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This open access book provides a comprehensive treatment of recent developments in kernel-based identification that are of interest to anyone engaged in learning dynamic systems from data. The reader is led step by step into understanding of a novel paradigm that leverages the power of machine learning without losing sight of the system-theoretical principles of black-box identification. The authors’ reformulation of the identification problem in the light of regularization theory not only offers new insight on classical questions, but paves the way to new and powerful algorithms for a variety of linear and nonlinear problems. Regression methods such as regularization networks and support vector machines are the basis of techniques that extend the function-estimation problem to the estimation of dynamic models. Many examples, also from real-world applications, illustrate the comparative advantages of the new nonparametric approach with respect to classic parametric prediction error methods. The challenges it addresses lie at the intersection of several disciplines so Regularized System Identification will be of interest to a variety of researchers and practitioners in the areas of control systems, machine learning, statistics, and data science. This is an open access book.
Machine learning --- Automatic control engineering --- Statistical physics --- Bayesian inference --- Probability & statistics --- Cybernetics & systems theory --- System Identification --- Machine Learning --- Linear Dynamical Systems --- Nonlinear Dynamical Systems --- Kernel-based Regularization --- Bayesian Interpretation of Regularization --- Gaussian Processes --- Reproducing Kernel Hilbert Spaces --- Estimation Theory --- Support Vector Machines --- Regularization Networks
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For the 250th birthday of Joseph Fourier, born in 1768 in Auxerre, France, this MDPI Special Issue will explore modern topics related to Fourier Analysis and Heat Equation. Modern developments of Fourier analysis during the 20th century have explored generalizations of Fourier and Fourier–Plancherel formula for non-commutative harmonic analysis, applied to locally-compact, non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups. One should add the developments, over the last 30 years, of the applications of harmonic analysis to the description of the fascinating world of aperiodic structures in condensed matter physics. The notions of model sets, introduced by Y. Meyer, and of almost periodic functions, have revealed themselves to be extremely fruitful in this domain of natural sciences. The name of Joseph Fourier is also inseparable from the study of the mathematics of heat. Modern research on heat equations explores the extension of the classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. In parallel, in geometric mechanics, Jean-Marie Souriau interpreted the temperature vector of Planck as a space-time vector, obtaining, in this way, a phenomenological model of continuous media, which presents some interesting properties. One last comment concerns the fundamental contributions of Fourier analysis to quantum physics: Quantum mechanics and quantum field theory. The content of this Special Issue will highlight papers exploring non-commutative Fourier harmonic analysis, spectral properties of aperiodic order, the hypoelliptic heat equation, and the relativistic heat equation in the context of Information Theory and Geometric Science of Information.
signal processing --- thermodynamics --- heat pulse experiments --- quantum mechanics --- variational formulation --- Wigner function --- nonholonomic constraints --- thermal expansion --- homogeneous spaces --- irreversible processes --- time-slicing --- affine group --- Fourier analysis --- non-equilibrium processes --- harmonic analysis on abstract space --- pseudo-temperature --- stochastic differential equations --- fourier transform --- Lie Groups --- higher order thermodynamics --- short-time propagators --- discrete thermodynamic systems --- metrics --- heat equation on manifolds and Lie Groups --- special functions --- poly-symplectic manifold --- non-Fourier heat conduction --- homogeneous manifold --- non-equivariant cohomology --- Souriau-Fisher metric --- Weyl quantization --- dynamical systems --- symplectization --- Weyl-Heisenberg group --- Guyer-Krumhansl equation --- rigged Hilbert spaces --- Lévy processes --- Born–Jordan quantization --- discrete multivariate sine transforms --- continuum thermodynamic systems --- interconnection --- rigid body motions --- covariant integral quantization --- cubature formulas --- Lie group machine learning --- nonequilibrium thermodynamics --- Van Vleck determinant --- Lie groups thermodynamics --- partial differential equations --- orthogonal polynomials
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