Incorporating Parametric Effects into Functional Principal Components Analysis

SUMMARY The ideas of functional principal component analysis are extended to deal with data that are hybrids of ‘functional’ and ‘parametric’ effects. The parametric effects may be more general than just the addition of a multiple of a given function. A detailed development is given in the case of shifts of the time axis for functions observed on a periodic interval, and some remarks are made for the extension to a far more general case. Given data, a Procrustes fitting method can be used to estimate the parametric effects. Several possible ways of treating the estimated parameter values are discussed. The methods are illustrated by reference to temperature data at 35 Canadian weather-stations.