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- Covariance_operator abstract "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 byfor 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 (the infinite-dimensional analogy of the transposition symmetry in the finite-dimensional case). 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 bywhere 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 called random process or random field) z iswhere 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.".
- Covariance_operator wikiPageID "33447667".
- Covariance_operator wikiPageRevisionID "521500407".
- Covariance_operator hasPhotoCollection Covariance_operator.
- Covariance_operator subject Category:Bilinear_forms.
- Covariance_operator subject Category:Covariance_and_correlation.
- Covariance_operator subject Category:Probability_theory.
- Covariance_operator type Abstraction100002137.
- Covariance_operator type BilinearForms.
- Covariance_operator type Form106290637.
- Covariance_operator type LanguageUnit106284225.
- Covariance_operator type Part113809207.
- Covariance_operator type Relation100031921.
- Covariance_operator type Word106286395.
- Covariance_operator comment "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 byfor 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 (the infinite-dimensional analogy of the transposition symmetry in the finite-dimensional case).".
- Covariance_operator label "Covariance operator".
- Covariance_operator sameAs m.0h972fb.
- Covariance_operator sameAs Q5178900.
- Covariance_operator sameAs Q5178900.
- Covariance_operator sameAs Covariance_operator.
- Covariance_operator wasDerivedFrom Covariance_operator?oldid=521500407.
- Covariance_operator isPrimaryTopicOf Covariance_operator.