Title: A sparse variable selection approach in multiscale local polynomial density estimation
Authors: Maarten Jansen - ULB Brussels (Belgium) [presenting]
Abstract: The multiscale local polynomial transform (MLPT, implemented in the Matlab Wavelet Toolbox) is a slightly overcomplete data representation that combines the sparsity of a wavelet decomposition and the smoothness properties of a local polynomial smoothing procedure on nonequispaced data points. The MLPT adopts the bandwidths as user controlled nondyadic resolution levels. Careful application of the MLPT enables us to perform density estimation without preprocessing and corresponding possible loss of information. The densities under consideration may have multiple singularities at unknown locations. The presence of singularities, as well as the intermittent nature of the density estimation problem itself, with intervals of low and high intensities, are natural arguments for a multiscale approach. Moreover, taking the sample values as data points, leads immediately to a nonequispaced problem. The MLPT basis functions can be used in the design matrix of a sparse high-dimensional regression model, with asymptotically exponential responses, where the responses are given by the spacings, i.e., the differences between successive values in the ordered sample. We explain how the Karush-Kuhn-Tucker conditions for the L1-regularised exponential regression model can be approximately solved by soft-thresholding the MLPT applied to the inverse spacings. Optimal thresholds can be chosen by (estimated) minimisation of the Kullback-Leibler distance.