Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur Graph Search.
Concept
Covariance function
Résumé
In probability theory and statistics, the covariance function describes how much two random variables change together (their covariance) with varying spatial or temporal separation. For a random field or stochastic process Z(x) on a domain D, a covariance function C(x, y) gives the covariance of the values of the random field at the two locations x and y:
The same C(x, y) is called the autocovariance function in two instances: in time series (to denote exactly the same concept except that x and y refer to locations in time rather than in space), and in multivariate random fields (to refer to the covariance of a variable with itself, as opposed to the cross covariance between two different variables at different locations, Cov(Z(x1), Y(x2))).
For locations x1, x2, ..., xN ∈ D the variance of every linear combination
can be computed as
A function is a valid covariance function if and only if this variance is non-negative for all possible choices of N and weights w1, ..., wN. A function with this property is called positive semidefinite.
In case of a weakly stationary random field, where
for any lag h, the covariance function can be represented by a one-parameter function
which is called a covariogram and also a covariance function. Implicitly the C(xi, xj) can be computed from Cs(h) by:
The positive definiteness of this single-argument version of the covariance function can be checked by Bochner's theorem.
For a given variance , a simple stationary parametric covariance function is the "exponential covariance function"
where V is a scaling parameter (correlation length), and d = d(x,y) is the distance between two points. Sample paths of a Gaussian process with the exponential covariance function are not smooth. The "squared exponential" (or "Gaussian") covariance function:
is a stationary covariance function with smooth sample paths.
The Matérn covariance function and rational quadratic covariance function are two parametric families of stationary covariance functions.
Source officielle
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
Proximité ontologique
Cours associés (6)
MICRO-455: Applied machine learning
Real-world engineering applications must cope with a large dataset of dynamic variables, which cannot be well approximated by classical or deterministic models. This course gives an overview of method
MICRO-570: Advanced machine learning
This course will present some of the core advanced methods in the field for structure discovery, classification and non-linear regression. This is an advanced class in Machine Learning; hence, student
FIN-417: Quantitative risk management
This course is an introduction to quantitative risk management that covers standard statistical methods, multivariate risk factor models, non-linear dependence structures (copula models), as well as p
Publications associées (34)