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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 ...
Any definition of expected value may be extended to define an expected value of a multidimensional random variable, i.e. a random vector X. It is defined component by component, as E[X] i = E[X i]. Similarly, one may define the expected value of a random matrix X with components X ij by E[X] ij = E[X ij].
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]
In this case the Fisher information matrix may be identified with the coefficient matrix of the normal equations of least squares estimation theory. Another special case occurs when the mean and covariance depend on two different vector parameters, say, β and θ. This is especially popular in the analysis of spatial data, which often uses a ...
Values of the concentration parameter above 1 prefer variates that are dense, evenly distributed distributions, i.e. all the values within a single sample are similar to each other. Values of the concentration parameter below 1 prefer sparse distributions, i.e. most of the values within a single sample will be close to 0, and the vast majority ...
The algebraic connectivity (also known as Fiedler value or Fiedler eigenvalue after Miroslav Fiedler) of a graph G is the second-smallest eigenvalue (counting multiple eigenvalues separately) of the Laplacian matrix of G. [1] This eigenvalue is greater than 0 if and only if G is a connected graph. This is a corollary to the fact that the number ...
(Here Θ is any matrix with the same dimensions as V, 1 indicates the identity matrix, and i is a square root of −1). [9] Properly interpreting this formula requires a little care, because noninteger complex powers are multivalued ; when n is noninteger, the correct branch must be determined via analytic continuation .
A matrix, has its column space depicted as the green line. The projection of some vector onto the column space of is the vector . From the figure, it is clear that the closest point from the vector onto the column space of , is , and is one where we can draw a line orthogonal to the column space of .