Comparison study of orthonormal representations of functional data in classification

• Explain why a functional sample can be seen as a point in corresponding Euclidean space.
• Answer how to select an orthonormal basis for a given functional data type.
• Discover orthogonal representation is better than non-orthogonal representation as per classification performance.

Functional data type, which is an important data type, is widely prevalent in many fields such as economics, biology, finance, and meteorology. Its underlying process is often seen as a continuous curve. The classification process for functional data is a basic data mining task. The common method is a two-stage learning process: first, by means of basis functions, the functional data series is converted into multivariate data; second, a machine learning algorithm is employed for performing the classification task based on the new representation. The problem is that a majority of learning algorithms are based on Euclidean distance, whereas the distance between functional samples is L2 distance. In this context, there are three very interesting problems. (1) Is seeing a functional sample as a point in the corresponding Euclidean space feasible? (2) How to select an orthonormal basis for a given functional data type? (3) Which one is better, orthogonal representation or non-orthogonal representation, under finite basis functions for the same number of basis? These issues are the main motivation of this study. For the first problem, theoretical studies show that seeing a functional sample as a point in the corresponding Euclidean space is feasible under the orthonormal representation. For the second problem, through experimental analysis, we find that Fourier basis is suitable for representing stable functions(especially, periodic functions), wavelet basis is good at differentiating functions with local differences, and data driven functional principal component basis could be the first preference especially when one does not have any prior knowledge on functional data types. For the third problem, experimental results show that orthogonal representation is better than non-orthogonal representation from the viewpoint of classification performance. These results have important significance for studying functional data classification.