As often happens, I usually do many thing in the same time, so during a break while I was working for a new post on applications of mutual information in data mining, I read the interesting paper suggested by Sandro Saitta on his blog (dataminingblog) related to the outlier detection.
...Usually such behavior is not proficient to obtain good results, but this time I think that the change of prospective has been positive!Chebyshev Theorem
In many real scenarios (under certain conditions) the Chebyshev Theorem provides a powerful algorithm to detect outliers.
The method is really easy to implement and it is based on the distance of Zeta-score values from k standard deviation.
...Surfing on internet you can find several explanations and theoretical explanation of this pillar of the Descriptive Statistic, so I don't want increase the Universe Entropy explaining once again something already available and better explained everywhere :)
Approach based on Mutual Information
Before to explain my approach I have to say that I have not had time to check in literature if this method has been already implemented (please drop a comment if someone find out a reference! ... I don't want take improperly credits).
The aim of the method is to remove iteratively the sorted Z-Scores till the mutual information between the Z-Scores and the candidates outlier I(Z|outlier) increases.
At each step the candidate outlier is the Z-score having the highest absolute value.
Basically, respect the Chebyschev method, there is no pre-fixed threshold.
Experiments
I compared the two methods through canonical distribution, and at a glance it seems that results are quite good.
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| Test on Normal Distribution |
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| Test on Normal Distribution having higher variance |
The following experiments have been done with Gamma Distribution and Negative Exponential
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| Results on Gamma seem comparable. |
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| Experiment done using Negative Exponential distribution |
Stay Tuned
Cristian
