Joint measures and cross-covariance operators

Joint Measures and Cross-covariance Operators Jon·it I"leasures and Cross-covariance Operators Joint L"leasljres and Cross-covariance Operators*

Let HI (resp., HZ) be a real and separable Hilbert space with Borel a-field f 1 (resp., f 2), and let (HIXH Z ' f 1 x f 2) be the product measurable space generated by the measurable rectangles. This paper develops relations between probability measures on (HIXH Z ' f 1 x f 2), i.e., joint measures, and the projections of such measures on (HI' f 1) and (HZ' f 2). In particular, the class of all...

JOINT MEASURES AND CROSS - COVARIANCE OPERATORS ( l )

Let H. (resp., H ) be a real and separable Hubert space with Borel O'field T (resp., rj, and let (H. X //-, T, X T.) be the product measurable space generated by the measurable rectangles. This paper develops relations between probability measures on (H. x H , V x T.), i.e., joint measures, and the projections of such measures on (H., T.) and (H , Y ). In particular, the class of all joint Gaus...

Visualization of Covariance and Cross-covariance Fields

We present a numerical technique to visualize covariance and cross-covariance fields of a stochastic simulation. The method is local in the sense that it demonstrates the covariance structure of the solution at a point with its neighboring locations. When coupled with an efficient stochastic simulation solver, our framework allows one to effectively concurrently visualize both the mean and (cro...

Notes on mean embeddings and covariance operators

Schrödinger operators and information measures

The Rényi R[ρ] = 1 1−q ∫ ρ(x)dx and Shannon S[ρ] = − ∫ ρ(x) log ρ(x)dx entropies are information-theoretic measures which have enabled to formulate the position-momentum uncertainty principle in a much more adequate and stringent way than the (variance-based) Heisenberg-like relation. They are also the basis for their associated spreading lengths, that measure the spreading of a probability den...