  A set of curves or images of similar shape is an increasingly common
functional data set collected in the sciences. Principal Component Analysis
(PCA) is the most widely used technique to decompose variation in functional
data. However, the linear modes of variation found by PCA are not always
interpretable by the experimenters. In addition, the modes of variation of
interest to the experimenter are not always linear. We present in this paper a
new analysis of variance for Functional Data. Our method was motivated by
decomposing the variation in the data into predetermined and interpretable
directions (i.e. modes) of interest. Since some of these modes could be
nonlinear, we develop a new defined ratio of sums of squares which takes into
account the curvature of the space of variation. We discuss, in the general
case, consistency of our estimates of variation, using mathematical tools from
differential geometry and shape statistics. We successfully applied our method
to a motivating example of biological data. This decomposition allows
biologists to compare the prevalence of different genetic tradeoffs in a
population and to quantify the effect of selection on evolution.
