The BAGIDIS Distance: About a Fractal Topology, with Applications to Functional Classification and Prediction

The BAGIDIS (semi-) distance of Timmermans and von Sachs (BAGIDIS: statistically investigating curves with sharp local patterns using a new functional measure of dissimilarity. Under revision. http://www.uclouvain.be/en-369695.html. ISBA Discussion Paper 2013-31, Universite catholique de Louvain, 2013) is the central building block of a nonparametric method for comparing curves with sharp local features, with the subsequent goal of classification or prediction. This semi-distance is data-driven and highly adaptive to the curves being studied. Its main originality is its ability to consider simultaneously horizontal and vertical variations of patterns. As such it can handle curves with sharp patterns which are possibly not well-aligned from one curve to another. The distance is based on the signature of the curves in the domain of a generalised wavelet basis, the Unbalanced Haar basis. In this note we give insights on the problem of stability of our proposed algorithm, in the presence of observational noise. For this we use theoretical investigations from Timmermans, Delsol and von Sachs (J Multivar Anal 115:421-444, 2013) on properties of the fractal topology behind our distance-based method. Our results are general enough to be applicable to any method using a distance which relies on a fractal topology.