ASYMPTOTICALLY OPTIMAL NONPARAMETRIC CLASSIFICATION RULES FOR SPIKE TRAIN DATA Miroslaw Pawlak, Mateusz Pabian, Dominik Rzepka Department of Measurement and Electronics AGH University of Science and Technology Kraków, Poland ABSTRACT Spike train data find a growing list of applications in com- putational neuroscience, streaming data and finance. Statis- tical analysis of spike trains is based on various probabilistic and neural network models. The statistical approach relies on parametric or nonparametric specifications of the underlying model. In this paper we consider the nonparametric classi- fication problem for a class of spike train data characterized by nonparametricaly specified intensity functions. We derive the optimal Bayes rule and next formulate the plug-in non- parametric kernel classifiers. Asymptotical properties of the rules are established including the limit with respect to the increasing observation time interval and the size of a train- ing set. The obtained results are supported by a finite sample simulation studies. Index Terms— spike train data, nonparametric learning, Bayes risk consistency 1. INTRODUCTION Event driven systems are often encountered in science and engineering. In such systems data are represented by point processes that define arrival times of events. In computa- tional neuroscience and machine learning this type of data are called spike trains [1], [2]. The mathematical theory of point processes has been extensively studied in the statistical and stochastic processes literature [3], [4]. On the other hand, the research on event type processes from the machine learning perspective has been initiated very recently [2], [5], [6]. Prob- abilistic spiking neural networks have been introduced [2], [5] for supervised and unsupervised learning problems. Various numerical results have been reported supporting their useful- ness without, however, any accuracy studies and fundamental limits. In this paper, we develop the Bayes strategy [7] for the spiking data supervised classification problem. This strategy can be applied to research problems where event ocurrence is Authors were supported by the Polish National Center of Science under Grant DEC-2017/27/B/ST7/03082 the primary information carrier [8], [9]. We consider a class of temporal spiking processes that are characterized by non- random intensity functions. The intensity function plays the central role in our theory as it describes the local rate of oc- currence of spikes. For such processes (Section 2) we derive the optimal Bayes rule in terms of class intensity functions and examine the behavior of the corresponding Bayes risk as the observation window increases. In Section 3.1 a wide class of plug-in nonparametric classification rules from mul- tiple replications of spiking processes is introduced. Under general conditions on the class intensity functions we estab- lish the Bayes risk consistency result, i.e., the convergence of any rule in this class to the Bayes optimal decision. In Sec- tion 3.2 we design the Bayes risk consistent kernel rule. The consistency and the rate of convergence results are presented. It is worth mentioning that the asymptotic optimality does not hold if one observes the long single realization of the un- derlying spiking process. In fact, in the intensity estimation problem increasing the observation window does not result in new observations at the left side of the window [10]. Hence, for a fixed observation interval one must increase the number of events. This can be achieved by either scaling the inten- sity function or by using the replicates of the spiking process. The former approach is based on the multiplicative intensity model due to Aalen [11], whereas the latter one (used in this paper) is the standard machine learning strategy, where the replicates form the training set. In this case the resulting ker- nel estimate will be obtained by aggregating kernel estimates from the single realizations. Our asymptotic results are sup- ported by simulation studies presented in Section 4. 2. OPTIMAL CLASSIFICATION RULES A temporal spiking process {N (t),t ≥ 0} consists of a sequence of random times {t i } of isolated events in time such that N (0) = 0. We assume that the process is ob- served on the time window [0,T ] and is characterized by the non-random intensity function λ(t). This is a non- negative function that describes the local arriving rate of events such that E[N (T )] =  T 0 λ(u)du is the average num- ber of events in [0,T ]. Hence, the observed on [0,T ] pro- ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) | 978-1-7281-6327-7/23/$31.00 ©2023 IEEE | DOI: 10.1109/ICASSP49357.2023.10096060