Cardinal Interpolation

A Bayesian probability density for an interpolating function is developed, and its desirable properties and practical potential are demonstrated. This density has an often needed but previously unachieved property, here called cardinal interpolation, which ensures extrapolation to the density of the...

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Bibliographic Details
Published in:IEEE transactions on pattern analysis and machine intelligence Vol. 29; no. 9; pp. 1538 - 1545
Main Authors: Gustafson, S.C., Parker, D.R., Martin, R.K.
Format: Journal Article
Language:English
Published: Los Alamitos, CA IEEE 01.09.2007
IEEE Computer Society
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Applied sciences
Artificial Intelligence
Autoregressive processes
Bayes Theorem
Bayesian analysis
Bayesian methods
Bayesian statistics
Computer science; control theory; systems
Computer Simulation
Density
Exact sciences and technology
Extrapolation
Gaussian
Gaussian processes
Interpolation
Least squares method
Mathematical models
Modeling and prediction
Models, Statistical
Numerical Analysis, Computer-Assisted
Optimization methods
Pattern Recognition, Automated - methods
Polynomials
Predictive models
Probability
Probability and statistics
Regression
Statistics
Studies
Variance
Bayes estimation
probability and statistics
Statistical analysis
Probabilistic approach
Non gaussian noise
Modeling
Linear model
Optimization
Variance
Probability density
Extrapolation
interpolation
modeling and prediction
Gaussian process
Least squares method
Bayesian statistics
regression
Smooth function
Density function
Pattern analysis
Artificial intelligence
Quadratic function
ISSN:0162-8828, 1939-3539
Online Access:Get full text
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Summary:A Bayesian probability density for an interpolating function is developed, and its desirable properties and practical potential are demonstrated. This density has an often needed but previously unachieved property, here called cardinal interpolation, which ensures extrapolation to the density of the least-squares linear model. In particular, the mean of the cardinal interpolation density is a smooth function that intersects given (x, y) points and which extrapolates to their least-squares line, and the variance of this density is a smooth function that is zero at the point x values, that increases with distance from the nearest point x value, and that extrapolates to the well-known quadratic variance function for the least-squares line. The new cardinal interpolation density is developed for Gaussian radial basis interpolators using fully Bayesian methods that optimize interpolator smoothness. This optimization determines the basis function widths and yields an interpolating density that is non-Gaussian except for large magnitude x and which is therefore not the outcome of a Gaussian process. Further, new development shows that the salient property of extrapolation to the density of the least- squares linear model can be achieved for more general approximating (not just interpolating) functions.
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ISSN:0162-8828
1939-3539
DOI:10.1109/TPAMI.2007.1170