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.

Dissipative nonlinear structures in optical fiber systems III : dissipative solitons via nonlinear mapping

In the third and final talk on dissipative structures in fiber applications, we discuss mathematical techniques that can be used to characterize modern laser systems that consist of several discrete elements. In particular, we use a nonlinear mapping technique to evaluate high power laser systems where significant changes in the pulse evolution per cavity round trip is observed. We demonstrate that dissipative soliton solutions might be effectively described using this Poincaré mapping approach.

Dissipative nonlinear structures in optical fiber systems III:dissipative solitons via nonlinear mapping

In the third and final talk on dissipative structures in fiber applications, we discuss mathematical techniques that can be used to characterize modern laser systems that consist of several discrete elements. In particular, we use a nonlinear mapping technique to evaluate high power laser systems where significant changes in the pulse evolution per cavity round trip is observed. We demonstrate that dissipative soliton solutions might be effectively described using this Poincaré mapping approach.

One-dimensional Fermi accelerator model with moving wall described by a nonlinear van der Pol oscillator

A modification of the one-dimensional Fermi accelerator model is considered in this work. The dynamics of a classical particle of mass m, confined to bounce elastically between two rigid walls where one is described by a nonlinear van der Pol type oscillator while the other one is fixed, working as a reinjection mechanism of the particle for a next collision, is carefully made by the use of a two-dimensional nonlinear mapping. Two cases are considered: (i) the situation where the particle has mass negligible as compared to the mass of the moving wall and does not affect the motion of it; and (ii) the case where collisions of the particle do affect the movement of the moving wall. For case (i) the phase space is of mixed type leading us to observe a scaling of the average velocity as a function of the parameter (χ) controlling the nonlinearity of the moving wall. For large χ, a diffusion on the velocity is observed leading to the conclusion that Fermi acceleration is taking place. On the other hand, for case (ii), the motion of the moving wall is affected by collisions with the particle. However, due to the properties of the van der Pol oscillator, the moving wall relaxes again to a limit cycle. Such kind of motion absorbs part of the energy of the particle leading to a suppression of the unlimited energy gain as observed in case (i). The phase space shows a set of attractors of different periods whose basin of attraction has a complicated organization. © 2013 American Physical Society.

Performance analysis using equivariant kernel density estimator in nonlinear mixture

This paper investigates the performance analysis of separation of mutually independent sources in nonlinear models. The nonlinear mapping constituted by an unsupervised linear mixture is followed by an unknown and invertible nonlinear distortion, are found in many signal processing cases. Generally, blind separation of sources from their nonlinear mixtures is rather difficult. We propose using a kernel density estimator incorporated with equivariant gradient analysis to separate the sources with nonlinear distortion. The kernel density estimator parameters of which are iteratively updated to minimize the output independence expressed as a mutual information criterion. The equivariant gradient algorithm has the form of nonlinear decorrelation to perform the convergence analysis. Experiments are proposed to illustrate these results.

Model reduction for nonlinear aerodynamics and aeroelasticity using a Discrete Empirical Interpolation Method

A novel surrogate model is proposed in lieu of Computational Fluid Dynamics (CFD) solvers, for fast nonlinear aerodynamic and aeroelastic modeling. A nonlinear function is identified on selected interpolation points by
a discrete empirical interpolation method (DEIM). The flow field is then reconstructed using a least square approximation of the flow modes extracted
by proper orthogonal decomposition (POD). The aeroelastic reduce order
model (ROM) is completed by introducing a nonlinear mapping function
between displacements and the DEIM points. The proposed model is investigated to predict the aerodynamic forces due to forced motions using
a N ACA 0012 airfoil undergoing a prescribed pitching oscillation. To investigate aeroelastic problems at transonic conditions, a pitch/plunge airfoil
and a cropped delta wing aeroelastic models are built using linear structural models. The presence of shock-waves triggers the appearance of limit
cycle oscillations (LCO), which the model is able to predict. For all cases
tested, the new ROM shows the ability to replicate the nonlinear aerodynamic forces, structural displacements and reconstruct the complete flow
field with sufficient accuracy at a fraction of the cost of full order CFD
model.

Ambulatory assessment of 3D ground reaction force using plantar pressure distribution.

This study aimed to use the plantar pressure insole for estimating the three-dimensional ground reaction force (GRF) as well as the frictional torque (T(F)) during walking. Eleven subjects, six healthy and five patients with ankle disease participated in the study while wearing pressure insoles during several walking trials on a force-plate. The plantar pressure distribution was analyzed and 10 principal components of 24 regional pressure values with the stance time percentage (STP) were considered for GRF and T(F) estimation. Both linear and non-linear approximators were used for estimating the GRF and T(F) based on two learning strategies using intra-subject and inter-subjects data. The RMS error and the correlation coefficient between the approximators and the actual patterns obtained from force-plate were calculated. Our results showed better performance for non-linear approximation especially when the STP was considered as input. The least errors were observed for vertical force (4%) and anterior-posterior force (7.3%), while the medial-lateral force (11.3%) and frictional torque (14.7%) had higher errors. The result obtained for the patients showed higher error; nevertheless, when the data of the same patient were used for learning, the results were improved and in general slight differences with healthy subjects were observed. In conclusion, this study showed that ambulatory pressure insole with data normalization, an optimal choice of inputs and a well-trained nonlinear mapping function can estimate efficiently the three-dimensional ground reaction force and frictional torque in consecutive gait cycle without requiring a force-plate.

Fast Approximation of Nonlinearities for improving inversion algorithms of PNL mixtures and Wiener systems

This paper proposes a very fast method for blindly approximating a nonlinear mapping which transforms a sum of random variables. The estimation is surprisingly good even when the basic assumption is not satisfied.We use the method for providing a good initialization for inverting post-nonlinear mixtures and Wiener systems. Experiments show that the algorithm speed is strongly improved and the asymptotic performance is preserved with a very low extra computational cost.

Initialisation of Nonlinearities for PNL and Wiener systems Inversion

This paper proposes a very fast method for blindly initial- izing a nonlinear mapping which transforms a sum of random variables. The method provides a surprisingly good approximation even when the basic assumption is not fully satis¯ed. The method can been used success- fully for initializing nonlinearity in post-nonlinear mixtures or in Wiener system inversion, for improving algorithm speed and convergence.

Estimación de funciones no lineales en mezclas post-no lineales

This paper proposes a new method for blindly inverting a nonlinear mapping which transforms a sum of random variables. This is the case of post-nonlinear (PNL) source separation mixtures. The importance of the method is based on the fact that it permits to decouple the estimation of the nonlinear part from the estimation of the linear one. Only the nonlinear part is inverted, without considering on the linear part. Hence the initial problem is transformed into a linear one that can then be solved with any convenient linear algorithm. The method is compared with other existing algorithms for blindly approximating nonlinear mappings. Experiments show that the proposed algorithm outperforms the results obtained with other algorithms and give a reasonably good linearized data

Wiener-System subspace identification for mobile wireless mm-wave networks

We discuss the feasibility of wireless terahertz communications links deployed in a metropolitan area and model the large-scale fading of such channels. The model takes into account reception through direct line of sight, ground and wall reflection, as well as diffraction around a corner. The movement of the receiver is modeled by an autonomous dynamic linear system in state space, whereas the geometric relations involved in the attenuation and multipath propagation of the electric field are described by a static nonlinear mapping. A subspace algorithm in conjunction with polynomial regression is used to identify a single-output Wiener model from time-domain measurements of the field intensity when the receiver motion is simulated using a constant angular speed and an exponentially decaying radius. The identification procedure is validated by using the model to perform q-step ahead predictions. The sensitivity of the algorithm to small-scale fading, detector noise, and atmospheric changes are discussed. The performance of the algorithm is tested in the diffraction zone assuming a range of emitter frequencies (2, 38, 60, 100, 140, and 400 GHz). Extensions of the simulation results to situations where a more complicated trajectory describes the motion of the receiver are also implemented, providing information on the performance of the algorithm under a worst case scenario. Finally, a sensitivity analysis to model parameters for the identified Wiener system is proposed.

Dissipative area-preserving one-dimensional Fermi accelerator model

The influence of dissipation on the simplified Fermi-Ulam accelerator model (SFUM) is investigated. The model is described in terms of a two-dimensional nonlinear mapping obtained from differential equations. It is shown that a dissipative SFUM possesses regions of phase space characterized by the property of area preservation.

Dynamical properties for the problem of a particle in an electric field of wave packet: Low velocity and relativistic approach

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

On the dynamical properties of an elliptical-oval billiard with static boundary

Some dynamical properties for a classical particle confined inside a closed region with an elliptical-oval-like shape are studied. The dynamics of the model is made by using a two-dimensional nonlinear mapping. The phase space of the system is of mixed kind and we have found the condition that breaks the invariant spanning curves in the phase space. We have discussed also some statistical properties of the phase space and obtained the behaviour of the positive Lyapunov exponent. (C) 2009 Elsevier B.V. All rights reserved.

Can Drag Force Suppress Fermi Acceleration in a Bouncer Model?

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)