Steven Boker

Local linear approximation (LLA) has been used to provide explicit estimates of derivatives from three dimension time-delay embeddings of repeated observations data in order to fit differential equations models. This talk presents a method for constructing weight matrices for approximation of derivatives using arbitrary time-delay embedding dimensions. The method is shown to be equivalent to fixed loading matrices used in latent differential equations (LDE) modeling and growth curve modeling. An example data set from a study of ovarian hormones and disordered eating is used to apply 3, 6, 9, and 14 dimensional time--delay embedding and to calculate generalized local linear approximation (GLLA) of derivatives. Choices of embedding dimension lend increased flexibility to deal with issues of signal to noise ratio and patterns of missingness that may characterize a particular data set. Higher embedding dimensions also appear to help reduce the attenuation of interindividual differences in estimated period that occurs in three dimensional embedding used in LLA. Finally, the talk will present some speculation in how convolution kernels used in image processing might be used to identify transition points (edges) rather than smooth growth curves.

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