Wavelet coherence for point processes

Analysing and understanding correlations is one of the most fundamental and insightful statistical tasks undertaken by scientists, engineers and analysts when presented with a data set. This project is concerned with developing methods for analysing correlations in point process data. Point processes are typically used to model random events in time, such as the times at which a computer sends traffic over a network, or the firing events of a neuron. A common question to ask when presented with a pair of point processes is; are they correlated? For example, does the firing of one neuron correlate with the firing of another?

Spectral analysis has long been established as a route for understanding underlying structure in random process, and the ordinary coherence gives a measure of the correlation that exists between a pair of processes due to oscillations at a particular frequency of interest. As well as its prevalent use in analysing ordinary time series, it is also a well established concept in the analysis of point processes for finding the frequencies at which the events of a pair of processes correlate. Its use, however, is reliant on the strong assumption that the random processes are stationary, i.e. their spectral properties remain constant in time. In truth, stationarity is a luxury that is often absent in real world processes and ordinary coherence is therefore inappropriate. Tools for analysing coherence in non-stationary processes is very much an open problem.

Wavelet coherence extends the notion of coherence to nonstationary processes to give a measure of correlation as a function of both time and frequency. For ordinary time series, its use as an exploratory tool is prevalent across many scientific and engineering disciplines. However, until now, its use in analysing non-stationary point processes is an open problem.

The project is concerned with extending the notion of wavelet coherence to point processes. A rigorous treatment of wavelet coherence will provide the first method for analysing coherence between non-stationary point processes. For a pair of point processes it will expose localised regions in time where the two processes correlate, and expose the common periodicities.

This project will consider both the continuous-time and discrete-time setting to maximise its relevance. In addition to the theoretical results, openly available Matlab code and an R-package will be produced that will take a pair of point processes (either a list of event times for continuous time processes, or event counts for a discrete time process), and provide a full wavelet coherence analysis, including hypothesis testing. It will produce visual representation of wavelet coherence as a function of time and frequency, clearly indicating where in time significant correlation between the two processes exist, and on which frequencies those correlations are occurring.

Scientific disciplines where this project will have impact are extensive, with neuroscience and cyber-security being two highlighted fields. In neuroscience it is typical to analyse neural signals, and specifically to understand how the activity of two neurons may be dependent. With the acknowledgement that these signals are intrinsically nonstationary, wavelet techniques are becoming more and more commonplace. In this area the proposed tools could have an immediate impact on fundamental biomedical research. Cyber-security is a relatively new field of research where novel statistical methodology to analyse large multivariate point process data sets is in huge demand. Detecting correlations between a pair of point processes, and specifically the periodicities at which these dependencies occur, has the potential to expose malicious activity and characterise its nature.

The methods developed in this project will be disseminated to the relevant communities through tutorials at their leading workshops and conferences. An open-source R-package will allow the output of this project to be promptly and easily accessible for all to use.

Mathworks have stated they would look to implement the output of the project in the Matlab Wavelet Toolbox, putting these methods directly into the hands of researchers and analysts around the world.