### SphericalCubature – Numerical integration of functions over spheres and balls

Defines several methods to integrate functions over the unit sphere and ball in n-dimensional Euclidean space. Routines for converting to/from multivariate polar/spherical coordinates are also provided.

The following code fragment integrates a function of 3 variables over the sphere and over the ball.

f1 <- function( x ) { return(x[1]^2+3*x[2]+exp(x[3])) }

adaptIntegrateSphere( f1, n=3 )

adaptIntegrateBall( f1, n=3 )

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### SimplicialCubature – Numerical integration of functions over simplices

Provides methods to integrate functions over m-dimensional simplices in n-dimensional Euclidean space. There are exact methods for polynomials and adaptive methods for integrating an arbitrary function.

The following code evaluates an integral over the canonical simplex in 4-dimensions.

S <- CanonicalSimplex( n=4 )

f1 <- function( x ) { x[1]^3 }

adaptIntegrateSimplex( f1, S )

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### mvmesh – Multivariate meshes

Define, manipulate and plot multivariate meshes/grids in n-dimensional Euclidean space. Multivariate histograms based on these meshes are provided.

The following code produces the equal volume edge 4-subdivision of the solid simplex in 3-dimensions shown on the right.

plot( SolidSimplex( n=3,k=4), col="blue", lwd=2 )

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### mvhist – Multivariate histograms

Compute and plot histograms in 2 and (sometimes) higher dimensions.

`x <- matrix( rnorm(8000), ncol=2 )`

`histRectangular( x, breaks=5 )`

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### gensphere – Generalized spherical distributions

Define and compute with generalized spherical distributions – multivariate probability laws that are specified by a star shaped contour (directional behavior) and a radial component.

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### ecdfHT – Empirical cdf for Heavy Tailed data

Computes and plots a transformed empirical CDF (ecdf) as a diagnostic for heavy tailed data, specifically data with power law decay on the tails. Routines for annotating the plot, comparing data to a model, fitting a nonparametric model, and some multivariate extensions are given.

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### R code from the paper “R as a tool in computational finance”, J. P. Nolan (2012)

(stored as a text file to get around Edspace file restrictions; rename to financetools.R)