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| {{Redirect|MVN|the airport with that [[International Air Transport Association airport code|IATA code]]|Mount Vernon Airport}}
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| {{Probability distribution
| |
| | name =
| |
| | type = multivariate
| |
| | pdf_image = [[Image:MultivariateNormal.png|300px]]<br/> <small>Many samples from a multivariate normal distribution, shown along with the 3-sigma ellipse, the two marginal distributions, and the two 1-d histograms.</small>
| |
| | cdf_image =
| |
| | notation = <math>\mathcal{N}(\boldsymbol\mu,\,\boldsymbol\Sigma)</math>
| |
| | parameters = '''''μ''''' ∈ '''R'''<sup>''k''</sup> — [[location parameter|location]]<br/>'''Σ''' ∈ '''R'''<sup>''k×k''</sup> — [[covariance matrix|covariance]] ([[nonnegative-definite matrix]])
| |
| | support = '''''x''''' ∈ '''μ'''+span('''Σ''') ⊆ '''R'''<sup>''k''</sup>
| |
| | pdf = <math>(2\pi)^{-\frac{k}{2}}|\boldsymbol\Sigma|^{-\frac{1}{2}}\, e^{ -\frac{1}{2}(\mathbf{x}-\boldsymbol\mu)'\boldsymbol\Sigma^{-1}(\mathbf{x}-\boldsymbol\mu) },</math><br/>exists only when '''Σ''' is [[positive-definite matrix|positive-definite]]
| |
| | cdf = (no analytic expression)
| |
| | mean = '''''μ'''''
| |
| | median =
| |
| | mode = '''''μ'''''
| |
| | variance = '''Σ'''
| |
| | skewness =
| |
| | kurtosis =
| |
| | entropy = <math>\frac{k}{2} (1 + \ln (2\pi)) + \frac{1}{2} \ln |\boldsymbol\Sigma |</math>
| |
| | mgf = <math>\exp\!\Big( \boldsymbol\mu'\mathbf{t} + \tfrac{1}{2} \mathbf{t}'\boldsymbol\Sigma \mathbf{t}\Big)</math>
| |
| | char = <math>\exp\!\Big( i\boldsymbol\mu'\mathbf{t} - \tfrac{1}{2} \mathbf{t}'\boldsymbol\Sigma \mathbf{t}\Big)</math>
| |
| }}
| |
| In [[probability theory]] and [[statistics]], the '''multivariate normal distribution''' or '''multivariate Gaussian distribution''', is a generalization of the one-dimensional ([[univariate]]) [[normal distribution]] to higher dimensions. One possible 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. However, its importance derives mainly from the [[Central limit theorem#Multivariate central limit theorem|multivariate central limit theorem]]. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued [[random variable]]s each of which clusters around a mean value.
| |
| | |
| == Notation and parametrization ==
| |
| The multivariate normal distribution of a ''k''-dimensional random vector {{nowrap|'''x''' {{=}} [''X''<sub>1</sub>, ''X''<sub>2</sub>, …, ''X<sub>k</sub>'']}} can be written in the following notation:
| |
| : <math>
| |
| \mathbf{x}\ \sim\ \mathcal{N}(\boldsymbol\mu,\, \boldsymbol\Sigma),
| |
| </math>
| |
| or to make it explicitly known that ''X'' is ''k''-dimensional,
| |
| : <math>
| |
| \mathbf{x}\ \sim\ \mathcal{N}_k(\boldsymbol\mu,\, \boldsymbol\Sigma).
| |
| </math>
| |
| with ''k''-dimensional [[mean vector]]
| |
| :<math> \boldsymbol\mu = [ \operatorname{E}[X_1], \operatorname{E}[X_2], \ldots, \operatorname{E}[X_k]] </math>
| |
| and ''k x k'' [[covariance matrix]]
| |
| :<math> \boldsymbol\Sigma = [\operatorname{Cov}[X_i, X_j]], i=1,2,\ldots,k; j=1,2,\ldots,k </math>
| |
| | |
| == Definition ==
| |
| | |
| A [[random vector]] {{nowrap|1='''x''' = (''X''<sub>1</sub>, …, ''X''<sub>''k''</sub>)'}} is said to have the multivariate normal distribution if it satisfies the following equivalent conditions.<ref>Gut, Allan (2009) ''An Intermediate Course in Probability'', Springer. ISBN 9781441901613 (Chapter 5)</ref>
| |
| | |
| *Every linear combination of its components ''Y'' = ''a''<sub>1</sub>''X''<sub>1</sub> + … + ''a<sub>k</sub>X<sub>k</sub>'' is [[normal distribution|normally distributed]]. That is, for any constant vector {{nowrap|'''a''' ∈ '''R'''<sup>''k''</sup>}}, the random variable {{nowrap|1=''Y'' = '''a′x'''}} has a univariate normal distribution.
| |
| | |
| *There exists a random ''ℓ''-vector '''z''', whose components are independent standard normal random variables, a ''k''-vector '''μ''', and a ''k×ℓ'' [[matrix (math)|matrix]] '''A''', such that {{nowrap|1='''x''' = '''Az''' + '''μ'''}}. Here ''ℓ'' is the [[rank (linear algebra)|rank]] of the [[covariance matrix]] {{nowrap|1='''Σ''' = '''AA′'''}}. Especially in the case of full rank, see the section below on [[#Geometric interpretation|Geometric interpretation]].
| |
| | |
| *There is a ''k''-vector '''μ''' and a symmetric, [[nonnegative-definite]] ''k×k'' matrix '''Σ''', such that the [[Characteristic function (probability theory)|characteristic function]] of '''x''' is
| |
| :: <math>
| |
| \varphi_\mathbf{x}(\mathbf{u}) = \exp\Big( i\mathbf{u}'\boldsymbol\mu - \tfrac{1}{2} \mathbf{u}'\boldsymbol\Sigma \mathbf{u} \Big).
| |
| </math>
| |
| | |
| The covariance matrix is allowed to be singular (in which case the corresponding distribution has no density). This case arises frequently in [[statistics]]; for example, in the distribution of the vector of [[errors and residuals in statistics|residuals]] in the [[ordinary least squares]] regression. Note also that the ''X''<sub>''i''</sub> are in general ''not'' independent; they can be seen as the result of applying the matrix '''A''' to a collection of independent Gaussian variables '''z'''.
| |
| | |
| == Properties ==
| |
| | |
| ===Density function===
| |
| | |
| ====Non-degenerate case====
| |
| The multivariate normal distribution is said to be "non-degenerate" when the symmetric covariance matrix <math>\boldsymbol\Sigma</math> is [[Positive-definite matrix|positive definite]]. In this case the distribution has [[probability density function|density]]<ref>[http://www.math.uiuc.edu/~r-ash/Stat/StatLec21-25.pdf UIUC, Lecture 21. ''The Multivariate Normal Distribution''], 21.5:"Finding the Density".</ref>
| |
| | |
| :<math>
| |
| f_{\mathbf x}(x_1,\ldots,x_k) =
| |
| \frac{1}{\sqrt{(2\pi)^k|\boldsymbol\Sigma|}}
| |
| \exp\left(-\frac{1}{2}({\mathbf x}-{\boldsymbol\mu})^T{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})
| |
| \right),
| |
| </math>
| |
| | |
| where <math>|\boldsymbol\Sigma|</math> is the [[determinant]] of <math>\boldsymbol\Sigma</math>. Note how the equation above reduces to that of the univariate normal distribution if <math>\boldsymbol\Sigma</math> is a <math>1 \times 1</math> matrix (i.e. a real number).
| |
| | |
| 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 distribution]]s.
| |
| | |
| ;Bivariate case
| |
| In the 2-dimensional nonsingular case ({{nowrap|1=''k'' = rank(Σ) = 2}}), the [[probability density function]] of a vector {{nowrap|[''X'' ''Y'']′}} is
| |
| : <math>
| |
| f(x,y) =
| |
| \frac{1}{2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}}
| |
| \exp\left(
| |
| -\frac{1}{2(1-\rho^2)}\left[
| |
| \frac{(x-\mu_x)^2}{\sigma_x^2} +
| |
| \frac{(y-\mu_y)^2}{\sigma_y^2} -
| |
| \frac{2\rho(x-\mu_x)(y-\mu_y)}{\sigma_x \sigma_y}
| |
| \right]
| |
| \right),
| |
| </math>
| |
| where ''ρ'' is the [[Pearson product-moment correlation coefficient|correlation]] between ''X'' and ''Y'' and
| |
| where <math> \sigma_x>0 </math> and <math> \sigma_y>0 </math>. In this case,
| |
| : <math>
| |
| \boldsymbol\mu = \begin{pmatrix} \mu_x \\ \mu_y \end{pmatrix}, \quad
| |
| \boldsymbol\Sigma = \begin{pmatrix} \sigma_x^2 & \rho \sigma_x \sigma_y \\
| |
| \rho \sigma_x \sigma_y & \sigma_y^2 \end{pmatrix}.
| |
| </math>
| |
| In the bivariate case, the first equivalent condition for multivariate normality can be made less restrictive: it is sufficient to verify that [[countably infinite|countably many]] distinct linear combinations of X and Y are normal in order to conclude that the vector {{nowrap|[X Y]′}} is bivariate normal.<ref name=HT/>
| |
| | |
| The bivariate iso-density loci plotted in the ''x,y''-plane are ellipses. As the correlation parameter ''ρ'' increases, these loci appear to be squeezed to the following line :
| |
| | |
| : <math>
| |
| y\left( x \right) = {\mathop{\rm sgn}} \left( {{\rho }} \right)\frac{{{\sigma _y}}}{{{\sigma _x}}}\left( {x - {\mu _x}} \right) + {\mu _y}.
| |
| </math>
| |
| | |
| This is because the above expression - but without the rho being inside a signum function - is the [[best linear unbiased prediction]] of ''Y'' given a value of ''X''.<ref name=wyattlms/>
| |
| | |
| ====Degenerate case====
| |
| If the covariance matrix <math>\boldsymbol\Sigma</math> 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 [[absolute continuity#Absolute continuity of measures|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 <math>\text{rank}(\boldsymbol\Sigma)</math> of the coordinates of <math>\mathbf{x}</math> such that the covariance matrix for this subset is positive definite; then the other coordinates may be thought of as an [[affine function]] of the selected coordinates.{{citation needed|date=July 2012}}
| |
| | |
| To talk about densities meaningfully in the singular case, then, we must select a different base measure. Using the [[disintegration theorem]] we can define a restriction of Lebesgue measure to the <math>\text{rank}(\boldsymbol\Sigma)</math>-dimensional affine subspace of <math>\mathbb{R}^k</math> where the Gaussian distribution is supported, i.e. <math>\{\boldsymbol\mu+\boldsymbol{\Sigma ^{1/2}}\mathbf{v} : \mathbf{v} \in \mathbb{R}^k \}</math>. With respect to this probability measure the distribution has density:
| |
| :<math>f(\mathbf{x})=(\text{det}^*(2\pi\boldsymbol\Sigma))^{-\frac{1}{2}}\, e^{ -\frac{1}{2}(\mathbf{x}-\boldsymbol\mu)'\boldsymbol\Sigma^+(\mathbf{x}-\boldsymbol\mu) }</math>
| |
| where <math>\boldsymbol\Sigma^+</math> is the [[generalized inverse]] and det* is the [[pseudo-determinant]].<ref name=rao/>
| |
| | |
| ===Higher moments===
| |
| {{Main|Isserlis’ theorem}}
| |
| The ''k''th-order [[moment (mathematics)|moments]] of '''x''' are defined by
| |
| | |
| :<math>
| |
| \mu _{1,\dots,N}(\mathbf{x})\ \stackrel{\mathrm{def}}{=}\ \mu _{r_{1},\dots,r_{N}}(\mathbf{x})\ \stackrel{\mathrm{def}}{=}\ E\left[
| |
| \prod\limits_{j=1}^{N}x_j^{r_{j}}\right]
| |
| </math>
| |
| | |
| where {{nowrap|''r''<sub>1</sub> + ''r''<sub>2</sub> + ⋯ + ''r<sub>N</sub>'' {{=}} ''k''.}}
| |
| | |
| The central ''k''-order central moments are given as follows
| |
| | |
| (a) If ''k'' is odd, {{nowrap|''μ''<sub>1, …, ''N''</sub>('''x''' − '''μ''') {{=}} 0}}.
| |
| | |
| (b) If ''k'' is even with {{nowrap|''k'' {{=}} 2''λ''}}, then
| |
| | |
| :<math>
| |
| \mu _{1,\dots,2\lambda }(\mathbf{x}-\boldsymbol\mu )=\sum \left( \Sigma _{ij}\Sigma _{k\ell}\cdots\Sigma _{XZ}\right)
| |
| </math>
| |
| | |
| where the sum is taken over all allocations of the set <math>\left\{ 1,\dots,2\lambda
| |
| \right\}</math> into ''λ'' (unordered) pairs. That is, if you have a ''k''th ({{nowrap| {{=}} 2''λ'' {{=}} 6}}) central moment, you will be summing the products of {{nowrap|''λ'' {{=}} 3}} covariances (the -'''μ''' notation has been dropped in the interests of parsimony):
| |
| | |
| :<math>\begin{align}
| |
| & {} E[x_1 x_2 x_3 x_4 x_5 x_6] \\
| |
| &{} = E[x_1 x_2 ]E[x_3 x_4 ]E[x_5 x_6 ] + E[x_1 x_2 ]E[x_3 x_5 ]E[x_4 x_6] + E[x_1 x_2 ]E[x_3 x_6 ]E[x_4 x_5] \\
| |
| &{} + E[x_1 x_3 ]E[x_2 x_4 ]E[x_5 x_6 ] + E[x_1 x_3 ]E[x_2 x_5 ]E[x_4 x_6 ] + E[x_1 x_3]E[x_2 x_6]E[x_4 x_5] \\
| |
| &+ E[x_1 x_4]E[x_2 x_3]E[x_5 x_6]+E[x_1 x_4]E[x_2 x_5]E[x_3 x_6]+E[x_1 x_4]E[x_2 x_6]E[x_3 x_5] \\
| |
| & + E[x_1 x_5]E[x_2 x_3]E[x_4 x_6]+E[x_1 x_5]E[x_2 x_4]E[x_3 x_6]+E[x_1 x_5]E[x_2 x_6]E[x_3 x_4] \\
| |
| & + E[x_1 x_6]E[x_2 x_3]E[x_4 x_5 ] + E[x_1 x_6]E[x_2 x_4 ]E[x_3 x_5] + E[x_1 x_6]E[x_2 x_5]E[x_3 x_4].
| |
| \end{align}</math>
| |
| | |
| This yields <math>(2\lambda -1)!/(2^{\lambda -1}(\lambda -1)!)</math> 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 <math>\left[ 1,\dots,2\lambda \right]</math> by the corresponding terms of the list consisting of ''r''<sub>1</sub> ones, then ''r''<sub>2</sub> twos, etc.. To illustrate this, examine the following 4th-order central moment case:
| |
| | |
| :<math>E\left[ x_i^4\right] = 3\Sigma _{ii}^2</math>
| |
| :<math>E\left[ x_i^3 x_j\right] = 3\Sigma _{ii} \Sigma _{ij}</math>
| |
| :<math>E\left[ x_i^2 x_j^2\right] = \Sigma _{ii}\Sigma_{jj}+2\left( \Sigma _{ij}\right) ^2</math>
| |
| :<math>E\left[ x_i^2x_jx_k\right] = \Sigma _{ii}\Sigma _{jk}+2\Sigma _{ij}\Sigma _{ik}</math>
| |
| :<math>E\left[ x_i x_j x_k x_n\right] = \Sigma _{ij}\Sigma _{kn}+\Sigma _{ik}\Sigma _{jn}+\Sigma _{in}\Sigma _{jk}.
| |
| </math>
| |
| | |
| where <math>\Sigma_{ij}</math> is the covariance of ''x<sub>i</sub>'' and ''x<sub>j</sub>''. The idea with the above method is you first find the general case for a ''k''th moment where you have ''k'' different ''x'' variables - <math>E\left[ x_i x_j x_k x_n\right]</math> and then you can simplify this accordingly. Say, you have <math>E\left[ x_i^2 x_k x_n\right]</math> then you simply let {{nowrap|''x<sub>i</sub>'' {{=}} ''x<sub>j</sub>''}} and realise that {{nowrap|''<math>\Sigma _{ii}</math>'' {{=}} ''σ<sub>i</sub>''<sup>2</sup>}}.
| |
| | |
| ===Likelihood function===
| |
| | |
| If the mean and variance matrix are unknown, a suitable log likelihood function for a single observation '''x''' would be:{{citation needed|date=March 2012}}
| |
| | |
| :<math>\ln(L)= -\frac{1}{2} \ln (|\boldsymbol\Sigma|\,) -\frac{1}{2}(\mathbf{x}-\boldsymbol\mu)^{\rm T}\boldsymbol\Sigma^{-1}(\mathbf{x}-\boldsymbol\mu) - \frac{k}{2}\ln(2\pi)</math>
| |
| | |
| where ''x'' is a vector of real numbers. The complex case, where ''z'' is a vector of complex numbers, would be
| |
| | |
| :<math>\ln(L) = -\frac{1}{2}\ln (|\boldsymbol\Sigma|\,) -\frac{1}{2}(\mathbf{z}-\boldsymbol\mu)^\dagger\boldsymbol\Sigma^{-1}(\mathbf{z}-\boldsymbol\mu) - \frac{k}{2}\ln(2\pi)</math>
| |
| | |
| i.e. with the [[conjugate transpose]] (indicated by <math>\dagger</math>) replacing the normal [[transpose]] (indicated by <math>{}^{\rm T}</math>). A similar notation is used for [[multiple linear regression]].<ref>Tong, T. (2010) [http://amath.colorado.edu/courses/7400/2010Spr/lecture9.pdf Multiple Linear Regression : MLE and Its Distributional Results], Lecture Notes</ref>
| |
| | |
| ===Entropy===
| |
| | |
| The [[differential entropy]] of the multivariate normal distribution is<ref>{{cite journal
| |
| | last1 = Gokhale | first1 = DV | authorlink1=
| |
| | last2 = Ahmed | first2 = NA
| |
| | last3 = Res |first3=BC
| |
| | last4 = Piscataway |first4=NJ
| |
| | date = May 1989
| |
| | title = Entropy Expressions and Their Estimators for Multivariate Distributions
| |
| | journal = Information Theory, IEEE Transactions on
| |
| | volume = 35 | issue = 3 | pages = 688–692
| |
| | doi =10.1109/18.30996
| |
| }}</ref>
| |
| | |
| :<math>
| |
| \begin{align}
| |
| h\left(f\right) & = -\int_{-\infty}^\infty \int_{-\infty}^\infty \cdots\int_{-\infty}^\infty f(\mathbf{x}) \ln f(\mathbf{x})\,d\mathbf{x},\\
| |
| & = \frac12 \ln\left|(2\pi e) \boldsymbol\Sigma \right|,\\
| |
| \end{align}
| |
| </math>
| |
| where the bars denote the [[determinant|matrix determinant]].
| |
| | |
| ===Kullback–Leibler divergence===
| |
| The [[Kullback–Leibler divergence]] from <math>\mathcal{N}_0(\boldsymbol\mu_0, \boldsymbol\Sigma_0)</math> to <math>\mathcal{N}_1(\boldsymbol\mu_1, \boldsymbol\Sigma_1)</math>, for non-singular matrices Σ<sub>0</sub> and Σ<sub>1</sub>, is:<ref>Penny & Roberts, PARG-00-12, (2000) [http://www.allisons.org/ll/MML/KL/Normal]. pp. 18</ref>
| |
| | |
| :<math>
| |
| D_\text{KL}(\mathcal{N}_0 \| \mathcal{N}_1) = { 1 \over 2 } \left\{ \mathrm{tr} \left( \boldsymbol\Sigma_1^{-1} \boldsymbol\Sigma_0 \right) + \left( \boldsymbol\mu_1 - \boldsymbol\mu_0\right)^{\rm T} \boldsymbol\Sigma_1^{-1} ( \boldsymbol\mu_1 - \boldsymbol\mu_0 ) - K -\ln { | \boldsymbol \Sigma_0 | \over | \boldsymbol\Sigma_1 | } \right\},
| |
| </math>
| |
| where <math>K</math> is the dimension of the vector space.
| |
| | |
| The [[logarithm]] must be taken to base ''[[e (mathematical constant)|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 [[nat (information)|nats]]. Dividing the entire expression above by log<sub>''e''</sub> 2 yields the divergence in [[bit]]s.
| |
| | |
| === Cumulative distribution function ===
| |
| | |
| The notion of [[cumulative distribution function]] (cdf) in dimension 1 can be extended in two ways to the multidimensional case.
| |
| The first way is to define the cumulative distribution function <math>F(r)</math> as the probability that a sample '''falls''' inside the ellipsoid determined by its [[Mahalanobis distance]] <math>r</math> from the Gaussian, a direct generalization of the standard deviation
| |
| .<ref name=Bensimhoun>[https://upload.wikimedia.org/wikipedia/commons/a/a2/Cumulative_function_n_dimensional_Gaussians_12.2013.pdf Bensimhoun Michael, ''N-Dimensional Cumulative Function, And Other Useful Facts About Gaussians and Normal Densities'' (2006)]</ref>
| |
| In order to compute the values of this function, closed analytic formulae exist.<ref name="Bensimhoun"/>
| |
|
| |
| Another way to extend the notion of cumulative distribution function is to define
| |
| the [[cumulative distribution function]] (cdf) ''F''('''x'''<sub>0</sub>) of a random vector '''x''' as the probability that all components of '''x''' are less than or equal to the corresponding values in the vector '''x'''<sub>0</sub>. Though there is no closed form for ''F''('''x'''), there are a number of algorithms that estimate it numerically.{{Citation needed|reason=Existence of algorithms not obvious but easily demonstrated with a citation|date=January 2014}}
| |
| | |
| ===Prediction Interval===
| |
| | |
| The [[prediction interval]] for the multivariate normal distribution yields a region consisting of those vectors '''x''' satisfying
| |
| | |
| :<math>({\mathbf x}-{\boldsymbol\mu})^T{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu}) \leq \chi^2_k(p).</math>
| |
| | |
| Here <math>{\mathbf x}</math> is a <math>k</math>-dimensional vector, <math>{\boldsymbol\mu}</math> is the known <math>k</math>-dimensional mean vector, <math>\boldsymbol\Sigma</math> is the known [[covariance matrix]] and <math>\chi^2_k(p)</math> is the [[quantile function]] for probability <math>p</math> of the [[chi-squared distribution]] with <math>k</math> degrees of freedom.<ref name=Siotani/>
| |
| | |
| When <math>k = 2,</math> the expression defines the interior of an ellipse and the chi-squared distribution simplifies to an [[exponential distribution]] with mean equal to two.
| |
| | |
| ==Joint normality==
| |
| | |
| ===Normally distributed and independent===
| |
| | |
| If ''X'' and ''Y'' are normally distributed and [[statistical independence|independent]], this implies they are "jointly normally distributed", i.e., the pair (''X'', ''Y'') must have multivariate normal distribution. However, a pair of jointly normally distributed variables need not be independent (would only be so if uncorrelated, <math> \rho = 0</math> ).
| |
| | |
| ===Two normally distributed random variables need not be jointly bivariate normal===
| |
| {{See also|normally distributed and uncorrelated does not imply independent}}
| |
| The fact that two random variables ''X'' and ''Y'' both have a normal distribution does not imply that the pair (''X'', ''Y'') 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 ''Y'' = ''X'' if |''X''| > ''c'' and ''Y'' = −''X'' if |''X''| < ''c'', where ''c'' > 0. There are similar counterexamples for more than two random variables. In general, they sum to a [[mixture model]].
| |
| | |
| ===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 [[statistical independence|independent]]. This implies that any two or more of its components that are [[pairwise independence|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. In the preceding example, clearly ''X'' and ''Y'' are not independent, yet [[normally distributed and uncorrelated does not imply independent|choosing ''c'' to be 1.54]] makes them uncorrelated.
| |
| | |
| ==Conditional distributions==
| |
| | |
| If '''μ''' and '''Σ''' are partitioned as follows
| |
| | |
| :<math>
| |
| \boldsymbol\mu
| |
| =
| |
| \begin{bmatrix}
| |
| \boldsymbol\mu_1 \\
| |
| \boldsymbol\mu_2
| |
| \end{bmatrix}
| |
| \text{ with sizes }\begin{bmatrix} q \times 1 \\ (N-q) \times 1 \end{bmatrix}</math>
| |
| | |
| :<math>
| |
| \boldsymbol\Sigma
| |
| =
| |
| \begin{bmatrix}
| |
| \boldsymbol\Sigma_{11} & \boldsymbol\Sigma_{12} \\
| |
| \boldsymbol\Sigma_{21} & \boldsymbol\Sigma_{22}
| |
| \end{bmatrix}
| |
| \text{ with sizes }\begin{bmatrix} q \times q & q \times (N-q) \\ (N-q) \times q & (N-q) \times (N-q) \end{bmatrix}</math>
| |
| | |
| then, the distribution of '''x'''<sub>1</sub> conditional on '''x'''<sub>2</sub> = ''a'' is multivariate normal {{nowrap|('''x'''<sub>1</sub>{{!}}'''x'''<sub>2</sub> {{=}} '''a''') ~ ''N''(<span style{{=}}"text-decoration:overline;">'''μ'''</span>, <span style{{=}}"text-decoration:overline;">'''Σ'''</span>)}} where
| |
| | |
| :<math>
| |
| \bar{\boldsymbol\mu}
| |
| =
| |
| \boldsymbol\mu_1 + \boldsymbol\Sigma_{12} \boldsymbol\Sigma_{22}^{-1}
| |
| \left(
| |
| \mathbf{a} - \boldsymbol\mu_2
| |
| \right)
| |
| </math>
| |
| | |
| and covariance matrix
| |
| | |
| :<math>
| |
| \overline{\boldsymbol\Sigma}
| |
| =
| |
| \boldsymbol\Sigma_{11} - \boldsymbol\Sigma_{12} \boldsymbol\Sigma_{22}^{-1} \boldsymbol\Sigma_{21}.
| |
| </math><ref name=eaton>{{cite book|last=Eaton|first=Morris L.|title=Multivariate Statistics: a Vector Space Approach|year=1983|publisher=John Wiley and Sons|isbn=0-471-02776-6|pages=116–117}}</ref>
| |
| | |
| This matrix is the [[Schur complement]] of '''Σ'''<sub>22</sub> in '''Σ'''. This means that to calculate the conditional covariance matrix, one inverts the overall covariance matrix, drops the rows and columns corresponding to the variables being conditioned upon, and then inverts back to get the conditional covariance matrix. Here <math>\boldsymbol\Sigma_{22}^{-1}</math> is the [[generalized inverse]] of <math>\boldsymbol\Sigma_{22}</math>.
| |
| | |
| Note that knowing that {{nowrap|'''x'''<sub>2</sub> {{=}} '''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 <math>\boldsymbol\Sigma_{12} \boldsymbol\Sigma_{22}^{-1} \left(\mathbf{a} - \boldsymbol\mu_2 \right)</math>; compare this with the situation of not knowing the value of '''a''', in which case '''x'''<sub>1</sub> would have distribution
| |
| <math>\mathcal{N}_q \left(\boldsymbol\mu_1, \boldsymbol\Sigma_{11} \right)</math>.
| |
| | |
| An interesting fact derived in order to prove this result, is that the random vectors <math>\mathbf{x}_2</math> and <math>\mathbf{y}_1=\mathbf{x}_1-\boldsymbol\Sigma_{12}\boldsymbol\Sigma_{22}^{-1}\mathbf{x}_2</math> are independent.
| |
| | |
| The matrix '''Σ'''<sub>12</sub>'''Σ'''<sub>22</sub><sup>−1</sup> is known as the matrix of [[regression analysis|regression]] coefficients.
| |
| | |
| === Bivariate case ===
| |
| In the bivariate case where '''x''' is partitioned into ''X''<sub>1</sub> and ''X''<sub>2</sub>, the conditional distribution of ''X''<sub>1</sub> given ''X''<sub>2</sub> is<ref>{{cite book|last=Jensen|first=J|title=Statistics for Petroleum Engineers and Geoscientists|year=2000|publisher=Elsevier|location=Amsterdam|pages=207}}</ref>
| |
| | |
| : <math>X_1|X_2=x_2 \ \sim\ \mathcal{N}\left(\mu_1+\frac{\sigma_1}{\sigma_2}\rho( x_2 - \mu_2),\, (1-\rho^2)\sigma_1^2\right). </math>
| |
| | |
| where <math>\rho</math> is the [[Pearson product-moment correlation coefficient|correlation coefficient]] between ''X''<sub>1</sub> and ''X''<sub>2</sub>.
| |
| | |
| === Bivariate conditional expectation ===
| |
| | |
| ====In the general case====
| |
| | |
| :<math>
| |
| \begin{pmatrix}
| |
| X_1 \\
| |
| X_2
| |
| \end{pmatrix} \sim \mathcal{N} \left( \begin{pmatrix}
| |
| \mu_1 \\
| |
| \mu_2
| |
| \end{pmatrix} , \begin{pmatrix}
| |
| \sigma^2_1 & \sigma_{12} \\
| |
| \sigma_{12} & \sigma^2_2
| |
| \end{pmatrix} \right)
| |
| </math>
| |
| | |
| The conditional expectation of X<sub>1</sub> given X<sub>2</sub> is:
| |
| | |
| <math>\operatorname{E}(X_1 | X_2=x_2) = \mu_1 + \rho \frac{\sigma_1}{\sigma_2}(x_2 - \mu_2)</math>
| |
| | |
| Proof: the result is simply obtained taking the expectation of the conditional distribution <math>X_1|X_2</math> above.
| |
| | |
| ====In the standard normal case====
| |
| | |
| :<math>
| |
| \begin{pmatrix}
| |
| X_1 \\
| |
| X_2
| |
| \end{pmatrix} \sim \mathcal{N} \left( \begin{pmatrix}
| |
| 0 \\
| |
| 0
| |
| \end{pmatrix} , \begin{pmatrix}
| |
| 1 & \rho \\
| |
| \rho & 1
| |
| \end{pmatrix} \right)
| |
| </math>
| |
| | |
| The conditional expectation of X<sub>1</sub> given X<sub>2</sub> is:
| |
| | |
| <math>\operatorname{E}(X_1 | X_2=x_2)= \rho x_2 </math>
| |
| | |
| and the conditional expectation of X<sub>1</sub> given that X<sub>2</sub> is smaller/bigger than z is (Maddala 1983, p. 367<ref name=Maddala83>{{cite book|last=Gangadharrao|first=Maddala|title=Limited Dependent and Qualitative Variables in Econometrics|year=1983|publisher=Cambridge University Press}}</ref>) :
| |
| | |
| :<math>
| |
| \operatorname{E}(X_1 | X_2 < z) = -\rho { \phi(z) \over \Phi(z) } ,
| |
| </math>
| |
| | |
| :<math>
| |
| \operatorname{E}(X_1 | X_2 > z) = \rho { \phi(z) \over (1- \Phi(z)) } ,
| |
| </math>
| |
| | |
| where the final ratio here is called the [[inverse Mills ratio]].
| |
| | |
| Proof: the last two results are obtained using the result <math>\operatorname{E}(X_1 | X_2=x_2)= \rho x_2 </math>, so that :<math>
| |
| \operatorname{E}(X_1 | X_2 < z) = \rho (X_2 | X_2 < z)</math> and then using the properties of the expectation of a [[truncated normal distribution]].
| |
| | |
| ==Marginal distributions==
| |
| 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.<ref>The formal proof for marginal distribution is shown here http://fourier.eng.hmc.edu/e161/lectures/gaussianprocess/node7.html</ref>
| |
| | |
| ''Example''
| |
| | |
| Let {{nowrap|'''x''' {{=}} [''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>]}} be multivariate normal random variables with mean vector {{nowrap|'''μ''' {{=}} [''μ''<sub>1</sub>, ''μ''<sub>2</sub>, ''μ''<sub>3</sub>]}} and covariance matrix '''Σ''' (standard parametrization for multivariate normal distributions). Then the joint distribution of {{nowrap|'''x′''' {{=}} [''X''<sub>1</sub>, ''X''<sub>3</sub>]}} is multivariate normal with mean vector {{nowrap|'''μ′''' {{=}} [''μ''<sub>1</sub>, ''μ''<sub>3</sub>]}} and covariance matrix
| |
| <math> \boldsymbol\Sigma' =
| |
| \begin{bmatrix}
| |
| \boldsymbol\Sigma_{11} & \boldsymbol\Sigma_{13} \\
| |
| \boldsymbol\Sigma_{31} & \boldsymbol\Sigma_{33}
| |
| \end{bmatrix}
| |
| </math>.
| |
| | |
| ==Affine transformation==
| |
| | |
| If {{nowrap|'''y''' {{=}} '''c''' + '''Bx'''}} is an [[affine transformation]] of <math>\mathbf{x}\ \sim \mathcal{N}(\boldsymbol\mu, \boldsymbol\Sigma),</math> where '''c''' is an <math>M \times 1</math> vector of constants and '''B''' is a constant <math>M \times N</math> matrix, then '''y''' has a multivariate normal distribution with expected value {{nowrap|'''c''' + '''Bμ'''}} and variance '''BΣB'''<sup>T</sup> i.e., <math>\mathbf{y} \sim \mathcal{N} \left(\mathbf{c} + \mathbf{B} \boldsymbol\mu, \mathbf{B} \boldsymbol\Sigma \mathbf{B}^{\rm T}\right)</math>. In particular, any subset of the ''x<sub>i</sub>'' has a marginal distribution that is also multivariate normal.
| |
| To see this, consider the following example: to extract the subset (''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>4</sub>)<sup>T</sup>, use
| |
| | |
| :<math>
| |
| \mathbf{B}
| |
| =
| |
| \begin{bmatrix}
| |
| 1 & 0 & 0 & 0 & 0 & \ldots & 0 \\
| |
| 0 & 1 & 0 & 0 & 0 & \ldots & 0 \\
| |
| 0 & 0 & 0 & 1 & 0 & \ldots & 0
| |
| \end{bmatrix}
| |
| </math>
| |
| | |
| which extracts the desired elements directly.
| |
| | |
| Another corollary is that the distribution of {{nowrap|'''Z''' {{=}} '''b''' · '''x'''}}, where '''b''' is a constant vector of the same length as '''x''' and the dot indicates a vector product, is univariate Gaussian with <math>Z\sim\mathcal{N}\left(\mathbf{b}\cdot\boldsymbol\mu, \mathbf{b}^{\rm T}\boldsymbol\Sigma \mathbf{b}\right)</math>. This result follows by using
| |
| | |
| :<math>
| |
| \mathbf{B}=\begin{bmatrix}
| |
| b_1 & b_2 & \ldots & b_n
| |
| \end{bmatrix}.
| |
| </math>
| |
| 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 2'''x''' is not the same as the [[Sum of normally distributed random variables|sum of two independent realisations]] of '''x'''.
| |
| | |
| ==Geometric interpretation==
| |
| | |
| The equidensity contours of a non-singular multivariate normal distribution are [[ellipsoid]]s (i.e. linear transformations of [[hypersphere]]s) centered at the mean.<ref>{{cite web|author=Nikolaus Hansen|title=The CMA Evolution Strategy: A Tutorial|url=http://www.lri.fr/~hansen/cmatutorial.pdf|format=PDF}}</ref> Hence the multivariate normal distribution is an example of the class of [[elliptical distribution]]s. The directions of the principal axes of the ellipsoids are given by the eigenvectors of the covariance matrix '''Σ'''. The squared relative lengths of the principal axes are given by the corresponding eigenvalues.
| |
| | |
| If {{nowrap|'''Σ''' {{=}} '''UΛU'''<sup>T</sup> {{=}} '''UΛ'''<sup>1/2</sup>('''UΛ'''<sup>1/2</sup>)<sup>T</sup>}} is an [[eigendecomposition]] where the columns of '''U''' are unit eigenvectors and '''Λ''' is a [[diagonal matrix]] of the eigenvalues, then we have
| |
| | |
| ::<math>\mathbf{x}\ \sim \mathcal{N}(\boldsymbol\mu, \boldsymbol\Sigma) \iff \mathbf{x}\ \sim \boldsymbol\mu+\mathbf{U}\boldsymbol\Lambda^{1/2}\mathcal{N}(0, \mathbf{I}) \iff \mathbf{x}\ \sim \boldsymbol\mu+\mathbf{U}\mathcal{N}(0, \boldsymbol\Lambda).</math>
| |
| | |
| 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'''<sup>1/2</sup>, rotated by '''U''' and translated by '''μ'''.
| |
| | |
| Conversely, any choice of '''μ''', full rank matrix '''U''', and positive diagonal entries Λ<sub>''i''</sub> yields a non-singular multivariate normal distribution. If any Λ<sub>''i''</sub> is zero and '''U''' is square, the resulting covariance matrix '''UΛU'''<sup>T</sup> is [[singular matrix|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.
| |
| | |
| ==Estimation of parameters==
| |
| | |
| The derivation of the [[maximum likelihood|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 a multivariate normal is
| |
| | |
| :<math>f(\mathbf{x})= \frac{1}{\sqrt { (2\pi)^k|\boldsymbol \Sigma| } } \exp\left(-{1 \over 2} (\mathbf{x}-\boldsymbol\mu)^{\rm T} \boldsymbol\Sigma^{-1} ({\mathbf x}-\boldsymbol\mu)\right)</math>
| |
| | |
| and the ML estimator of the covariance matrix from a sample of ''n'' observations is
| |
| | |
| :<math>\widehat{\boldsymbol\Sigma} = {1 \over n}\sum_{i=1}^n ({\mathbf x}_i-\overline{\mathbf x})({\mathbf x}_i-\overline{\mathbf x})^T</math>
| |
| | |
| which is simply the [[sample covariance matrix]]. This is a [[biased estimator]] whose expectation is
| |
| | |
| :<math>E[\widehat{\boldsymbol\Sigma}] = \frac{n-1}{n} \boldsymbol\Sigma.</math>
| |
| | |
| An unbiased sample covariance is
| |
| | |
| :<math>\widehat{\boldsymbol\Sigma} = {1 \over n-1}\sum_{i=1}^n (\mathbf{x}_i-\overline{\mathbf{x}})(\mathbf{x}_i-\overline{\mathbf{x}})^{\rm T}.</math>
| |
| | |
| 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#Multivariate normal distribution|Fisher information]] for more details.
| |
| | |
| ==Bayesian inference==
| |
| 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]] <math>\mathcal{W}^{-1}</math> . Suppose then that ''n'' observations have been made
| |
| :<math>\mathbf{X} = \{\mathbf{x}_1,\dots,\mathbf{x}_n\} \sim \mathcal{N}(\boldsymbol\mu,\boldsymbol\Sigma)</math>
| |
| and that a conjugate prior has been assigned, where
| |
| :<math>p(\boldsymbol\mu,\boldsymbol\Sigma)=p(\boldsymbol\mu\mid\boldsymbol\Sigma)\ p(\boldsymbol\Sigma),</math>
| |
| where
| |
| :<math>p(\boldsymbol\mu\mid\boldsymbol\Sigma) \sim\mathcal{N}(\boldsymbol\mu_0,m^{-1}\boldsymbol\Sigma) ,</math>
| |
| and
| |
| :<math>p(\boldsymbol\Sigma) \sim \mathcal{W}^{-1}(\boldsymbol\Psi,n_0).</math>
| |
| | |
| Then,{{citation needed|date=July 2012}}
| |
| | |
| :<math>
| |
| \begin{array}{rcl}
| |
| p(\boldsymbol\mu\mid\boldsymbol\Sigma,\mathbf{X}) & \sim & \mathcal{N}\left(\frac{n\bar{\mathbf{x}} + m\boldsymbol\mu_0}{n+m},\frac{1}{n+m}\boldsymbol\Sigma\right),\\
| |
| p(\boldsymbol\Sigma\mid\mathbf{X}) & \sim & \mathcal{W}^{-1}\left(\boldsymbol\Psi+n\mathbf{S}+\frac{nm}{n+m}(\bar{\mathbf{x}}-\boldsymbol\mu_0)(\bar{\mathbf{x}}-\boldsymbol\mu_0)', n+n_0\right),
| |
| \end{array}
| |
| </math>
| |
| where
| |
| :<math>
| |
| \begin{array}{rcl}
| |
| \bar{\mathbf{x}} & = & n^{-1}\sum_{i=1}^{n} \mathbf{x}_i ,\\
| |
| \mathbf{S} & = & n^{-1}\sum_{i=1}^{n} (\mathbf{x}_i - \bar{\mathbf{x}})(\mathbf{x}_i - \bar{\mathbf{x}})' .
| |
| \end{array}
| |
| </math>
| |
| | |
| == 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|''p''-value]] indicates non-normal data. Multivariate normality tests include the Cox-Small test<ref>{{cite doi | 10.1093/biomet/65.2.263}}</ref>
| |
| and Smith and Jain's adaptation<ref>{{cite doi | 10.1109/34.6789}}</ref> of the Friedman-Rafsky test.<ref>{{cite doi|10.1214/aos/1176344722}}</ref>
| |
| | |
| '''Mardia's test'''<ref name=Mardia/> is based on multivariate extensions of [[skewness]] and [[kurtosis]] measures. For a sample {'''x'''<sub>1</sub>, ..., '''x'''<sub>''n''</sub>} of ''k''-dimensional vectors we compute
| |
| : <math>\begin{align}
| |
| & \widehat{\boldsymbol\Sigma} = {1 \over n} \sum_{j=1}^n \left(\mathbf{x}_j - \bar{\mathbf{x}}\right)\left(\mathbf{x}_j - \bar{\mathbf{x}}\right)^T \\
| |
| | |
| & A = {1 \over 6n} \sum_{i=1}^n \sum_{j=1}^n \left[ (\mathbf{x}_i - \bar{\mathbf{x}})^T\;\widehat{\boldsymbol\Sigma}^{-1} (\mathbf{x}_j - \bar{\mathbf{x}}) \right]^3 \\
| |
| | |
| & B = \sqrt{\frac{n}{8k(k+2)}}\left\{{1 \over n} \sum_{i=1}^n \left[ (\mathbf{x}_i - \bar{\mathbf{x}})^T\;\widehat{\boldsymbol\Sigma}^{-1} (\mathbf{x}_i - \bar{\mathbf{x}}) \right]^2 - k(k+2) \right\}
| |
| \end{align}</math>
| |
| Under the null hypothesis of multivariate normality, the statistic ''A'' will have approximately a [[chi-squared distribution]] with {{nowrap|{{frac2|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 <math>(50 \le n < 400)</math>, the parameters of the asymptotic distribution of the kurtosis statistic are modified<ref>Rencher (1995), pages 112-113.</ref> For small sample tests (<math>n<50</math>) empirical critical values are used. Tables of critical values for both statistics are given by Rencher<ref>Rencher (1995), pages 493-495.</ref> 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.<ref>{{cite doi|10.1016/0047-259X(91)90031-V}}</ref>
| |
| | |
| The '''BHEP test'''<ref name=EP/> computes the norm of the difference between the empirical [[characteristic function (probability theory)|characteristic function]] and the theoretical characteristic function of the normal distribution. Calculation of the norm is performed in the [[Lp space|L<sup>2</sup>(''μ'')]] space of square-integrable functions with respect to the Gaussian weighting function <math>\scriptstyle \mu_\beta(\mathbf{t}) = (2\pi\beta^2)^{-k/2} e^{-|\mathbf{t}|^2/(2\beta^2)}</math>. The test statistic is
| |
| : <math>\begin{align}
| |
| T_\beta &= \int_{\mathbb{R}^k} \left| {1 \over n} \sum_{j=1}^n e^{i\mathbf{t}^T\widehat{\boldsymbol\Sigma}^{-1/2}(\mathbf{x}_j - \bar{\mathbf{x})}} - e^{-|\mathbf{t}|^2/2} \right|^2 \; \boldsymbol\mu_\beta(\mathbf{t}) d\mathbf{t} \\
| |
| &= {1 \over n^2} \sum_{i,j=1}^n e^{-{\beta \over 2}(\mathbf{x}_i-\mathbf{x}_j)^T\widehat{\boldsymbol\Sigma}^{-1}(\mathbf{x}_i-\mathbf{x}_j)} - \frac{2}{n(1 + \beta^2)^{k/2}}\sum_{i=1}^n e^{ -\frac{\beta^2}{2(1+\beta^2)} (\mathbf{x}_i-\bar{\mathbf{x}})^T\widehat{\boldsymbol\Sigma}^{-1}(\mathbf{x}_i-\bar{\mathbf{x}})} + \frac{1}{(1 + 2\beta^2)^{k/2}}
| |
| \end{align}</math>
| |
| The limiting distribution of this test statistic is a weighted sum of chi-squared random variables,<ref name=BH/> however in practice it is more convenient to compute the sample quantiles using the Monte-Carlo simulations.{{citation needed|date=July 2012}}
| |
| | |
| A detailed survey of these and other test procedures is available.<ref name=Henze/>
| |
| | |
| ==Drawing values from the distribution==
| |
| | |
| A widely used method for drawing a random vector '''x''' from the ''N''-dimensional multivariate normal distribution with mean vector '''μ''' and [[covariance matrix]] '''Σ''' works as follows:<ref name=Gentle/>
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| # Find any real matrix '''A''' such that {{nowrap|'''A''' '''A'''<sup>T</sup> {{=}} '''Σ'''}}. When '''Σ''' is positive-definite, the [[Cholesky decomposition]] is typically used, and the [[Cholesky decomposition#Avoiding taking square roots|extended form]] of this decomposition can always be used (as the covariance matrix may be only positive semi-definite) in both cases a suitable matrix '''A''' is obtained.{{citation needed|date=July 2012}} An alternative is to use the matrix '''A''' = '''UΛ'''<sup>½</sup> obtained from a [[Eigendecomposition of a matrix#Symmetric matrices|spectral decomposition]] '''Σ''' = '''UΛU'''<sup>T</sup> of '''Σ'''.{{citation needed|date=July 2012}} The former approach is more computationally straightforward but the matrices '''A''' change for different orderings of the elements of the random vector, while the latter approach gives matrices that are related by simple re-orderings. In theory both approaches give equally good ways of determining a suitable matrix '''A''', but there are differences in computation time.
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| # Let {{nowrap|'''z''' {{=}} (''z''<sub>1</sub>, …, ''z<sub>N</sub>'')<sup>T</sup>}} be a vector whose components are ''N'' [[statistical independence|independent]] [[normal distribution|standard normal]] variates (which can be generated, for example, by using the [[Box–Muller transform]]).
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| # Let '''x''' be {{nowrap|'''μ''' + '''Az'''}}. This has the desired distribution due to the affine transformation property.
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| == See also ==
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| * [[Chi distribution]], the [[probability density function|pdf]] of the [[Norm (mathematics)#p-norm|2-norm]] (or [[Euclidean norm]]) of a multivariate normally distributed vector (centered at zero).
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| * [[Complex normal distribution]], for the generalization to complex valued random variables.
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| * [[Multivariate stable distribution]] extension of the multivariate normal distribution, when the index (exponent in the characteristic function) is between zero to two.
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| * [[Mahalanobis distance]]
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| * [[Wishart distribution]]
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| == References ==
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| {{Reflist|refs=
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| <ref name = Siotani>{{cite journal
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| | author = Siotani, Minoru
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| | title = Tolerance regions for a multivariate normal population
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| | journal = Annals of the Institute of Statistical Mathematics
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| | year = 1964
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| | volume = 16
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| | number = 1
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| | pages = 135–153
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| | doi = 10.1007/BF02868568
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| | url = http://www.ism.ac.jp/editsec/aism/pdf/016_1_0135.pdf
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| }}</ref>
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| <ref name=Mardia>{{cite journal
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| | last = Mardia | first = K. V.
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| | year = 1970
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| | title = Measures of multivariate skewness and kurtosis with applications
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| | journal = Biometrika
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| | volume = 57 | issue = 3 | pages = 519–530
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| | doi = 10.1093/biomet/57.3.519
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| }}</ref>
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| <ref name=EP>{{cite journal
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| | last1 = Epps | first1 = Lawrence B.
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| | last2 = Pulley | first2 = Lawrence B.
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| | year = 1983
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| | title = A test for normality based on the empirical characteristic function
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| | journal = Biometrika
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| | volume = 70 | issue = 3 | pages = 723–726
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| | doi = 10.1093/biomet/70.3.723
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| }}</ref>
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| <ref name=BH>{{cite journal
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| | last1 = Baringhaus | first1 = L.
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| | last2 = Henze | first2 = N.
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| | year = 1988
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| | title = A consistent test for multivariate normality based on the empirical characteristic function
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| | journal = Metrika
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| | volume = 35 | issue = 1 | pages = 339–348
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| | doi = 10.1007/BF02613322
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| }}</ref>
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| <ref name=Henze>{{cite journal
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| | last = Henze | first = Norbert
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| | year = 2002
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| | title = Invariant tests for multivariate normality: a critical review
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| | journal = Statistical Papers
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| | volume = 43 | issue = 4 | pages = 467–506
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| | doi = 10.1007/s00362-002-0119-6
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| }}</ref>
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| <ref name=HT>{{cite journal
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| | last1 = Hamedani | first1 = G. G.
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| | last2 = Tata | first2 = M. N.
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| | year = 1975
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| | title = On the determination of the bivariate normal distribution from distributions of linear combinations of the variables
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| | journal = The American Mathematical Monthly
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| | volume = 82 | issue = 9 | pages = 913–915
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| | doi = 10.2307/2318494
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| }}</ref>
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| <ref name=wyattlms>{{cite web|last=Wyatt|first=John
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| |title=Linear least mean-squared error estimation
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| |url=http://web.mit.edu/6.041/www/LECTURE/lec22.pdf
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| |work=Lecture notes course on applied probability|accessdate=23 January 2012}}</ref>
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| <ref name=rao>{{cite book
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| | author = Rao, C.R.
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| | title = Linear Statistical Inference and Its Applications
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| |year = 1973
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| |publisher = Wiley
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| |location = New York
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| | pages = 527–528
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| }}</ref>
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| <ref name=Gentle>{{cite book
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| | author = Gentle, J.E.
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| | title = Computational Statistics
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| |year = 2009
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| |publisher = Springer
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| |location = New York
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| | pages = 315–316
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| | doi = 10.1007/978-0-387-98144-4
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| }}</ref>
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| }}
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| === Literature ===
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| {{refbegin}}
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| * {{cite book
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| | author = Rencher, A.C.
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| | title = Methods of Multivariate Analysis
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| |year = 1995
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| |publisher = Wiley
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| |location = New York
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| }}
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| {{refend}}
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| {{ProbDistributions|multivariate}}
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| {{DEFAULTSORT:Multivariate Normal Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Multivariate continuous distributions]]
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| [[Category:Normal distribution]]
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| [[Category:Exponential family distributions]]
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| [[Category:Stable distributions]]
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| [[Category:Probability distributions]]
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