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With any number of random variables in excess of 1, the variables can be stacked into a random vector whose i th element is the i th random variable. Then the variances and covariances can be placed in a covariance matrix, in which the (i, j) element is the covariance between the i th random variable and the j th one.
An entity closely related to the covariance matrix is the matrix of Pearson product-moment correlation coefficients between each of the random variables in the random vector , which can be written as = ( ()) ( ()), where is the matrix of the diagonal elements of (i.e., a diagonal matrix of the variances of for =, …,).
Simple cases, where observations are complete, can be dealt with by using the sample covariance matrix. The sample covariance matrix (SCM) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in R p×p; however, measured using the intrinsic geometry of positive ...
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
Computing this requires , the inverse of the covariance matrix which runs in () time (using the sample covariance matrix to obtain a sample partial correlation). Note that only a single matrix inversion is required to give all the partial correlations between pairs of variables in V {\displaystyle \mathbf {V} } .
Given a genetic covariance matrix, the genetic correlation is computed by standardizing this, i.e., by converting the covariance matrix to a correlation matrix. Generally, if Σ {\displaystyle \Sigma } is a genetic covariance matrix and D = diag ( Σ ) {\displaystyle D={\sqrt {\operatorname {diag} (\Sigma )}}} , then the correlation matrix ...
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 observation and the sample mean, but the sample mean is slightly correlated with each observation since it is defined in terms of all observations.
In probability theory and statistics, two real-valued random variables, , , are said to be uncorrelated if their covariance, [,] = [] [] [], is zero.If two variables are uncorrelated, there is no linear relationship between them.