Generalized Local Linear Approximation of Derivatives from Time Series

Abstract

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 timeindependent 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, submitted). 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.

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