Estimating the Coefficients of a Linear Differential Operator
Principal Differential Analysis (PDA; Ramsay, 1996) is used to obtain low dimensional representations of functional data, where each observation is represented as a curve. PDA seeks to identify a Linear Differential Operator (LDO) L = ω0I + ω 1D + ... + ωmDm, where I denotes the identity function and D j the jth derivative, that satisfies as closely as possible that Lx = 0 for each functional observation x. A theorem from analysis establishes that the coefficients of the LDO are in the Sobolev space, and thus can be approximated by B-splines. Current PDA software used to estimate the LDO assumes that the leading coefficient is 1, and approximates the LDO coefficients by B-splines, even though the Sobolev space is not closed under division. We present a method that eliminates the restriction on the leading coefficient of the LDO. The proposed resampling method is inspired by results in linear regression (Frees, 1991 and Wu, 1986) that show that the weighted average of all (n m) pairwise slopes between n data points is equivalent to the least squares estimator of the regression line slope. In analyzing data, least-squares estimates and robust estimates of the LDO coefficients are computed, and results based on each estimate are compared. The R language implementation of the proposed method is available at https://www.utep.edu/science/computational-science/research/resources.html.
Barraza, María Ivette, "Estimating the Coefficients of a Linear Differential Operator" (2017). ETD Collection for University of Texas, El Paso. AAI10681443.