Positive-definite function: Difference between revisions

Revision as of 13:53, 13 March 2018

Revision as of 15:05, 27 May 2018

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====Applications====

In [[statistics]], and especially [[Bayesian statistics]], the theorem is usually applied to real functions. Typically, one takes ''n'' scalar measurements of some scalar value at points in <math>R^d</math> and one requires that points that are mutually close have measurements that are highly correlated. In practice, one must be careful to ensure that the resulting covariance matrix (an ''n''-by-''n'' matrix) is always positive definite. One strategy is to define a correlation matrix ''A'' which is then multiplied by a scalar to give a [[covariance matrix]]: this must be positive definite. Bochner's theorem states that if the correlation between two points is dependent only upon the distance between them (via function ''f()''), then function ''f()'' must be positive definite to ensure the covariance matrix ''A'' to be positive definite. See [[Kriging]].

In this context, one does not usually use Fourier terminology and instead one states that ''f(x)'' is the [[characteristic function (probability theory)|characteristic function]] of a [[symmetric]] [[probability density function|probability density function (PDF)]].

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==See also==

* [[Positive-definite_kernel|Positive-definite kernel]]

==References==