Fast unsupervised online drift detection using incremental Kolmogorov-Smirnov test

Data stream research has grown rapidly over the last decade. Two major features distinguish data stream from batch learning: stream data are generated on the fly, possibly in a fast and variable rate; and the underlying data distribution can be non-stationary, leading to a phenomenon known as concept drift. Therefore, most of the research on data stream classification focuses on proposing effcient models that can adapt to concept drifts and maintain a stable performance over time. However, specifically for the classification task, the majority of such methods rely on the instantaneous availability of true labels for all already classified instances. This is a strong assumption that is rarely fulfilled in practical applications. Hence there is a clear need for effcient methods that can detect concept drifts in an unsupervised way. One possibility is the well-known Kolmogorov-Smirnov test, a statistical hypothesis test that checks whether two samples differ. This work has two main contributions. The first one is the Incremental Kolmogorov-Smirnov algorithm that allows performing the Kolmogorov-Smirnov hypothesis test instantly using two samples that change over time, where the change is an insertion and/or removal of an observation. Our algorithm employs a randomized tree and is able to perform the insertion and removal operations in O(log N) with high probability and calculate the Kolmogorov-Smirnov test in O(1), where N is the number of sample observations. This is a significant speed-up compared to the O(N log N) cost of the non-incremental implementation. The second contribution is the use of the Incremental Kolmogorov-Smirnov test to detect concept drifts without true labels. Classification algorithms adapted to use the test rely on a limited portion of those labels just to update the classification model after a concept drift is detected.