Parallel Hybrid Monte Carlo Algorithms for Matrix Computations

In this paper we consider hybrid (fast stochastic approximation and deterministic refinement) algorithms for Matrix Inversion (MI) and Solving Systems of Linear Equations (SLAE). Monte Carlo methods are used for the stochastic approximation, since it is known that they are very efficient in finding a quick rough approximation of the element or a row of the inverse matrix or finding a component of the solution vector. We show how the stochastic approximation of the MI can be combined with a deterministic refinement procedure to obtain MI with the required precision and further solve the SLAE using MI. We employ a splitting A = D – C of a given non-singular matrix A, where D is a diagonal dominant matrix and matrix C is a diagonal matrix. In our algorithm for solving SLAE and MI different choices of D can be considered in order to control the norm of matrix T = D –1C, of the resulting SLAE and to minimize the number of the Markov Chains required to reach given precision. Further we run the algorithms on a mini-Grid and investigate their efficiency depending on the granularity. Corresponding experimental results are presented.

Parallel Monte Carlo algorithms for information retrieval

In any data mining applications, automated text and text and image retrieval of information is needed. This becomes essential with the growth of the Internet and digital libraries. Our approach is based on the latent semantic indexing (LSI) and the corresponding term-by-document matrix suggested by Berry and his co-authors. Instead of using deterministic methods to find the required number of first "k" singular triplets, we propose a stochastic approach. First, we use Monte Carlo method to sample and to build much smaller size term-by-document matrix (e.g. we build k x k matrix) from where we then find the first "k" triplets using standard deterministic methods. Second, we investigate how we can reduce the problem to finding the "k"-largest eigenvalues using parallel Monte Carlo methods. We apply these methods to the initial matrix and also to the reduced one. The algorithms are running on a cluster of workstations under MPI and results of the experiments arising in textual retrieval of Web documents as well as comparison of the stochastic methods proposed are presented. (C) 2003 IMACS. Published by Elsevier Science B.V. All rights reserved.

EMG signal filtering based on Empirical Mode Decomposition

This paper introduces a procedure for filtering electromyographic (EMG) signals. Its key element is the Empirical Mode Decomposition, a novel digital signal processing technique that can decompose my time-series into a set of functions designated as intrinsic mode functions. The procedure for EMG signal filtering is compared to a related approach based on the wavelet transform. Results obtained from the analysis of synthetic and experimental EMG signals show that Our method can be Successfully and easily applied in practice to attenuation of background activity in EMG signals. (c) 2006 Elsevier Ltd. All rights reserved.

Empirical mode decomposition: a novel technique for the study of tremor time series

Tremor is a clinical feature characterized by oscillations of a part of the body. The detection and study of tremor is an important step in investigations seeking to explain underlying control strategies of the central nervous system under natural (or physiological) and pathological conditions. It is well established that tremorous activity is composed of deterministic and stochastic components. For this reason, the use of digital signal processing techniques (DSP) which take into account the nonlinearity and nonstationarity of such signals may bring new information into the signal analysis which is often obscured by traditional linear techniques (e.g. Fourier analysis). In this context, this paper introduces the application of the empirical mode decomposition (EMD) and Hilbert spectrum (HS), which are relatively new DSP techniques for the analysis of nonlinear and nonstationary time-series, for the study of tremor. Our results, obtained from the analysis of experimental signals collected from 31 patients with different neurological conditions, showed that the EMD could automatically decompose acquired signals into basic components, called intrinsic mode functions (IMFs), representing tremorous and voluntary activity. The identification of a physical meaning for IMFs in the context of tremor analysis suggests an alternative and new way of detecting tremorous activity. These results may be relevant for those applications requiring automatic detection of tremor. Furthermore, the energy of IMFs was visualized as a function of time and frequency by means of the HS. This analysis showed that the variation of energy of tremorous and voluntary activity could be distinguished and characterized on the HS. Such results may be relevant for those applications aiming to identify neurological disorders. In general, both the HS and EMD demonstrated to be very useful to perform objective analysis of any kind of tremor and can therefore be potentially used to perform functional assessment.

Complexity of Monte Carlo algorithms for a class of integral equations

In this work we study the computational complexity of a class of grid Monte Carlo algorithms for integral equations. The idea of the algorithms consists in an approximation of the integral equation by a system of algebraic equations. Then the Markov chain iterative Monte Carlo is used to solve the system. The assumption here is that the corresponding Neumann series for the iterative matrix does not necessarily converge or converges slowly. We use a special technique to accelerate the convergence. An estimate of the computational complexity of Monte Carlo algorithm using the considered approach is obtained. The estimate of the complexity is compared with the corresponding quantity for the complexity of the grid-free Monte Carlo algorithm. The conditions under which the class of grid Monte Carlo algorithms is more efficient are given.

Robustness and applicability of Markov chain Monte Carlo algorithms for eigenvalue problems

In this paper we analyse applicability and robustness of Markov chain Monte Carlo algorithms for eigenvalue problems. We restrict our consideration to real symmetric matrices. Almost Optimal Monte Carlo (MAO) algorithms for solving eigenvalue problems are formulated. Results for the structure of both - systematic and probability error are presented. It is shown that the values of both errors can be controlled independently by different algorithmic parameters. The results present how the systematic error depends on the matrix spectrum. The analysis of the probability error is presented. It shows that the close (in some sense) the matrix under consideration is to the stochastic matrix the smaller is this error. Sufficient conditions for constructing robust and interpolation Monte Carlo algorithms are obtained. For stochastic matrices an interpolation Monte Carlo algorithm is constructed. A number of numerical tests for large symmetric dense matrices are performed in order to study experimentally the dependence of the systematic error from the structure of matrix spectrum. We also study how the probability error depends on the balancing of the matrix. (c) 2007 Elsevier Inc. All rights reserved.

A neurofuzzy network knowledge extraction and extended gram-Schmidt algorithm for model subspace decomposition

This paper introduces a new neurofuzzy model construction and parameter estimation algorithm from observed finite data sets, based on a Takagi and Sugeno (T-S) inference mechanism and a new extended Gram-Schmidt orthogonal decomposition algorithm, for the modeling of a priori unknown dynamical systems in the form of a set of fuzzy rules. The first contribution of the paper is the introduction of a one to one mapping between a fuzzy rule-base and a model matrix feature subspace using the T-S inference mechanism. This link enables the numerical properties associated with a rule-based matrix subspace, the relationships amongst these matrix subspaces, and the correlation between the output vector and a rule-base matrix subspace, to be investigated and extracted as rule-based knowledge to enhance model transparency. The matrix subspace spanned by a fuzzy rule is initially derived as the input regression matrix multiplied by a weighting matrix that consists of the corresponding fuzzy membership functions over the training data set. Model transparency is explored by the derivation of an equivalence between an A-optimality experimental design criterion of the weighting matrix and the average model output sensitivity to the fuzzy rule, so that rule-bases can be effectively measured by their identifiability via the A-optimality experimental design criterion. The A-optimality experimental design criterion of the weighting matrices of fuzzy rules is used to construct an initial model rule-base. An extended Gram-Schmidt algorithm is then developed to estimate the parameter vector for each rule. This new algorithm decomposes the model rule-bases via an orthogonal subspace decomposition approach, so as to enhance model transparency with the capability of interpreting the derived rule-base energy level. This new approach is computationally simpler than the conventional Gram-Schmidt algorithm for resolving high dimensional regression problems, whereby it is computationally desirable to decompose complex models into a few submodels rather than a single model with large number of input variables and the associated curse of dimensionality problem. Numerical examples are included to demonstrate the effectiveness of the proposed new algorithm.

Experimental design and model construction algorithms for radial basis function networks

New construction algorithms for radial basis function (RBF) network modelling are introduced based on the A-optimality and D-optimality experimental design criteria respectively. We utilize new cost functions, based on experimental design criteria, for model selection that simultaneously optimizes model approximation, parameter variance (A-optimality) or model robustness (D-optimality). The proposed approaches are based on the forward orthogonal least-squares (OLS) algorithm, such that the new A-optimality- and D-optimality-based cost functions are constructed on the basis of an orthogonalization process that gains computational advantages and hence maintains the inherent computational efficiency associated with the conventional forward OLS approach. The proposed approach enhances the very popular forward OLS-algorithm-based RBF model construction method since the resultant RBF models are constructed in a manner that the system dynamics approximation capability, model adequacy and robustness are optimized simultaneously. The numerical examples provided show significant improvement based on the D-optimality design criterion, demonstrating that there is significant room for improvement in modelling via the popular RBF neural network.

Robust neurofuzzy rule base knowledge extraction and estimation using subspace decomposition combined with regularization and D-optimality

A new robust neurofuzzy model construction algorithm has been introduced for the modeling of a priori unknown dynamical systems from observed finite data sets in the form of a set of fuzzy rules. Based on a Takagi-Sugeno (T-S) inference mechanism a one to one mapping between a fuzzy rule base and a model matrix feature subspace is established. This link enables rule based knowledge to be extracted from matrix subspace to enhance model transparency. In order to achieve maximized model robustness and sparsity, a new robust extended Gram-Schmidt (G-S) method has been introduced via two effective and complementary approaches of regularization and D-optimality experimental design. Model rule bases are decomposed into orthogonal subspaces, so as to enhance model transparency with the capability of interpreting the derived rule base energy level. A locally regularized orthogonal least squares algorithm, combined with a D-optimality used for subspace based rule selection, has been extended for fuzzy rule regularization and subspace based information extraction. By using a weighting for the D-optimality cost function, the entire model construction procedure becomes automatic. Numerical examples are included to demonstrate the effectiveness of the proposed new algorithm.

A novel approach to the detection of synchronisation in EEG based on empirical mode decomposition

Transient neural assemblies mediated by synchrony in particular frequency ranges are thought to underlie cognition. We propose a new approach to their detection, using empirical mode decomposition (EMD), a data-driven approach removing the need for arbitrary bandpass filter cut-offs. Phase locking is sought between modes. We explore the features of EMD, including making a quantitative assessment of its ability to preserve phase content of signals, and proceed to develop a statistical framework with which to assess synchrony episodes. Furthermore, we propose a new approach to ensure signal decomposition using EMD. We adapt the Hilbert spectrum to a time-frequency representation of phase locking and are able to locate synchrony successfully in time and frequency between synthetic signals reminiscent of EEG. We compare our approach, which we call EMD phase locking analysis (EMDPL) with existing methods and show it to offer improved time-frequency localisation of synchrony.

Detection of neural correlates of self-paced motor activity using empirical mode decomposition phase locking analysis

Transient episodes of synchronisation of neuronal activity in particular frequency ranges are thought to underlie cognition. Empirical mode decomposition phase locking (EMDPL) analysis is a method for determining the frequency and timing of phase synchrony that is adaptive to intrinsic oscillations within data, alleviating the need for arbitrary bandpass filter cut-off selection. It is extended here to address the choice of reference electrode and removal of spurious synchrony resulting from volume conduction. Spline Laplacian transformation and independent component analysis (ICA) are performed as pre-processing steps, and preservation of phase synchrony between synthetic signals. combined using a simple forward model, is demonstrated. The method is contrasted with use of bandpass filtering following the same preprocessing steps, and filter cut-offs are shown to influence synchrony detection markedly. Furthermore, an approach to the assessment of multiple EEG trials using the method is introduced, and the assessment of statistical significance of phase locking episodes is extended to render it adaptive to local phase synchrony levels. EMDPL is validated in the analysis of real EEG data, during finger tapping. The time course of event-related (de)synchronisation (ERD/ERS) is shown to differ from that of longer range phase locking episodes, implying different roles for these different types of synchronisation. It is suggested that the increase in phase locking which occurs just prior to movement, coinciding with a reduction in power (or ERD) may result from selection of the neural assembly relevant to the particular movement. (C) 2009 Elsevier B.V. All rights reserved.

Thermal decomposition of potassium hexanitronickelate(II) hydrate

The thermal decomposition of the complex K-4[Ni(NO2)6]center dot H2O has been investigated over the temperature range 25-600 degrees C by a combination of infrared spectroscopy, powder X-ray diffraction, FAB-mass spectrometry and elemental analysis. The first stage of reaction is loss of water and isomerisation of one of the coordinated nitro groups to form the complex K-4 [Ni(NO2)(4) (ONO)]center dot NO2. At temperatures around 200 degrees C the remaining nitro groups within the complex isomerise to the chelating nitrite form and this process acts as a precursor to the loss of NO2 gas at temperatures above 270 degrees C. The product, which is stable up to 600 degrees C, is the complex K-4[Ni(ONO)(4)]center dot NO2, where the nickel atom is formally in the +1 oxidation state. (c) 2005 Elsevier B.V. All rights reserved.