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