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.
Probability density function | |||
Notation | |||
---|---|---|---|
Parameters |
μ ∈ R^{k} — location Σ ∈ R^{k × k} — covariance (positive semi-definite matrix) | ||
Support | x ∈ μ + span(Σ) ⊆ R^{k} | ||
exists only when Σ is positive-definite | |||
Mean | μ | ||
Mode | μ | ||
Variance | Σ | ||
Entropy | |||
MGF | |||
CF | |||
Kullback-Leibler divergence | See § Kullback–Leibler divergence |
The multivariate normal distribution of a k-dimensional random vector can be written in the following notation:
or to make it explicitly known that X is k-dimensional,
with k-dimensional mean vector
such that and . The inverse of the covariance matrix is called the precision matrix, denoted by .
A real random vector is called a standard normal random vector if all of its components are independent and each is a zero-mean unit-variance normally distributed random variable, i.e. if for all .^{[1]}^{: p. 454 }
A real random vector is called a centered normal random vector if there exists a deterministic matrix such that has the same distribution as where is a standard normal random vector with components.^{[1]}^{: p. 454 }
A real random vector is called a normal random vector if there exists a random -vector , which is a standard normal random vector, a -vector , and a matrix , such that .^{[2]}^{: p. 454 }^{[1]}^{: p. 455 }
Formally:
Here the covariance matrix is .
In the degenerate case where the covariance matrix is singular, the corresponding distribution has no density; see the section below for details. This case arises frequently in statistics; for example, in the distribution of the vector of residuals in the ordinary least squares regression. The are in general not independent; they can be seen as the result of applying the matrix to a collection of independent Gaussian variables .
The following definitions are equivalent to the definition given above. A random vector has a multivariate normal distribution if it satisfies one of the following equivalent conditions.
The spherical normal distribution can be characterised as the unique distribution where components are independent in any orthogonal coordinate system.^{[3]}^{[4]}
The multivariate normal distribution is said to be "non-degenerate" when the symmetric covariance matrix is positive definite. In this case the distribution has density^{[5]}
where is a real k-dimensional column vector and is the determinant of , also known as the generalized variance. The equation above reduces to that of the univariate normal distribution if is a matrix (i.e. a single real number).
The circularly symmetric version of the complex normal distribution has a slightly different form.
Each iso-density locus — the locus of points in k-dimensional space each of which gives the same particular value of the density — is an ellipse or its higher-dimensional generalization; hence the multivariate normal is a special case of the elliptical distributions.
The quantity is known as the Mahalanobis distance, which represents the distance of the test point from the mean . Note that in the case when , the distribution reduces to a univariate normal distribution and the Mahalanobis distance reduces to the absolute value of the standard score. See also Interval below.
In the 2-dimensional nonsingular case ( ), the probability density function of a vector is:
In the bivariate case, the first equivalent condition for multivariate reconstruction of normality can be made less restrictive as it is sufficient to verify that countably many distinct linear combinations of and are normal in order to conclude that the vector of is bivariate normal.^{[6]}
The bivariate iso-density loci plotted in the -plane are ellipses, whose principal axes are defined by the eigenvectors of the covariance matrix (the major and minor semidiameters of the ellipse equal the square-root of the ordered eigenvalues).
As the absolute value of the correlation parameter increases, these loci are squeezed toward the following line :
This is because this expression, with (where sgn is the Sign function) replaced by , is the best linear unbiased prediction of given a value of .^{[7]}
If the covariance matrix is not full rank, then the multivariate normal distribution is degenerate and does not have a density. More precisely, it does not have a density with respect to k-dimensional Lebesgue measure (which is the usual measure assumed in calculus-level probability courses). Only random vectors whose distributions are absolutely continuous with respect to a measure are said to have densities (with respect to that measure). To talk about densities but avoid dealing with measure-theoretic complications it can be simpler to restrict attention to a subset of of the coordinates of such that the covariance matrix for this subset is positive definite; then the other coordinates may be thought of as an affine function of these selected coordinates.^{[8]}
To talk about densities meaningfully in singular cases, then, we must select a different base measure. Using the disintegration theorem we can define a restriction of Lebesgue measure to the -dimensional affine subspace of where the Gaussian distribution is supported, i.e. . With respect to this measure the distribution has the density of the following motif:
where is the generalized inverse, is the rank of and is the pseudo-determinant.^{[9]}
The notion of cumulative distribution function (cdf) in dimension 1 can be extended in two ways to the multidimensional case, based on rectangular and ellipsoidal regions.
The first way is to define the cdf of a random vector as the probability that all components of are less than or equal to the corresponding values in the vector :^{[10]}
Though there is no closed form for , there are a number of algorithms that estimate it numerically.^{[10]}^{[11]}
Another way is to define the cdf as the probability that a sample lies inside the ellipsoid determined by its Mahalanobis distance from the Gaussian, a direct generalization of the standard deviation.^{[12]} In order to compute the values of this function, closed analytic formulae exist,^{[12]} as follows.
The interval for the multivariate normal distribution yields a region consisting of those vectors x satisfying
Here is a -dimensional vector, is the known -dimensional mean vector, is the known covariance matrix and is the quantile function for probability of the chi-squared distribution with degrees of freedom.^{[13]} When the expression defines the interior of an ellipse and the chi-squared distribution simplifies to an exponential distribution with mean equal to two (rate equal to half).
The complementary cumulative distribution function (ccdf) or the tail distribution is defined as . When , then the ccdf can be written as a probability the maximum of dependent Gaussian variables:^{[14]}
While no simple closed formula exists for computing the ccdf, the maximum of dependent Gaussian variables can be estimated accurately via the Monte Carlo method.^{[14]}^{[15]}
The probability content of the multivariate normal in a quadratic domain defined by (where is a matrix, is a vector, and is a scalar), which is relevant for Bayesian classification/decision theory using Gaussian discriminant analysis, is given by the generalized chi-squared distribution.^{[16]} The probability content within any general domain defined by (where is a general function) can be computed using the numerical method of ray-tracing ^{[16]} (Matlab code).
The kth-order moments of x are given by
where r_{1} + r_{2} + ⋯ + r_{N} = k.
The kth-order central moments are as follows
where the sum is taken over all allocations of the set into λ (unordered) pairs. That is, for a kth (= 2λ = 6) central moment, one sums the products of λ = 3 covariances (the expected value μ is taken to be 0 in the interests of parsimony):
This yields terms in the sum (15 in the above case), each being the product of λ (in this case 3) covariances. For fourth order moments (four variables) there are three terms. For sixth-order moments there are 3 × 5 = 15 terms, and for eighth-order moments there are 3 × 5 × 7 = 105 terms.
The covariances are then determined by replacing the terms of the list by the corresponding terms of the list consisting of r_{1} ones, then r_{2} twos, etc.. To illustrate this, examine the following 4th-order central moment case:
where is the covariance of X_{i} and X_{j}. With the above method one first finds the general case for a kth moment with k different X variables, , and then one simplifies this accordingly. For example, for , one lets X_{i} = X_{j} and one uses the fact that .
A quadratic form of a normal vector , (where is a matrix, is a vector, and is a scalar), is a generalized chi-squared variable. ^{[16]}
If is a general scalar-valued function of a normal vector, its probability density function, cumulative distribution function, and inverse cumulative distribution function can be computed with the numerical method of ray-tracing (Matlab code).^{[16]}
If the mean and covariance matrix are known, the log likelihood of an observed vector is simply the log of the probability density function:
The circularly symmetric version of the noncentral complex case, where is a vector of complex numbers, would be
i.e. with the conjugate transpose (indicated by ) replacing the normal transpose (indicated by ). This is slightly different than in the real case, because the circularly symmetric version of the complex normal distribution has a slightly different form for the normalization constant.
A similar notation is used for multiple linear regression.^{[17]}
Since the log likelihood of a normal vector is a quadratic form of the normal vector, it is distributed as a generalized chi-squared variable.^{[16]}
The differential entropy of the multivariate normal distribution is^{[18]}
where the bars denote the matrix determinant and k is the dimensionality of the vector space.
The Kullback–Leibler divergence from to , for non-singular matrices Σ_{1} and Σ_{0}, is:^{[19]}
where is the dimension of the vector space.
The logarithm must be taken to base e since the two terms following the logarithm are themselves base-e logarithms of expressions that are either factors of the density function or otherwise arise naturally. The equation therefore gives a result measured in nats. Dividing the entire expression above by log_{e} 2 yields the divergence in bits.
When ,
The mutual information of a distribution is a special case of the Kullback–Leibler divergence in which is the full multivariate distribution and is the product of the 1-dimensional marginal distributions. In the notation of the Kullback–Leibler divergence section of this article, is a diagonal matrix with the diagonal entries of , and . The resulting formula for mutual information is:
where is the correlation matrix constructed from .^{[citation needed]}
In the bivariate case the expression for the mutual information is:
If and are normally distributed and independent, this implies they are "jointly normally distributed", i.e., the pair must have multivariate normal distribution. However, a pair of jointly normally distributed variables need not be independent (would only be so if uncorrelated, ).
The fact that two random variables and both have a normal distribution does not imply that the pair has a joint normal distribution. A simple example is one in which X has a normal distribution with expected value 0 and variance 1, and if and if , where . There are similar counterexamples for more than two random variables. In general, they sum to a mixture model.^{[citation needed]}
In general, random variables may be uncorrelated but statistically 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, as pointed out just above, it is not true that two random variables that are (separately, marginally) normally distributed and uncorrelated are independent.
If N-dimensional x is partitioned as follows
and accordingly μ and Σ are partitioned as follows
then the distribution of x_{1} conditional on x_{2} = a is multivariate normal (x_{1} | x_{2} = a) ~ N(μ, Σ) where
and covariance matrix
Here is the generalized inverse of . The matrix is the Schur complement of Σ_{22} in Σ. That is, the equation above is equivalent to inverting the overall covariance matrix, dropping the rows and columns corresponding to the variables being conditioned upon, and inverting back to get the conditional covariance matrix.
Note that knowing that x_{2} = a alters the variance, though the new variance does not depend on the specific value of a; perhaps more surprisingly, the mean is shifted by ; compare this with the situation of not knowing the value of a, in which case x_{1} would have distribution .
An interesting fact derived in order to prove this result, is that the random vectors and are independent.
The matrix Σ_{12}Σ_{22}^{−1} is known as the matrix of regression coefficients.
In the bivariate case where x is partitioned into and , the conditional distribution of given is^{[21]}
where is the correlation coefficient between and .
The conditional expectation of X_{1} given X_{2} is:
Proof: the result is obtained by taking the expectation of the conditional distribution above.
The conditional expectation of X_{1} given X_{2} is
and the conditional variance is
thus the conditional variance does not depend on x_{2}.
The conditional expectation of X_{1} given that X_{2} is smaller/bigger than z is:^{[22]}^{: 367 }
where the final ratio here is called the inverse Mills ratio.
Proof: the last two results are obtained using the result , so that
To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. The proof for this follows from the definitions of multivariate normal distributions and linear algebra.^{[23]}
Example
Let X = [X_{1}, X_{2}, X_{3}] be multivariate normal random variables with mean vector μ = [μ_{1}, μ_{2}, μ_{3}] and covariance matrix Σ (standard parametrization for multivariate normal distributions). Then the joint distribution of X′ = [X_{1}, X_{3}] is multivariate normal with mean vector μ′ = [μ_{1}, μ_{3}] and covariance matrix .
If Y = c + BX is an affine transformation of where c is an vector of constants and B is a constant matrix, then Y has a multivariate normal distribution with expected value c + Bμ and variance BΣB^{T} i.e., . In particular, any subset of the X_{i} has a marginal distribution that is also multivariate normal. To see this, consider the following example: to extract the subset (X_{1}, X_{2}, X_{4})^{T}, use
which extracts the desired elements directly.
Another corollary is that the distribution of Z = b · X, where b is a constant vector with the same number of elements as X and the dot indicates the dot product, is univariate Gaussian with . This result follows by using
Observe how the positive-definiteness of Σ implies that the variance of the dot product must be positive.
An affine transformation of X such as 2X is not the same as the sum of two independent realisations of X.
The equidensity contours of a non-singular multivariate normal distribution are ellipsoids (i.e. affine transformations of hyperspheres) centered at the mean.^{[24]} Hence the multivariate normal distribution is an example of the class of elliptical distributions. The directions of the principal axes of the ellipsoids are given by the eigenvectors of the covariance matrix . The squared relative lengths of the principal axes are given by the corresponding eigenvalues.
If Σ = UΛU^{T} = UΛ^{1/2}(UΛ^{1/2})^{T} is an eigendecomposition where the columns of U are unit eigenvectors and Λ is a diagonal matrix of the eigenvalues, then we have
Moreover, U can be chosen to be a rotation matrix, as inverting an axis does not have any effect on N(0, Λ), but inverting a column changes the sign of U's determinant. The distribution N(μ, Σ) is in effect N(0, I) scaled by Λ^{1/2}, rotated by U and translated by μ.
Conversely, any choice of μ, full rank matrix U, and positive diagonal entries Λ_{i} yields a non-singular multivariate normal distribution. If any Λ_{i} is zero and U is square, the resulting covariance matrix UΛU^{T} is singular. Geometrically this means that every contour ellipsoid is infinitely thin and has zero volume in n-dimensional space, as at least one of the principal axes has length of zero; this is the degenerate case.
"The radius around the true mean in a bivariate normal random variable, re-written in polar coordinates (radius and angle), follows a Hoyt distribution."^{[25]}
In one dimension the probability of finding a sample of the normal distribution in the interval is approximately 68.27%, but in higher dimensions the probability of finding a sample in the region of the standard deviation ellipse is lower.^{[26]}
Dimensionality | Probability |
---|---|
1 | 0.6827 |
2 | 0.3935 |
3 | 0.1987 |
4 | 0.0902 |
5 | 0.0374 |
6 | 0.0144 |
7 | 0.0052 |
8 | 0.0018 |
9 | 0.0006 |
10 | 0.0002 |
The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is straightforward.
In short, the probability density function (pdf) of a multivariate normal is
and the ML estimator of the covariance matrix from a sample of n observations is
which is simply the sample covariance matrix. This is a biased estimator whose expectation is
An unbiased sample covariance is
The Fisher information matrix for estimating the parameters of a multivariate normal distribution has a closed form expression. This can be used, for example, to compute the Cramér–Rao bound for parameter estimation in this setting. See Fisher information for more details.
In Bayesian statistics, the conjugate prior of the mean vector is another multivariate normal distribution, and the conjugate prior of the covariance matrix is an inverse-Wishart distribution . Suppose then that n observations have been made
and that a conjugate prior has been assigned, where
where
and
Then,^{[citation needed]}
where
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^{[27]} and Smith and Jain's adaptation^{[28]} of the Friedman–Rafsky test created by Larry Rafsky and Jerome Friedman.^{[29]}
Mardia's test^{[30]} is based on multivariate extensions of skewness and kurtosis measures. For a sample {x_{1}, ..., x_{n}} of k-dimensional vectors we compute
Under the null hypothesis of multivariate normality, the statistic A will have approximately a chi-squared distribution with 1/6⋅k(k + 1)(k + 2) degrees of freedom, and B will be approximately standard normal N(0,1).
Mardia's kurtosis statistic is skewed and converges very slowly to the limiting normal distribution. For medium size samples , the parameters of the asymptotic distribution of the kurtosis statistic are modified^{[31]} For small sample tests ( ) empirical critical values are used. Tables of critical values for both statistics are given by Rencher^{[32]} for k = 2, 3, 4.
Mardia's tests are affine invariant but not consistent. For example, the multivariate skewness test is not consistent against symmetric non-normal alternatives.^{[33]}
The BHEP test^{[34]} computes the norm of the difference between the empirical characteristic function and the theoretical characteristic function of the normal distribution. Calculation of the norm is performed in the L^{2}(μ) space of square-integrable functions with respect to the Gaussian weighting function . The test statistic is
The limiting distribution of this test statistic is a weighted sum of chi-squared random variables,^{[34]} however in practice it is more convenient to compute the sample quantiles using the Monte-Carlo simulations.^{[citation needed]}
A detailed survey of these and other test procedures is available.^{[35]}
Suppose that observations (which are vectors) are presumed to come from one of several multivariate normal distributions, with known means and covariances. Then any given observation can be assigned to the distribution from which it has the highest probability of arising. This classification procedure is called Gaussian discriminant analysis. The classification performance, i.e. probabilities of the different classification outcomes, and the overall classification error, can be computed by the numerical method of ray-tracing ^{[16]} (Matlab code).
A widely used method for drawing (sampling) a random vector x from the N-dimensional multivariate normal distribution with mean vector μ and covariance matrix Σ works as follows:^{[36]}
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