A unified framework on defining depth for point process using function smoothing

The notion of statistical depth has been extensively studied in multivariate and functional data over the past few decades. In contrast, on temporal point process is still under-explored. problem challenging because a two types randomness: 1) number events process, 2) distribution these events. Recent studies proposed depths weighted product terms, describing above randomness, respectively. Under new framework through smoothing procedure, randomnesses can be unified. Basically, observations are transformed into functions using conventional kernel methods, then well-known h -depth its modified, center-based version adopted to describe center-outward rank original data. To do so, proper metric defined processes with smoothed functions. Then an efficient algorithm provided estimate “center”. mathematical properties newly explored asymptotic theories studied. Simulation results show that properly observations. Finally, methods demonstrated classification task real neuronal spike train dataset.