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In this paper, the mathematical analysis of the Isolation Random ForestMethod (IRF Method) for anomaly detection is presented. We show that the IRFspace can be endowed with a probability induced by the Isolation Tree algorithm(iTree). In this setting, the convergence of the IRF method is proved using theLaw of Large Numbers. A couple of counterexamples are presented to show thatthe original method is inconclusive and no quality certificate can be given,when using it as a means to detect anomalies. Hence, an alternative version ofIRF is proposed, whose mathematical foundation, as well as its limitations, arefully justified. Finally, numerical experiments are presented to compare theperformance of the classic IRF with the proposed one.

A Mathematical Assessment of the Isolation Tree Method for Outliers Detection in Big Data