Brief summary of post contents Generalized Local Linear Approximation is a generalization of local linear approximation of derivatives (LLA, Boker & Graham, 1998). While LLA allows for estimation up to second-order derivatives (acceleration) with three observations, GLLA allows for any order of derivative with additional flexibility as to the number of observations used.
Recent developments in dynamical systems modeling of repeated observations data have led to first- and second-order differential equations being fit to psychological data using local linear approximation (LLA) of derivatives (Boker & Graham, 1998; Boker, 2001; Butner, Amazeen, & Mulvey, 2005). The LLA method provides simplified explicit estimation of derivatives from repeated observations using a three dimensional time delay embedding (Abarbanel, Carroll, Pecora, Sidorowich, & Tsimring, 1994; Noakes, 1991; Sauer, Yorke, & Casdagli, 1991; Takens, 1981; Whitney, 1936) in a manner similar to Savitzky–Golay filtering (Savitzky & Golay, 1964), but has several weaknesses. Three-dimensional embedding is sensitive to time–independent noise, which can bias estimates of the differential equations parameters unless the delay time, τ, used to create the time–delay embedded matrix is chosen correctly (Boker & Nesselroade, 2002; Deboeck, Boker, & Bergeman, 2009). In addition, as it is currently used, LLA is only able to estimate first and second derivatives. Some differential equation models may require higher order derivatives.
Code and Other Content
Boker, S. M., Deboeck, P. R., Edler, C. & Keel, P. K. (2009). Generalized Local Linear Approximation of Derivatives from Time Series. In S. Chow, E. Ferrer & F. Hsieh (Eds.), Statistical Methods for Modeling Human Dynamics: An Interdisciplinary Dialogue, 161-178. New York, NY: Taylor & Francis Group.