In physics, a covariance group is a group of coordinate transformations between frames of reference (see for example Ryckman (2005)[1]). A frame of reference provides a set of coordinates for an observer moving with that frame to make measurements and define physical quantities. The covariance principle states the laws of physics should transform from one frame to another covariantly, that is, according to a representation of the covariance group.
Special relativity considers observers in inertial frames, and the covariance group consists of rotations, velocity boosts, and the parity transformation. It is denoted as O(1,3) and is often referred to as Lorentz group.
For example, the Maxwell equation with sources,
transforms as a four-vector, that is, under the (1/2,1/2) representation of the O(1,3) group.
The Dirac equation,
transforms as a bispinor, that is, under the (1/2,0)⊕(0,1/2) representation of the O(1,3) group.
The covariance principle, unlike the relativity principle, does not imply that the equations are invariant under transformations from the covariance group. In practice the equations for electromagnetic and strong interactions are invariant, while the weak interaction is not invariant under the parity transformation. For example, the Maxwell equation is invariant, while the corresponding equation for the weak field explicitly contains left currents and thus is not invariant under the parity transformation.
In general relativity the covariance group consists of all arbitrary (invertible and differentiable) coordinate transformations.
Covariance in probability theory and statistics is a measure of the joint variability of two random variables. The sign of the covariance, therefore,...
In physics, a covariancegroup is a group of coordinate transformations between frames of reference (see for example Ryckman (2005)). A frame of reference...
and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square...
Analysis of covariance (ANCOVA) is a general linear model that blends ANOVA and regression. ANCOVA evaluates whether the means of a dependent variable...
group of coordinate transformations between admissible frames of reference of the physical theory. This group is referred to as the covariancegroup....
statistics, sometimes the covariance matrix of a multivariate random variable is not known but has to be estimated. Estimation of covariance matrices then deals...
In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the form of physical...
through space". Lorentz covariance, a related concept, is a property of the underlying spacetime manifold. Lorentz covariance has two distinct, but closely...
Multivariate analysis of covariance (MANCOVA) is an extension of analysis of covariance (ANCOVA) methods to cover cases where there is more than one dependent...
the covariance of two variables and the product of their standard deviations; thus, it is essentially a normalized measurement of the covariance, such...
eigenvectors of the data's covariance matrix. Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or singular...
vectors in general are covariant since they can be pushed forward. See also Covariance and contravariance of vectors. Every functor F : C → D {\displaystyle...
that the class covariances are identical, so Σ 0 = Σ 1 = Σ {\displaystyle \Sigma _{0}=\Sigma _{1}=\Sigma } ) and that the covariances have full rank....
representations of the general covariancegroup. However, there are of course spinorial representations of the Lorentz group. This fact is utilized by employing...
\operatorname {E} [X_{k}])^{\mathrm {T} },} and k × k {\displaystyle k\times k} covariance matrix Σ i , j = E [ ( X i − μ i ) ( X j − μ j ) ] = Cov [ X i , X...
well defined. Subtracting the mean before multiplication yields the auto-covariance function between times t 1 {\displaystyle t_{1}} and t 2 {\displaystyle...
to the value of x {\displaystyle x} . More generally, if the variance-covariance matrix of disturbance ε i {\displaystyle \varepsilon _{i}} across i {\displaystyle...
average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by σ 2...
in a multi-group model. A model's specification is not complete until the researcher specifies: which effects and/or correlations/covariances are to be...
equation and the covariance of spinors. In other words, the (restricted) Lorentz group is isomorphic to SL(2, C) / Z2 The symplectic group Sp(2, C) is isomorphic...
representations of the general covariancegroup. However, there are of course spinorial representations of the Lorentz group. This fact is utilized by employing...
respective random variables.: p.333 The covariance matrix (also called second central moment or variance-covariance matrix) of an n × 1 {\displaystyle n\times...
combination follows a multivariate normal distribution, multivariate variance-covariance matrix homogeneity, and linear relationship, no multicollinearity, and...
{\displaystyle \{z_{t}\}} is a white noise in the weak sense (the mean and cross-covariances are zero, and the variances are all the same), however it is not strictly...
modern mathematical methods, in league with the idea of invariance and covariancegroups, is to try to interpret historical views of space and time in modern...
mean-covariance approach of Markowitz. Markowitz assumed that asset covariance matrix Σ {\displaystyle \Sigma } can be observed. The covariance matrix...
value in roughly 68% of cases. If the uncertainties are correlated then covariance must be taken into account. Correlation can arise from two different sources...
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