Object Oriented Data Analysis: A few methodological challenges

We congratulate the authors for their excellent work that provides a clear and farsighted overview of the current research in Object Oriented Data Analysis. OODA is a field of research where new statistical tools are developed urged by new challenges that require a strong interplay of statistics with other scientific disciplines, including maths (analysis, scientific computing, geometry, and algebra), engineering, scientific communication design, computer graphics, computer science, and information technology. This mix of expertises clearly profiles a new type of statistician whose skills go far beyond the classical statistical academic education in the direction of a more eclectic kind of background in which statistics is not the unique piece of the puzzle but the one that all other pieces are connected to. In their inspiring paper, Marron and Alonso focus on different kinds of objects/atoms of increasing complexity: functional data, images, covariance matrices, and trees. We hereby want to complement their excursion along another direction—say, the “second principal component”—addressing a few methodological issues: phase and amplitude variability, sufficiency, incorporation of prior knowledge, dependence, inference, and visualization. We base our discussion mainly on the contributions to OODA developed by our research group at the MOX laboratory of the Politecnico di Milano.