I have a set of means and covariance matrices of a number of multivariant normal distributions, each having a class label. Then I get single data points, one after the other, to which I want to assign one of the class labels or the label 'None', if it is unlikely that the point belongs to any of the given classes (could be drawn from the corresponding distribution).
Simplified to the one-dimensional case I would compute the distance of the point to each mean (in terms of corresponding standard deviation units) and assign the label of the closest mean or 'None' if all distances exceed the e.g. 2-sigma boundary (to catch 95% of the probability).
What I have so far:
All I got so far comes from this book. There the same idea is applied to multi-dimensional data using the Mahalanobis distance, but with the difference that you cannot say 'outlier if distance is greater than some fixed value' because you have different deviations in (and between) each dimension. Instead the authors use the set of points (or to be more precise the size of the set of), for which they want to decide which points are outliers, to estimate the rejection distance threshold using a chi-squared distribution. The problem is that this only seems to be reasonable for large sets of points, what directly leads to my questions:
Can I apply the same approach if my point set is of size 1 and still get good results?
If not, can the approach be adapted to the situation where you have single points?
If not, is there a way to label single observation given the parameters of multiple multinomial distributions with an additional rejection class?
Thanks in advance.