Covariance Matrix

In probability theory and statistics, a covariance matrix (also known as dispersion matrix or variance covariance matrix) is a matrix whose element in the i, j position is the covariance between the i th and j th elements of a random vector (that is, of a vector of random variables). Each element of the vector is a scalar random variable, either with a finite number of observed empirical values or with a finite or infinite number of potential values specified by a theoretical joint probability distribution of all the random variables.

Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the x and y directions contain all of the necessary information; a 2×2 matrix would be necessary to fully characterize the two-dimensional variation.

Read more about Covariance MatrixDefinition, Conflicting Nomenclatures and Notations, Properties, As A Linear Operator, Which Matrices Are Covariance Matrices?, How To Find A Valid Covariance Matrix, Complex Random Vectors, Estimation, As A Parameter of A Distribution

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Sample Mean And Sample Covariance - Discussion
... The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector, a row vector whose jth ... The sample covariance matrix has in the denominator rather than due to a variant of Bessel's correction In short, the sample covariance relies on the difference between each ... The maximum likelihood estimate of the covariance for the Gaussian distribution case has N in the denominator as well ...
... CMA-ES stands for Covariance Matrix Adaptation Evolution Strategy ... The covariance matrix adaptation (CMA) is a method to update the covariance matrix of this distribution ... Adaptation of the covariance matrix amounts to learning a second order model of the underlying objective function similar to the approximation of the inverse Hessian matrix in the Quasi-Newton method ...
Covariance Matrix - As A Parameter of A Distribution
... If a vector of n possibly correlated random variables is jointly normally distributed, or more generally elliptically distributed, then its probability density function can be expressed in terms of the covariance matrix. ...
CMA-ES - Theoretical Foundations - Interpretation As Coordinate System Transformation
... Using a non-identity covariance matrix for the multivariate normal distribution in evolution strategies is equivalent to a coordinate system transformation of the solution vectors, mainly because the ... procedure applied to a simple evolution strategy with identity covariance matrix ...
Spatial Descriptive Statistics - Measures of Spatial Dispersion
... simple measures of spatial dispersion for a point set can be defined using the covariance matrix of the coordinates of the points ... The trace, the determinant, and the largest eigenvalue of the covariance matrix can be used as measures of spatial dispersion ... A measure of spatial dispersion that is not based on the covariance matrix is the average distance between nearest neighbors ...

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