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

Abstract

The data functions that are studied in the course of functional data analysis are assembled from discrete data, and the level of smoothing that is used is generally that which is appropriate for accurate approximation of the conceptually smooth functions that were not actually observed. Existing literature shows that this approach is effective, and even optimal, when using functional data methods for prediction or hypothesis testing. However, in the present paper we show that this approach is not effective in classification problems. There a useful rule of thumb is that undersmoothing is often desirable, but there are several surprising qualifications to that approach. First, the effect of smoothing the training data can be more significant than that of smoothing the new data set to be classified; second, undersmoothing is not always the right approach, and in fact in some cases using a relatively large bandwidth can be more effective; and third, these perverse results are the consequence of very unusual properties of error rates, expressed as functions of smoothing parameters. For example, the orders of magnitude of optimal smoothing parameter choices depend on the signs and sizes of terms in an expansion of error rate, and those signs and sizes can vary dramatically from one setting to another, even for the same classifier.

Clustering multivariate functional data using unsupervised binary trees

“…In this section, we develop estimators for the mean and the covariance functions of a component X (p) , 1 ≤ p ≤ P from the process X. These estimators are used to compute estimators of eigenvalues and eigenfunctions of X (p) for the Karhunen-Loève expansion (7). It is worthwhile to notice that, because of Lemma 4, we do not need to estimate the covariance between X (p) and X (q) for p = q.…”

“…Clustering procedures for functional data have been widely studied in the last two decades, see for instance, [11,12,7,27] and references therein. See also Bouveyron et al [6] for a recent textbook.…”

In-sample forecasting applied to reserving and mesothelioma mortality

“…It is therefore encouraging that our purely continuous approach adapts to aggregated data in a simple, efficient and robust way. Carroll et al (2013) concluded in a slightly different context that cross-validation methods seem more robust to discretization than methods based on asymptotic expansions such as plug-in methods and they give some theoretical background for this conclusion. We follow their advice and stick to cross-validation when choosing the level of smoothing.…”

Bayesian Fisher's Discriminant for Functional Data

“…However, applying smoothing filters to each observations require lots of computation efforts, since the tuning parameters for smoothing filters need to be selected for each smoothing. Moreover, [6] showed that the tuning parameters should be selected to under-smooth each observations for the pre-smoothing approaches to achieve reasonble results (compared to the other classifiers for functional data). However, choosing such tuning parameters that satisfy the requirements in [6] is non-trivial in practice.…”