Principal Differential Analysis with a Continuous Covariate: Low Dimensional Approximations for Functional Data

Given a collection of n curves that are independent realizations of a functional variable, we are interested in finding patterns in the curve data by exploring low dimensional approximations to the curves. It is assumed that the data curves are noisy samples from the vector space span{f₁, …, f }, where f₁, …, f are unknown functions on the real interval (0, T) with square-integrable derivatives of all orders m or less, and m < n. Ramsay (1996) first proposed the method of regularized principal differential analysis as an alternative to principal component analysis for finding low dimensional approximations to curves. Principal differential analysis (PDA) is based on the following theorem: there exists an annihilating linear differential operator [Formula: see text] of order m such that [Formula: see text], i = 1, …, m (Coddington and Levinson 1955, Theorem 6.2). Principal differential analysis specifies m, then uses the data to estimate an annihilating linear differential operator (LDO). Smooth estimates of the coefficients of the LDO are obtained by minimizing a penalized sum of the squared norm of the residuals. In this context, the residual is that part of the data curve that is not annihilated by the LDO. PDA obtains the smooth low dimensional approximation to the data curves by projecting onto the null space of the estimated annihilating LDO; PDA is thus useful for obtaining low dimensional approximations to the data curves whether or not the interpretation of the annihilating LDO is intuitive or obvious from the context of the data. This paper extends PDA to allow for the coefficients in the linear differential operator to smoothly depend upon a single continuous covariate. The estimating equations for the coefficients allowing for a continuous covariate are derived; the penalty of Eilers and Marx (1996) is used to impose smoothness. The results of a small computer simulation study investigating the bias and variance properties of the estimator are reported.