The hardware and bandwidth for this mirror is donated by dogado GmbH, the Webhosting and Full Service-Cloud Provider. Check out our Wordpress Tutorial.
If you wish to report a bug, or if you are interested in having us mirror your free-software or open-source project, please feel free to contact us at mirror[@]dogado.de.
link to CRAN: https://cran.r-project.org/web/packages/mvQuad/
This package provides a collection of methods for (potentially) multivariate quadrature in R. It’s especially designed for use in statistical problems, characterized by integrals of the form \(\int_a^b g(x)p(x) \ dx\), where \(p(x)\) denotes a density-function. Furthermore the methods are also applicable to standard integration problems with finite, semi-finite and infinite intervals.
In general quadrature (also named: numerical integration, numerical quadrature) work this way: The integral of interests is approximated by a weighted sum of function values.
\[ I = \int_a^b h(x) \ dx \approx A = \sum_{i=1}^n w_i \cdot h(x_i) \]
The so called nodes (\(x_i\)) and weights(\(w_i\)) are defined by the chosen quadrature rule, which should be appropriate (better: optimal) for the integration problem in hand1.
This principle can be transferred directly to the multivariate case.
The methods provided in this package cover the following tasks:
This section shows a typical workflow for quadrature with the
mvQuad-package. More details and additional features of
this package are provided in the subsequent sections. In this
illustration the following two-dimensional integral will be
approximated: \[ I = \int_1^2 \int_1^2 x
\cdot e^y \ dx dy \]
```{r } library(mvQuad)
# create grid nw <- createNIGrid(dim=2, type=“GLe”, level=6)
# rescale grid for desired domain rescale(nw, domain = matrix(c(1, 1, 2, 2), ncol=2))
# define the integrand myFun2d <- function(x){ x[,1]*exp(x[,2]) }
# compute the approximated value of the integral A <- quadrature(myFun2d, grid = nw) ```
Explanation Step-by-Step
mvQuad-package is loadedcreateNIGrid-command a two-dimensional
(dim=2) grid, based on Gauss-Legendre quadrature rule
(type="GLe") with a given accuracy level
(level=6) is created and stored in the variable
nwThe grid created above is designed for the domain \([0, 1]^2\) but we need one for the domain \([1, 2]^2\)
the command rescale changes the domain-feature of
the grid; the new domain is passed in a matrix
(domain=...)
the integrand is defined
the quadrature-command computes the weighted sum of
function values as mentioned in the introduction
The choice of quadrature rule is heavily related to the integration problem. Especially the domain/support (\([a, b]\) (finite), \([a, \infty)\) (semi-finite), \((-\infty, \infty)\) (infinite)) determines the choice.
The mvQuad-packages provides the following quadrature
rules.
cNC1, cNC2, ..., cNC6: closed
Newton-Cotes Formulas of degree 1-6 (1=trapezoidal-rule;
2=Simpson’s-rule; …), finite domain
oNC0, oNC1, ..., oNC3: open
Newton-Cote Formula of degree 0-3 (0=midpoint-rule; …), finite
domain
GLe, GKr: Gauss-Legendre and
Gauss-Kronrod rule, finite domain
nLe: nested Gauss-Legendre rule
(finite domain) [@Petras2003]
Leja: Leja-Points (finite
domain)
GLa: Gauss-Laguerre rule
(semi-finite domain)
GHe: Gauss-Hermite rule (infinite
domain)
nHe: nested Gauss-Hermite rule
(infinite domain) [@GenzKeister1996]
GHN, nHN: (nested)
Gauss-Hermite rule as before but weights are pre-multiplied by the
standard normal density (\(\hat{w}_i = w_i *
\phi(x_i)\)).2
For each rule grids can be created of different accuracy. The
adjusting screw in the createNIGrid is the
level-option. In general, the higher level the
more precise the approximation. For the Newton-Cotes rules an arbitrary
level can be chosen. The other rules uses lookup-tables for the nodes
and weights and are therefore restricted to a maximum level (see
help(QuadRules))
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.