A differential geometry perspective about multiple data streams preprocessing

In the Multiple Data Streams (MDS) environment, data sources generate data with no end in sight. Because of the difference of data sources, transaction numbers of MDS are not always equal to each other during a same period. Preprocessing MDS to obtain same number of samples for each stream is an essential step for lots of mining tasks. All existing preprocessing methods assume that data arrive simultaneously. However, this assumption may not be true in many real environments due to multiple data sources and different ways of data generating. This asynchronous issue is explored in this paper, by introducing the differential geometry as a trick. First, we establish a novel stream model called POLAR. The POLAR is an intrinsic surface spanned by time, probability and value. And then, we propose a preprocessing approach, called COPOLAR, to obtain same number of samples for each stream of MDS. COPOLAR first projects original observations onto POLAR; and then merges points with shortest geodesic distances along a geodesic on surface into mid-point on the same geodesic iteratively and incrementally until the number of points which we hope to obtain is met. Experimental results on synthetic and real data show that COPOLAR is effective in terms of maintaining characteristics of both statistics and vector.