naginterfaces.library.stat.prob_​multi_​normal

naginterfaces.library.stat.prob_multi_normal(xmu, sig, a=None, b=None, tol=0.0001)

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/flhtml/g01/g01hbf.html

Parameters

xmufloat, array-like, shape $\left(n\right)$

$\mu$, the mean vector of the multivariate Normal distribution.

sigfloat, array-like, shape $(n,n)$

$\Sigma$, the variance-covariance matrix of the multivariate Normal distribution. Only the lower triangle is referenced.

aNone or float, array-like, shape $\left(n\right)$, optional

If upper tail or central probablilities are to be returned, $a$ should supply the lower bounds, $a_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

bNone or float, array-like, shape $\left(n\right)$, optional

If lower tail or central probablilities are to be returned, $b$ should supply the upper bounds, $b_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

tolfloat, optional

If $n>2$ the relative accuracy required for the probability, and if the upper or the lower tail probability is requested then $tol$ is also used to determine the cut-off points, see Accuracy.

If $n=1$, $tol$ is not referenced.

Returns

pfloat

The upper tail, lower tail or central probability associated with then multivariate Normal distribution.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $1\le n\le 10$.

(errno $1$)

On entry, $\text{tail}=⟨value⟩$.

Constraint: $\text{tail}=\text{'L'}$, $\text{'U'}$ or $\text{'C'}$.

(errno $1$)

On entry, $tol=⟨value⟩$.

Constraint: $tol>0.0$.

(errno $2$)

On entry, the $⟨value⟩$$⟨value⟩$ value in $b$ is less than or equal to the corresponding value in $a$.

(errno $3$)

On entry, $sig$ is not positive definite.

Warns

NagAlgorithmicWarning

(errno $4$)

Full accuracy not achieved, relative accuracy $=⟨value⟩$.

(errno $5$)

Accuracy requested by $tol$ is too strict: $tol=⟨value⟩$.

Notes

Let the vector random variable $X={(X_1,X_2,\dots ,X_n)}^T$ follow an $n$-dimensional multivariate Normal distribution with mean vector $\mu$ and $n\times n$ variance-covariance matrix $\Sigma$, then the probability density function, $f(X:\mu ,\Sigma )$, is given by

$$f(X:\mu ,\Sigma )={\left(2\pi \right)}^{-\left(1/2\right)n}{\left|\Sigma \right|}^{-1/2}exp(-\frac{1}{2}{(X-\mu )}^T{\Sigma }^{-1}(X-\mu ))\text{.}$$

The lower tail probability is defined by:

$$P(X_1\le b_1,\dots ,X_n\le b_n:\mu ,\Sigma )={\int }_{-\infty }^{b_1}\cdots {\int }_{-\infty }^{b_n}f(X:\mu ,\Sigma )d{X}_n\cdots d{X}_1\text{.}$$

The upper tail probability is defined by:

$$P(X_1\ge a_1,\dots ,X_n\ge a_n:\mu ,\Sigma )={\int }_{a_1}^{\infty }\cdots {\int }_{a_n}^{\infty }f(X:\mu ,\Sigma )d{X}_n\cdots d{X}_1\text{.}$$

The central probability is defined by:

$$P(a_1\le X_1\le b_1,\dots ,a_n\le X_n\le b_n:\mu ,\Sigma )={\int }_{a_1}^{b_1}\cdots {\int }_{a_n}^{b_n}f(X:\mu ,\Sigma )d{X}_n\cdots d{X}_1\text{.}$$

To evaluate the probability for $n\ge 3$, the probability density function of $X_1,X_2,\dots ,X_n$ is considered as the product of the conditional probability of $X_1,X_2,\dots ,X_{n-2}$ given $X_{n-1}$ and $X_n$ and the marginal bivariate Normal distribution of $X_{n-1}$ and $X_n$. The bivariate Normal probability can be evaluated as described in prob_bivariate_normal() and numerical integration is then used over the remaining $n-2$ dimensions. In the case of $n=3$, quad.dim1_fin_bad is used and for $n>3$ quad.md_adapt is used.

To evaluate the probability for $n=1$ a direct call to prob_normal() is made and for $n=2$ calls to prob_bivariate_normal() are made.

References

Kendall, M G and Stuart, A, 1969, The Advanced Theory of Statistics (Volume 1), (3rd Edition), Griffin