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Throughout this article, boldfaced unsubscripted and are used to refer to random vectors, and Roman subscripted and are used to refer to scalar random variables.. If the entries in the column vector = (,, …,) are random variables, each with finite variance and expected value, then the covariance matrix is the matrix whose (,) entry is the covariance [1]: 177 ...
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-definite matrices, the SCM is a biased and inefficient estimator. [1]
The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector, a vector whose jth element (=, …,) is one of the random variables.
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.
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.
The covariance-free approach avoids the np 2 operations of explicitly calculating and storing the covariance matrix X T X, instead utilizing one of matrix-free methods, for example, based on the function evaluating the product X T (X r) at the cost of 2np operations.
Precisely which covariance matrix is of concern is a matter of context. Alternative estimators have been proposed in MacKinnon & White (1985) that correct for unequal variances of regression residuals due to different leverage. [11] Unlike the asymptotic White's estimator, their estimators are unbiased when the data are homoscedastic.
The covariance matrix (also called second central moment or variance-covariance matrix) of an random vector is an matrix whose (i,j) th element is the covariance between the i th and the j th random variables.