Modeling Multivariate Time Series on Manifolds with Skew Radial Basis Functions

• Published in: Neural computation Vol. 23; no. 1; pp. 97 - 123
• Main Authors: Jamshidi, Arta A., Kirby, Michael J.
• Format: Journal Article
• Language: English
• Published: One Rogers Street, Cambridge, MA 02142-1209, USA MIT Press 01.01.2011; MIT Press Journals, The
• Subjects: Algorithms; Applied sciences; Artificial intelligence; Biological and medical sciences; Computer science; control theory; systems; Computer Simulation; Convergence; Data processing; Exact sciences and technology; Fundamental and applied biological sciences. Psychology; General aspects; Inference from stochastic processes; time series analysis; Learning and adaptive systems; Letters; Mapping; Mathematics; Mathematics in biology. Statistical analysis. Models. Metrology. Data processing in biology (general aspects); Miscellaneous; Models, Neurological; Multivariate Analysis; Neural Networks (Computer); Nonlinear Dynamics; Probability and statistics; Sciences and techniques of general use; Statistics; Time Factors; Autocorrelation function; Neural computation; Correlation; Methodology; Prediction; Time series; Neural network; Multivariate analysis; Algorithm; Modeling; Convergence; Radial basis function; Hypothesis test; Statistical test; Manifold; Scale parameter; Correlation analysis; Step function
• ISSN: 0899-7667; 1530-888X
• DOI: 10.1162/NECO_a_00060

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.