Conformal Prediction: a Unified Review of Theory and New Challenges

In this work we provide a review of basic ideas and novel developments about Conformal Prediction -- an innovative distribution-free, non-parametric forecasting method, based on minimal assumptions -- that is able to yield in a very straightforward way predictions sets that are valid in a statistical sense also in in the finite sample case. The in-depth discussion provided in the paper covers the theoretical underpinnings of Conformal Prediction, and then proceeds to list the more advanced developments and adaptations of the original idea.Comment: arXiv admin note: text overlap with arXiv:0706.3188, arXiv:1604.04173, arXiv:1709.06233, arXiv:1203.5422 by other author

Hierarchical independent component analysis: A multi-resolution non-orthogonal data-driven basis

A new method named Hierarchical Independent Component Analysis is presented, particularly suited for dealing with two problems regarding the analysis of high-dimensional and complex data: dimensional reduction and multi-resolution analysis. It takes into account the Blind Source Separation framework, where the purpose is the research of a basis for a dimensional reduced space to represent data, whose basis elements represent physical features of the phenomenon under study. In this case orthogonal basis could be not suitable, since the orthogonality introduces an artificial constraint not related to the phenomenological properties of the analyzed problem. For this reason this new approach is introduced. It is obtained through the integration between Treelets and Independent Component Analysis, and it is able to provide a multi-scale non-orthogonal data-driven basis. Furthermore a strategy to perform dimensional reduction with a non orthogonal basis is presented and the theoretical properties of Hierarchical Independent Component Analysis are analyzed. Finally HICA algorithm is tested both on synthetic data and on a real dataset regarding electroencephalographic traces

Local inference for functional data on manifold domains using permutation tests

Pini and Vantini (2017) introduced the interval-wise testing procedure which performs local inference for functional data defined on an interval domain, where the output is an adjusted p-value function that controls for type I errors. We extend this idea to a general setting where domain is a Riemannian manifolds. This requires new methodology such as how to define adjustment sets on product manifolds and how to approximate the test statistic when the domain has non-zero curvature. We propose to use permutation tests for inference and apply the procedure in three settings: a simulation on a "chameleon-shaped" manifold and two applications related to climate change where the manifolds are a complex subset of $S^2$ and $S^2 \times S^1$, respectively. We note the tradeoff between type I and type II errors: increasing the adjustment set reduces the type I error but also results in smaller areas of significance. However, some areas still remain significant even at maximal adjustment