The Nyquist Issue in Linear Inverse Modeling

Abstract

Abstract Linear inverse modeling (LIM) is a statistical technique based on covariance statistics that estimates the best-fit linear Markov process to a multivariate time series. An integral, often-ignored part of the technique is a test of whether or not the linear assumptions are valid. One test for linearity is the so-called tau test. While this test can be trusted when it passes, it sometimes fails when it ought to pass. In this article, we discuss one of the reasons for spurious failure, the “Nyquist issue,” which occurs when the lagged covariance matrix used in the analysis is numerically performed at a lag greater than or nearly equal to half the period of a natural mode of variability represented in the time series. As an illustration relevant to a system with many degrees of freedom, but simple enough to solve analytically, we consider a four-dimensional system consisting of two modal pairs. Within this framework, we suggest one solution that can be applied if the time series are long enough. It is hoped that awareness of this issue can prevent misinterpretation of LIM results.