Covariance operator (original) (raw)
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.
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| dbo: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) |
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| 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) |
| rdfs:label | Kovarianzoperator (de) Covariance operator (en) |
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