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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Vydáno v:	IEEE transactions on pattern analysis and machine intelligence Ročník 29; číslo 9; s. 1538 - 1545
Hlavní autoři:	Gustafson, S.C., Parker, D.R., Martin, R.K.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Los Alamitos, CA IEEE 01.09.2007 IEEE Computer Society The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:	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
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Popis
Shrnutí:	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.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23 ObjectType-Article-2 ObjectType-Feature-1
ISSN:	0162-8828 1939-3539
DOI:	10.1109/TPAMI.2007.1170