entries) in Σ, making the spread of the data apparently larger. Points outside this envelope are considered anomalies/outliers.įigure 1: Gaussian distribution for a 2-dimensional data set (image by author). In Figure 1, a Gaussian distribution for some two-dimensional sensor data set X is plotted just like with the Z-score (e.g., ☑.96 σ), an envelope around the data set can be constructed by choosing a critical value of the Mahalanobis distance. However, in the presence of outliers, both are distorted and the Mahalanobis distance is rendered useless, hence a robust method, i.e., a method that is little affected by outliers, would be desired. Given a data set X with n samples in p variables (i.e, data from p sensors), mean and covariance matrix are easily computed. It is useful in predictive maintenance as it enables finding unusual behavior of a system upcoming down-times and malfunction can be targeted and identified, which is particularly important since finding faults in equipment, machines, and devices early can reduce the extent of the damage. An extreme observation has a large distance from the center of a distribution. This metric assesses how many standard deviations σ away xᵢ is from μ, thereby being dimensionless. Equation 1: Mahalonobis distance (image by author).
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