Post-Nonlinear Mixtures and Beyond

Although sources in general nonlinear mixturm arc not separable iising only statistical independence, a special and realistic case of nonlinear mixtnres, the post nonlinear (PNL) mixture is separable choosing a suited separating system. Then, a natural approach is based on the estimation of tho separating Bystem parameters by minimizing an indcpendence criterion, like estimated mwce mutual information. This class of methods requires higher (than 2) order statistics, and cannot separate Gaarsian sources. However, use of [weak) prior, like source temporal correlation or nonstationarity, leads to other source separation Jgw rithms, which are able to separate Gaussian sourra, and can even, for a few of them, works with second-order statistics. Recently, modeling time correlated s011rces by Markov models, we propose vcry efficient algorithms hmed on minimization of the conditional mutual information. Currently, using the prior of temporally correlated sources, we investigate the fesihility of inverting PNL mixtures with non-bijectiw non-liacarities, like quadratic functions. In this paper, we review the main ICA and BSS results for riunlinear mixtures, present PNL models and algorithms, and finish with advanced resutts using temporally correlated snu~sm

Simulating maize phenology as a function of air temperature with a linear and a nonlinear model

The objective of this study was to adapt a nonlinear model (Wang and Engel - WE) for simulating the phenology of maize (Zea mays L.), and to evaluate this model and a linear one (thermal time), in order to predict developmental stages of a field-grown maize variety. A field experiment, during 2005/2006 and 2006/2007 was conducted in Santa Maria, RS, Brazil, in two growing seasons, with seven sowing dates each. Dates of emergence, silking, and physiological maturity of the maize variety BRS Missões were recorded in six replications in each sowing date. Data collected in 2005/2006 growing season were used to estimate the coefficients of the two models, and data collected in the 2006/2007 growing season were used as independent data set for model evaluations. The nonlinear WE model accurately predicted the date of silking and physiological maturity, and had a lower root mean square error (RMSE) than the linear (thermal time) model. The overall RMSE for silking and physiological maturity was 2.7 and 4.8 days with WE model, and 5.6 and 8.3 days with thermal time model, respectively.

Robust Nonlinear Mapping of Soil Contamination Using Support Regression

Spatial data analysis mapping and visualization is of great importance in various fields: environment, pollution, natural hazards and risks, epidemiology, spatial econometrics, etc. A basic task of spatial mapping is to make predictions based on some empirical data (measurements). A number of state-of-the-art methods can be used for the task: deterministic interpolations, methods of geostatistics: the family of kriging estimators (Deutsch and Journel, 1997), machine learning algorithms such as artificial neural networks (ANN) of different architectures, hybrid ANN-geostatistics models (Kanevski and Maignan, 2004; Kanevski et al., 1996), etc. All the methods mentioned above can be used for solving the problem of spatial data mapping. Environmental empirical data are always contaminated/corrupted by noise, and often with noise of unknown nature. That's one of the reasons why deterministic models can be inconsistent, since they treat the measurements as values of some unknown function that should be interpolated. Kriging estimators treat the measurements as the realization of some spatial randomn process. To obtain the estimation with kriging one has to model the spatial structure of the data: spatial correlation function or (semi-)variogram. This task can be complicated if there is not sufficient number of measurements and variogram is sensitive to outliers and extremes. ANN is a powerful tool, but it also suffers from the number of reasons. of a special type ? multiplayer perceptrons ? are often used as a detrending tool in hybrid (ANN+geostatistics) models (Kanevski and Maignank, 2004). Therefore, development and adaptation of the method that would be nonlinear and robust to noise in measurements, would deal with the small empirical datasets and which has solid mathematical background is of great importance. The present paper deals with such model, based on Statistical Learning Theory (SLT) - Support Vector Regression. SLT is a general mathematical framework devoted to the problem of estimation of the dependencies from empirical data (Hastie et al, 2004; Vapnik, 1998). SLT models for classification - Support Vector Machines - have shown good results on different machine learning tasks. The results of SVM classification of spatial data are also promising (Kanevski et al, 2002). The properties of SVM for regression - Support Vector Regression (SVR) are less studied. First results of the application of SVR for spatial mapping of physical quantities were obtained by the authorsin for mapping of medium porosity (Kanevski et al, 1999), and for mapping of radioactively contaminated territories (Kanevski and Canu, 2000). The present paper is devoted to further understanding of the properties of SVR model for spatial data analysis and mapping. Detailed description of the SVR theory can be found in (Cristianini and Shawe-Taylor, 2000; Smola, 1996) and basic equations for the nonlinear modeling are given in section 2. Section 3 discusses the application of SVR for spatial data mapping on the real case study - soil pollution by Cs137 radionuclide. Section 4 discusses the properties of the modelapplied to noised data or data with outliers.

Chaotic systems with a null conditional Lyapunov exponent under nonlinear driving.

Control of a chaotic system by homogeneous nonlinear driving, when a conditional Lyapunov exponent is zero, may give rise to special and interesting synchronizationlike behaviors in which the response evolves in perfect correlation with the drive. Among them, there are the amplification of the drive attractor and the shift of it to a different region of phase space. In this paper, these synchronizationlike behaviors are discussed, and demonstrated by computer simulation of the Lorentz model [E. N. Lorenz, J. Atmos. Sci. 20 130 (1963)] and the double scroll [T. Matsumoto, L. O. Chua, and M. Komuro, IEEE Trans. CAS CAS-32, 798 (1985)].

Nonlinear prediction based on score function

The linear prediction coding of speech is based in the assumption that the generation model is autoregresive. In this paper we propose a structure to cope with the nonlinear effects presents in the generation of the speech signal. This structure will consist of two stages, the first one will be a classical linear prediction filter, and the second one will model the residual signal by means of two nonlinearities between a linear filter. The coefficients of this filter are computed by means of a gradient search on the score function. This is done in order to deal with the fact that the probability distribution of the residual signal still is not gaussian. This fact is taken into account when the coefficients are computed by a ML estimate. The algorithm based on the minimization of a high-order statistics criterion, uses on-line estimation of the residue statistics and is based on blind deconvolution of Wiener systems [1]. Improvements in the experimental results with speech signals emphasize on the interest of this approach.

Covalidation of Integral Transforms and Method of Lines in Nonlinear Convection-Diffusion with Mathematica

The Mathematica system (version 4.0) is employed in the solution of nonlinear difusion and convection-difusion problems, formulated as transient one-dimensional partial diferential equations with potential dependent equation coefficients. The Generalized Integral Transform Technique (GITT) is first implemented for the hybrid numerical-analytical solution of such classes of problems, through the symbolic integral transformation and elimination of the space variable, followed by the utilization of the built-in Mathematica function NDSolve for handling the resulting transformed ODE system. This approach ofers an error-controlled final numerical solution, through the simultaneous control of local errors in this reliable ODE's solver and of the proposed eigenfunction expansion truncation order. For covalidation purposes, the same built-in function NDSolve is employed in the direct solution of these partial diferential equations, as made possible by the algorithms implemented in Mathematica (versions 3.0 and up), based on application of the method of lines. Various numerical experiments are performed and relative merits of each approach are critically pointed out.

Estimating Attractor Dimension on the Nonlinear Pendulum Time Series

Chaotic dynamical systems exhibit trajectories in their phase space that converges to a strange attractor. The strangeness of the chaotic attractor is associated with its dimension in which instance it is described by a noninteger dimension. This contribution presents an overview of the main definitions of dimension discussing their evaluation from time series employing the correlation and the generalized dimension. The investigation is applied to the nonlinear pendulum where signals are generated by numerical integration of the mathematical model, selecting a single variable of the system as a time series. In order to simulate experimental data sets, a random noise is introduced in the time series. State space reconstruction and the determination of attractor dimensions are carried out regarding periodic and chaotic signals. Results obtained from time series analyses are compared with a reference value obtained from the analysis of mathematical model, estimating noise sensitivity. This procedure allows one to identify the best techniques to be applied in the analysis of experimental data.

Linear and nonlinear analysis of heart rate variability in healthy subjects and after acute myocardial infarction in patients

The objectives of this study were to evaluate and compare the use of linear and nonlinear methods for analysis of heart rate variability (HRV) in healthy subjects and in patients after acute myocardial infarction (AMI). Heart rate (HR) was recorded for 15 min in the supine position in 10 patients with AMI taking β-blockers (aged 57 ± 9 years) and in 11 healthy subjects (aged 53 ± 4 years). HRV was analyzed in the time domain (RMSSD and RMSM), the frequency domain using low- and high-frequency bands in normalized units (nu; LFnu and HFnu) and the LF/HF ratio and approximate entropy (ApEn) were determined. There was a correlation (P < 0.05) of RMSSD, RMSM, LFnu, HFnu, and the LF/HF ratio index with the ApEn of the AMI group on the 2nd (r = 0.87, 0.65, 0.72, 0.72, and 0.64) and 7th day (r = 0.88, 0.70, 0.69, 0.69, and 0.87) and of the healthy group (r = 0.63, 0.71, 0.63, 0.63, and 0.74), respectively. The median HRV indexes of the AMI group on the 2nd and 7th day differed from the healthy group (P < 0.05): RMSSD = 10.37, 19.95, 24.81; RMSM = 23.47, 31.96, 43.79; LFnu = 0.79, 0.79, 0.62; HFnu = 0.20, 0.20, 0.37; LF/HF ratio = 3.87, 3.94, 1.65; ApEn = 1.01, 1.24, 1.31, respectively. There was agreement between the methods, suggesting that these have the same power to evaluate autonomic modulation of HR in both AMI patients and healthy subjects. AMI contributed to a reduction in cardiac signal irregularity, higher sympathetic modulation and lower vagal modulation.

Long-term correlation of the electrocorticogram as a bioindicator of brain exposure to ionizing radiation

Understanding the effects of radiation and its possible influence on the nervous system are of great clinical interest. However, there have been few electrophysiological studies on brain activity after exposure to ionizing radiation (IR). A new methodological approach regarding the assessment of the possible effects of IR on brain activity is the use of linear and nonlinear mathematical methods in the analysis of complex time series, such as brain oscillations measured using the electrocorticogram (ECoG). The objective of this study was to use linear and nonlinear mathematical methods as biomarkers of gamma radiation regarding cortical electrical activity. Adult Wistar rats were divided into 3 groups: 1 control and 2 irradiated groups, evaluated at 24 h (IR24) and 90 days (IR90) after exposure to 18 Gy of gamma radiation from a cobalt-60 radiotherapy source. The ECoG was analyzed using power spectrum methods for the calculation of the power of delta, theta, alpha and beta rhythms and by means of the α-exponent of the detrended fluctuation analysis (DFA). Using both mathematical methods it was possible to identify changes in the ECoG, and to identify significant changes in the pattern of the recording at 24 h after irradiation. Some of these changes were persistent at 90 days after exposure to IR. In particular, the theta wave using the two methods showed higher sensitivity than other waves, suggesting that it is a possible biomarker of exposure to IR.

STUDIES ON THE STRESS FLUCTUATIONS IN SHEARED STOKESIAN SUSPENSIONS USING CHAOS THEORY AND NONLINEAR DYNAMICS

The thesis report results obtained from a detailed analysis of the fluctuations of the rheological parameters viz. shear and normal stresses, simulated by means of the Stokesian Dynamics method, of a macroscopically homogeneous sheared suspension of neutrally buoyant non-Brownian suspension of identical spheres in the Couette gap between two parallel walls in the limit of vanishingly small Reynolds numbers using the tools of non-linear dynamics and chaos theory for a range of particle concentration and Couette gaps. The thesis used the tools of nonlinear dynamics and chaos theory viz. average mutual information, space-time separation plots, visual recurrence analysis, principal component analysis, false nearest-neighbor technique, correlation integrals, computation of Lyapunov exponents for a range of area fraction of particles and for different Couette gaps. The thesis observed that one stress component can be predicted using another stress component at the same area fraction. This implies a type of synchronization of one stress component with another stress component. This finding suggests us to further analysis of the synchronization of stress components with another stress component at the same or different area fraction of particles. The different model equations of stress components for different area fraction of particles hints at the possible existence a general formula for stress fluctuations with area fraction of particle as a parameter

Speech Analysis using Modern Techniques of Nonlinear Dynamics

Medical fields requires fast, simple and noninvasive methods of diagnostic techniques. Several methods are available and possible because of the growth of technology that provides the necessary means of collecting and processing signals. The present thesis details the work done in the field of voice signals. New methods of analysis have been developed to understand the complexity of voice signals, such as nonlinear dynamics aiming at the exploration of voice signals dynamic nature. The purpose of this thesis is to characterize complexities of pathological voice from healthy signals and to differentiate stuttering signals from healthy signals. Efficiency of various acoustic as well as non linear time series methods are analysed. Three groups of samples are used, one from healthy individuals, subjects with vocal pathologies and stuttering subjects. Individual vowels/ and a continuous speech data for the utterance of the sentence "iruvarum changatimaranu" the meaning in English is "Both are good friends" from Malayalam language are recorded using a microphone . The recorded audio are converted to digital signals and are subjected to analysis.Acoustic perturbation methods like fundamental frequency (FO), jitter, shimmer, Zero Crossing Rate(ZCR) were carried out and non linear measures like maximum lyapunov exponent(Lamda max), correlation dimension (D2), Kolmogorov exponent(K2), and a new measure of entropy viz., Permutation entropy (PE) are evaluated for all three groups of the subjects. Permutation Entropy is a nonlinear complexity measure which can efficiently distinguish regular and complex nature of any signal and extract information about the change in dynamics of the process by indicating sudden change in its value. The results shows that nonlinear dynamical methods seem to be a suitable technique for voice signal analysis, due to the chaotic component of the human voice. Permutation entropy is well suited due to its sensitivity to uncertainties, since the pathologies are characterized by an increase in the signal complexity and unpredictability. Pathological groups have higher entropy values compared to the normal group. The stuttering signals have lower entropy values compared to the normal signals.PE is effective in charaterising the level of improvement after two weeks of speech therapy in the case of stuttering subjects. PE is also effective in characterizing the dynamical difference between healthy and pathological subjects. This suggests that PE can improve and complement the recent voice analysis methods available for clinicians. The work establishes the application of the simple, inexpensive and fast algorithm of PE for diagnosis in vocal disorders and stuttering subjects.

Systematic study of the static electrical properties of the CO molecule: influence of the basis set size and correlation energy

The influence of the basis set size and the correlation energy in the static electrical properties of the CO molecule is assessed. In particular, we have studied both the nuclear relaxation and the vibrational contributions to the static molecular electrical properties, the vibrational Stark effect (VSE) and the vibrational intensity effect (VIE). From a mathematical point of view, when a static and uniform electric field is applied to a molecule, the energy of this system can be expressed in terms of a double power series with respect to the bond length and to the field strength. From the power series expansion of the potential energy, field-dependent expressions for the equilibrium geometry, for the potential energy and for the force constant are obtained. The nuclear relaxation and vibrational contributions to the molecular electrical properties are analyzed in terms of the derivatives of the electronic molecular properties. In general, the results presented show that accurate inclusion of the correlation energy and large basis sets are needed to calculate the molecular electrical properties and their derivatives with respect to either nuclear displacements or/and field strength. With respect to experimental data, the calculated power series coefficients are overestimated by the SCF, CISD, and QCISD methods. On the contrary, perturbation methods (MP2 and MP4) tend to underestimate them. In average and using the 6-311 + G(3df) basis set and for the CO molecule, the nuclear relaxation and the vibrational contributions to the molecular electrical properties amount to 11.7%, 3.3%, and 69.7% of the purely electronic μ, α, and β values, respectively

Variational calculation of vibrational linear and nonlinear optical properties

A variational approach for reliably calculating vibrational linear and nonlinear optical properties of molecules with large electrical and/or mechanical anharmonicity is introduced. This approach utilizes a self-consistent solution of the vibrational Schrödinger equation for the complete field-dependent potential-energy surface and, then, adds higher-level vibrational correlation corrections as desired. An initial application is made to static properties for three molecules of widely varying anharmonicity using the lowest-level vibrational correlation treatment (i.e., vibrational Møller-Plesset perturbation theory). Our results indicate when the conventional Bishop-Kirtman perturbation method can be expected to break down and when high-level vibrational correlation methods are likely to be required. Future improvements and extensions are discussed

Correlation-type structure activity relationships for the kinetics of gas-phase RO2 self-reactions and reaction with HO2

Structure activity relationships (SARs) are presented for the gas-phase reactions of RO2 with HO2, and the self- and cross-reactions of RO2. For RO2+HO2 the SAR is based upon a correlation between the logarithm of the measured rate coefficient and a calculated ionisation potential for the molecule R-CH=CH2, R being the same group in both the radical and molecular analogue. The correlation observed is strong and only for one RO2 species does the measured rate coefficient deviate by more than a factor of two from the linear least-squares regression line. For the self- and cross-reactions of RO2 radicals, the SAR is based upon a correlation between the logarithm of the measured rate coefficient and the calculated electrostatic potential (ESP) at the equivalent carbon atom in the RH molecule to which oxygen is attached in RO2, again R being the same group in the molecule and the radical. For cases where R is a simple alkyl-group, a strong linear correlation observed. For RO2 radicals which contain lone pair-bearing substituents and for which the calculated ESP<-0.05 self-reaction rate coefficients appear to be insensitive to the value of the ESP. For RO2 of this type with ESP>-0.05 a linear relationship between log k and the ESP is again observed. Using the relationships, 84 out of the 85 rate coefficients used to develop the SARs are predicted to within a factor of three of their measured values. A relationship is also presented that allows the prediction of the Arrhenius parameters for the self-reactions of simple alkyl RO2 radicals. On the basis of the correlations, predictions of room-temperature rate coefficients are made for a number of atmospherically important peroxyl-peroxyl radical reactions. (C) 2003 Elsevier Ltd. All rights reserved.

Gas-phase rate coefficients for the reactions of NO3, OH and O-3 with alpha,beta-unsaturated esters and ketones: structure-activity relations (SARs)

Gas-phase rate coefficients for the atmospherically important reactions of NO3, OH and O-3 are predicted for 55 alpha,beta-unsaturated esters and ketones. The rate coefficients were calculated using a correlation described previously [Pfrang, C., King, M.D., C. E. Canosa-Mas, C.E., Wayne, R.P., 2006. Atmospheric Environment 40, 1170-1179]. These rate coefficients were used to extend structure-activity relations for predicting the rate coefficients for the reactions of NO3, OH or O-3 with alkenes to include alpha,beta-unsaturated esters and ketones. Conjugation of an alkene with an alpha,beta-keto or alpha,beta-ester group will reduce the value of a rate coefficient by a factor of similar to 110, similar to 2.5 and similar to 12 for reaction with NO3, OH or O-3, respectively. The actual identity of the alkyl group, R, in -C(O)R or -C(O)OR has only a small influence. An assessment of the reliability of the SAR is given that demonstrates that it is useful for reactions involving NO3 and OH, but less valuable for those of O-3 or peroxy nitrate esters. (c) 2006 Elsevier Ltd. All rights reserved.