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The design matrix has dimension n-by-p, where n is the number of samples observed, and p is the number of variables measured in all samples. [4] [5]In this representation different rows typically represent different repetitions of an experiment, while columns represent different types of data (say, the results from particular probes).
For the special case of correlation matrices, we know that = and = /.This bounds the probability mass over the interval defined by = (). Since this distribution describes the spectrum of random matrices with mean 0, the eigenvalues of correlation matrices that fall inside of the aforementioned interval could be considered spurious or noise.
Suppose G is a p × n matrix, each column of which is independently drawn from a p-variate normal distribution with zero mean: = (, …,) (,). Then the Wishart distribution is the probability distribution of the p × p random matrix [4]
where x is an axis (i.e., a unit vector), M is an orthogonal orientation matrix, Z is a diagonal concentration matrix, and (;,) is a confluent hypergeometric function of matrix argument. The matrices M and Z are the result of diagonalizing the positive-definite covariance matrix of the Gaussian distribution that underlies the Bingham distribution.
English: A selection of Normal Distribution Probability Density Functions (PDFs). Both the mean, μ , and variance, σ² , are varied. The key is given on the graph.
For a discrete probability space, that is, a probability space on a finite set of objects, the Fisher metric can be understood to simply be the Euclidean metric restricted to a positive orthant (e.g. "quadrant" in ) of a unit sphere, after appropriate changes of variable.
In probability theory and statistics, the Rademacher distribution (which is named after Hans Rademacher) is a discrete probability distribution where a random variate X has a 50% chance of being +1 and a 50% chance of being −1.
A substochastic matrix is a real square matrix whose row sums are all ; In the same vein, one may define a probability vector as a vector whose elements are nonnegative real numbers which sum to 1. Thus, each row of a right stochastic matrix (or column of a left stochastic matrix) is a probability vector.