Stochastic Hyperkernel Convolution Trains and <italic>h</italic>-Counting Processes

Abstract

The paper presents two new families of stochastic processes called hyperkernel convolution train and $h$ -counting processes. These models generalize respectively the spike train and counting process models. The convolution train model is designed to encompass both continuous and singular spiking activities. The $h$ -counting process can be used to model counting phenomena for which the increments are not necessarily instantaneous. This $h$ -counting model can also be used to represent uncertainties on the exact locations of state transitions of a standard discrete event system. The paper also highlights some statistical properties of the provided convolution train model, in addition to a framework based on wavelet packets for simulating or learning such a process from multiple observations of disturbed input trains.

Keywords

• Convolution train
• h-counting process
• spike train
• counting process
• hyperkenel train
• wavelet train