Fuzzy c-means (FCM) is a generalization of the classical k-means clustering algorithm to the case where an observation can belong to several clusters at the same time. The algorithm was previously observed to have initialization problems when the number of desired clusters or the number of dimensions of the data are high. We have tested FCM against clustering problems with functional data, generated from stationary Gaussian processes, and thus in principle infinite-dimensional. We observed that when the data is more functional in nature, which can be obtained by tuning the length-scale parameter of the Gaussian process, the aforementioned problems do not appear. This not only indicates that FCM is suitable as a clustering method for functional data, but also illustrates how functional data differs from traditional multivariate data. In addition this seems to suggest a qualitative way to measure the latent dimensionality of the functional distribution itself.

In functional classification problems the data available for learning are characterized by functions, rather than vectors of attributes. In consequence, multivariate classifiers need to be adapted, and new types of classifiers designed to take into account the special characteristics of these types of data. In this work, an empirical evaluation of different classification methods is carried out using a variety of functional classification problems from different areas of application. The classifiers considered include nearest centroids with functional means as class prototypes and functional distances, standard multivariate classifiers used in combination with a variable selection method, classifiers based on the notion of functional depth, a functional version of k-nearest neighbors (k-NN), and random forest. From the results of this comparative study one concludes that random forest is among the best off-the-shelf classifiers not only for multivariate but also for functional classification problems. The variable selection method used in combination with a quadratic discriminant has fairly good overall accuracy using only a small set of impact points. This dimensionality reduction leads to improvements both in efficiency and interpretability. Finally, a functional version of k-NN that uses the α-Mahalanobis distance exhibits consistently good predictive performance in all the problems considered. This robustness makes k-NN a good benchmark for functional classification.

Machine learning from functional data poses particular challenges that require specific computational tools that take into account their structure. In this work, we present scikit-fda, a Python library for functional data analysis, visualization, preprocessing, and machine learning. The library is designed for smooth integration in the Python scientific ecosystem. In particular, it complements and can be used in combination with scikit-learn, the reference Python library for machine learning. The functionality of scikit-fda is illustrated in clustering, regression, and classification problems from different areas of application.