Point-process models have been shown to be useful in characterizing neural spiking activity as a function of extrinsic and intrinsic factors. Most point-process models of neural activity are parametric, as they are often efficiently computable. However, if the actual point process does not lie in the assumed parametric class of functions, misleading inferences can arise. Nonparametric methods are attractive due to fewer assumptions, but computation in general grows with the size of the data. We propose a computationally efficient method for nonparametric maximum likelihood estimation when the conditional intensity function, which characterizes the point process in its entirety, is assumed to be a Lipschitz continuous function but otherwise arbitrary. We show that by exploiting much structure, the problem becomes efficiently solvable. We next demonstrate a model selection procedure to estimate the Lipshitz parameter from data, akin to the minimum description length principle and demonstrate consistency of our estimator under appropriate assumptions. Finally, we illustrate the effectiveness of our method with simulated neural spiking data, goldfish retinal ganglion neural data, and activity recorded in CA1 hippocampal neurons from an awake behaving rat. For the simulated data set, our method uncovers a more compact representation of the conditional intensity function when it exists. For the goldfish and rat neural data sets, we show that our nonparametric method gives a superior absolute goodness-of-fit measure used for point processes than the most common parametric and splines-based approaches.
The parameter estimation of conditional intensity function temporal point process as renewal process using Bayesian method and its application on the data of earthquake in East Nusa Tenggara
