Title: The role of lag time in spectral estimation for Markov processes

Abstract: Spectral estimation algorithms attempt to identify the slowly-changing patterns of Markov processes using data from simulations or observations. They include popular algorithms such as Ulam’s Method, (Extended) Dynamic Mode Decomposition, Time-lagged Independent Component Analysis, the Variational Approach in Conformation Dynamics (VAC), and others. However, the error properties of these schemes remains poorly understood.

Here we analyze the error in VAC, an algorithm specific to time-reversible dynamics. VAC approximates the eigenvalues and eigenvectors of the transition operator. We establish the convergence of VAC eigenvectors as the data set increases and the library of basis functions increases. Moreover, we investigate the role of the lag time parameter which controls the conditioning of the VAC problem. At short and long lag times, we find that the conditioning is especially poor, giving rise to excess error in eigenvector estimates. Following this analysis, we explore the possibility of reducing error by selecting an intermediate lag time.