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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 Kalman filter tracks the average state of a system as a vector x of length N and covariance as an N × N matrix P. The matrix P is always positive semi-definite and can be decomposed into LL T. The columns of L can be added and subtracted from the mean x to form a set of 2N vectors called sigma points. These sigma points completely capture ...
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 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.
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 "almost" triangular matrix, for example, an upper Hessenberg matrix has zero entries below the first subdiagonal. Hollow matrix: A square matrix whose main diagonal comprises only zero elements. Integer matrix: A matrix whose entries are all integers. Logical matrix: A matrix with all entries either 0 or 1.
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.