naginterfaces.library.stat.prob_multi_normal¶
- naginterfaces.library.stat.prob_multi_normal(xmu, sig, a=None, b=None, tol=0.0001)[source]¶
prob_multi_normal
returns the upper tail, lower tail or central probability associated with a multivariate Normal distribution of up to ten dimensions.For full information please refer to the NAG Library document for g01hb
https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g01/g01hbf.html
- Parameters
- xmufloat, array-like, shape
, the mean vector of the multivariate Normal distribution.
- sigfloat, array-like, shape
, the variance-covariance matrix of the multivariate Normal distribution. Only the lower triangle is referenced.
- aNone or float, array-like, shape , optional
If upper tail or central probablilities are to be returned, should supply the lower bounds, , for .
- bNone or float, array-like, shape , optional
If lower tail or central probablilities are to be returned, should supply the upper bounds, , for .
- tolfloat, optional
If the relative accuracy required for the probability, and if the upper or the lower tail probability is requested then is also used to determine the cut-off points, see Accuracy.
If , is not referenced.
- Returns
- pfloat
The upper tail, lower tail or central probability associated with then multivariate Normal distribution.
- Raises
- NagValueError
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: , or .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, the value in is less than or equal to the corresponding value in .
- (errno )
On entry, is not positive definite.
- Warns
- NagAlgorithmicWarning
- (errno )
Full accuracy not achieved, relative accuracy .
- (errno )
Accuracy requested by is too strict: .
- Notes
Let the vector random variable follow an -dimensional multivariate Normal distribution with mean vector and variance-covariance matrix , then the probability density function, , is given by
The lower tail probability is defined by:
The upper tail probability is defined by:
The central probability is defined by:
To evaluate the probability for , the probability density function of is considered as the product of the conditional probability of given and and the marginal bivariate Normal distribution of and . The bivariate Normal probability can be evaluated as described in
prob_bivariate_normal()
and numerical integration is then used over the remaining dimensions. In the case of ,quad.dim1_fin_bad
is used and forquad.md_adapt
is used.To evaluate the probability for a direct call to
prob_normal()
is made and for calls toprob_bivariate_normal()
are made.
- References
Kendall, M G and Stuart, A, 1969, The Advanced Theory of Statistics (Volume 1), (3rd Edition), Griffin