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The data functions that are studied in the course of functional data analysisare assembled from discrete data, and the level of smoothing that is used isgenerally that which is appropriate for accurate approximation of theconceptually smooth functions that were not actually observed. Existingliterature shows that this approach is effective, and even optimal, when usingfunctional data methods for prediction or hypothesis testing. However, in thepresent paper we show that this approach is not effective in classificationproblems. There a useful rule of thumb is that undersmoothing is oftendesirable, but there are several surprising qualifications to that approach.First, the effect of smoothing the training data can be more significant thanthat of smoothing the new data set to be classified; second, undersmoothing isnot always the right approach, and in fact in some cases using a relativelylarge bandwidth can be more effective; and third, these perverse results arethe consequence of very unusual properties of error rates, expressed asfunctions of smoothing parameters. For example, the orders of magnitude ofoptimal smoothing parameter choices depend on the signs and sizes of terms inan expansion of error rate, and those signs and sizes can vary dramaticallyfrom one setting to another, even for the same classifier.

Unexpected properties of bandwidth choice when smoothing discrete data for constructing a functional data classifier