An extended orthogonal forward regression algorithm for system identification using entropy

In this paper, a fast identification algorithm for nonlinear dynamic stochastic system identification is presented. The algorithm extends the classical Orthogonal Forward Regression (OFR) algorithm so that instead of using the Error Reduction Ratio (ERR) for term selection, a new optimality criterion —Shannon’s Entropy Power Reduction Ratio(EPRR) is introduced to deal with both Gaussian and non-Gaussian signals. It is shown that the new algorithm is both fast and reliable and examples are provided to illustrate the effectiveness of the new approach

M-Power Regularized Least Squares Regression

Regularization is used to find a solution that both fits the data and is sufficiently smooth, and thereby is very effective for designing and refining learning algorithms. But the influence of its exponent remains poorly understood. In particular, it is unclear how the exponent of the reproducing kernel Hilbert space~(RKHS) regularization term affects the accuracy and the efficiency of kernel-based learning algorithms. Here we consider regularized least squares regression (RLSR) with an RKHS regularization raised to the power of m, where m is a variable real exponent. We design an efficient algorithm for solving the associated minimization problem, we provide a theoretical analysis of its stability, and we compare its advantage with respect to computational complexity, speed of convergence and prediction accuracy to the classical kernel ridge regression algorithm where the regularization exponent m is fixed at 2. Our results show that the m-power RLSR problem can be solved efficiently, and support the suggestion that one can use a regularization term that grows significantly slower than the standard quadratic growth in the RKHS norm

Exact reconstruction with directional wavelets on the sphere

A new formalism is derived for the analysis and exact reconstruction of band-limited signals on the sphere with directional wavelets. It represents an evolution of the wavelet formalism developed by Antoine & Vandergheynst (1999) and Wiaux et al. (2005). The translations of the wavelets at any point on the sphere and their proper rotations are still defined through the continuous three-dimensional rotations. The dilations of the wavelets are directly defined in harmonic space through a new kernel dilation, which is a modification of an existing harmonic dilation. A family of factorized steerable functions with compact harmonic support which are suitable for this kernel dilation is firstly identified. A scale discretized wavelet formalism is then derived, relying on this dilation. The discrete nature of the analysis scales allows the exact reconstruction of band-limited signals. A corresponding exact multi-resolution algorithm is finally described and an implementation is tested. The formalism is of interest notably for the denoising or the deconvolution of signals on the sphere with a sparse expansion in wavelets. In astrophysics, it finds a particular application for the identification of localized directional features in the cosmic microwave background (CMB) data, such as the imprint of topological defects, in particular cosmic strings, and for their reconstruction after separation from the other signal components.Comment: 22 pages, 2 figures. Version 2 matches version accepted for publication in MNRAS. Version 3 (identical to version 2) posted for code release announcement - "Steerable scale discretised wavelets on the sphere" - S2DW code available for download at http://www.mrao.cam.ac.uk/~jdm57/software.htm