Parameter estimation for correlated Ornstein-Uhlenbeck time-series

In this article, we develop a Maximum likelihood (ML) approach to estimate parameters from correlated time traces that originate from coupled Ornstein-Uhlenbeck processes. The most common technique to characterize the correlation between time-series is to calculate the Pearson correlation coefficient. Here we show that for time series with memory (or a characteristic relaxation time), our method gives more reliable results, but also results in coupling coefficients and their uncertainties given the data. We investigate how these uncertainties depend on the number of samples, the relaxation times and sampling time. To validate our analytic results, we performed simulations over a wide range of correlation coefficients both using our maximum likelihood solutions and Markov-Chain Monte-Carlo (MCMC) simulations. We found that both ML and MCMC result in the same parameter estimations. We also found that when analyzing the same data, the ML and MCMC uncertainties are strongly correlated, while ML underestimates the uncertainties by a factor of 1.5 to 3 over a large range of parameters. For large datasets, we can therfore use the less computationally expensive maximum likelihood method to run over the whole dataset, and then we can use MCMC on a few samples to determine the factor by which the ML method underestimates the uncertainties. To illustrate the application of our method, we apply it to time series of brain activation using fMRI measurements of the human default mode network. We show that our method significantly improves the interpretation of multi-subject measurements of correlations between brain regions by providing parameter confidence intervals for individual measurements, which allows for distinguishing between the variance from differences between subjects from variance due to measurement error.