| Title: | Sparse grid integration in R |
|---|---|
| Description: | SparseGrid is a package to create sparse grids for numerical integration, based on code from www.sparse-grids.de |
| Authors: | Jelmer Ypma |
| Maintainer: | Jelmer Ypma <[email protected]> |
| License: | GPL |
| Version: | 0.8.2 |
| Built: | 2024-10-31 21:23:47 UTC |
| Source: | CRAN |
This function creates nodes and weights that can be used for integration. It is a convenience function that calls createSparseGrid and createProductRuleGrid and returns the grid with the least number of nodes. Typically, a grid created by the product rule will only contain fewer nodes than a sparse grid for very low dimensions.
createIntegrationGrid(type, dimension, k, sym = FALSE)createIntegrationGrid(type, dimension, k, sym = FALSE)
type |
String or function for type of 1D integration rule, can take on values
|
dimension |
Dimension of the integration problem. |
k |
Accuracy level. The rule will be exact for polynomials up to total order 2k-1. |
sym |
(optional) only used for own 1D quadrature rule (type not "KPU",...). If sym is supplied and not FALSE, the code will run faster but will produce incorrect results if 1D quadrature rule is asymmetric. |
The return value contains a list with nodes and weights
nodes |
matrix with a node in each row |
weights |
vector with corresponding weights |
Jelmer Ypma
Florian Heiss, Viktor Winschel, Likelihood approximation by numerical integration on sparse grids, Journal of Econometrics, Volume 144, Issue 1, May 2008, Pages 62-80, http://www.sparse-grids.de
createSparseGrid
createProductRuleGrid
createMonteCarloGrid
integrate
pmvnorm
# load library library('SparseGrid') # create integration grid int.grid <- createIntegrationGrid( 'GQU', dimension=3, k=5 )# load library library('SparseGrid') # create integration grid int.grid <- createIntegrationGrid( 'GQU', dimension=3, k=5 )
Simulate nodes using a random number generator supplied by the user, and combine these with a vector of equal weights into a list. Sparse grids can be created with the function createSparseGrid.
createMonteCarloGrid( rng, dimension, num.sim, ... )createMonteCarloGrid( rng, dimension, num.sim, ... )
rng |
function that generates random numbers. The first argument of this function should be called |
dimension |
dimension of the integration problem. |
num.sim |
number of simulated integration nodes. |
... |
arguments that will be passed to the random number generator |
The return value contains a list with nodes and weights
nodes |
matrix with a node in each row |
weights |
vector with corresponding weights |
Jelmer Ypma
createSparseGrid
createProductRuleGrid
createIntegrationGrid
integrate
pmvnorm
# load library library('SparseGrid') # set random seed set.seed( 3141 ) # Create Monte Carlo integration grids # 1. with draws from a uniform distribution mc.grid <- createMonteCarloGrid( runif, dimension=2, num.sim=10 ) mc.grid # 2. with draws from a standard normal distribution mc.grid <- createMonteCarloGrid( rnorm, dimension=3, num.sim=1000 ) # 3. with draws from a normal distribution with mean=2 and sd=5 mc.grid <- createMonteCarloGrid( rnorm, dimension=3, num.sim=1000, mean=2, sd=5 )# load library library('SparseGrid') # set random seed set.seed( 3141 ) # Create Monte Carlo integration grids # 1. with draws from a uniform distribution mc.grid <- createMonteCarloGrid( runif, dimension=2, num.sim=10 ) mc.grid # 2. with draws from a standard normal distribution mc.grid <- createMonteCarloGrid( rnorm, dimension=3, num.sim=1000 ) # 3. with draws from a normal distribution with mean=2 and sd=5 mc.grid <- createMonteCarloGrid( rnorm, dimension=3, num.sim=1000, mean=2, sd=5 )
Creates nodes and weights according to the product rule, combining 1D nodes and weights. Sparse grids can be created with the function createSparseGrid.
createProductRuleGrid(type, dimension, k, sym = FALSE)createProductRuleGrid(type, dimension, k, sym = FALSE)
type |
String or function for type of 1D integration rule, can take on values
|
dimension |
dimension of the integration problem. |
k |
Accuracy level. The rule will be exact for polynomial up to total order 2k-1. |
sym |
(optional) only used for own 1D quadrature rule (type not "KPU",...). If sym is supplied and not FALSE, the code will run faster but will produce incorrect results if 1D quadrature rule is asymmetric. |
The return value contains a list with nodes and weights
nodes |
matrix with a node in each row |
weights |
vector with corresponding weights |
Jelmer Ypma
Florian Heiss, Viktor Winschel, Likelihood approximation by numerical integration on sparse grids, Journal of Econometrics, Volume 144, Issue 1, May 2008, Pages 62-80, http://www.sparse-grids.de
createSparseGrid
createMonteCarloGrid
createIntegrationGrid
integrate
pmvnorm
# load library library('SparseGrid') # define function to be integrated # g(x) = x[1] * x[2] * ... * x[n] g <- function( x ) { return( prod( x ) ) } # # Create sparse integration grid to approximate integral of a function with uniform weights # sp.grid <- createSparseGrid( 'KPU', dimension=3, k=5 ) # number of nodes and weights length( sp.grid$weights ) # evaluate function g in nodes gx.sp <- apply( sp.grid$nodes, 1, g ) # take weighted sum to get approximation for the integral val.sp <- gx.sp %*% sp.grid$weights # # Create integration grid to approximate integral of a function with uniform weights # pr.grid <- createProductRuleGrid( 'KPU', dimension=3, k=5 ) # number of nodes and weights length( pr.grid$weights ) # evaluate function g in nodes gx.pr <- apply( pr.grid$nodes, 1, g ) # take weighted sum to get approximation for the integral val.pr <- gx.pr %*% pr.grid$weights # # Create integration grid to approximation integral using Monte Carlo simulation # set.seed( 3141 ) mc.grid <- createMonteCarloGrid( runif, dimension=3, num.sim=1000 ) # number of nodes and weights length( mc.grid$weights ) # evaluate function g in MC nodes gx.mc <- apply( mc.grid$nodes, 1, g ) # take weighted sum to get approximation for the integral # the weights are all equal to 1/1000 in this case val.mc <- gx.mc %*% mc.grid$weights val.sp val.pr val.mc# load library library('SparseGrid') # define function to be integrated # g(x) = x[1] * x[2] * ... * x[n] g <- function( x ) { return( prod( x ) ) } # # Create sparse integration grid to approximate integral of a function with uniform weights # sp.grid <- createSparseGrid( 'KPU', dimension=3, k=5 ) # number of nodes and weights length( sp.grid$weights ) # evaluate function g in nodes gx.sp <- apply( sp.grid$nodes, 1, g ) # take weighted sum to get approximation for the integral val.sp <- gx.sp %*% sp.grid$weights # # Create integration grid to approximate integral of a function with uniform weights # pr.grid <- createProductRuleGrid( 'KPU', dimension=3, k=5 ) # number of nodes and weights length( pr.grid$weights ) # evaluate function g in nodes gx.pr <- apply( pr.grid$nodes, 1, g ) # take weighted sum to get approximation for the integral val.pr <- gx.pr %*% pr.grid$weights # # Create integration grid to approximation integral using Monte Carlo simulation # set.seed( 3141 ) mc.grid <- createMonteCarloGrid( runif, dimension=3, num.sim=1000 ) # number of nodes and weights length( mc.grid$weights ) # evaluate function g in MC nodes gx.mc <- apply( mc.grid$nodes, 1, g ) # take weighted sum to get approximation for the integral # the weights are all equal to 1/1000 in this case val.mc <- gx.mc %*% mc.grid$weights val.sp val.pr val.mc
Creates nodes and weights that can be used for sparse grid integration. Based on Matlab code by Florian Heiss and Viktor Winschel, available from http://www.sparse-grids.de
createSparseGrid(type, dimension, k, sym = FALSE)createSparseGrid(type, dimension, k, sym = FALSE)
type |
String or function for type of 1D integration rule, can take on values
|
dimension |
dimension of the integration problem. |
k |
Accuracy level. The rule will be exact for polynomial up to total order 2k-1. |
sym |
(optional) only used for own 1D quadrature rule (type not "KPU",...). If sym is supplied and not FALSE, the code will run faster but will produce incorrect results if 1D quadrature rule is asymmetric. |
The return value contains a list with nodes and weights
nodes |
matrix with a node in each row |
weights |
vector with corresponding weights |
Jelmer Ypma
Florian Heiss, Viktor Winschel, Likelihood approximation by numerical integration on sparse grids, Journal of Econometrics, Volume 144, Issue 1, May 2008, Pages 62-80, http://www.sparse-grids.de
createProductRuleGrid
createMonteCarloGrid
createIntegrationGrid
integrate
pmvnorm
# load library library('SparseGrid') # define function to be integrated # g(x) = x[1] * x[2] * ... * x[n] g <- function( x ) { return( prod( x ) ) } # # Create sparse integration grid to approximate integral of a function with uniform weights # sp.grid <- createSparseGrid( 'KPU', dimension=3, k=5 ) # number of nodes and weights length( sp.grid$weights ) # evaluate function g in nodes gx.sp <- apply( sp.grid$nodes, 1, g ) # take weighted sum to get approximation for the integral val.sp <- gx.sp %*% sp.grid$weights # # Create integration grid to approximate integral of a function with uniform weights # pr.grid <- createProductRuleGrid( 'KPU', dimension=3, k=5 ) # number of nodes and weights length( pr.grid$weights ) # evaluate function g in nodes gx.pr <- apply( pr.grid$nodes, 1, g ) # take weighted sum to get approximation for the integral val.pr <- gx.pr %*% pr.grid$weights # # Create integration grid to approximation integral using Monte Carlo simulation # set.seed( 3141 ) mc.grid <- createMonteCarloGrid( runif, dimension=3, num.sim=1000 ) # number of nodes and weights length( mc.grid$weights ) # evaluate function g in MC nodes gx.mc <- apply( mc.grid$nodes, 1, g ) # take weighted sum to get approximation for the integral # the weights are all equal to 1/1000 in this case val.mc <- gx.mc %*% mc.grid$weights val.sp val.pr val.mc# load library library('SparseGrid') # define function to be integrated # g(x) = x[1] * x[2] * ... * x[n] g <- function( x ) { return( prod( x ) ) } # # Create sparse integration grid to approximate integral of a function with uniform weights # sp.grid <- createSparseGrid( 'KPU', dimension=3, k=5 ) # number of nodes and weights length( sp.grid$weights ) # evaluate function g in nodes gx.sp <- apply( sp.grid$nodes, 1, g ) # take weighted sum to get approximation for the integral val.sp <- gx.sp %*% sp.grid$weights # # Create integration grid to approximate integral of a function with uniform weights # pr.grid <- createProductRuleGrid( 'KPU', dimension=3, k=5 ) # number of nodes and weights length( pr.grid$weights ) # evaluate function g in nodes gx.pr <- apply( pr.grid$nodes, 1, g ) # take weighted sum to get approximation for the integral val.pr <- gx.pr %*% pr.grid$weights # # Create integration grid to approximation integral using Monte Carlo simulation # set.seed( 3141 ) mc.grid <- createMonteCarloGrid( runif, dimension=3, num.sim=1000 ) # number of nodes and weights length( mc.grid$weights ) # evaluate function g in MC nodes gx.mc <- apply( mc.grid$nodes, 1, g ) # take weighted sum to get approximation for the integral # the weights are all equal to 1/1000 in this case val.mc <- gx.mc %*% mc.grid$weights val.sp val.pr val.mc
This function reads nodes and weights with the format of the .asc files available from http://www.sparse-grids.de
readASCGrid(filename, dimension)readASCGrid(filename, dimension)
filename |
name of the file that you want to read. The extension should be included. |
dimension |
dimension of the grid that you want to read. |
The return value contains a list with nodes and weights
nodes |
matrix with a node in each row |
weights |
vector with corresponding weights |
Jelmer Ypma
Florian Heiss, Viktor Winschel, Likelihood approximation by numerical integration on sparse grids, Journal of Econometrics, Volume 144, Issue 1, May 2008, Pages 62-80, http://www.sparse-grids.de
createSparseGrid
createProductRuleGrid
createIntegrationGrid
integrate
pmvnorm
# load library library('SparseGrid') ## Not run: # read file (e.g. after downloading from www.sparse-grids.de) ReadASCFile(filename='GQU_d3_l5.asc', dimension=3) ## End(Not run)# load library library('SparseGrid') ## Not run: # read file (e.g. after downloading from www.sparse-grids.de) ReadASCFile(filename='GQU_d3_l5.asc', dimension=3) ## End(Not run)