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Multivariate normal cumulative distribution function
y = mvncdf(X)
y = mvncdf(X,mu,SIGMA)
y = mvncdf(xl,xu,mu,SIGMA)
[y,err] = mvncdf(...)
[...] = mvncdf(...,options)
y = mvncdf(X) returns the cumulative probability of the multivariate normal distribution with zero mean and identity covariance matrix, evaluated at each row of X. Rows of the n-by-d matrix X correspond to observations or points, and columns correspond to variables or coordinates. y is an n-by-1 vector.
y = mvncdf(X,mu,SIGMA) returns the cumulative probability of the multivariate normal distribution with mean mu and covariance SIGMA, evaluated at each row of X. mu is a 1-by-d vector, and SIGMA is a d-by-d symmetric, positive definite matrix. mu can also be a scalar value, which mvncdf replicates to match the size of X. If the covariance matrix is diagonal, containing variances along the diagonal and zero covariances off the diagonal, SIGMA may also be specified as a 1-by-d vector containing just the diagonal. Pass in the empty matrix [] for mu to use as its default value when you want to only specify SIGMA.
The multivariate normal cumulative probability at X is defined as the probability that a random vector V, distributed as multivariate normal, will fall within the semi-infinite rectangle with upper limits defined by X, for example, Pr{V(1) ≤ X(1),V(2) ≤ X(2),...,V(d) ≤ X(d)}.
y = mvncdf(xl,xu,mu,SIGMA) returns the multivariate normal cumulative probability evaluated over the rectangle with lower and upper limits defined by xl and xu, respectively.
[y,err] = mvncdf(...) returns an estimate of the error in y. For bivariate and trivariate distributions, mvncdf uses adaptive quadrature on a transformation of the t density, based on methods developed by Drezner and Wesolowsky and by Genz, as described in the references. The default absolute error tolerance for these cases is 1e-8. For four or more dimensions, mvncdf uses a quasi-Monte Carlo integration algorithm based on methods developed by Genz and Bretz, as described in the references. The default absolute error tolerance for these cases is 1e-4.
[...] = mvncdf(...,options) specifies control parameters for the numerical integration used to compute y. This argument can be created by a call to statset. Choices of statset parameters:
'TolFun' — Maximum absolute error tolerance. Default is 1e-8 when d < 4, or 1e-4 when d ≥ 4.
'MaxFunEvals' — Maximum number of integrand evaluations allowed when d ≥ 4. Default is 1e7. 'MaxFunEvals' is ignored when d < 4.
'Display' — Level of display output. Choices are 'off' (the default), 'iter', and 'final'. 'Display' is ignored when d < 4.
[1] Drezner, Z. "Computation of the Trivariate Normal Integral." Mathematics of Computation. Vol. 63, 1994, pp. 289–294.
[2] Drezner, Z., and G. O. Wesolowsky. "On the Computation of the Bivariate Normal Integral." Journal of Statistical Computation and Simulation. Vol. 35, 1989, pp. 101–107.
[3] Genz, A. "Numerical Computation of Rectangular Bivariate and Trivariate Normal and t Probabilities." Statistics and Computing. Vol. 14, No. 3, 2004, pp. 251–260.
[4] Genz, A., and F. Bretz. "Numerical Computation of Multivariate t Probabilities with Application to Power Calculation of Multiple Contrasts." Journal of Statistical Computation and Simulation. Vol. 63, 1999, pp. 361–378.
[5] Genz, A., and F. Bretz. "Comparison of Methods for the Computation of Multivariate t Probabilities." Journal of Computational and Graphical Statistics. Vol. 11, No. 4, 2002, pp. 950–971.