Specialized Session 5: Distributional and Functional Data Analysis

This paper discusses approaches for measuring distances between probability density functions within the framework of Bayes spaces. One particular choice of distance function is a Mahalanobis distance (MD). While the notion of the MD for densities has been very recently introduced, it neglected to leverage the smoothness of the underlying curves. We, therefore, propose a regularized version of MD tailored for density data in Bayes spaces, which accounts for compositional nature, smoothness, and noise reduction. Due to the fact that MD (3) intrinsically smoothens the underlying density, the approach is expected to be particularly advantageous in very noisy scenarios.

The annual mortality age-specific death rate for French males from 1816 to 2006 shows different curves that appear as shape outliers. We propose to detect shape outliers studying the complexity of the data. In particular, it is assumed that the process is contaminated by some higher complexity process that is, the whole process is a mixture of processes with different complexities. The components of the mixture and the contamination are identified and studied through the use of the small-ball probability of the original process and a heuristic procedure is proposed.

This paper presents a novel approach to cluster-wise regression for distributional data. The proposed method is based on a new regression technique for distributional data, which maps density functions into a Hilbert space using a logarithmic transformation of the derivative of quantile functions (LDQ). We represent LDQ functions as functional data and develop a cluster-wise regression algorithm that partitions the dataset into K clusters by optimizing a new objective function to evaluate the fit of data to local regression models. The application of these methods to real-world datasets demonstrates the effectiveness of the proposed approach.