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Probability density function | |
Cumulative distribution function | |
Parameters | location (real vector) covariance matrix (positive definite real matrix) |
Support | |
cdf | |
Mean | |
Median | |
Mode | |
Variance | |
Skewness | 0 |
Kurtosis | 0 |
Entropy | |
mgf | |
Char. func. |
In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution).
General case
A random vector follows a multivariate normal distribution if it satisfies the following equivalent conditions:
- every linear combination is normally distributed
- there is a random vector , whose components are independent standard normal random variables, a vector and an matrix such that .
- there is a vector and a symmetric, positive semi-definite matrix such that the characteristic function of is
The following is not quite equivalent to the conditions above, since it fails to allow for a singular matrix as the variance:
- there is a vector and a symmetric, positive definite covariance matrix ( matrix) such that has density
where is the determinant of . Note how the equation above reduces to that of the univariate normal distribution if is a scalar (i.e., a real number).
The vector in these conditions is the expected value of and the matrix is the covariance matrix of the components .
It is important to realize that the covariance matrix must be allowed to be singular. That case arises frequently in statistics; for example, in the distribution of the vector of residuals in ordinary linear regression problems. Note also that the are in general not independent; they can be seen as the result of applying the linear transformation to a collection of independent Gaussian variables .
The multivariate normal can be written in the following notation:
or to make it explicitly known is N-dimensional
Cumulative distribution function
The cumulative distribution function (cdf) is defined as the probability that all values in a random vector are less than or equal to the corresponding values in vector . Though there is no closed form for , there are a number of algorithms that estimate it numerically. For example, see MVNDST under [1] (includes FORTRAN code) or [2] (includes MATLAB code).
A counterexample
The fact that two or more random variables X and Y are normally distributed does not imply that the pair (X, Y) has a joint normal distribution. A simple example is one in which Y = X if |X| > 1 and Y = −X if |X| < 1.
Normally distributed and independent
If X and Y are normally distributed and independent, then they are "jointly normally distributed", i.e., the pair (X, Y) has a bivariate normal distribution. There are of course also many bivariate normal distributions in which the components are correlated.
Bivariate case
In the 2-dimensional nonsingular case, the probability density function (with mean (0,0)) is
where is the correlation between and . In this case,
Linear transformation
If is a linear transformation of where is an matrix then has a multivariate normal distribution with expected value and variance (i.e., .
Corollary: any subset of the has a marginal distribution that is also multivariate normal. To see this consider the following example: to extract the subset , use
which extracts the desired elements directly.
Correlations and independence
In general, random variables may be uncorrelated but highly dependent. But if a random vector has a multivariate normal distribution then any two or more of its components that are uncorrelated are independent. This implies that any two or more of its components that are pairwise independent are independent.
But it is not true that two random variables that are (separately, marginally) normally distributed and uncorrelated are independent. Two random variables that are normally distributed may fail to be jointly normally distributed, i.e., the vector whose components they are may fail to have a multivariate normal distribution. For an example of two normally distributed random variables that are uncorrelated but not independent, see normally distributed and uncorrelated does not imply independent.
Higher moments
The -order moments of are defined by
where
The central -order moments are given as follows
(a) If is odd, .
(b) If is even with , then
where the sum is taken over all allocations of the set into (unordered) pairs, giving terms in the sum, each being the product of covariances. The covariances are determined by replacing the terms of the list by the corresponding terms of the list consisting of ones, then twos, etc, after each of the possible allocations of the former list into pairs.
In particular, the 4-order moments are
Conditional distributions
If and are partitioned as follows
- with sizes
- with sizes
then the distribution of conditional on is multivariate normal where
and covariance matrix
This matrix is the Schur complement of in .
Note that knowing the value of to be alters the variance; perhaps more surprisingly, the mean is shifted by ; compare this with the situation of not knowing the value of , in which case would have distribution .
The matrix is known as the matrix of regression coefficients.
Fisher information matrix
The Fisher information matrix (FIM) for a normal distribution takes a special formulation. The element of the FIM for is
where
- is the trace function
Kullback-Leibler divergence
The Kullback-Leibler divergence from to is:
Estimation of parameters
The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is perhaps surprisingly subtle and elegant. See estimation of covariance matrices.
In short, the probability density function (pdf) of an N-dimensional multivariate normal is
and the ML estimator of the covariance matrix is
which is simply the sample covariance matrix for sample size n. This is a biased estimator whose expectation is
An unbiased sample covariance is
Entropy
The differential entropy of the multivariate normal distribution is ^{[1]}
where is the determinant of the covariance matrix .
Multivariate normality tests
Multivariate normality tests check a given set of data for similarity to the multivariate normal distribution. The null hypothesis is that the data set is similar to the normal distribution, therefore a sufficiently small p-value indicates non-normal data. Multivariate normality tests include the Cox-Small test ^{[2]} and Smith and Jain's adaptation of the Friedman-Rafsky test ^{[3]}.
Drawing values from the distribution
A widely used method for drawing a random vector from the -dimensional multivariate normal distribution with mean vector and covariance matrix (required to be symmetric and positive definite) works as follows:
- Compute the Cholesky decomposition (matrix square root) of , that is, find the unique lower triangular matrix such that .
- Let be a vector whose components are independent standard normal variates (which can be generated, for example, by using the Box-Muller transform).
- Let be .
References
- ↑ Gokhale, DV, NA Ahmed, BC Res, NJ Piscataway (May 1989). Entropy Expressions and Their Estimators for Multivariate Distributions. Information Theory, IEEE Transactions on 35 (3): 688-692.
- ↑ Cox, D. R., N. J. H. Small (August 1978). Testing multivariate normality. Biometrika 65 (2): 263–272.
- ↑ Smith, Stephen P., Anil K. Jain (September 1988). A test to determine the multivariate normality of a dataset. IEEE Transactions on Pattern Analysis and Machine Intelligence 10 (5): 757–761. DOI:10.1109/34.6789.
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