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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 ...
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 ...
The matrix ¯ is the Schur complement of Σ 22 in Σ. That is, the equation above is equivalent to inverting the overall covariance matrix, dropping the rows and columns corresponding to the variables being conditioned upon, and inverting back to get the conditional covariance matrix.
PCA of a multivariate Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the (0.866, 0.5) direction and of 1 in the orthogonal direction. . The vectors shown are the eigenvectors of the covariance matrix scaled by the square root of the corresponding eigenvalue, and shifted so their tails are at the m
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.
When these assumptions are satisfied, the following covariance matrix K applies for the 1D profile parameters , , and under i.i.d. Gaussian noise and under Poisson noise: [9] = , = , where is the width of the pixels used to sample the function, is the quantum efficiency of the detector, and indicates the standard deviation of the measurement noise.
When N(t) is colored (correlated in time) Gaussian noise with zero mean and covariance function (,) = [() ()], we cannot sample independent discrete observations by evenly spacing the time. Instead, we can use K–L expansion to decorrelate the noise process and get independent Gaussian observation 'samples'.
These methods approximate the true model in a way the covariance matrix is sparse. Typically, each method proposes its own algorithm that takes the full advantage of the sparsity pattern in the covariance matrix. Two prominent members of this class of approaches are covariance tapering and domain partitioning.