The Experts below are selected from a list of 130323 Experts worldwide ranked by ideXlab platform
Shinichi Aihara - One of the best experts on this subject based on the ideXlab platform.
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consistency property of extended least squares parameter estimation for stochastic diffusion equation
Systems & Control Letters, 1998Co-Authors: Shinichi AiharaAbstract:The extended least-squares parameter estimate for stochastic heat diffusion equations is considered. The unknown parameter is a heat diffusion coefficient which is a function of a Spatial Variable. Almost sure convergence for the estimated parameter is proved. A numerical example is demonstrated for supporting the theoretical results developed here.
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consistency of extended least square parameter estimate for stochastic distributed parameter systems
European Control Conference, 1997Co-Authors: Shinichi AiharaAbstract:The extended least square parameter estimate for stochastic heat diffusion equations is considered. The unknown parameter is a heat diffusion coefficient which is a function of a Spatial Variable. Almost sure convergence for the estimated parameter is proved. A numerical example is demonstrated for supporting the theoretical resluts developed here.
Thomas Nauss - One of the best experts on this subject based on the ideXlab platform.
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importance of Spatial predictor Variable selection in machine learning applications moving from data reproduction to Spatial prediction
Ecological Modelling, 2019Co-Authors: Hanna Meyer, Christoph Reudenbach, Stephan Wöllauer, Thomas NaussAbstract:Abstract Machine learning algorithms find frequent application in Spatial prediction of biotic and abiotic environmental Variables. However, the characteristics of Spatial data, especially Spatial autocorrelation, are widely ignored. We hypothesize that this is problematic and results in models that can reproduce training data but are unable to make Spatial predictions beyond the locations of the training samples. We assume that not only Spatial validation strategies but also Spatial Variable selection is essential for reliable Spatial predictions. We introduce two case studies that use remote sensing to predict land cover and the leaf area index for the “Marburg Open Forest”, an open research and education site of Marburg University, Germany. We use the machine learning algorithm Random Forests to train models using non-Spatial and Spatial cross-validation strategies to understand how Spatial Variable selection affects the predictions. Our findings confirm that Spatial cross-validation is essential in preventing overoptimistic model performance. We further show that highly autocorrelated predictors (such as geolocation Variables, e.g. latitude, longitude) can lead to considerable overfitting and result in models that can reproduce the training data but fail in making Spatial predictions. The problem becomes apparent in the visual assessment of the Spatial predictions that show clear artefacts that can be traced back to a misinterpretation of the Spatially autocorrelated predictors by the algorithm. Spatial Variable selection could automatically detect and remove such Variables that lead to overfitting, resulting in reliable Spatial prediction patterns and improved statistical Spatial model performance. We conclude that in addition to Spatial validation, a Spatial Variable selection must be considered in Spatial prediction models of ecological data to produce reliable results.
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Importance of Spatial predictor Variable selection in machine learning applications -- Moving from data reproduction to Spatial prediction
arXiv: Applications, 2019Co-Authors: Hanna Meyer, Christoph Reudenbach, Stephan Wöllauer, Thomas NaussAbstract:Machine learning algorithms find frequent application in Spatial prediction of biotic and abiotic environmental Variables. However, the characteristics of Spatial data, especially Spatial autocorrelation, are widely ignored. We hypothesize that this is problematic and results in models that can reproduce training data but are unable to make Spatial predictions beyond the locations of the training samples. We assume that not only Spatial validation strategies but also Spatial Variable selection is essential for reliable Spatial predictions. We introduce two case studies that use remote sensing to predict land cover and the leaf area index for the "Marburg Open Forest", an open research and education site of Marburg University, Germany. We use the machine learning algorithm Random Forests to train models using non-Spatial and Spatial cross-validation strategies to understand how Spatial Variable selection affects the predictions. Our findings confirm that Spatial cross-validation is essential in preventing overoptimistic model performance. We further show that highly autocorrelated predictors (such as geolocation Variables, e.g. latitude, longitude) can lead to considerable overfitting and result in models that can reproduce the training data but fail in making Spatial predictions. The problem becomes apparent in the visual assessment of the Spatial predictions that show clear artefacts that can be traced back to a misinterpretation of the Spatially autocorrelated predictors by the algorithm. Spatial Variable selection could automatically detect and remove such Variables that lead to overfitting, resulting in reliable Spatial prediction patterns and improved statistical Spatial model performance. We conclude that in addition to Spatial validation, a Spatial Variable selection must be considered in Spatial predictions of ecological data to produce reliable predictions.
Hanna Meyer - One of the best experts on this subject based on the ideXlab platform.
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importance of Spatial predictor Variable selection in machine learning applications moving from data reproduction to Spatial prediction
Ecological Modelling, 2019Co-Authors: Hanna Meyer, Christoph Reudenbach, Stephan Wöllauer, Thomas NaussAbstract:Abstract Machine learning algorithms find frequent application in Spatial prediction of biotic and abiotic environmental Variables. However, the characteristics of Spatial data, especially Spatial autocorrelation, are widely ignored. We hypothesize that this is problematic and results in models that can reproduce training data but are unable to make Spatial predictions beyond the locations of the training samples. We assume that not only Spatial validation strategies but also Spatial Variable selection is essential for reliable Spatial predictions. We introduce two case studies that use remote sensing to predict land cover and the leaf area index for the “Marburg Open Forest”, an open research and education site of Marburg University, Germany. We use the machine learning algorithm Random Forests to train models using non-Spatial and Spatial cross-validation strategies to understand how Spatial Variable selection affects the predictions. Our findings confirm that Spatial cross-validation is essential in preventing overoptimistic model performance. We further show that highly autocorrelated predictors (such as geolocation Variables, e.g. latitude, longitude) can lead to considerable overfitting and result in models that can reproduce the training data but fail in making Spatial predictions. The problem becomes apparent in the visual assessment of the Spatial predictions that show clear artefacts that can be traced back to a misinterpretation of the Spatially autocorrelated predictors by the algorithm. Spatial Variable selection could automatically detect and remove such Variables that lead to overfitting, resulting in reliable Spatial prediction patterns and improved statistical Spatial model performance. We conclude that in addition to Spatial validation, a Spatial Variable selection must be considered in Spatial prediction models of ecological data to produce reliable results.
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Importance of Spatial predictor Variable selection in machine learning applications -- Moving from data reproduction to Spatial prediction
arXiv: Applications, 2019Co-Authors: Hanna Meyer, Christoph Reudenbach, Stephan Wöllauer, Thomas NaussAbstract:Machine learning algorithms find frequent application in Spatial prediction of biotic and abiotic environmental Variables. However, the characteristics of Spatial data, especially Spatial autocorrelation, are widely ignored. We hypothesize that this is problematic and results in models that can reproduce training data but are unable to make Spatial predictions beyond the locations of the training samples. We assume that not only Spatial validation strategies but also Spatial Variable selection is essential for reliable Spatial predictions. We introduce two case studies that use remote sensing to predict land cover and the leaf area index for the "Marburg Open Forest", an open research and education site of Marburg University, Germany. We use the machine learning algorithm Random Forests to train models using non-Spatial and Spatial cross-validation strategies to understand how Spatial Variable selection affects the predictions. Our findings confirm that Spatial cross-validation is essential in preventing overoptimistic model performance. We further show that highly autocorrelated predictors (such as geolocation Variables, e.g. latitude, longitude) can lead to considerable overfitting and result in models that can reproduce the training data but fail in making Spatial predictions. The problem becomes apparent in the visual assessment of the Spatial predictions that show clear artefacts that can be traced back to a misinterpretation of the Spatially autocorrelated predictors by the algorithm. Spatial Variable selection could automatically detect and remove such Variables that lead to overfitting, resulting in reliable Spatial prediction patterns and improved statistical Spatial model performance. We conclude that in addition to Spatial validation, a Spatial Variable selection must be considered in Spatial predictions of ecological data to produce reliable predictions.
Jerome A. Goldstein - One of the best experts on this subject based on the ideXlab platform.
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Burgers and Black–Merton–Scholes equations with real time Variable and complex Spatial Variable
Applicable Analysis, 2013Co-Authors: Ciprian G. Gal, Sorin G. Gal, Jerome A. GoldsteinAbstract:The purpose of this article is to study the Burgers and Black–Merton–Scholes equations with real time Variable and complex Spatial Variable. The complexification of the Spatial Variable in these equations is made by two different methods which produce different equations: first, one complexifies the Spatial Variable in the corresponding (real) solution by replacing the usual sum of Variables (translation) by an exponential product (rotation) and secondly, one complexifies the Spatial Variable in the corresponding evolution equation and then one searches for analytic and non-analytic solutions. By both methods, new kinds of evolution equations (or systems of equations) in two dimensional Spatial Variables are generated and their solutions are constructed.
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Schrödinger type equations with real-time Variable and complex Spatial Variables
Complex Variables and Elliptic Equations, 2013Co-Authors: Ciprian G. Gal, Sorin G. Gal, Jerome A. GoldsteinAbstract:In recent work, heat and Laplace equations, (un)damped wave equations, the Burgers and the Black–Merton–Scholes equations with real-time Variable and complex Spatial Variable were studied. The purpose of this article is to make a similar study for the Schrodinger equation with real-time Variable and complex Spatial Variable. The complexification of the Spatial Variable in the case of the Schrodinger equation is made by two different methods which produce different equations: first, one complexifies the Spatial Variable in the corresponding convolution formula by replacing the usual sum of Variables (translation) by an exponential product (rotation) and second, one complexifies the Spatial Variable in the corresponding evolution equation and then one searches for non-analytic and for analytic solutions. By both methods, new kinds of evolution equations (or systems of equations) in two-dimensional Spatial Variables are generated and their solutions are constructed. It is of interest to note that in the case...
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Wave and telegraph equations with real time Variable and complex Spatial Variables
Complex Variables and Elliptic Equations, 2012Co-Authors: Ciprian G. Gal, Sorin G. Gal, Jerome A. GoldsteinAbstract:In two recent papers [C.G. Gal, S.G. Gal and J.A. Goldstein, Evolution equations with real time Variable and complex Spatial Variables, Complex Var. Elliptic Eqns. 53 (2008), pp. 753–774; C.G. Gal, S.G. Gal and J.A. Goldstein, Higher order heat and Laplace type equations with real time Variable and complex Spatial Variable, Complex Var. Elliptic Eqns., 55 (2010), pp. 357–373, the classical heat and Laplace equations with real time Variable and complex Spatial Variable are studied. The purpose of this article is to make a similar study for the classical wave and telegraph equations with real time Variable and complex Spatial Variable. The complexification of the Spatial Variable in the wave and telegraph equations is made by two different methods which produce different equations. By the former method, we complexify the Spatial Variable in the corresponding formulas by replacing the usual translations x ± ct, c is the speed of propagation, by the rotations ze ±ict and, by the latter, we complexify the spat...
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Evolution equations with real time Variable and complex Spatial Variables
Complex Variables and Elliptic Equations, 2008Co-Authors: Ciprian G. Gal, Sorin G. Gal, Jerome A. GoldsteinAbstract:It is known that if the time Variable in the heat and wave equations is complex and belongs to a sector in ℂ, then the theory of analytic semigroups becomes a powerful tool of study. Also, it is known that if both Variables, time and Spatial, are complex, then e.g. the Cauchy problem for the heat equation admits as solution, only a formal power series which, in general, converges nowhere. The purpose of this article is, in a sense, complementary: to study the complex versions of the classical heat and Laplace equations, obtained by ‘complexifying’ now the Spatial Variable only (and keeping the time Variable real). This ‘complexification’ is studied by two different methods, which produce different equations: first, one complexifies the Spatial Variable in the corresponding semigroups of operators and secondly, one complexifies the Spatial Variable in the corresponding evolution equation and then one searches for analytic and non-analytic solutions. It is of interest to note that in the case of the first m...
Ciprian G. Gal - One of the best experts on this subject based on the ideXlab platform.
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Burgers and Black–Merton–Scholes equations with real time Variable and complex Spatial Variable
Applicable Analysis, 2013Co-Authors: Ciprian G. Gal, Sorin G. Gal, Jerome A. GoldsteinAbstract:The purpose of this article is to study the Burgers and Black–Merton–Scholes equations with real time Variable and complex Spatial Variable. The complexification of the Spatial Variable in these equations is made by two different methods which produce different equations: first, one complexifies the Spatial Variable in the corresponding (real) solution by replacing the usual sum of Variables (translation) by an exponential product (rotation) and secondly, one complexifies the Spatial Variable in the corresponding evolution equation and then one searches for analytic and non-analytic solutions. By both methods, new kinds of evolution equations (or systems of equations) in two dimensional Spatial Variables are generated and their solutions are constructed.
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ON FOKKER-PLANCK AND LINEARIZED KORTEWEG-DE VRIES TYPE EQUATIONS WITH COMPLEX Spatial VariableS
Cubo (Temuco), 2013Co-Authors: Ciprian G. Gal, Sorin G. GalAbstract:In the present work, we construct solutions to a Fokker-Planck type equation with real time Variable and complex Spatial Variable, and prove some properties. The equations are obtained from the complexification of the Spatial Variable by two different methods. Firstly, one complexifies the Spatial Variable in the corresponding convolution integral in the solution, by replacing the usual sum of Variables (translation) by an exponential product (rotation). Secondly, one complexifies the Spatial Variable directly in the corresponding evolution equation and then one searches for analytic solutions. These methods are also applied to a linear evolution equation related to the Korteweg-de Vries equation.
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Schrödinger type equations with real-time Variable and complex Spatial Variables
Complex Variables and Elliptic Equations, 2013Co-Authors: Ciprian G. Gal, Sorin G. Gal, Jerome A. GoldsteinAbstract:In recent work, heat and Laplace equations, (un)damped wave equations, the Burgers and the Black–Merton–Scholes equations with real-time Variable and complex Spatial Variable were studied. The purpose of this article is to make a similar study for the Schrodinger equation with real-time Variable and complex Spatial Variable. The complexification of the Spatial Variable in the case of the Schrodinger equation is made by two different methods which produce different equations: first, one complexifies the Spatial Variable in the corresponding convolution formula by replacing the usual sum of Variables (translation) by an exponential product (rotation) and second, one complexifies the Spatial Variable in the corresponding evolution equation and then one searches for non-analytic and for analytic solutions. By both methods, new kinds of evolution equations (or systems of equations) in two-dimensional Spatial Variables are generated and their solutions are constructed. It is of interest to note that in the case...
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Wave and telegraph equations with real time Variable and complex Spatial Variables
Complex Variables and Elliptic Equations, 2012Co-Authors: Ciprian G. Gal, Sorin G. Gal, Jerome A. GoldsteinAbstract:In two recent papers [C.G. Gal, S.G. Gal and J.A. Goldstein, Evolution equations with real time Variable and complex Spatial Variables, Complex Var. Elliptic Eqns. 53 (2008), pp. 753–774; C.G. Gal, S.G. Gal and J.A. Goldstein, Higher order heat and Laplace type equations with real time Variable and complex Spatial Variable, Complex Var. Elliptic Eqns., 55 (2010), pp. 357–373, the classical heat and Laplace equations with real time Variable and complex Spatial Variable are studied. The purpose of this article is to make a similar study for the classical wave and telegraph equations with real time Variable and complex Spatial Variable. The complexification of the Spatial Variable in the wave and telegraph equations is made by two different methods which produce different equations. By the former method, we complexify the Spatial Variable in the corresponding formulas by replacing the usual translations x ± ct, c is the speed of propagation, by the rotations ze ±ict and, by the latter, we complexify the spat...
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Evolution equations with real time Variable and complex Spatial Variables
Complex Variables and Elliptic Equations, 2008Co-Authors: Ciprian G. Gal, Sorin G. Gal, Jerome A. GoldsteinAbstract:It is known that if the time Variable in the heat and wave equations is complex and belongs to a sector in ℂ, then the theory of analytic semigroups becomes a powerful tool of study. Also, it is known that if both Variables, time and Spatial, are complex, then e.g. the Cauchy problem for the heat equation admits as solution, only a formal power series which, in general, converges nowhere. The purpose of this article is, in a sense, complementary: to study the complex versions of the classical heat and Laplace equations, obtained by ‘complexifying’ now the Spatial Variable only (and keeping the time Variable real). This ‘complexification’ is studied by two different methods, which produce different equations: first, one complexifies the Spatial Variable in the corresponding semigroups of operators and secondly, one complexifies the Spatial Variable in the corresponding evolution equation and then one searches for analytic and non-analytic solutions. It is of interest to note that in the case of the first m...
