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Yingwei Lu, N. Sundararajan, P. Saratchandran (1998)
Performance evaluation of a sequential minimal radial basis function (RBF) neural network learning algorithmIEEE transactions on neural networks, 9 2
P. Yee, S. Haykin (2001)
Regularized radial basis functional networks: theory and applications
Jooyoung Park, I. Sandberg (1991)
Universal Approximation Using Radial-Basis-Function NetworksNeural Computation, 3
N. Packard, J. Crutchfield, J. Farmer, Robert Shaw (1980)
Geometry from a Time SeriesPhysical Review Letters, 45
Stéphane Lafon, Weiqiang Jing (2006)
Diffusion maps
M. Anderle, M. Kirby (2001)
Correlation feedback resource allocation RBFIJCNN'01. International Joint Conference on Neural Networks. Proceedings (Cat. No.01CH37222), 3
S. Roweis, L. Saul (2000)
Nonlinear dimensionality reduction by locally linear embedding.Science, 290 5500
Douglas Hundley, Michael Kirby, Rick Miranda (1999)
Empirical dynamical system reduction II: Neural charts
Yingwei Lu, N. Sundararajan, P. Saratchandran (1997)
A Sequential Learning Scheme for Function Approximation Using Minimal Radial Basis Function Neural NetworksNeural Computation, 9
M. Brand (2002)
Charting a Manifold
Y. Teh, S. Roweis (2002)
Automatic Alignment of Local Representations
B. Scholkopf, Alex Smola, K. Müller (1998)
Nonlinear Component Analysis as a Kernel Eigenvalue ProblemNeural Computation, 10
A. Jamshidi, M. Kirby (2007)
Towards a Black Box Algorithm for Nonlinear Function Approximation over High-Dimensional DomainsSIAM J. Sci. Comput., 29
N. Karayiannis, Glenn Mi (1997)
Growing radial basis neural networks: merging supervised and unsupervised learning with network growth techniquesIEEE transactions on neural networks, 8 6
D. Cook, A. Buja, Javier Cabrera, C. Hurley (1995)
Grand tour and projection pursuitJournal of Computational and Graphical Statistics, 4
M. Belkin, P. Niyogi (2003)
Laplacian Eigenmaps for Dimensionality Reduction and Data RepresentationNeural Computation, 15
Jooyoung Park, I. Sandberg (1993)
Approximation and Radial-Basis-Function NetworksNeural Computation, 5
M. Mackey, L. Glass (1977)
Oscillation and chaos in physiological control systems.Science, 197 4300
A. Jamshidi, M. Kirby (2009)
Skew-Radial Basis Function Expansions for Empirical ModelingSIAM J. Sci. Comput., 31
A. Jamshidi, M. Kirby (2006)
Examples of Compactly Supported Functions for Radial Basis Approximations
D. Broomhead, M. Kirby (2000)
A New Approach to Dimensionality Reduction: Theory and AlgorithmsSIAM J. Appl. Math., 60
V. Vapnik (2000)
The Nature of Statistical Learning Theory
R. Harder, R. Desmarais (1972)
Interpolation using surface splines.Journal of Aircraft, 9
J. Farmer, J. SIDorowich (1987)
Predicting chaotic time series.Physical review letters, 59 8
C. Holmes, B. Mallick (1998)
Bayesian Radial Basis Functions of Variable DimensionNeural Computation, 10
(1998)
Local nonlinear modeling via neural charts. Unpublished doctoral dissertation
J. Tenenbaum, V. Silva, J. Langford (2000)
A global geometric framework for nonlinear dimensionality reduction.Science, 290 5500
John Platt (1991)
A Resource-Allocating Network for Function InterpolationNeural Computation, 3
M. Kennel, Reggie Brown, H. Abarbanel (1992)
Determining embedding dimension for phase-space reconstruction using a geometrical construction.Physical review. A, Atomic, molecular, and optical physics, 45 6
(1992)
The theory of radial basis functions in 1990
R. Hardy (1971)
Multiquadric equations of topography and other irregular surfacesJournal of Geophysical Research, 76
E. Ziegel (2012)
Time Series: Theory and Methods (2nd ed,)Technometrics
M. Kirby, Rick Miranda (1996)
Circular Nodes in Neural NetworksNeural Computation, 8
W. Liebert, K. Pawelzik, H. Schuster (1991)
Optimal Embeddings of Chaotic Attractors from Topological ConsiderationsEPL, 14
B. Nadler, Stéphane Lafon, R. Coifman, I. Kevrekidis (2005)
Diffusion maps, spectral clustering and reaction coordinates of dynamical systemsApplied and Computational Harmonic Analysis, 21
J. Verbeek, S. Roweis, N. Vlassis (2003)
Non-linear CCA and PCA by Alignment of Local Models
A. Jamshidi (2008)
Modeling Spatio-Temporal Systems with Skew Radial Basis Functions : Theory , Algorithms and Applications
D. Broomhead, D. Lowe (1988)
Multivariable Functional Interpolation and Adaptive NetworksComplex Syst., 2
D. Broomhead, M. Kirby (2001)
The Whitney Reduction Network: A Method for Computing Autoassociative GraphsNeural Computation, 13
P. Brockwell, R. Davis (2013)
Time Series: Theory and Methods
M. Çek (2004)
Analysis of observed chaotic data
Lianfen Qian (2002)
Regularized Radial Basis Function Networks: Theory and ApplicationsTechnometrics, 44
I. Nabney, A. Mclachlan, D. Lowe (1996)
Practical methods of tracking of nonstationary time series applied to real-world data, 2760
S. Roweis, L. Saul, Geoffrey Hinton (2001)
Global Coordination of Local Linear Models
F. Takens (1981)
Detecting strange attractors in turbulence, 898
N. Karayiannis, Glenn Mi (1997)
Growing radial basis neural networksProceedings of International Conference on Neural Networks (ICNN'97), 3
D. Donoho, C. Grimes (2003)
Hessian eigenmaps: Locally linear embedding techniques for high-dimensional dataProceedings of the National Academy of Sciences of the United States of America, 100
D. Broomhead, D. Lowe (1988)
Radial Basis Functions, Multi-Variable Functional Interpolation and Adaptive Networks
V. Kadirkamanathan, M. Niranjan (1993)
A Function Estimation Approach to Sequential Learning with Neural NetworksNeural Computation, 5
Rong Chen, P. Brockwell, R. Davis (1992)
Time Series: Theory and Methods (2nd ed.).Journal of the American Statistical Association, 87
Nicholas Baron, Andrew Philippides, Nicolas Rojas (1889)
AMERICAN SOCIETY OF MECHANICAL ENGINEERSScience, ns-14
R. Schaback, H. Wendland (2001)
Characterization and construction of radial basis functions
(2004)
A new spatio-temporal resource allocation network (STRAN)
(1974)
A Projection Pursuit Algorithm for Exploratory Data AnalysisIEEE Transactions on Computers, C-23
D. Broomhead, M. Kirby (2005)
Dimensionality Reduction Using Secant-Based Projection Methods: The Induced Dynamics in Projected SystemsNonlinear Dynamics, 41
We present an approach for constructing nonlinear empirical mappings from high-dimensional domains to multivariate ranges. We employ radial basis functions and skew radial basis functions for constructing a model using data that are potentially scattered or sparse. The algorithm progresses iteratively, adding a new function at each step to refine the model. The placement of the functions is driven by a statistical hypothesis test that accounts for correlation in the multivariate range variables. The test is applied on training and validation data and reveals nonstatistical or geometric structure when it fails. At each step, the added function is fit to data contained in a spatiotemporally defined local region to determine the parameters—in particular, the scale of the local model. The scale of the function is determined by the zero crossings of the autocorrelation function of the residuals. The model parameters and the number of basis functions are determined automatically from the given data, and there is no need to initialize any ad hoc parameters save for the selection of the skew radial basis functions. Compactly supported skew radial basis functions are employed to improve model accuracy, order, and convergence properties. The extension of the algorithm to higher-dimensional ranges produces reduced-order models by exploiting the existence of correlation in the range variable data. Structure is tested not just in a single time series but between all pairs of time series. We illustrate the new methodologies using several illustrative problems, including modeling data on manifolds and the prediction of chaotic time series.
Neural Computation – MIT Press
Published: Jan 1, 2011
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