A new Ensemble Empirical Mode Decomposition (EEMD) is presented. This new approach consists of sifting an ensemble of white noise-added signal (data) and treats the mean as the final true result. Finite, not infinitesimal, amplitude white noise is necessary to force the ensemble to exhaust all possible solutions in the sifting process, thus making the different scale signals to collate in the proper intrinsic mode functions (IMF) dictated by the dyadic filter banks. As EEMD is a time–space analysis method, the added white noise is averaged out with sufficient number of trials; the only persistent part that survives the averaging process is the component of the signal (original data), which is then treated as the true and more physical meaningful answer. The effect of the added white noise is to provide a uniform reference frame in the time–frequency space; therefore, the added noise collates the portion of the signal of comparable scale in one IMF. With this ensemble mean, one can separate scales naturall...

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Ensemble empirical mode decomposition: a noise-assisted data analysis method

The organization of behavior: A neuropsychological theory

Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.

http://www.eeci-institute.eu/GSC2011/Photos-EECI/EECI-GSC-2011-M5/book_AMS.pdf

Optimization Algorithms on Matrix Manifolds

When a parameter space has a certain underlying structure, the ordinary gradient of a function does not represent its steepest direction, but the natural gradient does. Information geometry is used for calculating the natural gradients in the parameter space of perceptrons, the space of matrices (for blind source separation), and the space of linear dynamical systems (for blind source deconvolution). The dynamical behavior of natural gradient online learning is analyzed and is proved to be Fisher efficient, implying that it has asymptotically the same performance as the optimal batch estimation of parameters. This suggests that the plateau phenomenon, which appears in the backpropagation learning algorithm of multilayer perceptrons, might disappear or might not be so serious when the natural gradient is used. An adaptive method of updating the learning rate is proposed and analyzed.

Natural gradient works efficiently in learning

Independent Component Analysis.

Adaptive algorithms for the rejection of sinusoidal disturbances with unknown frequency

Multichannel deconvolution and equalization is an important task for numerous applications in communications, signal processing, and control. We extend the efficient natural gradient search method of Amari, Cichocki and Yang (see Advances in Neural Information Processing Systems, p.752-63, 1995) to derive a set of on-line algorithms for combined multichannel blind source separation and time-domain deconvolution/equalization of additive, convolved signal mixtures. We prove that the doubly-infinite multichannel equalizer based on the maximum entropy cost function with natural gradient possesses the so-called "equivariance property" such that its asymptotic performance depends on the normalized stochastic distribution of the source signals and not on the characteristics of the unknown channel. Simulations indicate the ability of the algorithm to perform efficient simultaneous multichannel signal deconvolution and source separation.

Multichannel blind deconvolution and equalization using the natural gradient

In some adaptive filtering applications, the least-mean-square (LBIS) algorithm may be too computationally- and memory-intensive to implement. In this paper, we analyze two adaptive algorithms that update only a portion of the coefficients of the adaptive filter per iteration. These algorithms use decimated versions of the error and regressor signals, respectively. Simulations verify the accuracy of the analyzes, and the robustness of the algorithms is also explored.

Adaptive filters employing partial updates

Gradient adaptation is a useful technique for adjusting a set of parameters to minimize a cost function. While often easy to implement, the convergence speed of gradient adaptation can be slow when the slope of the cost function varies widely for small changes in the parameters. In this paper, we outline an alternative technique, termed natural gradient adaptation, that overcomes the poor convergence properties of gradient adaptation in many cases. The natural gradient is based on differential geometry and employs knowledge of the Riemannian structure of the parameter space to adjust the gradient search direction. Unlike Newton's method, natural gradient adaptation does not assume a locally-quadratic cost function. Moreover, for maximum likelihood estimation tasks, natural gradient adaptation is asymptotically Fisher-efficient. A simple example illustrates the desirable properties of natural gradient adaptation.

Why natural gradient

A derivation of the normalized LMS algorithm is generalized, resulting in a family of projection-like algorithms based on an L/sub p/-minimized filter coefficient change. The resulting algorithms include the simplified NLMS algorithm of Nagumo and Noda (1967) and an even simpler single-coefficient update algorithm based on the maximum absolute value datum of the input data vector. A complete derivation of the algorithm family is given, and simulations are performed to show the convergence behaviors of the algorithms. >

Scott C. Douglas

Papers

Adaptive algorithms for the rejection of sinusoidal disturbances with unknown frequency

Multichannel blind deconvolution and equalization using the natural gradient

Adaptive filters employing partial updates

Why natural gradient

A family of normalized LMS algorithms