Principal Components Analysis of Sampled Functions

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Abstract
This paper describes a technique for principal components analysis of data consisting of n functions each observed at p argument values. This problem arises particularly in the analysis of longitudinal data in which some behavior of a number of subjects is measured at a number of points in time. In such cases information about the behavior of one or more derivatives of the function being sampled can often be very useful, as for example in the analysis of growth or learning curves. It is shown that the use of derivative information is equivalent to a change of metric for the row space in classical principal components analysis. The reproducing kernel for the Hilbert space of functions plays a central role, and defines the best interpolating functions, which are generalized spline functions. An example is offered of how sensitivity to derivative information can reveal interesting aspects of the data.

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