About: Covariance operator

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In probability theory, for a probability measure P on a Hilbert space H with inner product , the covariance of P is the bilinear form Cov: H × H → R given by for all x and y in H. The covariance operator C is then defined by (from the Riesz representation theorem, such operator exists if Cov is bounded). Since Cov is symmetric in its arguments, the covariance operator isself-adjoint. When P is a centred Gaussian measure, C is also a nuclear operator. In particular, it is a compact operator of trace class, that is, it has finite trace. * v * t * e

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rdf:type	yago:WikicatBilinearForms yago:Abstraction100002137 yago:Form106290637 yago:LanguageUnit106284225 yago:Part113809207 yago:Relation100031921 yago:Word106286395
rdfs:label	Kovarianzoperator (de) Covariance operator (en)
rdfs:comment	Der Kovarianzoperator bezeichnet in der Stochastik einen linearen Operator, der den Begriff der Kovarianz auf unendlich-dimensionale Räume erweitert. Der Begriff wird in der Theorie der und der stochastischen Analysis auf Banach- und Hilberträumen verwendet. (de) In probability theory, for a probability measure P on a Hilbert space H with inner product , the covariance of P is the bilinear form Cov: H × H → R given by for all x and y in H. The covariance operator C is then defined by (from the Riesz representation theorem, such operator exists if Cov is bounded). Since Cov is symmetric in its arguments, the covariance operator isself-adjoint. When P is a centred Gaussian measure, C is also a nuclear operator. In particular, it is a compact operator of trace class, that is, it has finite trace. * v * t * e (en)
dcterms:subject	Covariance and correlation Probability theory Bilinear forms
Wikipage page ID	33447667 (xsd:integer)
Wikipage revision ID	1095682517 (xsd:integer)
Link from a Wikipage to another Wikipage	Bilinear form Riesz representation theorem Covariance and correlation Gaussian measure Nuclear operator Compact operator Banach space Probability theory Trace (mathematics) Random field Bilinear forms Linear functional Probability theory Hilbert space Covariance function Trace class Inner product Random process Self-adjoint operator Random element Probability measure Algebraic dual
sameAs	Covariance operator Covariance operator Covariance operator Covariance operator Covariance operator
dbp:wikiPageUsesTemplate	dbt:Short_description dbt:Probability-stub
has abstract	Der Kovarianzoperator bezeichnet in der Stochastik einen linearen Operator, der den Begriff der Kovarianz auf unendlich-dimensionale Räume erweitert. Der Begriff wird in der Theorie der und der stochastischen Analysis auf Banach- und Hilberträumen verwendet. (de) In probability theory, for a probability measure P on a Hilbert space H with inner product , the covariance of P is the bilinear form Cov: H × H → R given by for all x and y in H. The covariance operator C is then defined by (from the Riesz representation theorem, such operator exists if Cov is bounded). Since Cov is symmetric in its arguments, the covariance operator isself-adjoint. When P is a centred Gaussian measure, C is also a nuclear operator. In particular, it is a compact operator of trace class, that is, it has finite trace. Even more generally, for a probability measure P on a Banach space B, the covariance of P is the bilinear form on the algebraic dual B#, defined by where is now the value of the linear functional x on the element z. Quite similarly, the covariance function of a function-valued random element (in special cases is called random process or random field) z is where z(x) is now the value of the function z at the point x, i.e., the value of the linear functional evaluated at z. * v * t * e (en)
prov:wasDerivedFrom	wikipedia-en:Covariance_operator?oldid=1095682517&ns=0
page length (characters) of wiki page	2022 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Covariance_operator
is Link from a Wikipage to another Wikipage of	Downside beta Preconditioned Crank–Nicolson algorithm Covariance Functional data analysis Feldman–Hájek theorem Functional principal component analysis Upside beta
is foaf:primaryTopic of	wikipedia-en:Covariance_operator

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