In multivariate statistics and probability theory, the scatter matrix is a statistic that is used to make estimates of the covariance matrix, for instance of the multivariate normal distribution.
Given n samples of m-dimensional data, represented as the m-by-n matrix, , the sample mean is
where is the j-th column of . [1]
The scatter matrix is the m-by-m positive semi-definite matrix
where denotes matrix transpose, [2] and multiplication is with regards to the outer product. The scatter matrix may be expressed more succinctly as
where is the n-by-n centering matrix.
The maximum likelihood estimate, given n samples, for the covariance matrix of a multivariate normal distribution can be expressed as the normalized scatter matrix
When the columns of are independently sampled from a multivariate normal distribution, then has a Wishart distribution.
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by , , , , or .
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value.
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve.
Covariance in probability theory and statistics is a measure of the joint variability of two random variables.
In probability theory and statistics, a covariance matrix is a square matrix giving the covariance between each pair of elements of a given random vector.
In statistics, the Pearson correlation coefficient (PCC) is a correlation coefficient that measures linear correlation between two sets of data. It is the ratio between the covariance of two variables and the product of their standard deviations; thus, it is essentially a normalized measurement of the covariance, such that the result always has a value between −1 and 1. As with covariance itself, the measure can only reflect a linear correlation of variables, and ignores many other types of relationships or correlations. As a simple example, one would expect the age and height of a sample of teenagers from a high school to have a Pearson correlation coefficient significantly greater than 0, but less than 1.
In statistics, the Wishart distribution is a generalization of the gamma distribution to multiple dimensions. It is named in honor of John Wishart, who first formulated the distribution in 1928. Other names include Wishart ensemble, or Wishart–Laguerre ensemble, or LOE, LUE, LSE.
In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
In statistics, propagation of uncertainty is the effect of variables' uncertainties on the uncertainty of a function based on them. When the variables are the values of experimental measurements they have uncertainties due to measurement limitations which propagate due to the combination of variables in the function.
The Mahalanobis distance is a measure of the distance between a point and a distribution , introduced by P. C. Mahalanobis in 1936. Mahalanobis's definition was prompted by the problem of identifying the similarities of skulls based on measurements in 1927.
In statistics, particularly in hypothesis testing, the Hotelling's T-squared distribution (T2), proposed by Harold Hotelling, is a multivariate probability distribution that is tightly related to the F-distribution and is most notable for arising as the distribution of a set of sample statistics that are natural generalizations of the statistics underlying the Student's t-distribution. The Hotelling's t-squared statistic (t2) is a generalization of Student's t-statistic that is used in multivariate hypothesis testing.
In statistics, sometimes the covariance matrix of a multivariate random variable is not known but has to be estimated. Estimation of covariance matrices then deals with the question of how to approximate the actual covariance matrix on the basis of a sample from the multivariate distribution. 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 Rp×p; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator. In addition, if the random variable has a normal distribution, the sample covariance matrix has a Wishart distribution and a slightly differently scaled version of it is the maximum likelihood estimate. Cases involving missing data, heteroscedasticity, or autocorrelated residuals require deeper considerations. Another issue is the robustness to outliers, to which sample covariance matrices are highly sensitive.
In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of a multivariate normal distribution.
In probability theory and statistics, partial correlation measures the degree of association between two random variables, with the effect of a set of controlling random variables removed. When determining the numerical relationship between two variables of interest, using their correlation coefficient will give misleading results if there is another confounding variable that is numerically related to both variables of interest. This misleading information can be avoided by controlling for the confounding variable, which is done by computing the partial correlation coefficient. This is precisely the motivation for including other right-side variables in a multiple regression; but while multiple regression gives unbiased results for the effect size, it does not give a numerical value of a measure of the strength of the relationship between the two variables of interest.
The sample mean or empirical mean, and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.
In probability theory, the family of complex normal distributions, denoted or , characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: location parameter μ, covariance matrix , and the relation matrix . The standard complex normal is the univariate distribution with , , and .
In probability theory and statistics, the normal-Wishart distribution is a multivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a multivariate normal distribution with unknown mean and precision matrix.
In statistics, the matrix t-distribution is the generalization of the multivariate t-distribution from vectors to matrices. The matrix t-distribution shares the same relationship with the multivariate t-distribution that the matrix normal distribution shares with the multivariate normal distribution. For example, the matrix t-distribution is the compound distribution that results from sampling from a matrix normal distribution having sampled the covariance matrix of the matrix normal from an inverse Wishart distribution.
In statistics, the complex Wishart distribution is a complex version of the Wishart distribution. It is the distribution of times the sample Hermitian covariance matrix of zero-mean independent Gaussian random variables. It has support for Hermitian positive definite matrices.
The complex inverse Wishart distribution is a matrix probability distribution defined on complex-valued positive-definite matrices and is the complex analog of the real inverse Wishart distribution. The complex Wishart distribution was extensively investigated by Goodman while the derivation of the inverse is shown by Shaman and others. It has greatest application in least squares optimization theory applied to complex valued data samples in digital radio communications systems, often related to Fourier Domain complex filtering.
 | This statistics-related article is a stub. You can help Wikipedia by expanding it. |
 | This article about matrices is a stub. You can help Wikipedia by expanding it. |
