Positive-definite function
Encyclopedia
In mathematics
, the term positive-definite function may refer to a couple of different concepts.
-valued, continuously differentiable function
f is positive definite on a neighborhood of the origin, D, if
and 
for every non-zero 
.
A function is negative definite if the inequality is reversed. A function is semidefinite if the strong inequality is replaced with a weak (
or 
) one.
-valued function f:R → C such that for any real numbers x1, ..., xn the n×n matrix
is positive semi-definite
(in particular, A should be Hermitian
, therefore f(-x) is the complex conjugate
of f(x)).
In particular, it is necessary (but not sufficient) that
(these inequalities follow from the condition for n=1,2.)
; it is easy to see directly that to be positive-definite it is sufficient for f to be the Fourier transform of a function g on the real line with g(y) ≥ 0.
The converse result is Bochner's theorem
, stating that any continuous positive-definite function on the real line is the Fourier transform of a (positive) measure
.
, 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
and one requires that points that are closely separated 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 is 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
of a symmetric PDF
.
of groups on Hilbert space
s (i.e. the theory of unitary representation
s).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the term positive-definite function may refer to a couple of different concepts.
In dynamical systems
A realReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
-valued, continuously differentiable function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
f is positive definite on a neighborhood of the origin, D, if
and for every non-zero .A function is negative definite if the inequality is reversed. A function is semidefinite if the strong inequality is replaced with a weak (
or ) one.In analysis
A positive-definite function of a real variable x is a complexComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
-valued function f:R → C such that for any real numbers x1, ..., xn the n×n matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
is positive semi-definite
Positive-definite matrix
In linear algebra, a positive-definite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form ....
(in particular, A should be Hermitian
Hermitian
A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite:*Hermitian adjoint*Hermitian connection, the unique connection on a Hermitian manifold that satisfies specific conditions...
, therefore f(-x) is the complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
of f(x)).
In particular, it is necessary (but not sufficient) that
(these inequalities follow from the condition for n=1,2.)
Bochner's theorem
Positive-definiteness arises naturally in the theory of the Fourier transformFourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
; it is easy to see directly that to be positive-definite it is sufficient for f to be the Fourier transform of a function g on the real line with g(y) ≥ 0.
The converse result is Bochner's theorem
Bochner's theorem
In mathematics, Bochner's theorem characterizes the Fourier transform of a positive finite Borel measure on the real line.- Background :...
, stating that any continuous positive-definite function on the real line is the Fourier transform of a (positive) measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
.
Applications
In statisticsStatistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, and especially Bayesian statistics
Bayesian statistics
Bayesian statistics is that subset of the entire field of statistics in which the evidence about the true state of the world is expressed in terms of degrees of belief or, more specifically, Bayesian probabilities...
, the theorem is usually applied to real functions. Typically, one takes n scalar measurements of some scalar value at points in
and one requires that points that are closely separated 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 matrixCovariance matrix
In probability theory and statistics, a covariance matrix is a matrix whose element in the i, j position is the covariance between the i th and j th elements of a random vector...
: 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 is positive definite. See Kriging
Kriging
Kriging is a group of geostatistical techniques to interpolate the value of a random field at an unobserved location from observations of its value at nearby locations....
.
In this context, one does not usually use Fourier terminology and instead one states that f(x) is the characteristic function
Characteristic function
In mathematics, characteristic function can refer to any of several distinct concepts:* The most common and universal usage is as a synonym for indicator function, that is the function* In probability theory, the characteristic function of any probability distribution on the real line is given by...
of a symmetric PDF
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
.
Generalisation
One can define positive-definite functions on any locally compact abelian topological group; Bochner's theorem extends to this context. Positive-definite functions on groups occur naturally in the representation theoryRepresentation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
of groups on Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
s (i.e. the theory of unitary representation
Unitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...
s).