Positive-definite function: Difference between revisions

Revision as of 16:15, 9 June 2020

Revision as of 16:48, 9 June 2020

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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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==In dynamical systems==

A [[real number|real]]-valued, continuously differentiable [[function (mathematics)|function]] ''f'' is '''positive -definite''' on a neighborhood of the origin, ''D'', if <math>f(0) = 0</math> and <math>f(x) > 0</math> for every non-zero <math>x \in D</math>.<ref>{{cite book|last=Verhulst|first=Ferdinand|title=Nonlinear Differential Equations and Dynamical Systems|edition=2nd|publisher=Springer|year=1996|isbn=3-540-60934-2}}</ref><ref>{{cite book|last=Hahn|first=Wolfgang|title=Stability of Motion|url=https://archive.org/details/stabilityofmotio0000hahn|url-access=registration|publisher=Springer|year=1967}}</ref> This definition is in conflict with the one above.

==See also==