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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 eigenvalue is approximated by r T (X T X) r, which is the Rayleigh quotient on the unit vector r for the covariance matrix X T X . If the largest singular value is well separated from the next largest one, the vector r gets close to the first principal component of X within the number of iterations c , which is small relative to p , at the ...
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.
In multivariate statistics and probability theory, the scatter matrix is a statistic that is used to make estimates of the covariance matrix, for instance of the multivariate normal distribution. Definition
A similar canonical set of sigma points can be generated in any number of dimensions by taking the zero vector and the points comprising the rows of the identity matrix, computing the mean of the set of points, subtracting the mean from each point so that the resulting set has a mean of zero, then computing the covariance of the zero-mean set ...
In probability theory and statistics, a cross-covariance matrix is a matrix whose element in the i, j position is the covariance between the i-th element of a random vector and j-th element of another random vector. When the two random vectors are the same, the cross-covariance matrix is referred to as covariance matrix.
In either case: (i) estimates of missing data points are produced iteratively, and are then used to compute a self-consistent lag-covariance matrix and its EOFs ; and (ii) cross-validation is used to optimize the window width and the number of leading SSA modes to fill the gaps with the iteratively estimated "signal," while the noise is discarded.