Package ‘mc2d’

March 6, 2017
Type Package
Title Tools for Two-Dimensional Monte-Carlo Simulations
Version 0.1-18
Date 2017-03-03
Author Regis Pouillot [aut, cre],
Marie-Laure Delignette-Muller [ctb],
Jean-Baptiste Denis [ctb]
Maintainer Regis Pouillot <rpouillot@yahoo.fr>
Suggests fitdistrplus, survival
Depends R (>= 2.10.0), mvtnorm
Imports stats, grDevices, graphics, utils
Description A complete framework to build and study Two-Dimensional Monte-Carlo simula-
tions, aka Second-Order Monte-Carlo simulations. Also includes various distributions (pert, tri-
angular, Bernoulli, empirical discrete and continuous).
License GPL (>= 2)
Repository CRAN
Repository/R-Forge/Project riskassessment
Repository/R-Forge/Revision 512
Repository/R-Forge/DateTimeStamp 2017-03-03 15:01:19
Date/Publication 2017-03-06 13:53:06
NeedsCompilation no

R topics documented:
bernoulli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
betagen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
converg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
cornode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
dimmcnode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1
2 bernoulli

dmultinomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
ec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
empiricalC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
empiricalD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
evalmcmod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
extractvar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
hist.mc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
is.mc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
lhs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
mc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
mc.control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
mcapply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
mccut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
mcmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
mcnode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
mcprobtree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
mcratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
mcstoc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
multinormal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
NA.mcnode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Ops.mcnode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
outm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
pert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
plot.mc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
plot.tornado . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
pmin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
print.mc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
quantile.mc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
rtrunc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
summary.mc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
tornado . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
tornadounc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
total . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
triangular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
typemcnode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
unmc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Index 60

bernoulli The Bernoulli Distribution

Description
Density, distribution function, quantile function and random generation for the Bernoulli distribu-
tion with probability equals to ‘prob’.
betagen 3

Usage
dbern(x, prob=.5, log=FALSE)
pbern(q, prob=.5, lower.tail=TRUE, log.p=FALSE)
qbern(p, prob=.5, lower.tail=TRUE, log.p=FALSE)
rbern(n, prob=.5)

Arguments
x,q vector of quantiles.
p vector of probabilities.
n number of observations. If ‘length(n) > 1’, the length is taken to be the
number required.
prob vector of probabilities of success of each trial.
log, log.p logical; if ‘TRUE’, probabilities ‘p’ are given as ‘log(p)’.
lower.tail logical; if ‘TRUE’ (default), probabilities are ‘P[X <= x]’, otherwise, ‘P[X > x]’.

Details
These fonctions use the corresponding functions from the binomial distribution with argument
‘size = 1’. Thus, 1 is for success, 0 is for failure.

Value
‘dbern’ gives the density, ‘pbern’ gives the distribution function, ‘qbern’ gives the quantile func-
tion, and ‘rbern’ generates random deviates.

See Also
Binomial

Examples
rbern(n=10, prob=.5)
rbern(n=3, prob=c(0, .5, 1))

betagen The Generalised Beta Distribution

Description
Density, distribution function, quantile function and random generation for the Beta distribution
defined on the ‘[min, max]’ domain with parameters ‘shape1’ and ‘shape2’ ( and optional non-
centrality parameter ‘ncp’).
4 betagen

Usage
dbetagen(x, shape1, shape2, min=0, max=1, ncp=0, log=FALSE)
pbetagen(q, shape1, shape2, min=0, max=1, ncp=0, lower.tail=TRUE,
log.p=FALSE)
qbetagen(p, shape1, shape2, min=0, max=1, ncp=0, lower.tail=TRUE,
log.p=FALSE)
rbetagen(n, shape1, shape2, min=0, max=1, ncp=0)

Arguments
x,q Vector of quantiles.
p Vector of probabilities.
n Number of observations. If ‘length(n) > 1’, the length is taken to be the
number required.
shape1, shape2 Positive parameters of the Beta distribution.
min Vector of minima.
max Vector of maxima.
ncp Non-centrality parameter of the Beta distribution.
log, log.p Logical; if ‘TRUE’, probabilities ‘p’ are given as ‘log(p)’.
lower.tail Logical; if ‘TRUE’ (default), probabilities are ‘P[X <= x]’, otherwise, ‘P[X > x]’.

Details

x ∼ betagen(shape1, shape2, min, max, ncp)

if
x − min
∼ beta(shape1, shape2, ncp)
max − min
These functions use the Beta distribution functions after correct parametrisation.

Value
‘dbetagen’ gives the density, ‘pbetagen’ gives the distribution function, ‘qbetagen’ gives the
quantile function, and ‘rbetagen’ generates random deviates.

See Also
Beta

Examples
curve(dbetagen(x, shape1=3, shape2=5, min=1, max=6), from = 0, to = 7)
curve(dbetagen(x, shape1=1, shape2=1, min=2, max=5), from = 0, to = 7, lty=2, add=TRUE)
curve(dbetagen(x, shape1=.5, shape2=.5, min=0, max=7), from = 0, to = 7, lty=3, add=TRUE)
converg 5

converg Graph of Running Statistics in the Variability or in the Uncertainty

Dimension.

Description
This function provides basic graphs to evaluate the convergence of a node of a mc or a mccut object
in the variability or in the uncertainty dimension.

Usage
converg(x, node=length(x), margin=c("var", "unc"), nvariates=1, iter=1,
probs=c(0.025, 0.975), lim=c(0.025, 0.975), griddim=NULL,
log=FALSE)

Arguments
x A mcnode object, a mc object or a mccut object.
node The node to be considered in a ‘mc’ object or a ‘mccut’ object, displayed either
as the order number or the name of the node. By default: the last node of the
object.The corresponding node should not be of type ‘"0"’ in a ‘mc’ object or of
type ‘"0"’ or ‘"V"’ in a ‘mccut’ object.
margin The margin used to plot the graph. ‘margin’ is used only if the node is a
‘"VU" mcnode’.
nvariates The variates to be considered. ‘nvariates’ is used only for multivariates nodes.
iter If ‘margin == "var"’ and the node is a ‘"VU" mcnode’, ‘iter’ specify the
iteration in the uncertainty dimension to be used for the graph.
probs The quantiles to be provided in the variability dimension.
lim The quantiles to be used in the uncertainty dimension.
griddim A vector of two integers, indicating the size of the grid of the graph. If ‘NULL’,
the grid is calculated to produce a "nice" graph.
log If ‘TRUE’, the data will be log transformed.

Details
If the node is of type ‘"V"’, the running mean, median and ‘probs’ quantiles according to the
variability dimension will be provided. If the node is of type ‘"VU"’ and ‘margin="var"’, this
graph will be provided on one simulation in the uncertainty dimension (chosen by ‘iter’).
If the node is of type ‘"U"’ the running mean, median and ‘lim’ quantiles according to the uncer-
tainty dimension will be provided.
If the node is of type ‘"VU"’ (with ‘margin="unc"’ or from a ‘mccut’ object), one graph are pro-
vided for each of the mean, median and ‘probs’ quantiles calculated in the variability dimension.
6 cornode

Note
This function may be used on a ‘mccut’ object only if a ‘summary.mc’ function was used in the
third block of the evalmccut call. The values used as ‘probs’ arguments in ‘converg’ should have
been used in the ‘summary.mc’ function of this third block.

Examples
data(total)
converg(xVU, margin="var")
converg(xVU, margin="unc")

cornode Builds a Rank Correlation using the Iman and Connover Method.

Description
This function builds a rank correlation structure between columns of a matrix or between ‘mcnode’
objects using the Iman and Connover method (1982).

Usage
cornode(..., target, outrank=FALSE, result=FALSE, seed=NULL)

Arguments
... A matrix (each of its ‘n’ columns but the first one will be reordered) or ‘n mcnode’
objects (each elements but the first one will be reordered).
target A scalar (only if ‘n=2’) or a ‘(n x n)’ matrix of correlation.
outrank Should the order be returned?
result Should the correlation eventually obtained be printed?
seed The random seed used for building the correlation. If ‘NULL’ the ‘seed’ is un-
changed.

Details
The arguments should be named.
The function accepts for ‘data’ a matrix or:

• some ‘"V" mcnode’ objects separated by a comma;

• some ‘"U" mcnode’ objects separated by a comma;
• some ‘"VU" mcnode’ objects separated by a comma. In that case, the structure is built columns
by colums (the first column of each ‘"VU" mcnode’ will have a correlation structure, the
second ones will have a correlation structure, ....).
• one ‘"V" mcnode’ as a first element and some ‘"VU" mcnode’ objects, separated by a comma.
In that case, the structure is built between the ‘"V" mcnode’ and each column of the ‘"VU"
mcnode’ objects. The correlation result (‘result = TRUE’) is not provided in that case.
cornode 7

The number of variates of the elements should be equal.

‘target’ should be a scalar (two columns only) or a real symmetric positive-definite square matrix.
Only the upper triangular part of ‘target’ is used (see chol).
The final correlation structure should be checked because it is not always possible to build the target
correlation structure.
In a Monte-Carlo simulation, note that the order of the values within each ‘mcnode’ will be changed
by this function (excepted for the first one of the list). As a consequence, previous links between
variables will be broken. The ‘outrank’ option may help to rebuild these links (see the Examples).

Value
If ‘rank = FALSE’: the matrix or a list of rearranged ‘mcnode’s.
If ‘rank = TRUE’: the order to be used to rearranged the matrix or the ‘mcnodes’ to build the desired
correlation structure.

References
Connover W., Iman R. (1982). A distribution-free approach to inducing rank correlation among
input variables. Technometric, 3, 311-334.

Examples
x1 <- rnorm(1000)
x2 <- rnorm(1000)
x3 <- rnorm(1000)
mat <- cbind(x1, x2, x3)
## Target
(corr <- matrix(c(1, 0.5, 0.2, 0.5, 1, 0.2, 0.2, 0.2, 1), ncol=3))
## Before
cor(mat, method="spearman")
matc <- cornode(mat, target=corr, result=TRUE)
## The first row is unchanged
all(matc[, 1] == mat[, 1])

##Using mcnode and outrank

cook <- mcstoc(rempiricalD, values=c(0, 1/5, 1/50), prob=c(0.027, 0.373, 0.600), nsv=1000)
serving <- mcstoc(rgamma, shape=3.93, rate=0.0806, nsv=1000)
roundserv <- mcdata(round(serving), nsv=1000)
## Strong relation between roundserv and serving (of course)
cor(cbind(cook, roundserv, serving), method="spearman")

##The classical way to build the correlation structure

matcorr <- matrix(c(1, 0.5, 0.5, 1), ncol=2)
matc <- cornode(cook=cook, roundserv=roundserv, target=matcorr)
## The structure between cook and roundserv is OK but ...
## the structure between roundserv and serving is lost
cor(cbind(cook=matc$cook, serv=matc$roundserv, serving), method="spearman")

##An alternative way to build the correlation structure

matc <- cornode(cook=cook, roundserv=roundserv, target=matcorr, outrank=TRUE)
8 dimmcnode

## Rebuilding the structure

roundserv[] <- roundserv[matc$roundserv, , ]
serving[] <- serving[matc$roundserv, , ]
## The structure between cook and roundserv is OK and ...
## the structure between roundserv and serving is preserved
cor(cbind(cook, roundserv, serving), method="spearman")

dimmcnode Dimension of mcnode and mc Objects

Description

Provides the dimension (i.e. the number of simulations in the variability dimension, the number of
simulations in the uncertainty dimension and the maximum number of variates of a ‘mcnode’ or a
‘mc’ object.

Usage

dimmcnode(x)
dimmc(x)

Arguments

x a ‘mcnode’ or a ‘mc’ object.

Value

A vector of three scalars: the dimension of variability (1 for ‘"0"’ and ‘"U" mcnode’), the dimension
of uncertainty (1 for ‘"0"’ and ‘"V" mcnode’) and the number of variates (the maximal number of
variates for an ‘mc’ object.

Note

This function does not test if the object is correctly built. See is.mcnode and is.mc .

Examples
data(total)
dimmcnode(xVUM2)
dimmc(total)
dirichlet 9

dirichlet The Dirichlet Distribution

Description
Density function and random generation from the Dirichlet distribution.

Usage
ddirichlet(x, alpha)
rdirichlet(n, alpha)

Arguments
x A vector containing a single deviate or a matrix containing one random deviate
per row.
alpha A vector of shape parameters, or a matrix of shape parameters by rows. Recy-
cling (by row) is permitted.
n Number of random vectors to generate. If length(n) > 1, the length is taken to
be the number required.

Details
The Dirichlet distribution is the multidimensional generalization of the beta distribution. The orig-
inal code was adapted to provide a kind of "vectorization" used in multivariates ‘mcnode’.

Value
‘ddirichlet’ gives the density. ‘rdirichlet’ returns a matrix with ‘n’ rows, each containing a
single Dirichlet random deviate.

Author(s)
Code is adapted from ‘MCMCpack’. It originates from Greg’s Miscellaneous Functions (gregmisc).

See Also
Beta

Examples
dat <- c(1, 10, 100, 1000, 1000, 100, 10, 1)
(alpha <- matrix(dat, nrow=4, byrow=TRUE))
round(x <- rdirichlet(4, alpha), 2)
ddirichlet(x, alpha)

## rdirichlet used with mcstoc

mcalpha <- mcdata(dat, type="V", nsv=4, nvariates=2)
10 dmultinomial

(x <- mcstoc(rdirichlet, type="V", alpha=mcalpha, nsv=4, nvariates=2))

unclass(x)
x <- mcstoc(rdirichlet, type="VU", alpha=mcalpha, nsv=4, nsu=10, nvariates=2)
unclass(x)

dmultinomial The Vectorized Multinomial Distribution

Description
Generate multinomially distributed random number vectors and compute multinomial probabilities.

Usage
dmultinomial(x, size=NULL, prob, log=FALSE)
rmultinomial(n, size, prob)

Arguments
x vector or matrix of length (or ncol) K of integers in ‘0:size’.
n number of random vectors to draw.
size a vector of integers, say N, specifying the total number of objects that are put
into K boxes in the typical multinomial experiment. For ‘dmultinom’, it defaults
to ‘sum(x)’. The first element correspond to the vector ‘prob’ or the first row
of ‘prob’, ...
prob Numeric non-negative vector of length K, or matrix of size ‘(x x K)’ specifying
the probability for the K classes; is internally normalized to sum 1.
log Logical; if TRUE, log probabilities are computed.

Details
These functions are the vectorized versions of rmultinom and dmultinom. Recycling is permitted.

Examples
x <- c(100, 200, 700)
x1 <- matrix(c(100, 200, 700, 200, 100, 700, 700, 200, 100), byrow=TRUE, ncol=3)
p <- c(1, 2, 7)
p1 <- matrix(c(1, 2, 7, 2, 1, 7, 7, 2, 1), byrow=TRUE, ncol=3)
dmultinomial(x1, prob=p)
## is equivalent to
c( dmultinom(x1[1, ], prob=p),
dmultinom(x1[2, ], prob=p),
dmultinom(x1[3, ], prob=p))

dmultinomial(x1, prob=p1, log=TRUE)

## is equivalent to
ec 11

c( dmultinom(x1[1, ], prob=p1[1, ], log=TRUE),

dmultinom(x1[2, ], prob=p1[2, ], log=TRUE),
dmultinom(x1[3, ], prob=p1[3, ], log=TRUE))

dmultinomial(x, prob=p1, log=TRUE)

## is equivalent to
c( dmultinom(x, prob=p1[1, ], log=TRUE),
dmultinom(x, prob=p1[2, ], log=TRUE),
dmultinom(x, prob=p1[3, ], log=TRUE))

prob <- c(1, 2, 7)

rmultinomial(4, 1000, prob)
rmultinomial(4, c(10, 100, 1000, 10000), prob)

## rmultinomial used with mcstoc

## (uncertain size and prob)
s <- mcstoc(rpois, "U", lambda=50)
p <- mcstoc(rdirichlet, "U", nvariates=3, alpha=c(4, 10, 20))
mcstoc(rmultinomial, "VU", nvariates=3, size=s, prob=p)

ec An exemple on Escherichia coli in ground beef

Description
The fictive example is as following:
A batch of ground beef is contaminated with E. coli, with a mean concentration ‘conc’.
Consumers may eat the beef "rare", "medium rare" or "well cooked". If "rare", no bacteria is killed.
If "medium rare", 1/5 of bacteria survive. If "well cooked", 1/50 of bacteria survive.
The serving size is variable.
The risk of infection follows an exponential model.
For the one-dimensional model, it is assumed that:
conc <- 10
cook <- sample(n, x=c(1,1/5,1/50),replace=TRUE,prob=c(0.027,0.373,0.600))
serving <- rgamma(n, shape=3.93,rate=0.0806)
expo <- conc * cook * serving
dose <- rpois(n, lambda=expo)
risk <- 1-(1-0.001)^dose
For the two-dimensional model, it is assumed moreover that the concentration and the ‘r’ parameter
of the dose response are uncertain.
conc <- rnorm(n,mean=10,sd=2)
r <- runif(n ,min=0.0005,max=0.0015)
12 empiricalC

Usage
ec

Format
A list of two expression to be passed in mcmodel

Source
Fictive example

References
None

empiricalC The Continuous Empirical Distribution

Description
Density, distribution function and random generation for a continuous empirical distribution.

Usage
dempiricalC(x, min, max, values, prob=NULL, log=FALSE)
pempiricalC(q, min, max, values, prob=NULL, lower.tail=TRUE, log.p=FALSE)
qempiricalC(p, min, max, values, prob=NULL, lower.tail=TRUE, log.p=FALSE)
rempiricalC(n, min, max, values, prob=NULL)

Arguments
x, q Vector of quantiles.
p Vector of probabilities.
n Number of random values. If ‘length(n) > 1’, the length is taken to be the
number required.
min A finite minimal value.
max A finite maximal value.
values Vector of numerical values.
prob Optionnal vector of count or probabilities.
log, log.p logical; if ‘TRUE’, probabilities ‘p’ are given as ‘log(p)’.
lower.tail logical; if ‘TRUE’ (default), probabilities are ‘P[X <= x]’, otherwise, ‘P[X > x]’.
empiricalC 13

Details
Given pi , the distribution value for xi with ‘i’ the rank i = 0, 1, 2, . . . , N + 1, x0 = min and
xN +1 = max the density is:
x − xi
f (x) = pi + ( )(pi+1 − pi )
xi+1 − xi

The ‘p’ values being normalized to give the distribution a unit area.
‘min’ and/or ‘max’ and/or ‘values’ and/or ‘prob’ may vary: in that case, ‘min’ and/or ‘max’ should
be vector(s). ‘values’ and/or ‘prob’ should be matrixes, the first row being used for the first
element of ‘x’, ‘q’, ‘p’ or the first random value, the second row for the second element of ‘x’, ‘q’,
‘p’ or random value, ... Recycling is permitted if the number of elements of ‘min’ or ‘max’ or the
number of rows of ‘prob’ and ‘values’ are equal or equals one.

Value
‘dempiricalC’ gives the density, ‘pempiricalC’ gives the distribution function, ‘qempiricalC’
gives the quantile function and ‘rempiricalC’ generates random deviates.

See Also
empiricalD

Examples
prob <- c(2, 3, 1, 6, 1)
values <- 1:5
par(mfrow=c(1, 2))
curve(dempiricalC(x, min=0, max=6, values, prob), from=-1, to=7, n=1001)
curve(pempiricalC(x, min=0, max=6, values, prob), from=-1, to=7, n=1001)

## Varying values
(values <- matrix(1:10, ncol=5))
## the first x apply to the first row
## the second x to the second one
dempiricalC(c(1, 1), values, min=0, max=11)

##Use with mc2d

val <- c(100, 150, 170, 200)
pr <- c(6, 12, 6, 6)
out <- c("min", "mean", "max")
##First Bootstrap in the uncertainty dimension
##with rempirical D
(x <- mcstoc(rempiricalD, type = "U", outm = out, nvariates = 30, values = val, prob = pr))
##Continuous Empirical distribution in the variability dimension
mcstoc(rempiricalC, type = "VU", values = x, min=90, max=210)
14 empiricalD

empiricalD The Discrete Empirical Distribution

Description
Density, distribution function and random generation for a discrete empirical distribution. This
function is vectorized to accept different sets of ‘values’ or ‘prob’.

Usage
dempiricalD(x, values, prob=NULL, log=FALSE)
pempiricalD(q, values, prob=NULL, lower.tail=TRUE, log.p=FALSE)
qempiricalD(p, values, prob=NULL, lower.tail=TRUE, log.p=FALSE)
rempiricalD(n, values, prob=NULL)

Arguments
x, q Vector of quantiles.
p Vector of probabilities.
n Number of random values. If length(n) > 1, the length is taken to be the number
required.
values Vector or matrix of numerical values. See details.
prob Optionnal vector or matrix of count or probabilities. See details.
log, log.p logical; if ‘TRUE’, probabilities ‘p’ are given as ‘log(p)’.
lower.tail logical; if ‘TRUE’ (default), probabilities are ‘P[X <= x]’, otherwise, ‘P[X > x]’.

Details
If ‘prob’ is missing, the discrete distribution is obtained directly from the vector of ‘values’, oth-
erwise ‘prob’ is used to weight the values. ‘prob’ is normalized before use. Thus, ‘prob’ may be
the count of each ‘values’. ‘prob’ values should be non negative and their sum should not be 0.
‘values’ and/or ‘prob’ may vary: in that case, ‘values’ and/or ‘prob’ should be sent as matrixes,
the first row being used for the first element of ‘x’, ‘q’, ‘p’ or the first random value, the second row
for the second element of ‘x’, ‘q’, ‘p’ or random value, ... Recycling is permitted if the number of
rows of ‘prob’ and ‘values’ are equal or if the number of rows of ‘prob’ and/or ‘values’ are one.
‘rempiricalD(n, values, prob)’ with ‘values’ and ‘prob’ as vectors is equivalent to ‘sample(x=values,size=n, repla

Value
‘dempiricalD’ gives the density, ‘pempiricalD’ gives the distribution function, ‘qempiricalD’
gives the quantile function and ‘rempiricalD’ generates random deviates.

Note
In the future, the fonctions should be written for non numerical values.
evalmcmod 15

See Also
sample. empiricalC.

Examples
dempiricalD(1:6, 2:6, prob=c(10, 10, 70, 0, 10))
pempiricalD(1:6, 2:6, prob=c(10, 10, 70, 0, 10))
qempiricalD(seq(0, 1, 0.1), 2:6, prob=c(10, 10, 70, 0, 10))
table(rempiricalD(10000, 2:6, prob=c(10, 10, 70, 0, 10)))

## Varying values
(values <- matrix(1:10, ncol=5))
## the first x apply to the first row : p = 0.2
## the second x to the second one: p = 0
dempiricalD(c(1, 1), values)

##Use with mc2d

##Non Parameteric Bootstrap
val <- c(100, 150, 170, 200)
pr <- c(6, 12, 6, 6)
out <- c("min", "mean", "max")
##First Bootstrap in the uncertainty dimension
(x <- mcstoc(rempiricalD, type = "U", outm = out, nvariates = 30, values = val, prob = pr))
##Second one in the variability dimension
mcstoc(rempiricalD, type = "VU", values = x)

evalmcmod Evaluates a Monte-Carlo model

Description
Evaluates a mcmodel object (or a valid expression) using a specified number of simulations and with
(or without) a specified seed.

Usage
evalmcmod(expr, nsv=ndvar(), nsu=ndunc(), seed=NULL)

Arguments
expr A model of class mcmodel or a valid expression.
nsv The number of simulations in the dimension of variability used in the evaluation.
nsu The number of simulations in the dimension of uncertainty used in the evalua-
tion.
seed The random seed used for the evaluation. If ‘NULL’ the ‘seed’ is unchanged.
16 extractvar

Details
The model is evaluated. The intermediate variables used to build the ‘mc’ object are not stored.

Value
The results of the evaluation. It should be a ‘mc’ object.

Note
The seed is set at the beginning of the evaluation. Thus, the complete similarity of two evaluations
with similar seed is not certain, depending on the structure of your model.

See Also
mcmodel
evalmccut to evaluate high dimension Monte Carlo Model in a loop.

Examples
data(ec)
ec$modEC1
evalmcmod(ec$modEC1, nsv=100, nsu=100, seed=666)

extractvar Utilities for multivariate nodes

Description
‘extractvar’ extracts one variate from a multivariate node.
‘addvar’ adds consistent ‘mcnode’s to build a multivariate ‘mcnode’ .

Usage
extractvar(x, which=1)
addvar(...)

Arguments
x a multivariates ‘mcnode’.
which a vector. which variate(s) should be extracted?
... ‘mcnode’s to be gathered in a multivariate ‘mcnode’. These ‘mcnode’s should be
of same type and dimension.

Details
The ‘outm’ attribute of the output of ‘addvar’ will be the one of the first element.
hist.mc 17

Value
The new built ‘mcnode’.

See Also
mcnode for ‘mcnode’ objects.

Examples
x <- mcdata(0:3, "0", nvariates = 4)
y <- extractvar(x, c(1, 3))
y
addvar(x, y)

hist.mc Histogram of a Monte Carlo Simulation

Description
Shows histogram of a ‘mcnode’ or a ‘mc’ object.

Usage
## S3 method for class 'mc'
hist(x, griddim=NULL, xlab=names(x), ylab="Frequency", main="", ...)
## S3 method for class 'mcnode'
hist(x, ...)

Arguments
x An ‘mcnode’ or an ‘mc’ object.
griddim A vector of two integers, indicating the size of the grid of plots. If ‘NULL’, the
grid is calculated to produce a "nice" graph.
xlab A vector of labels for the x-axis for drawn histograms (those whose ‘outm(x)!="none"’).
May be recycled.
ylab A vector of labels for the y-axis for drawn histograms. May be recycled.
main A vector of main title of histograms for drawn histograms. May be recycled.
... Other arguments to be passed to all calls of ‘hist’.

Note
For Two-dimensional ‘mc’, the histogram is based on all data (variability and uncertainty) pooled
together.
18 is.mc

Examples
data(total)
hist(xVUM3)
hist(total)

is.mc Tests mc and mcnode Objects

Description

‘is.mc’ tests ‘mc’ objects and ‘is.mcnode’ tests ‘mcnode’ objects.

Usage

is.mc(x)
is.mcnode(x)

Arguments

x An ‘mc’ or a ‘mcnode’ object.

Details

‘is.mc’ tests if ‘x’ is a list of ‘mcnode’, each elements being of compatible dimension. It tests if the
class ‘"mc"’ is affected to the object.
‘is.mcnode’ tests if ‘x’ is an array of numeric or logical, if it has a "type" attribute and compatible
dimensions, and if the class ‘"mcnode"’ is affected to the object.

Value

‘TRUE’ or ‘FALSE’

Examples
data(total)
is.mcnode(xVU)
is.mcnode(total)
is.mc(total)
lhs 19

lhs Random Latin Hypercube Sampling

Description
Creates a Latin Hypercube Sample (LHS) of the specified distribution.

Usage
lhs(distr="runif", nsv=ndvar(), nsu=ndunc(), nvariates=1, ...)

Arguments
distr The function for generating random sample or its name. If ‘distr’ is "rdist",
the function "qdist" must be the quantile function of this distribution with argu-
ment ‘p’ as a vector of probabilities, as all univariates distributions of the ‘stat’
library.
nsv The number of raws of the final matrix.
nsu The number of columns of the final matrix
nvariates The number of variates
... All arguments to be passed to ‘distr’ except the size of the sample.

Value
A ‘nsv x nsu’ matrix of random variates.

Note
The resulting lhs is in fact a latin hypersquare sampling: the lhs is provided only in the first 2
dimensions.
It is not possible to send truncated distribution with rtrunc. Use mcstoc for this purpose, with
‘lhs=TRUE’ and ‘rtrunc=TRUE’.
The . . . arguments will be recycled.

Author(s)
adapted from a code of Rob Carnell (library ‘lhs’)

See Also
mcstoc

Examples
ceiling(lhs(runif, nsu=10, nsv=10)*10)
20 mc

mc Monte Carlo Object

Description
Creates ‘mc’ objects from mcnode or ‘mc’ objects.

Usage
mc(..., name=NULL, devname=FALSE)

Arguments
... ‘mcnode’ and/or ‘mc’ object(s) to be gathered in a ‘mc’ object separated by a
coma.
name Vector of character of the same length of the final ‘mc’ object. If NULL, the
name will be given from the name of the elements.
devname Develop the name from the name of the ‘mc’ objects, if any.

Details
A ‘mc’ object is a list of mcnode objects. ‘mcnode’ objects must be of coherent dimensions.
If one of the arguments is a ‘mc’ object, the name of the elements of this ‘mc’ object are used.
‘devname = TRUE’ will develop the name, using as a prefix the name of the ‘mc’ object.
Finally, names are transformed to be unique.

Value
An object of class ‘mc’.

See Also
mcnode, the basic element of a ‘mc’ object.
To evaluate ‘mc’ objects: mcmodel, evalmcmod, evalmccut
Informations about an ‘mc’ object: is.mc, dimmc
To study ‘mc’ objects: print.mc, summary.mc, plot.mc, converg, hist.mc, tornado, tornadounc.mc

Examples

x <- mcstoc(runif)
y <- mcdata(3, type="0")
z <- x * y
(m <- mc(x, y, z, name=c('n1', 'n2', 'n3')))
mc(m, x, devname=TRUE)
mc.control 21

mc.control Sets or Gets the Default Number of Simulations.

Description

Sets or retrieves the default number of simulations.

Usage

ndvar(n)
ndunc(n)

Arguments

n Number of simulations.

Details

‘ndvar()’ gets and ‘ndvar(n)’ sets the default number of simulation in the 1D simulations or the
number of simulation in the variability dimension in the 2D simulations.
‘ndunc()’ gets and ‘ndunc(n)’ sets the number of simulations in the uncertainty dimension in the
2D simulations.
‘n’ is rounded to its ceiling value.
The default values when loaded are 1001 for ‘ndvar’ and 101 for ‘ndunc’.

Value

The current value, AFTER modification if ‘n’ is present (!= ‘options’).

Examples
(oldvar <- ndvar())
(oldunc <- ndunc())
mcstoc(runif, type="VU")
ndvar(12)
ndunc(21)
mcstoc(runif, type="VU")
ndvar(oldvar)
ndunc(oldunc)
22 mcapply

mcapply Apply Functions Over mc or mcnode Objects

Description

Apply a function on all values or over a given dimension of an ‘mcnode’ object. May be used for
all ‘mcnode’ of an ‘mc’ object.

Usage

mcapply(x, margin=c("all", "var", "unc", "variates"), fun, ...)

Arguments

x A ‘mc’ or a ‘mcnode’ object.

margin The dimension on which applying the function. Maybe ‘"all"’ (default) to
apply the function on all values, ‘"var"’ to apply the function on the variabil-
ity dimension, ‘"unc"’ to apply the function on the uncertainty dimension, or
‘"variates"’ to apply the function on the variates. Watch out: do not use ’var’
for ’variates’
fun The function to be applied. When applied to a vector of length ‘n’, ‘fun’ should
return a vector of length ‘n’ or ‘1’.
... Optionnal arguments to ‘fun’.

Value

If ‘fun’ returns a function of length ‘n’ or if ‘margin="all"’, the returned ‘mcnode’s are of type
and dimension of ‘x’. In other cases, the type of ‘mcnode’ is changed.

See Also

apply, mc, mcnode.

Examples
data(total)
xVUM
mcapply(xVUM, "unc", sum)
mcapply(xVUM, "var", sum)
mcapply(xVUM, "all", sum)
mcapply(xVUM, "variates", sum)
mcapply(total, "all", exp)
mccut 23

mccut Evaluates a Two-Dimensional Monte Carlo Model in a Loop.

Description
‘evalmccut’ evaluates a Two-Dimensional Monte Carlo model using a loop on the uncertainty
dimension. Within each loop, it calculates statistics in the variability dimension and stores them for
further analysis. It allows to evaluate very high dimension models using (unlimited?) time instead
of (limited) memory.
‘mcmodelcut’ builds a ‘mcmodelcut’ object that can be sent to ‘evalmccut’.

Usage
evalmccut(model, nsv=ndvar(), nsu=ndunc(), seed=NULL, ind="index")
## S3 method for class 'mccut'
print(x, lim=c(0.025, 0.975), digits=3, ...)
mcmodelcut(x, is.expr=FALSE)

Arguments
model a ‘mcmodelcut’ object obtained using ‘mcmodelcut’ function or (directly) a
valid call including three blocks. See Details and Examples for the structure
of the call.
x a call or an expression (if ‘is.expr=TRUE’) including three blocks. See Details
and Examples for the structure of the call.
nsv The number of simulations for variability used in the evaluation.
nsu The number of simulations for uncertainty used in the evaluation.
seed The random seed used for the evaluation. If ‘NULL’ the ‘seed’ is unchanged.
ind The variable name used in ‘model’ to refers to the uncertainty. see Details and
Example.
is.expr ‘FALSE’ to send a call, ‘TRUE’ to send an expression (see mcmodel examples)
lim A vector of values used for the quantile function (uncertainty dimension).
digits Number of digits in the print.
... Additional arguments to be passed in the final print function.

Details
This function should be used for high dimension Two-Dimensional Monte-Carlo simulations, when
the memory limits of R are attained. The use of a loop will take (lots of) time, but less memory.
‘x’ (or ‘model’ if a call is used directly in ‘evalmccut’) should be built as three blocks, separated
by ‘{’.

1. The first block is evaluated once (and only once) before the first loop (step 1).
2. The second block, which should lead to an ‘mc’ object, is evaluated using ‘nsu = 1’ (step 2).
24 mccut

3. The third block is evaluated on the ‘mc’ object. All resulting statistics are stored (step 3).
4. The steps 2 and 3 are repeated ‘nsu’ times. At each iteration, the values of the loop index
(from 1 to ‘nsu’) is given to the variable specified in ‘ind’.
5. Finally, the ‘nsu’ statistics are returned in an invisible object of class ‘mccut’.

Understanding this, the call should be built like this: ‘{{block 1}{block 2}{block 3}}’

1. The first block (maybe empty) is an expression that will be evaluated only once. This block
should evaluate all ‘"V" mcnode’ and ‘"0" mcnode’s. It may evaluate and ‘"U" mcnode’
that will be sent in the second and third block by column, and, optionnaly, some other codes
(even ‘"VU" mcnode’, sent by columns) that can not be evaluated if ‘ndunc=1’ (e.g. sampling
without replacement in the uncertainty dimension).
2. The second block is an expression that leads to the ‘mc’ object. It must end with an expres-
sion as ‘mymc <- mc(...)’. The variable specified as ‘ind’ may be helpful to refer to the
uncertainty dimension in this step
3. The last block should build a list of statistics refering to the ‘mc’ object. The function ‘summary’
should be used if a summary, a tornado on uncertainty (tornadounc.mccut) or a convergence
diagnostic converg is needed, the function plot.mc should be used if a plot is needed, the
function tornado should be used if a tornado is needed. Moreover, any other function that
leads to a vector, a matrix, or a list of vector/matrix of statistics evaluated from the ‘mc’ object
may be used. list are time consuming.

IMPORTANT WARNING: do not forget to affect the results, since the print method provide only a
summary of the results while all data may be stored in an ‘mccut’ object.

Value
An object of class ‘mccut’. This is a list including statistics evaluated within the third block. Each
list consists of all the ‘nsu’ values obtained. The ‘print.mccut’ method print the median, the
mean, the ‘lim’ quantiles estimated on each statistics on the uncertainty dimension.

Note
The methods and functions available on the ‘mccut’ object is function of the statistics evaluated
within the third block:

• a print.mccut is available as soon as one statistic is evaluated within the third block;
• a summary.mccut and a tornadounc.mccut are available if a summary.mc is evaluated within
the third block;
• converg may be used if a summary.mc is evaluated within the third block;
• a plot.mccut is available if a plot.mc is evaluated within the third block. (Do not forget to
use the argument ‘draw = FALSE’ in the third block);
• a tornado is available if a tornado is evaluated within the third block.

The seed is set at the beginning of the evaluation. Thus, the complete similarity of two evaluations is
not certain, depending of the structure of your model. Moreover, with a similar seed, the simulation
will not be equal to the one obtained with evalmcmod since the random samples will not be obtained
in the same order.
mcmodel 25

In order to avoid conflicts between the ‘model’ evaluation and the function, the function uses upper
case variables. Do not use upper case variables in your model.
The function should be re-adapted if a new function to be applied on ‘mc’ objects is written.

See Also
evalmcmod

Examples
modEC3 <- mcmodelcut({

## First block:
## Evaluates all the 0, V and U nodes.
{ cook <- mcstoc(rempiricalD, type = "V", values = c(0, 1/5,
1/50), prob = c(0.027, 0.373, 0.6))
serving <- mcstoc(rgamma, type = "V", shape = 3.93, rate = 0.0806)
conc <- mcstoc(rnorm, type = "U", mean = 10, sd = 2)
r <- mcstoc(runif, type = "U", min = 5e-04, max = 0.0015)
}
## Second block:
## Evaluates all the VU nodes
## Leads to the mc object.
{
expo <- conc * cook * serving
dose <- mcstoc(rpois, type = "VU", lambda = expo)
risk <- 1 - (1 - r)^dose
res <- mc(conc, cook, serving, expo, dose, r, risk)
}
## Third block:
## Leads to a list of statistics: summary, plot, tornado
## or any function leading to a vector (et), a list (minmax),
## a matrix or a data.frame (summary)
{
list(
sum = summary(res),
plot = plot(res, draw=FALSE),
minmax = lapply(res, range)
)
}
})

x <- evalmccut(modEC3, nsv = 101, nsu = 101, seed = 666)

summary(x)

mcmodel Monte Carlo model

26 mcmodel

Description

Specify a ‘mcmodel’, without evaluating it, for a further evaluation using evalmcmod.

Usage

mcmodel(x, is.expr=FALSE)

Arguments

x An R call or an expression.
is.expr ‘FALSE’ to send a call, ‘TRUE’ to send an expression (see Examples)

Details

The model should be put between ‘{’ and the last line should be of the form ‘mc(...)’. Any
reference to the number of simulation in the dimension of variability should be done via ‘ndvar()’
or (preferred) ‘nsv’. Any reference to the number of simulations in the dimension of uncertainty
should be done via ‘ndunc()’ or (preferred) ‘nsu’.

Value

an R expression, with class ‘mcmodel’

See Also

expression.
evalmcmod to evaluate the model.
mcmodelcut to evaluate high Dimension Monte Carlo Model in a loop.

Examples

modEC1 <- mcmodel({

conc <- mcdata(10, "0")
cook <- mcstoc(rempiricalD, values=c(0, 1/5, 1/50), prob=c(0.027, 0.373, 0.600))
serving <- mcstoc(rgamma, shape=3.93, rate=0.0806)
expo <- conc * cook * serving
dose <- mcstoc(rpois, lambda=expo)
risk <- 1-(1-0.001)^dose
mc(conc, cook, serving, expo, dose, risk)
})
evalmcmod(modEC1, nsv=100, nsu=100)
mcnode 27

mcnode Build mcnode Objects from Data or other mcnode Objects

Description
Creates a ‘mcnode’ object from a vector, an array or a ‘mcnode’.

Usage
mcdata(data, type=c("V", "U", "VU", "0"), nsv=ndvar(), nsu=ndunc(),
nvariates=1, outm="each")
mcdatanocontrol(data, type=c("V", "U", "VU", "0"), nsv=ndvar(), nsu=ndunc(),
nvariates=1, outm="each")

Arguments
data The numeric/logical vector/matrix/array of data or the ‘mcnode’ object.
type The type of node to be built. By default, a ‘"V"’ node.
nsv The variability dimension (‘type="V"’ or ‘type="VU"’) of the node. By default:
the current value in mc.control
nsu The uncertainty dimension (‘type="U"’ or ‘type="VU"’) of the node. By de-
fault: the current value in mc.control
nvariates The number of variates. By default: 1
outm The output of the ‘mcnode’ for multivariates nodes. May be "each" (default)
if output should be provided for each variates considered independently, "none"
for no output or a vector of name of function(s) (as a character string) that will be
applied on the variates dimension before any output (ex: ‘"mean"’, ‘"median"’,
‘c("min", "max")’). The function should have no other arguments and send
one value per vector of values (ex. do not use ‘"range"’). Note that the ‘outm’
attribute may be changed at any time using the outm function.

Details
A ‘mcnode’ object is the basic element of a mc object. It is an array of dimension ‘(nsv x nsu x nvariates)’.
Four types of ‘mcnode’ exists:

• ‘"V" mcnode’, for "Variability", are arrays of dimension ‘(nsv x 1 x nvariates)’. The
alea in the data should reflect variability of the parameter.
• ‘"U" mcnode’, for "Uncertainty", are arrays of dimension ‘c(1 x nsu x nvariates)’. The
alea in the data should reflect uncertainty of the parameter.
• ‘"VU" mcnode’, for "Variability and Uncertainty", are arrays of dimension ‘(nsv x nsu x nvariates)’.
The alea in the data reflects separated variability (in rows) and uncertainty (in columns) of the
parameter.
• ‘"0" mcnode’, for "Neither Variability or Uncertainty", are arrays of dimension ‘(1 x 1 x nvariates)’.
No alea is considered for these nodes. ‘"0" mcnode’ are not necessary in the univariate context
(use scalar instead) but may be useful for operations on multivariate nodes.
28 mcnode

Multivariate nodes (i.e. ‘nvariates != 1’) should be used for multivariate distributions imple-
mented in ‘mc2d’ (rmultinomial, rmultinormal, rempiricalD and rdirichlet).
For security, recycling rules are limited to fill the array using ‘data’. The general rules is that
recycling is only permitted to fill a dimension from 1 to the final size of the dimension.
If the final dimension of the node is ‘(nsv x nsu x nvariates)’ (with ‘nsv = 1’ and ‘nsu = 1’
for ‘"0"’ nodes, ‘nsu = 1’ for ‘"V"’ nodes and ‘nsv = 1’ for ‘"U"’ nodes), ‘mcdata’ accepts :

• Vectors of length ‘1’ (recycled on all dimensions), vectors of length ‘(nsv * nsu)’ (filling first
the dimension of variability, then the dimension of uncertainty then recycling on nvariates), or
vectors of length ‘(nsv * nsu * nvariates)’ (filling first the dimension of variability, then
the uncertainty, then the variates).
• Matrixes of dimensions ‘(nsv x nsu)’, recycling on variates.
• Arrays of dimensions ‘(nsv x nsu x nvariates)’ or ‘(nsv x nsu x 1)’, recycling on
variates.
• For ‘data’ as ‘mcnode’, recycling is dealt to proper fill the array:
1. a ‘"V"’ node accepts a ‘"0"’ node of dimension ‘(1 x 1 x nvariates)’ (recycling on
variability) or of dimension ‘(1 x 1 x 1)’ (recycling on variability and variates), or a
‘"V"’ node of dimension ‘(nsv x 1 x nvariates)’ or ‘(nsv x 1 x 1)’ (recycling on
variates),
2. a ‘"U"’ node accepts a ‘"0"’ node of dimension ‘(1 x 1 x nvariates)’ (recycling on
uncertainty) or of dimension ‘(1 x 1 x 1)’ (recycling on uncertainty and variates), or a
‘"U"’ node of dimension ‘(1 x nsu x nvariates)’, or ‘(1 x nsu x 1)’ (recycling on
variates),
3. a ‘"VU"’ node accepts a ‘"0"’ node of dimension ‘(1 x 1 x nvariates)’ (recycling on
varaiability and uncertainty) or of dimension ‘(1 x 1 x 1)’ (recycling on variability, un-
certainty and variates), a ‘"U"’ node of dimension ‘(1 x nsu x nvariates)’(recycling
"by row" on the variability dimension), or of dimension ‘(1 x nsu x 1)’(recycled
"by row" on the variability dimension then on variates), a ‘"V"’ node of dimension
‘(nsv x 1 x nvariates)’(recycling on the uncertainty dimension) or of dimension
‘(nsv x 1 x 1)’(recycled on the uncertainty dimension then on variates), and a ‘"VU"’
node of dimension ‘(nsv x nsu x nvariates)’ or of dimension ‘(nsv x nsu x 1)’
(recycling on variates).
4. a ‘"0"’ node accepts a ‘"0"’ node of dimension ‘(1 x 1 x nvariates)’ or ‘(1 x 1 x 1)’
(recycling on variates).

‘mcdatanocontrol’ is a dangerous version of ‘mcnode’ which forces the dimension of data to be

‘(nsv x nsu x nvariates)’ and gives the atributes and the class without any control. This
function is useful when your model is tested since it is much more quicker.

Value
An ‘mcnode’ object.

See Also
mcstoc to build a stochastic ‘mcnode’ object, mcprobtree to build a stochastic node fro a probabil-
ity tree.
mcnode 29

Ops.mcnode for operations on ‘mcnode’ objects.

mc to build a Monte-Carlo object.
Informations about an mcnode: is.mcnode, dimmcnode, typemcnode.
To build a correlation structure between ‘mcnode’: cornode.
To study ‘mcnode’ objects: print.mcnode, summary.mcnode, plot.mcnode, converg, hist.mcnode
To modify ‘mcnode’ objects: NA.mcnode

Examples
oldvar <- ndvar()
oldunc <- ndunc()
ndvar(3)
ndunc(5)

(x0 <- mcdata(100, type="0"))

mcdata(matrix(100), type="0")

(xV <- mcdata(1:ndvar(), type="V"))

mcdata(matrix(1:ndvar(), ncol=1), type="V")

(xU <- mcdata(10*1:ndunc(), type="U"))

mcdata(matrix(10*1:ndunc(), nrow=1), type="U")

(xVU <- mcdata(1:(ndvar()*ndunc()), type="VU"))

mcdata(matrix(1:(ndvar()*ndunc()), ncol=5, nrow=3), type="VU")

##Do not use

## Not run:
mcdata(matrix(1:5, nrow=1), type="VU")

## End(Not run)
##use instead
mcdata(mcdata(matrix(1:ndunc(), nrow=1), type="U"), "VU")
##or
mcdata(matrix(1:ndunc(), nrow=1), type="U") + mcdata(0, "VU")

mcdata(x0, type="0")

mcdata(x0, type="V")
mcdata(xV, type="V")

mcdata(x0, type="U")
mcdata(xU, type="U")

mcdata(x0, type="VU")
mcdata(xU, type="VU")
mcdata(xV, type="VU")

##Multivariates
(x0M <- mcdata(1:2, type="0", nvariates=2))
mcdata(1, type="0", nvariates=2)
30 mcprobtree

(xVM <- mcdata(1:(2*ndvar()), type="V", nvariates=2))

mcdata(1:ndvar(), type="V", nvariates=2)
mcdata(array(1:(2*ndvar()), dim=c(3, 1, 2)), type="V", nvariates=2)

mcdata(1, type="V", nvariates=2)

mcdata(x0, type="V", nvariates=2)
mcdata(x0M, type="V", nvariates=2)
mcdata(xV, type="V", nvariates=2)
mcdata(xVM, type="V", nvariates=2)

(xUM <- mcdata(10*(1:(2*ndunc())), type="U", nvariates=2))

mcdata(array(10*(1:(2*ndunc())), dim=c(1, 5, 2)), type="U", nvariates=2)

mcdata(1, type="U", nvariates=2)

mcdata(x0, type="U", nvariates=2)
mcdata(x0M, type="U", nvariates=2)
mcdata(xU, type="U", nvariates=2)
mcdata(xUM, type="U", nvariates=2)

(xVUM <- mcdata(1:(ndvar()*ndunc()), type="VU", nvariates=2))

mcdata(array(1:(ndvar()*ndunc()), dim=c(3, 5, 2)), type="VU", nvariates=2)

mcdata(1, type="VU", nvariates=2)

mcdata(x0, type="VU", nvariates=2)
mcdata(x0M, type="VU", nvariates=2)
mcdata(xV, type="VU", nvariates=2)
mcdata(xVM, type="VU", nvariates=2)
mcdata(xU, type="VU", nvariates=2)
mcdata(xUM, type="VU", nvariates=2)
mcdata(xVU, type="VU", nvariates=2)
mcdata(xVUM, type="VU", nvariates=2)

ndvar(oldvar)
ndunc(oldunc)

mcprobtree Creates a Stochastic mcnode Object using a Probability Tree

Description
This function builds an ‘mcnode’ as a mixture ‘mcnode’ objects.

Usage
mcprobtree(mcswitch, mcvalues, type=c("V", "U", "VU", "0"), nsv=ndvar(),
nsu=ndunc(), nvariates=1, outm="each", seed=NULL)
mcprobtree 31

Arguments
mcswitch A vector of probabilities/weights or an ‘mcnode’.
mcvalues A named list of ‘mcnode’s, ‘mcdata’ functions or ‘mcstoc’ functions, or a com-
bination of those objects. Each element should be or lead to a compatible
‘mcnode’ (see Details).
type The type of ‘mcnode’ to be built. By default, a ‘"V"’ node. see mcnode for
details.
nsv The number of simulations in the variability dimension of the final node.
nsu The number of simulations in the uncertainty dimension of the final node.
nvariates The number of variates of the final ‘mcnode’.
outm The default output of the ‘mcnode’ for multivariates nodes. see outm.
seed The random seed used for the evaluation. If ‘NULL’ the ‘seed’ is unchanged.

Details
‘mcswitch’ may be either:

• a vector of weights. They need not sum to one, but they should be nonnegative and not all
zero. The length of this vector should equal the number of elements in the list ‘mcvalues’.
Each elements of ‘mcvalues’ will appear in the final sample a random number of times with
probability as specified by this vector.
• a ‘"0 mcnode"’ to build any type of node.
• a ‘"V mcnode"’ to build a ‘"V mcnode"’ or a ‘"VU mcnode"’.
• a ‘"U mcnode"’ to build a ‘"U mcnode"’ or a ‘"VU mcnode"’.
• a ‘"VU mcnode"’ to build a ‘"VU mcnode"’.

Each elements of ‘mcvalues’ may be either:

• a ‘"0 mcnode"’ to build any type of node.

• a ‘"V mcnode"’ to build a ‘"V mcnode"’ or a ‘"VU mcnode"’.
• a ‘"U mcnode"’ to build a ‘"U mcnode"’ or a ‘"VU mcnode"’.
• a ‘"VU mcnode"’ to build a ‘"VU mcnode"’.

Their name should correspond to the values in ‘mcswitch’, specified as character (See Examples).
These elements will be evaluated only if needed : if the corresponding value is not present in
‘mcswitch’, the element will not be evaluated.

Value
An ‘mcnode’ object.

See Also
mcdata, mcstoc, switch.
32 mcratio

Examples
## A mixture of normal (prob=0.75), uniform (prob=0.20) and constant (prob=0.05)
conc1 <- mcstoc(rnorm, type="VU", mean=10, sd=2)
conc2 <- mcstoc(runif, type="VU", min=-6, max=-5)
conc3 <- mcdata(0, type="VU")

## Randomly in the cells

whichdist <- mcstoc(rempiricalD, type="VU", values=1:3, prob= c(.75, .20, .05))
mcprobtree(whichdist, list("1"=conc1, "2"=conc2, "3"=conc3), type="VU")
## Which is equivalent to
mcprobtree(c(.75, .20, .05), list("1"=conc1, "2"=conc2, "3"=conc3), type="VU")
## Not that there is no control on the exact number of occurences.

## Randomly by colums (Uncertainty)

whichdist <- mcstoc(rempiricalD, type="U", values=1:3, prob= c(.75, .20, .05))
mcprobtree(whichdist, list("1"=conc1, "2"=conc2, "3"=conc3), type="VU")

## Randomly by line (Variability)

whichdist <- mcstoc(rempiricalD, type="V", values=1:3, prob= c(.75, .20, .05))
mcprobtree(whichdist, list("1"=conc1, "2"=conc2, "3"=conc3), type="VU")

## The elements of mcvalues may be of various (but compatible) type

conc1 <- mcstoc(rnorm, type="V", mean=10, sd=2)
conc2 <- mcstoc(runif, type="U", min=-6, max=-5)
conc3 <- mcdata(0, type="0")
whichdist <- mcstoc(rempiricalD, type="VU", values=1:3, prob= c(.75, .20, .05))
mcprobtree(whichdist, list("1"=conc1, "2"=conc2, "3"=conc3), type="VU")

mcratio Ratio of uncertainty and the variability

Description
Provides measures of variability, uncertainty, and both combined for an ‘mc’ or an ‘mcnode’ object.

Usage
mcratio(x, pcentral=.5, pvar=.975, punc=.975, na.rm=FALSE)

Arguments
x an ‘mc’ or an ‘mcnode’ object
pcentral the quantile for the central tendency. .
pvar the quantile for the measure of variability.
punc the quantile for the measure of uncertainty.
na.rm a logical value indicating whether NA values should be stripped before the com-
putation proceeds.
mcstoc 33

Details
The function evaluates three ratios for each ‘mcnode’. Given:

A the ‘(100 * pcentral)’th percentile of uncertainty (by default the median) for the ‘(100 * pcentral)’th
percentile of variability
B the ‘(100 * pcentral)’th percentile of uncertainty for the ‘(100 * pvar)’th percentile of
variability
C the ‘(100 * punc)’th percentile of uncertainty for the ‘(100 * pcentral)’th percentile of
variability
D the ‘(100 * punc)’th percentile of uncertainty for the ‘(100 * pvar)’th percentile of variability

The following ratio are estimated

• Variability Ratio: B / A
• Uncertainty Ratio: C / A
• Overall Uncertainty Ratio: D / A

For multivariate nodes, the statistics are evaluate on each dimension or on statistics according to the
corresponding ‘outm’ value.

Value
A matrix.

References
Ozkaynak, H., Frey, H.C., Burke, J., Pinder, R.W. (2009) "Analysis of coupled model uncertainties
in source-to-dose modeling of human exposures to ambient air pollution: A PM2.5 case study",
Atmospheric environment, Volume 43, Issue 9, March 2009, Pages 1641-1649.

Examples
data(total)
mcratio(total, na.rm=TRUE)

mcstoc Creates Stochastic mcnode Objects

Description
Creates a mcnode object using a random generating function.

Usage
mcstoc(func=runif, type=c("V", "U", "VU", "0"), ..., nsv=ndvar(),
nsu=ndunc(), nvariates=1, outm="each", nsample="n",
seed=NULL, rtrunc=FALSE, linf=-Inf, lsup=Inf, lhs=FALSE)
34 mcstoc

Arguments
func A function providing random data or its name as character.
type The type of ‘mcnode’ to be built. By default, a ‘"V"’ node. see mcnode for
details.
... All other arguments but the size of the sample to be passed to ‘func’. These
arguments should be vectors or ‘mcnode’s (arrays prohibited).
nsv The number of simulations in the variability dimension.
nsu The number of simulations in the uncertainty dimension.
nvariates The number of variates of the output.
outm The output of the ‘mcnode’ for multivariates nodes. May be "each" (default) if
an output should be provided for each variates considered independently, "none"
for no output or a vector of functions (as a character string) that will be ap-
plied on the variates dimension before any output (ex: ‘"mean"’, ‘"median"’,
‘c("min","max")’). Each function should return 1 value when applied to 1
value (ex. do not use ‘"range"’). Note that the ‘outm’ attribute may be changed
further using the outm function.
nsample The name of the parameter of the function giving the size of the vector. By
default, ‘n’, as in most of the random sampling distributions of the ‘stats’
library (with the exceptions of ‘rhyper’ and ‘rwilcox’ where ‘nsample="nn"’
should be used).
seed The random seed used for the evaluation. If ‘NULL’ the ‘seed’ is unchanged.
rtrunc Should the distribution be truncated? See rtrunc.
linf If truncated: lower limit. May be a scalar, an array or a mcnode.
lsup If truncated: upper limit. May be a scalar, an array or a mcnode. ‘lsup’ should
be pairwise strictly greater then ‘linf’
lhs Should a Random Latin Hypercube Sampling be used? see lhs

Details
Note that arguments after . . . must match exactly.
Any function who accepts vectors/matrix as arguments may be used (notably: all current random
generator of the ‘stats’ package). The arguments may be sent classically but it is STRONGLY
recommended to use consistant ‘mcnode’s if arguments should be recycled, since a very complex
recycling is handled for ‘mcnode’ and not for vectors. The rules for compliance of ‘mcnode’ argu-
ments are as following (see below for special functions):

type="V" accepts ‘"0" mcnode’ of dimension ‘(1 x 1 x nvariates)’ or of dimension ‘(1 x 1 x 1)’
(recycled) and ‘"V" mcnode’ of dimension ‘(nsv x 1 x nvariates)’ or ‘(nsv x 1 x 1)’
(recycled).
type="U" accepts ‘"0" mcnode’ of dimension ‘(1 x 1 x nvariates)’ or of dimension ‘(1 x 1 x 1)’
(recycled) and ‘"U" mcnode’ of dimension ‘(1 x nsu x nvariates)’ or of dimension
‘(1 x nsu x 1)’ (recycled).
mcstoc 35

type="VU" accepts ‘"0" mcnode’ of dimension ‘(1 x 1 x nvariates)’ or of dimension

‘(1 x 1 x 1)’ (recycled), ‘"V" mcnode’ of dimension ‘(nsv x 1 x nvariates)’
(recycled classicaly) or ‘(nsv x 1 x 1)’ (recycled classically), ‘"U" mcnode’ of dimen-
sion ‘(1 x nsu x nvariates)’ (recycled by rows) or ‘(1 x nsu x 1)’ (recycled by
row on the uncertainty dimension and classicaly on variates), ‘"VU" mcnode’ of dimension
‘(nsv x nsu x nvariates)’ or of dimension ‘(nsv x nsu x 1)’ (recycled).
type="0" accepts ‘"0" mcnode’ of dimension ‘(1 x 1 x nvariates)’ or ‘(1 x 1 x 1)’
(recycled).

Multivariate nodes and multivariate distributions:

The number of variates should be provided (not guesses by the function). A multivariates node may
be built using a univariate distribution and ‘nvariates!=1’. See examples.
rdirichlet needs for ‘alpha’ a vector or a multivariates nodes and returns a multivariate node.
rmultinomial needs for ‘size’ and ‘prob’ vectors and/or multivariate nodes and return a univariate
or a multivariate node. rmultinormal needs for ‘mean’ and ‘sigma’ vectors and/or multivariate
nodes and return a multivariate node. rempiricalD needs for ‘values’ and ‘prob’ vectors and/or
multivariate nodes and return a a univariate or a multivariate node. See examples.
‘trunc=TRUE’ is valid for univariates distributions only. The distribution will be truncated on
‘(linf, lsup]’. The function ’func’ should have a ’q’ form (with first argument ’p’) and a ’p’
form, as all current random generator of the ‘stats’ library. Example : ’rnorm’ (has a ’qnorm’ and
a ’pnorm’ form), ’rbeta’, ’rbinom’, ’rgamma’, ...
If ‘lhs=TRUE’, a Random Hypercube Sampling will be used on ‘nsv’ and ‘nsu’ The function ’func’
should have a ’q’ form (with argument ’p’). ‘lhs=TRUE’ is thus not allowed on multivariates distri-
butions.

Value
An ‘mcnode’ object.

See Also
mcnode for a description of ‘mcnode’ object, methods and functions on ‘mcnode’ objects.
Ops.mcnode for operations on ‘mcnode’ objects. rtrunc for important warnings on the use of the
‘trunc’ option.

Examples
Oldnvar <- ndvar()
Oldnunc <- ndunc()
ndvar(5)
ndunc(4)

## compatibility with mcdata as arguments

x0 <- mcstoc(runif, type="0")
xV <- mcstoc(runif, type="V")
xU <- mcstoc(runif, type="U")
xVU <- mcstoc(runif, type="VU")
36 mcstoc

## "0" accepts mcdata "0"

mcstoc(runif, type="0", min=-10, max=x0)

## "V" accepts "0" mcdata and "V" mcdata

mcstoc(rnorm, type="V", mean=x0, sd=xV)

## "U" accepts "0" mcdata and "U" mcdata

mcstoc(rnorm, type="U", mean=x0, sd=xU)

## "VU" accepts "0" mcdata, "U" mcdata

## "V" mcdata and "U" mcdata with correct recycling
mcstoc(rnorm, type="VU", mean=x0, sd=xVU)
mcstoc(rnorm, type="VU", mean=xV, sd=xU)

## any function giving a set (vector/matrix) of value of length 'size' works

f <- function(popi) 1:popi
mcstoc(f, type="V", nsample="popi")

##Multivariates

ndvar(2)
ndunc(5)
##Build a multivariate node with univariate distribution
mcstoc(rnorm, "0", nvariates=3)
mcstoc(rnorm, "V", nvariates=3)
mcstoc(rnorm, "U", nvariates=3)
mcstoc(rnorm, "VU", nvariates=3)

##Build a multivariate node with multivariates distribution

alpha <- mcdata(c(1, 1000, 10, 100, 100, 10, 1000, 1), "V", nvariates=4)
(p <- mcstoc(rdirichlet, "V", alpha=alpha, nvariates=4))
mcstoc(rmultinomial, "VU", size=10, p, nvariates=4)

##Build a univariates node with "multivariates" distribution

size <- mcdata(c(1:5), "U")
mcstoc(rmultinomial, "VU", size, p, nvariates=1) #since a multinomial return one value

##Build a multivariates node with "multivariates" distribution

mcstoc(rmultinomial, "VU", size, p, nvariates=4) #sent 4 times to fill the array

##Use of rempiricalD with nodes

##A bootstrap
ndunc(5)
ndvar(5)
dataset <- c(1:9)
(b <- mcstoc(rempiricalD, "U", nvariates=9, values=dataset))
unclass(b)
##Then we build a VU node by sampling in each set of bootstrap
(node <- mcstoc(rempiricalD, "VU", values=b))
unclass(node)

## truncated
ndvar(2)
multinormal 37

ndunc(5)
linf <- mcdata(-1:3, "U")
x <- mcstoc(rnorm, "VU", rtrunc=TRUE, linf=linf)
unclass(round(x))
linf <- mcdata(1:5, "U")
mcstoc(rnorm, "VU", nsv=100, rtrunc=TRUE, linf=linf, lhs=TRUE)

ndvar(Oldnvar)
ndunc(Oldnunc)

multinormal The Vectorized Multivariate Random Deviates

Description
This function is the vectorized version of the ‘rmvnorm’ from the ‘mvtnorm’ library. It provides a
random number generator for the multivariate normal distribution with varying vectors of means
and varying covariance matrixes.

Usage
rmultinormal(n, mean, sigma, method=c("eigen", "svd", "chol"))
dmultinormal(x, mean, sigma, log=FALSE)

Arguments
x Vector or matrix of quantiles. If x is a matrix, each row is taken to be a quantile.
n Number of observations. If ‘length(n) > 1’, the length is taken to be the
number required.
mean Vector or matrix of means. If a matrix, each row is taken to be a quantile. Default
is a vector of 0 of convenient length.
sigma Covariance vector corresponding to the coercion of the covariance matrix into
a vector (if unique for all ‘n’ or ‘x’) or array of covariance vectors (if varying
according to ‘n’ or ‘x’). default is a diagonal matrix of convenient sizee.
method Matrix decomposition used to determine the matrix root of sigma, possible
methods are eigenvalue decomposition ("eigen", default), singular value decom-
position ("svd"), and Cholesky decomposition ("chol").
log Logical; if ‘TRUE’, densities d are given as log(d).

Details
‘rmvnorm(n, m, s)’ is equivalent to ‘rmultinormal(n, m,as.vector(s))’. ‘dmvnorm(x, m, s)’
is equivalent to ‘dmultinormal(x, m, as.vector(s))’.
If ‘mean’ and/or ‘sigma’ is a matrix, the first random deviate will use the first row of ‘mean’ and/or
‘sigma’, the second random deviate will use the second row of ‘mean’ and/or ‘sigma’, ... recycling
being permitted by raw. If ‘mean’ is a vector of length ‘l’ or is a matrix with ‘l’ columns, ‘sigma’
should be a vector of length ‘l x l’ or a matrix of number of ‘l x 2’ columns.
38 NA.mcnode

Note
The use of a varying sigma may be very time consumming.

Examples
## including equivalence with dmvnorm
## mean and sigma as vectors
(mean <- c(10, 0))
(sigma <- matrix(c(1, 2, 2, 10), ncol=2))
sigma <- as.vector(sigma)
(x <- matrix(c(9, 8, 1, -1), ncol=2))
round(rmultinormal(10, mean, sigma))
dmultinormal(x, mean, sigma)
## Eq
dmvnorm(x, mean, matrix(sigma, ncol=2))

## mean as matrix
(mean <- matrix(c(10, 0, 0, 10), ncol=2))
round(rmultinormal(10, mean, sigma))
dmultinormal(x, mean, sigma)
## Eq
dmvnorm(x[1, ], mean[1, ], matrix(sigma, ncol=2))
dmvnorm(x[2, ], mean[2, ], matrix(sigma, ncol=2))

## sigma as matrix
(mean <- c(10, 0))
(sigma <- matrix(c(1, 2, 2, 10, 10, 2, 2, 1), nrow=2, byrow=TRUE))
round(rmultinormal(10, mean, sigma))
dmultinormal(x, mean, sigma)
## Eq
dmvnorm(x[1, ], mean, matrix(sigma[1, ], ncol=2))
dmvnorm(x[2, ], mean, matrix(sigma[2, ], ncol=2))

## mean and sigma as matrix

(mean <- matrix(c(10, 0, 0, 10), ncol=2))
(sigma <- matrix(c(1, 2, 2, 10, 10, 2, 2, 1), nrow=2, byrow=TRUE))
round(rmultinormal(10, mean, sigma))
dmultinormal(x, mean, sigma)
## Eq
dmvnorm(x[1, ], mean[1, ], matrix(sigma[1, ], ncol=2))
dmvnorm(x[2, ], mean[2, ], matrix(sigma[2, ], ncol=2))

(mean <- c(10, 0))

(sigma <- matrix(c(1, 2, 2, 10, 10, 2, 2, 1), nrow=2, byrow=TRUE))
x <- rmultinormal(1000, mean, sigma)
plot(x)

NA.mcnode Finite, Infinite, NA and NaN Numbers in mcnode.

Ops.mcnode 39

Description
‘is.na’, ‘is.nan’, ‘is.finite’ and ‘is.infinite’ return a logical ‘mcnode’ of the same dimen-
sion as ‘x’.

Usage
## S3 method for class 'mcnode'
is.na(x)
## S3 method for class 'mcnode'
is.nan(x)
## S3 method for class 'mcnode'
is.finite(x)
## S3 method for class 'mcnode'
is.infinite(x)

Arguments
x A ‘mcnode’ object.

Value
A logical ‘mcnode’ object.

See Also
is.finite, NA

Examples
x <- log(mcstoc(rnorm, nsv=1001))
x
is.na(x)

Ops.mcnode Operations on mcnode Objects

Description
This function alters the way operations are performed on ‘mcnode’ objects for a better consistancy
of the theory.

Usage
## S3 method for class 'mcnode'
Ops(e1, e2)
40 Ops.mcnode

Arguments

e1 An ‘mcnode’ object, a vector or an array.

e2 An optionnal ‘mcnode’ object, a vector or a matrix with at least one of both
objects as an ‘mcnode’.

Details

This method will be used for any of the Group Ops functions.
The rules are as following (illustrated with a ‘+’ function and ignoring the ‘nvariates’ dimension):

• ‘0 + 0 = 0’;
• ‘0 + V = V’: classical recycling of the scalar;
• ‘0 + U = U’: classical recycling of the scalar;
• ‘0 + VU = VU’: classical recycling of the scalar;
• ‘V + V = V’: if both of the same ‘(nsv)’ dimension;
• ‘V + U = VU’: the ‘U’ object will be recycled "by row". The ‘V’ object will be recycled
classically "by column";
• ‘V + VU = VU’: if the dimension of the ‘V’ is ‘(nsv)’ and the dimension of the ‘VU’ is ‘(nsv x
nsu)’. The ‘V’ object will be recycled classically "by column";
• ‘U + U = U’: if both of the same ‘(nsu)’ dimension;
• ‘U + VU = VU’: if the dimension of the ‘U’ is ‘(nsu)’ and the dimension of the ‘VU’ is ‘(nsv x
nsu)’. The ‘U’ object will be recycled "by row";
• ‘VU + VU = VU’: if the dimension of the ‘VU’ nodes is ‘(nsu x nsv)’;

A vector or an array may be combined with an ‘mcnode’ of size ‘(nsv x nsu)’ if an ‘mcnode’ of
this dimension may be built from this vector/array using the ‘mcdata’ function. See mcdata for the
rules.
The ‘outm’ attribute is transferred as following: ‘each + each = each’; ‘none + other = other’;
‘other1 + other2 = other1’. The ‘outm’ attribute of the resulting node may be changed using
the outm function.
For multivariate nodes, a recycling on the ‘nvariates’ dimension is done if a ‘(nsu x nsv x nvariates)’
node is combined with a ‘(nsu x nsv x 1)’ node.

Value

The results as a ‘mcnode’ object.

See Also

mcdata, mcstoc
Ops.mcnode 41

Examples

oldvar <- ndvar()

oldunc <- ndunc()
ndvar(30)
ndunc(20)

## Given
x0 <- mcdata(3, type="0")
xV <- mcdata(1:ndvar(), type="V")
xU <- mcdata(1:ndunc(), type="U")
xVU <- mcdata(1:(ndunc()*ndvar()), type="VU")
x0M <- mcdata(c(5, 10), type="0", nvariates=2)
xVM <- mcdata(1:(2*ndvar()), type="V", nvariates=2)
xUM <- mcdata(1:(2*ndunc()), type="U", nvariates=2)
xVUM <- mcdata(1:(2*(ndunc()*ndvar())), type="VU", nvariates=2)

## All possible combinations

## "0"
-x0
x0 + 3

## "V"
-xV
3 + xV
xV * (1:ndvar())
xV * x0
xV - xV

## "U"
-xU
xU + 3
(1:ndunc()) * xU
xU * x0
xU - xU

## Watch out the resulting type

xV + xU
xU + xV

## "VU"
-xVU
3 + xVU
(1:(ndunc()*ndvar())) * xVU
xVU + xV
x0 + xVU
xU + xVU
xVU - xVU

## Some Multivariates
x0M+3
xVM * (1:ndvar())
xVM - xV
42 outm

xUM - xU
xVUM - xU

outm Output of Nodes

Description

Changes the output of Nodes

Usage

outm(x, value="each", which.node=1)

Arguments

x A ‘mcnode’ or a ‘mc’ object.

value The output of the ‘mcnode’ for multivariates nodes. May be "each" (default)
if output should be provided for each variates considered independently, "none"
for no output or a vector of name of function(s) (as a character string) that will be
applied on the variates dimension before any output (ex: ‘"mean"’, ‘"median"’,
‘c("min","max")’). The function should have no other arguments and send one
value per vector of values (ex. do not use ‘"range"’).
which.node which node should be changed in a ‘mc’ object

Value

‘x’ with a modified ‘outm’ attribute.

Examples

data(total)
total$xVUM2
## since outm = NULL
summary(total$xVUM2)
x <- outm(total$xVUM2, c("min"))
summary(x)
pert 43

pert The (Modified) PERT Distribution

Description
Density, distribution function, quantile function and random generation for the PERT (aka Beta
PERT) distribution with minimum equals to ‘min’, mode equals to ‘mode’ and maximum equals to
‘max’.

Usage
dpert(x, min=-1, mode=0, max=1, shape=4, log=FALSE)
ppert(q, min=-1, mode=0, max=1, shape=4, lower.tail=TRUE, log.p=FALSE)
qpert(p, min=-1, mode=0, max=1, shape=4, lower.tail=TRUE, log.p=FALSE)
rpert(n, min=-1, mode=0, max=1, shape=4)

Arguments
x,q Vector of quantiles.
p Vector of probabilities.
n Number of observations. If length(n) > 1, the length is taken to be the number
required.
min Vector of minima.
mode Vector of modes.
max Vector of maxima.
shape Vector of scaling parameters. Default value: 4.
log, log.p Logical; if ‘TRUE’, probabilities ‘p’ are given as ‘log(p)’.
lower.tail Logical; if ‘TRUE’ (default), probabilities are ‘P[X <= x]’, otherwise, ‘P[X > x]’.

Details
The PERT distribution is a beta distribution extended to the domain ‘[min, max]’ with mean
min + shape × mode + max
µ=
shape + 2

The underlying beta distribution is specified by α1 and α2 defined as

(µ − min)(2 × mode − min − max)

α1 =
(mode − µ)(max − min)

α1 × (max − µ)
α2 =
mu − min

If µ = mode, α1 is set to 1 + ν/2.

44 plot.mc

David Vose (See reference) proposed a modified PERT distribution with a shape parameter different
from 4.
The PERT distribution is frequently used (with the triangular distribution) to translate expert esti-
mates of the min, max and mode of a random variable in a smooth parametric distribution.

Value

‘dpert’ gives the density, ‘ppert’ gives the distribution function, ‘qpert’ gives the quantile func-
tion, and ‘rpert’ generates random deviates.

Author(s)

Regis Pouillot and Matthew Wiener

References

Vose D. Risk Analysis - A Quantitative Guide (2nd and 3rd editions, John Wiley and Sons, 2000,
2008).

See Also

Beta

Examples

curve(dpert(x, min=3, mode=5, max=10, shape=6), from = 2, to = 11, lty=3)

curve(dpert(x, min=3, mode=5, max=10), from = 2, to = 11, add=TRUE)
curve(dpert(x, min=3, mode=5, max=10, shape=2), from = 2, to = 11, add=TRUE, lty=2)
legend(x = 8, y = 2, c("Default", "shape:2", "shape:6"), lty=1:3)

plot.mc Plots Results of a Monte Carlo Simulation

Description

Plots the empirical cumulative distribution function of a ‘mcnode’ or a ‘mc’ object ("0" and "V"
nodes) or the empirical cumulative distribution function of the estimate of a ‘mcnode’ or ‘mc’ object
("U" and "VU" nodes).
plot.mc 45

Usage
## S3 method for class 'mc'
plot(x, prec=0.001, stat=c("median", "mean"), lim=c(0.025, 0.25, 0.75,
0.975), na.rm=TRUE, griddim=NULL, xlab=NULL, ylab="Fn(x)",
main="", draw=TRUE, paint=TRUE, xlim=NULL, ylim=NULL, ...)
## S3 method for class 'mcnode'
plot(x, ...)
## S3 method for class 'plotmc'
plot(x, ...)
## S3 method for class 'mccut'
plot(x, stat=c("median", "mean"), lim=c(0.025, 0.25, 0.75, 0.975),
griddim=NULL, xlab=names(x), ylab="Fn(x)", main="",
draw=TRUE, ...)

Arguments
x a ‘mcnode’ or a ‘mc’ objects
prec the precision of the plot. 0.001 will provide an ecdf from the 0.000, 0.001, .002,
..., 1.000 quantiles.
stat the function used for estimates (2D ‘mc’ or ‘mcnode’). By default the median.
lim a vector of numbers (between 0 and 1) indicating the enveloppe (2D ‘mc’ or
‘mcnode’) . Maybe ‘NULL’ or empty.
na.rm Should NA values be discarded
griddim a vector of two integers, indicating the size of the grid of the graph. If ‘NULL’,
the grid is calculated to produce a "nice" graph.
xlab vector of labels for the x-axis. If ‘NULL’, use the name of the node.
ylab vector of labels for the y-axis.
main vector of main titles of the graph.
draw Should the plot be drawn?
paint Should the enveloppes be filled?
xlim x coordinate range. ‘xlim’ is either a vector of length 2, used for each graph,
or a list of vectors of length 2, whose ith element is used for the ith graph. By
default, the data range is used as ‘xlim’.
ylim y coordinate range. ‘ylim’ is either a vector of length 2, used for each graph,
or a list of vectors of length 2, whose ith element is used for the ith graph. By
default, the data range is 0-1.
... further arguments to be passed to ‘plot.stepfun’.

Details
‘plot.mcnode’ is a user-friendly function that send the ‘mcnode’ to ‘plot.mc’.
For ‘"VU"’ and ‘"U"’ ‘mcnode’s, quantiles are calculated using quantile.mc within each of the
‘nsu’ simulations (i.e. by columns of each ‘mcnode’). The medians (but may be the means using
‘stat="mean"’) calculated from the ‘nsu’ values are plotted. The 0.025 and 0.975 quantiles, and
the 0.25 and 0.75 quantiles (default values of ‘lim’) of these quantiles are used as the enveloppe.
46 plot.tornado

Value

A ‘plot.mc’ object, list of the quantiles used to plot the draw.

References

Cullen AC and Frey HC (1999) Probabilistic techniques in exposure assessment. Plenum Press,
USA, pp. 81-155.

See Also

ecdf, plot, quantile.mc

Examples
data(total)

plot(xVUM3)
## only one enveloppe corresponding to quantiles 0.025 and 0.975
plot(xVUM3, lim=c(0.025, 0.975))
## only one enveloppe not painted
plot(xVUM3, lim=c(0.025, 0.975), paint=FALSE)

def.par <- par(no.readonly = TRUE)

par(mar=c(4, 4, 1, 1))
plot(total)
par(def.par)

plot.tornado Draws a Tornado chart.

Description

Draws a Tornado chart as provided by ‘tornado’.

Usage

## S3 method for class 'tornado'

plot(x, which=1, name=NULL, stat=c("median", "mean"), xlab="method",
ylab="", ...)
## S3 method for class 'tornadounc'
plot(x, which=1, stat="median", name=NULL, xlab="method", ylab="", ...)
pmin 47

Arguments
x A tornado object or a tornadounc object.
which Which output to print -for multivariates output-.
name Vector of name of input variables. If NULL, the name will be given from the
name of the elements.
stat The name of the statistics of the output to be considered. For a ‘tornado’ object:
"median" or "mean". For a ‘tornadounc’ object: the value should match one
row name of the ‘tornadounc’ object. Alternatively, for a ‘tornadounc’ object,
the number of the row may be used.
xlab Label of the x axis. if "method", use the correlation method used in the ‘tornado’
object.
ylab Label of the y axis.
... Further arguments to be passed to the ‘plot’ function.

Details
A point is drawn at the estimate and the segment reflects the uncertainty around this estimate.

Value
NULL

See Also
tornado

Examples
data(ec)
x <- evalmcmod(ec$modEC2, nsv=100, nsu=100, seed=666)
tor <- tornado(x, 7)
plot(tor)

pmin Maxima and Minima for mcnodes

Description
Returns the parallel maxima and minima of the input values.

Usage
## S3 method for class 'mcnode'
pmin(..., na.rm=FALSE)
## S3 method for class 'mcnode'
pmax(..., na.rm=FALSE)
48 print.mc

Arguments
... One or more ‘mcnodes’s or one or more ‘mcnode’s and vector(s) of compatible
size. Note that one ‘mcnode’ must be at the first place.
na.rm a logical indicating whether missing values should be removed.

Details
‘pmax’ and ‘pmin’ take one or more ‘mcnode’ and/or vectors as arguments and return a ‘mcnode’
of adequate type and size giving the "parallel" maxima (or minima) of the ‘mcnode’ and/or vectors.
Note that the first element of ... should be an ‘mcnode’. The resulting type of ‘mcnode’ is variable
according to the elements that are passed. The same rules as in Ops.mcnode are applied.

Value
an ‘mcnode’ of adequate type and dimension.

See Also
min, Ops.mcnode

Examples
ndvar(10);ndunc(21)
x <- mcstoc(rnorm, "V")
pmin(x, 0)
y <- mcdata(rep(c(-1, 1), length=ndunc()), "U")
unclass(pmin(x, y))

print.mc Prints a mcnode or a mc Object

Description
Print a description of the structure of the ‘mc’ or the ‘mcnode’ object.

Usage
## S3 method for class 'mc'
print(x, digits=3, ...)
## S3 method for class 'mcnode'
print(x, ...)
quantile.mc 49

Arguments
x a ‘mcnode’ or a ‘mc’ object.
digits Number of digits to be used.
... Further arguments to be passed to the print function.

Value
An invisible data frame.

See Also
mcnode for ‘mcnode’ objects. mc for ‘mc’ objects.

quantile.mc Quantiles of a mc Object

Description
Evaluates quantiles of a ‘mc’ object. This function is used by ‘plot.mc’

Usage
## S3 method for class 'mc'
quantile(x, probs=seq(0, 1, 0.01), lim=c(0.025, 0.975), na.rm=TRUE, ...)
## S3 method for class 'mcnode'
quantile(x, ...)

Arguments
x a ‘mc’ objects
probs the quantiles to be calculated
na.rm TRUE or FALSE
lim a vector of numbers (between 0 and 1) indicating the enveloppe. Maybe ‘NULL’
or empty.
... For generic method consistancy.

Details
The quantiles are evaluated in the variability dimension. Then, the median, the mean and the ‘lim’
quantiles are evaluated for each of these quantiles.

Value
A list of quantiles.
50 rtrunc

See Also
plot.mc, quantile.

Examples
data(total)
quantile(total$xVUM3)
quantile(total)

rtrunc Random Truncated Distributions

Description
Provides samples from classical R distributions and ‘mc2d’ specific distributions truncated between
‘linf’ (excluded) and ‘lsup’ (included).

Usage
rtrunc(distr=runif, n, linf=-Inf, lsup=Inf, ...)

Arguments
distr A function providing random data or its name as character. The function ’rdistr’
should have a ’qdistr’ form (with argument ’p’) and a ’pdistr’ form (with ar-
gument ’q’). Example : ’rnorm’ (has a ’qnorm’ and a ’pnorm’ form), ’rbeta’,
’rbinom’, ’rgamma’, ...
n The size of the sample. .
linf A vector of lower bounds.
lsup A vector of upper bounds, with ‘lsup < linf’ (strictly).
... All arguments to be passed to ‘pdistr’ and ‘qdistr’.

Details
The function 1) evaluates the ‘p’ values corresponding to ‘linf’ and ‘lsup’ using ‘pdistr’; 2)
samples ‘n’ values using ‘runif(n, min=pinf, max=psup)’, and 3) takes the ‘n’ corresponding
quantiles from the specified distribution using ‘qdistr’.
All distributions (but sample) implemented in the stats library could be used. The arguments in
. . . should be named. Do not use ’log’ or ’log.p’ or ’lower.tail’. For discrete dictribution, rtrunc
sample within ‘(linf, lsup]’. See example.

Value
A vector of ‘n’ values.
summary.mc 51

Note
The inversion of the quantile function leads to time consuming functions for some distributions.
WARNING: The method is flexible, but can lead to problems linked to rounding errors in some
extreme situations. The function checks that the values are in the expected range and returns an error
if not. It also warns some extreme situation that could lead to unexpected results. See Examples.

Examples
rtrunc("rnorm", n=10, linf=0)
range(rtrunc(rnorm, n=1000, linf=3, lsup=5, sd=10))
## Discrete distributions
range(rtrunc(rpois, 1000, linf=2, lsup=4, lambda=1))
##Examples of rounding problems.
##The first one will provide a warning while the results are unexpected,
##The second will provide an error.
## Not run:
table(rtrunc(rbinom, n=1000, size=10, prob=1-1E-20, lsup=9))
table(rtrunc(rbinom, n=1000, size=10, prob=1E-14, linf=0))

## End(Not run)

summary.mc Summary of mcnode and mc Object

Description
Provides a summary of a ‘mcnode’, a ‘mc’ or a ‘mccut’ object.

Usage
## S3 method for class 'mc'
summary(object, probs=c(0, 0.025, 0.25, 0.5, 0.75, 0.975, 1), lim=c(0.025,
0.975), ...)
## S3 method for class 'mcnode'
summary(object, probs=c(0, 0.025, 0.25, 0.5, 0.75, 0.975, 1), lim=c(0.025,
0.975), digits=3, ...)
## S3 method for class 'mc'
print.summary(x, digits=3, ...)
## S3 method for class 'mccut'
summary(object, lim=c(0.025, 0.975), ...)

Arguments
object a ‘mcnode’ or a ‘mc’ object or a ‘mccut’ object.
x A ‘summary.mc’ object as provided by the ‘summary.mc’ function.
probs A vector of values used for the quantile function (variability dimension).
digits Number of digits in the print.
52 tornado

lim A vector of values used for the quantile function (uncertainty dimension).
... For generic functions consistancy.

Details
The mean, the standard deviation and the ‘probs’ quantiles will be evaluated in the variability
dimension. The median, the mean and the ‘lim’ quantiles will then be evaluated on these statistics
in the uncertainty dimension.
Multivariate nodes:
If the ‘"outm"’ attributes of the mcnode is "none", the node is not evaluated, if it is "each" the
variates are evaluated one by one, if it is a function (e.g. "mean"), the function is applied on the
‘nvariates’ dimension before providing a classical output.

Value
a list.

See Also
mcnode for mcnode objects, mc for mc objects, mccut for mccut objects, quantile

Examples
data(total)
summary(xVUM3)
summary(total)

tornado Computes Correlation between Inputs and Output in a mc Object (tor-

nado) in the Variability Dimension;

Description
Provides statistics for a tornado chart. Evaluates correlations between output and inputs of a ‘mc’
object.

Usage
tornado(mc, output=length(mc), use="all.obs", method=c("spearman",
"kendall", "pearson"), lim=c(0.025, 0.975))
## S3 method for class 'tornado'
print(x, ...)
tornado 53

Arguments

mc a mc object or a mccut object.

x A ‘tornado’ object as provided by the ‘tornado’ function.
output (for ‘mc’ objects only). The rank or the name of the output to be considered. By
default: the last element of the ‘mc’.
use (for ‘mc’ objects only). An optional character string giving a method for comput-
ing covariances in the presence of missing values. This must be (an abbreviation
of) one of the strings "all.obs", "complete.obs" or "pairwise.complete.obs" (see
cor).
method (for ‘mc’ objects only). A character string indicating which correlation coeffi-
cient (or covariance) is to be computed. One of "spearman" (default), "kendall"
or "pearson", can be abbreviated (see cor). Warning : the default is not the same
in cor.
lim A vector of quantiles used to compute the credible interval in two-dimensional
models.
... Further arguments to be passed to the final print function.

Details

The tornado function computes the spearman’s rho statistic. It is used to estimate a rank-based
measure of association between one set of random variable of a ‘mc’ object (the output) and the
others (the inputs).
‘tornado’ may be applied on a ‘mccut’ object if a ‘tornado’ function was used in the third block
of the evalmccut call.
If "output" refers to a ‘"0" mcnode’, it is an error. If "output" refers to a ‘"V" mcnode’, correlations
are only provided for other ‘"V" mcnode’s. If "output" refers to a ‘"U" mcnode’, correlations are
only provided for other ‘"U" mcnode’s. If "output" refers to a ‘"VU" mcnode’, correlations are only
provided for other ‘"VU" mcnode’s and ‘"V" mcnode’s.
If use is "all.obs", then the presence of missing observations will produce an error. If use is "com-
plete.obs" then missing values are handled by casewise deletion. Finally, if use has the value "pair-
wise.complete.obs" then the correlation between each pair of variables is computed using all com-
plete pairs of observations on those variables.

Value

An invisible object of class tornado. A tornado object is a list of objects containing the following
objects:

value the value of correlation coefficients

output the name of the output
method the method used
use the use parameter
54 tornadounc

See Also
cor.
plot.tornado to draw the results.

Examples
data(total)
tornado(total, 2, "complete.obs", "spearman", c(0.025, 0.975))
tornado(total, 4, "pairwise.complete.obs", "spearman", c(0.025, 0.975))
tornado(total, 6, "complete.obs", "kendall", c(0.025, 0.975))
tornado(total, 8, "complete.obs", "spearman", c(0.025, 0.975))
(y <- tornado(total, 10, "complete.obs", "spearman", c(0.025, 0.975)))
plot(y)

tornadounc Computes Correlation between Inputs and Output in a mc Object (tor-

nado) in the Uncertainty Dimension

Description
Provides statistics for a tornado chart. Evaluates correlations between output and inputs of a ‘mc’
object in the uncertainty dimension.

Usage
## S3 method for class 'mc'
tornadounc(mc, output=length(mc), quant=c(0.5, 0.75, 0.975), use="all.obs",
method=c("spearman", "kendall", "pearson"), ...)
## Default S3 method:
tornadounc(mc, ...)
## S3 method for class 'tornadounc'
print(x, ...)
## S3 method for class 'mccut'
tornadounc(mc, output=length(mc), quant=c(0.5, 0.75, 0.975), use="all.obs",
method=c("spearman", "kendall", "pearson"), ...)

Arguments
mc a ‘mc’ object.
x a ‘tornadounc’ object.
output The rank or the name of the output to be considered. Should be a ‘"VU"’ or a
‘"U" type mcnode’. By default: the last element of ‘mc’.
quant The vector of quantiles used in the variability dimension.
use An optional character string giving a method for computing covariances in the
presence of missing values. This must be (an abbreviation of) one of the strings
"all.obs", "complete.obs" or "pairwise.complete.obs" (see cor).
tornadounc 55

method A character string indicating which correlation coefficient (or covariance) is to

be computed. One of "spearman" (default), "kendall" or "pearson", can be ab-
breviated (see cor). Warning : "pearson" is the default for cor).
... Further arguments to be passed to the final print function.

Details

The ‘tornadounc.mc’ function computes the spearman’s rho statistic between

• values (‘"U" type mcnode’) or statistics calculated in the variability dimension (‘"VU" type mcnode’)
of inputs and
• values (‘"U" type mcnode’) or statistics calculated in the variability dimension (‘"VU" type mcnode’)
of one output.

The statistics are the mean, the median and the quantiles specified by ‘quant’.
It is useful to estimate a rank-based measure of association between one set of random variable of a
‘mc’ object (the output) and the others in the uncertainty dimension.
‘tornadounc.mccut’ may be applied on a mccut object if a ‘summary.mc’ function was used in the
third block of the evalmccut call.
If output refers to a ‘"0"’ or ‘"V" mcnode’, it is an error.
If use is "all.obs", then the presence of missing observations will produce an error. If use is "com-
plete.obs" then missing values are handled by casewise deletion. Finally, if use has the value "pair-
wise.complete.obs" then the correlation between each pair of variables is computed using all com-
plete pairs of observations on those variables.

Value

An invisible object of class ‘tornadounc’. A ‘tornadounc’ object is a list of objects containing the
following objects:

value a matrix of values of correlation coefficients. Each row are the value or the
statistics of inputs, each columns the value or the statistics of outputs.
output the name of the output
method the method used
use the ‘use’ parameter

See Also

cor.
tornado for tornado in the variability dimension.
plot.tornadounc to draw the results.
56 total

Examples
data(total)
tornadounc(total, 3)
tornadounc(total, 4, use="complete")
tornadounc(total, 7, use="complete.obs")
tornadounc(total, 8, use="complete.obs")
(y <- tornadounc(total, 10, use="complete.obs"))
plot(y, 1, 1)

total An Exemple of all Kind of mcnode

Description
An exemple for each kind of ‘mcnode’s. They are used in some ‘mc2d’ examples. They have been
built using the following code:
ndvar(101) ndunc(51)
x0 <- mcstoc(type="0")
xV <- mcstoc(type="V")
xU <- mcstoc(type="U")
xVU <- mcstoc(type="VU")
x0M <- mcstoc(type="0",nvariates=2)
xVM <- mcstoc(type="V",nvariates=2)
xUM <- mcstoc(type="U",nvariates=2)
xVUM <- mcstoc(type="VU",nvariates=2)
xVUM[c(1,12,35)] <- NA
xVUM2 <- mcstoc(type="VU",nvariates=2,outm="none")
xVUM3 <- mcstoc(type="VU",nvariates=2,outm=c("mean","min"))
total <- mc(x0,xV,xU,xVU,x0M,xVM,xUM,xVUM,xVUM2,xVUM3)

Usage
total

Format
Some ‘mcnode’ objects and one ‘mc’ object.
triangular 57

Source
None

References
None

triangular The Triangular Distribution

Description
Density, distribution function, quantile function and random generation for the triangular distribu-
tion with minimum equal to ‘min’, mode equal ‘mode’ and maximum equal to ‘max’.

Usage
dtriang(x, min=-1, mode=0, max=1, log=FALSE)
ptriang(q, min=-1, mode=0, max=1, lower.tail=TRUE, log.p=FALSE)
qtriang(p, min=-1, mode=0, max=1, lower.tail=TRUE, log.p=FALSE)
rtriang(n, min=-1, mode=0, max=1)

Arguments
x,q vector of quantiles.
p vector of probabilities.
n number of observations. If length(n) > 1, the length is taken to be the number
required.
min vector of minima.
mode vector of modes.
max vector of maxima.
log, log.p logical; if ‘TRUE’, probabilities ‘p’ are given as ‘log(p)’.
lower.tail logical; if ‘TRUE’ (default), probabilities are ‘P[X <= x]’, otherwise, ‘P[X > x]’.

Details
For the case of u := min == mode == max, there is no density in that case and dtriang will return
NaN (the error condition) (Similarity with dunif).

Value
‘dtriang’ gives the density, ‘ptriang’ gives the distribution function, ‘qtriang’ gives the quantile
function, and ‘rtriang’ generates random deviates.
58 typemcnode

Examples

curve(dtriang(x, min=3, mode=5, max=10), from = 2, to = 11)

##no density when min == mode == max
dtriang(c(1, 2, 3), min=2, mode=2, max=2)

typemcnode Provides the Type of a mcnode Object

Description

Provide the type of a ‘mcnode’ object.

Usage

typemcnode(x, index=FALSE)

Arguments

x a ‘mcnode’ object
index if ‘TRUE’ give the index of the type rather than the type.

Value

‘"0", "V","U" or "VU"’ or the corresponding index if ‘index=TRUE’.

‘NULL’ if none of this element is found.

Note

This function does not test if the object is correct. See is.mcnode.

Examples

data(total)
typemcnode(total$xVUM2)
unmc 59

unmc Unclasses the mc or the mcnode Object

Description
Unclasses the ‘mc’ object in a list of arrays or the ‘mcnode’ object in an array.

Usage
unmc(x, drop=TRUE)

Arguments
x A ‘mc’ or a ‘mcnode’ object.
drop Should the dimensions of size 1 be dropped (see drop).

Value
if x is an ‘mc’ object: a list of arrays. If ‘drop=TRUE’, a list of vectors, matrixes and arrays. if x is
an ‘mcnode’ object: an array. If ‘drop=TRUE’, a vector, matrix or array.

Examples
data(total)
## A vector
unmc(total$xV, drop=TRUE)
## An array
unmc(total$xV, drop=FALSE)
Index

∗Topic NA ∗Topic multivariate

NA.mcnode, 38 cornode, 6
∗Topic datasets ∗Topic print
ec, 11 print.mc, 48
total, 56 ∗Topic univar
∗Topic design quantile.mc, 49
lhs, 19 summary.mc, 51
∗Topic distribution tornado, 52
bernoulli, 2 tornadounc, 54
betagen, 3 ∗Topic utilities
dirichlet, 9 dimmcnode, 8
dmultinomial, 10 is.mc, 18
empiricalC, 12 Ops.mcnode, 39
empiricalD, 14 pmin, 47
mcratio, 32 typemcnode, 58
multinormal, 37
pert, 43 addvar (extractvar), 16
rtrunc, 50 apply, 22
triangular, 57
bernoulli, 2
∗Topic hplot
Beta, 4, 9, 44
converg, 5
betagen, 3
hist.mc, 17
Binomial, 3
plot.mc, 44
binomial, 3
plot.tornado, 46
∗Topic manip chol, 7
unmc, 59 converg, 5, 20, 24, 29
∗Topic methods cor, 53–55
evalmcmod, 15 cornode, 6, 29
extractvar, 16
mc, 20 dbern (bernoulli), 2
mccut, 23 dbetagen (betagen), 3
mcmodel, 25 ddirichlet (dirichlet), 9
mcnode, 27 dempiricalC (empiricalC), 12
mcprobtree, 30 dempiricalD (empiricalD), 14
mcstoc, 33 dimmc, 20
∗Topic misc dimmc (dimmcnode), 8
mc.control, 21 dimmcnode, 8, 29
mcapply, 22 dirichlet, 9
outm, 42 dmultinom, 10

60
INDEX 61

dmultinomial, 10 NA, 39
dmultinormal (multinormal), 37 NA.mcnode, 29, 38
dpert (pert), 43 ndunc (mc.control), 21
drop, 59 ndvar (mc.control), 21
dtriang (triangular), 57
Ops, 40
ec, 11 Ops.mcnode, 29, 35, 39, 48
ecdf, 46 outm, 27, 31, 34, 40, 42
empiricalC, 12, 15
empiricalD, 13, 14 pbern (bernoulli), 2
evalmccut, 6, 16, 20, 53, 55 pbetagen (betagen), 3
evalmccut (mccut), 23 pempiricalC (empiricalC), 12
evalmcmod, 15, 20, 24–26 pempiricalD (empiricalD), 14
expression, 26 pert, 43
extractvar, 16 plot, 46
plot.mc, 20, 24, 44, 50
hist.mc, 17, 20 plot.mccut, 24
hist.mcnode, 29 plot.mccut (plot.mc), 44
hist.mcnode (hist.mc), 17 plot.mcnode, 29
plot.mcnode (plot.mc), 44
is.finite, 39 plot.plotmc (plot.mc), 44
is.finite.mcnode (NA.mcnode), 38 plot.tornado, 46, 54
is.infinite.mcnode (NA.mcnode), 38 plot.tornadounc, 55
is.mc, 8, 18, 20 plot.tornadounc (plot.tornado), 46
is.mcnode, 8, 29, 58 pmax (pmin), 47
is.mcnode (is.mc), 18 pmin, 47
is.na.mcnode (NA.mcnode), 38 ppert (pert), 43
is.nan.mcnode (NA.mcnode), 38 print.mc, 20, 48
print.mccut, 24
lhs, 19, 34 print.mccut (mccut), 23
print.mcnode, 29
mc, 5, 20, 22, 27, 29, 49, 52, 53 print.mcnode (print.mc), 48
mc.control, 21, 27 print.summary.mc (summary.mc), 51
mcapply, 22 print.tornado (tornado), 52
mccut, 5, 23, 52, 53, 55 print.tornadounc (tornadounc), 54
mcdata, 31, 40 ptriang (triangular), 57
mcdata (mcnode), 27
mcdatanocontrol (mcnode), 27 qbern (bernoulli), 2
mcmodel, 15, 16, 20, 23, 25 qbetagen (betagen), 3
mcmodelcut, 26 qempiricalC (empiricalC), 12
mcmodelcut (mccut), 23 qempiricalD (empiricalD), 14
mcnode, 5, 17, 20, 22, 27, 31, 33–35, 49, 52 qpert (pert), 43
mcprobtree, 28, 30 qtriang (triangular), 57
mcratio, 32 quantile, 50, 52
mcstoc, 19, 28, 31, 33, 40 quantile.mc, 45, 46, 49
min, 48 quantile.mcnode (quantile.mc), 49
modEC1 (ec), 11
modEC2 (ec), 11 rbern (bernoulli), 2
multinormal, 37 rbetagen (betagen), 3
62 INDEX

rdirichlet, 28, 35
rdirichlet (dirichlet), 9
rempiricalC (empiricalC), 12
rempiricalD, 28, 35
rempiricalD (empiricalD), 14
rmultinom, 10
rmultinomial, 28, 35
rmultinomial (dmultinomial), 10
rmultinormal, 28, 35
rmultinormal (multinormal), 37
rpert (pert), 43
rtriang (triangular), 57
rtrunc, 19, 34, 35, 50

sample, 15
summary.mc, 20, 24, 51
summary.mccut, 24
summary.mccut (summary.mc), 51
summary.mcnode, 29
summary.mcnode (summary.mc), 51
switch, 31

tornado, 20, 24, 47, 52, 55

tornadounc, 47, 54
tornadounc.mc, 20
tornadounc.mccut, 24
total, 56
triangular, 44, 57
typemcnode, 29, 58

unmc, 59

x0 (total), 56
x0M (total), 56
xU (total), 56
xUM (total), 56
xV (total), 56
xVM (total), 56
xVU (total), 56
xVUM (total), 56
xVUM2 (total), 56
xVUM3 (total), 56