Nonlinear function approximation over high-dimensional domains
First Claim
1. A method for modeling a relationship between first and second data collections, wherein for each member of the first data collection, there is a corresponding member of the second collection, comprising:
- determining residuals between the first and second collections;
determining, using the residuals, a position of one of the members of the first data collection;
determining proximity data relative to a value (V) for the one member of the first data collection at the position;
determining a subcollection of the first collection, wherein each member of the subcollection has a value that is in proximity to the value V according to the proximity data; and
generating the basis function from the subcollection for obtaining a model of the relationship; and
outputting model information for presentation to a user or a predetermined process for affecting or identifying a physical event, wherein the model information includes at least one of;
(a) data indicative of a correspondence between the model and the relationship, (b) data indicative of a variance between the model and the relationship, (c) an extrapolation of the relationship, (d) an interpolation of the relationship, and (e) notification of the physical event.
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Abstract
An algorithm is disclosed for constructing nonlinear models from high-dimensional scattered data. The algorithm progresses iteratively adding a new basis function at each step to refine the model. The placement of the basis functions is driven by a statistical hypothesis test that reveals geometric structure when it fails. At each step the added function is fit to data contained in a spatio-temporally defined local region to determine the parameters, in particular, the scale of the local model. The proposed method requires no ad hoc parameters. Thus, the number of basis functions required for an accurate fit is determined automatically by the algorithm. The approach may be applied to problems including modeling data on manifolds and the prediction of financial time-series. The algorithm is presented in the context of radial basis functions but in principle can be employed with other methods for function approximation such as multi-layer perceptrons.
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Citations
11 Claims
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1. A method for modeling a relationship between first and second data collections, wherein for each member of the first data collection, there is a corresponding member of the second collection, comprising:
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determining residuals between the first and second collections; determining, using the residuals, a position of one of the members of the first data collection; determining proximity data relative to a value (V) for the one member of the first data collection at the position; determining a subcollection of the first collection, wherein each member of the subcollection has a value that is in proximity to the value V according to the proximity data; and generating the basis function from the subcollection for obtaining a model of the relationship; and outputting model information for presentation to a user or a predetermined process for affecting or identifying a physical event, wherein the model information includes at least one of;
(a) data indicative of a correspondence between the model and the relationship, (b) data indicative of a variance between the model and the relationship, (c) an extrapolation of the relationship, (d) an interpolation of the relationship, and (e) notification of the physical event. - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11)
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Specification