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- 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.
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- 2022 (xsd:nonNegativeInteger)
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- 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.
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- Kovarianzoperator (de)
- Covariance operator (en)
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