How to use PCA to detect outliers? A PCA will reduce the dimensionality of the original data and construct a subspace generated by eigenvectors of which each represents the (next) highest variance to explain the data.
Let's start at this subspace: A PCA has been performed and we have a look at the according subspace now:

Now let's assume there are outliers (however where exactly). How can they be detected now?
So far, I know there are two methods:


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*Track the angle(s ?) between the PCs

*Check the number of PCs
I think both are not robust, because new or more data will probably change the angles without providing an outlier. The number of axes makes more sense but still I can construct situations in my head where new data might cause introducing a new axis without making all the data there outliers.
I thought of 


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*using a distance/defined radius to scan for new outliers but I can hardly find according approaches?
On 

*Why is PCA sensitive to outliers? it is explained why it is sensitive to Outliers, this can probably used as a tool, as well.
In other words: How exactly is PCA used to detect outliers respectively how are they detected after performing the PCA?
 A: One approach is to consider outliers those points that can not be well reconstructed using the principal vectors that you have selected . 
The procedure goes like this:
1.Fix two positive numbers , a and b (see the next steps for there meaning an to understand how to select them; to be refined using cross-validation)
2.Compute PCA 
3.Keep the principal vectors that are associated with principal values greater than a,
say $v_1,v_2,..,v_k$ (this are orthonormal vectors)
4.For each data point compute the reconstruction error using the  principal vectors from step 3 . For a data point x, the reconstruction error is:
$e = ||x-\sum_{i=1}^{k}w_iv_i||_2$ , where $w_i = v_i^Tx$
5.Output as outliers those data points that have an reconstruction error greater than b. 
Update: The procedure capture only "direction" outliers . Additionally , before the first step , a "norm" outliers detection step can be included . This consist in computing the norms of the data points and labeling as outliers those that have a too small or too big norm. 
It depends on what an outlier is in your context .
