The Effects of Irregular Sampling and Missing Data on Largest Lyapunov Exponents

David M. Kreindler, University of Toronto

Charles J. Lumsden, University of Toronto

**Abstract: **Human self-report time series data are typically marked
by irregularities in sampling rates arising from the data generation
process. The largest Lyapunov exponent lamda_1 is an indicator of
chaos in time series data. Relatively little has been published to
assist the calculation of lamda_1”s using irregularly sampled data. We
report the results of a series of computational experiments on synthetic
data sets assessing techniques for handling irregular time series data in
the calculation of lamda_1 . Regularly sampled data sets were disrupted by data
point removal using an empirically motivated data gap distribution of
either uniform random or power law form. Missing data segments were
patched using segment concatenation, segment filling with average data
values, or local interpolation in phase space. We compared results of lamda_1
calculations using complete and patched sets. The greatest proportion of
missing data possible that will allow an accurate estimate of lamda_1 depends on the
nature of the underlying system and the patching technique used. Self-similar
data patched with segment concatenation was particularly robust. Local
interpolation in phase space was successful in many cases, but required
potentially impractical quantities of intact data as a primer. Optimally,
estimates of lamda_1 can readily be recovered with 15%–20% or greater
amounts of missing data.

*Keywords: *largest Lyapunov exponent, simulation, nonlinear analysis, nonlinear dynamics, scaling, missing data