The effectiveness of Principal component analysis in detecting process abnormalities requires that the process data to follow a Gaussian distribution
This is what I read over a thread. Can anyone explain why process data need to follow a Gaussian distribution?
In general, Gaussian data are neither necessary nor sufficient for successful PCA-based anomaly detection.
As described here, PCA-based anomaly detection identifies a point as anomalous if its PCA reconstruction error exceeds a chosen threshold. The reconstruction error is typically defined as the squared Euclidean distance between the point and its projection onto a linear subspace fit by PCA (which is a $k$-dimensional plane). For this procedure to succeed, anomalies must therefore lie further from this plane than 'regular' data, with high probability.
Notice that Gaussian data aren't required. For example, imagine a 3d space where the regular data are distributed arbitrarily over a 2d plane, and anomalies are distributed diffusely over all 3 dimensions. The data aren't Gaussian, but the anomaly detection procedure above will work well (provided we choose a good threshold, and the PCA training set isn't too heavily contaminated by anomalies so the 2d plane is properly identified).
On the other hand, notice that 'in-plane' anomalies can't be detected. This is because the PCA reconstruction error is small for all points lying near the plane, regardless of how far they're perturbed along the plane (as illustrated here). This implies that anomaly detection might not succeed, even if the data are Gaussian. For example, imagine that regular data are drawn from a 3d Gaussian distribution with small variance along one direction, so they're concentrated near a 2d plane. Suppose that anomalies are distributed around the same plane, but can lie arbitrarily far from the regular data. E.g. they might be Gaussian too, but have enormous variance along the plane. PCA-based anomaly detection will fail in this case.
