Parameter estimation for correlated Ornstein-Uhlenbeck processes

In many fields of science, we observe time-series of a fluctuating quantity: local density fluctuations of a liquid, or fluctuations of neural activity as measured by fMRI. Here, we are exploring how to characterize correlations between two such time-series taking into account that the time series exhibit a relaxation time (memory). In particular, we will restrict ourselves to the simplest random process that results in a fluctuating time series with a characteristic relaxation time: the Ornstein-Uhlenbeck (OU) process. Here we show that by coupling two OU processes, we can create correlated time series with Pearson correlation coefficients from -1 to 1. We express the likelihood probability of an equally spaced time-series and can numerically solve for the maximum likelihood with respect to the parameters. From the Hessian of log(p) we can estimate the variance of the estimated parameters. To test our approach, we simulated correlated time-series of different lengths and degree of correlation and validated our maximum likelihood against Markov-Chain Monte-Carlo methods (pymc3). Our method is better suited to estimate correlations in time-series with memory than the often-used Pearson correlation.