The smooth complex logarithm and quasi-periodic models

Abstract

Quasi-periodic signals, which look like sine waves with variable frequency and amplitude, are common in nature and society. Examples that will be analyzed in this paper are sounds of crickets, counts of sunspots, movements of ocean currents, and brightness of variable stars. Euler's formula for the complex logarithm, combined with smoothly changing real and imaginary components, provides a powerful model. It is highly non-linear and special care is needed to get starting values for an iterative estimating algorithm. The model is extended with a trend and harmonics. A cascaded link function allows modeling of quasi-periodic series of counts. The model and real-world applications are described in an expository style. © 2010 Springer-Verlag Berlin Heidelberg.