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The Marsaglia polar method [1] is a pseudo-random number sampling method for generating a pair of independent standard normal random variables. [ 2 ] Standard normal random variables are frequently used in computer science , computational statistics , and in particular, in applications of the Monte Carlo method .
A matrix normal form or matrix canonical form describes the transformation of a matrix to another with special properties. Pages in category "Matrix normal forms" The following 10 pages are in this category, out of 10 total.
Different from linear dimensionality reduction methods such as principal component analysis (PCA), diffusion maps are part of the family of nonlinear dimensionality reduction methods which focus on discovering the underlying manifold that the data has been sampled from. By integrating local similarities at different scales, diffusion maps give ...
If the matrix X T X is well-conditioned and positive definite, implying that it has full rank, the normal equations can be solved directly by using the Cholesky decomposition R T R, where R is an upper triangular matrix, giving:
In the simplest cases, normalization of ratings means adjusting values measured on different scales to a notionally common scale, often prior to averaging. In more complicated cases, normalization may refer to more sophisticated adjustments where the intention is to bring the entire probability distributions of adjusted values into alignment.
In mathematics, a complex square matrix A is normal if it commutes with its conjugate transpose A *: =. The concept of normal matrices can be extended to normal operators on infinite-dimensional normed spaces and to normal elements in C*-algebras. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the ...
In mathematics, the Smith normal form (sometimes abbreviated SNF [1]) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal , and can be obtained from the original matrix by multiplying on the left and right by invertible square ...
where [] is the augmented matrix with E and F side by side and ‖ ‖ is the Frobenius norm, the square root of the sum of the squares of all entries in a matrix and so equivalently the square root of the sum of squares of the lengths of the rows or columns of the matrix. This can be rewritten as