Fisher-Rao distance and pullback SPD cone distances between multivariate normal distributions

Data sets of multivariate normal distributions abound in many scientific areas like diffusion tensor imaging, structure tensor computer vision, radar signal processing, machine learning, just to name a few. In order to process those normal data sets for downstream tasks like filtering, classification or clustering, one needs to define proper notions of dissimilarities between normals and paths joining them. The Fisher-Rao distance defined as the Riemannian geodesic distance induced by the Fisher information metric is such a principled metric distance which however is not known in closed-form excepts for a few particular cases. In this work, we first report a fast and robust method to approximate arbitrarily finely the Fisher-Rao distance between multivariate normal distributions. Second, we introduce a class of distances based on diffeomorphic embeddings of the normal manifold into a submanifold of the higher-dimensional symmetric positive-definite cone corresponding to the manifold of centered normal distributions. We show that the projective Hilbert distance on the cone yields a metric on the embedded normal submanifold and we pullback that cone distance with its associated straight line Hilbert cone geodesics to obtain a distance and smooth paths between normal distributions. Compared to the Fisher-Rao distance approximation, the pullback Hilbert cone distance is computationally light since it requires to compute only the extreme minimal and maximal eigenvalues of matrices. Finally, we show how to use those distances in clustering tasks.Comment: 25 page

Portfolio Analysis with Multivariate Normal Tempered Stable Processes and Distributions

Using the construction approach of Brownian subordination, the univariate framework of Normal Tempered Stable Lévy processes is extended to an arbitrary number of dimensions. A thorough study of the mathematical properties of the multivariate stochastic process is followed by various applications of its distributions in financial econometrics and portfolio analysis. Moreover, these distributions are employed in ARMA-GARCH models for capturing volatility clustering effects in financial markets

A hill-sliding strategy for initialization of Gaussian clusters in the multidimensional space

A hill sliding technique was devised to extract Gaussian clusters from the multivariate probability density estimate of sample data for the first step of iterative unsupervised classification. Each cluster was assumed to posses a unimodal normal distribution. A clustering function proposed distinguished elements of a cluster under formation from the rest in the feature space. Initial clusters were extracted one by one according to the hill sliding tactics. A dimensionless cluster compactness parameter was proposed as a universal measure of cluster goodness and used satisfactorily in test runs with LANDSAT multispectral scanner data. The normalized divergence, defined by the cluster divergence divided by the entropy of the entire sample data, was utilized as a general separability measure between clusters. An overall clustering objective function was set forth in terms of cluster covariance matrices, from which the cluster compactness measure could be deduced. Minimal improvement of initial data partitioning was evaluated by this objective function in eliminating scattered sparse data points. The hill sliding clustering technique developed herein has the potential applicability to decomposition any multivariate mixture distribution into a number of unimodal distributions when an appropriate distribution function to the data set is employed