Package ‘dlm’

June 13, 2018

Title Bayesian and Likelihood Analysis of Dynamic Linear Models
Version 1.1-5
Date 2018-05-30
Suggests MASS
Imports stats, utils, methods, grDevices, graphics
Description Provides routines for Maximum likelihood,
Kalman filtering and smoothing, and Bayesian
analysis of Normal linear State Space models, also known as
Dynamic Linear Models.
License GPL (>= 2)
NeedsCompilation yes
Author Giovanni Petris [aut, cre],
Wally Gilks [ctb] (Author of original C code for ARMS)
Maintainer Giovanni Petris <GPetris@uark.edu>
Repository CRAN
Date/Publication 2018-06-13 19:16:46 UTC

R topics documented:
arms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
ARtransPars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
bdiag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
convex.bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
dlm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
dlmBSample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
dlmFilter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
dlmForecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
dlmGibbsDIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
dlmLL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
dlmMLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
dlmModARMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
dlmModPoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1
2 arms

dlmModReg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
dlmModSeas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
dlmModTrig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
dlmRandom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
dlmSmooth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
dlmSum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
dlmSvd2var . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
dropFirst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
FF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
mcmc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
NelPlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
residuals.dlmFiltered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
rwishart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
USecon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Index 36

arms Function to perform Adaptive Rejection Metropolis Sampling

Description
Generates a sequence of random variables using ARMS. For multivariate densities, ARMS is used
along randomly selected straight lines through the current point.

Usage
arms(y.start, myldens, indFunc, n.sample, ...)

Arguments
y.start initial point
myldens univariate or multivariate log target density
indFunc indicator function of the convex support of the target density
n.sample desired sample size
... parameters passed to myldens and indFunc

Details
Strictly speaking, the support of the target density must be a bounded convex set. When this is not
the case, the following tricks usually work. If the support is not bounded, restrict it to a bounded set
having probability practically one. A workaround, if the support is not convex, is to consider the
convex set generated by the support and define myldens to return log(.Machine$double.xmin)
outside the true support (see the last example.)
The next point is generated along a randomly selected line through the current point using arms.
Make sure the value returned by myldens is never smaller than log(.Machine$double.xmin), to
avoid divisions by zero.
arms 3

Value
An n.sample by length(y.start) matrix, whose rows are the sampled points.

Note
The function is based on original C code by W. Gilks for the univariate case.

Author(s)
Giovanni Petris <GPetris@uark.edu>

References
Gilks, W.R., Best, N.G. and Tan, K.K.C. (1995) Adaptive rejection Metropolis sampling within
Gibbs sampling (Corr: 97V46 p541-542 with Neal, R.M.), Applied Statistics 44:455–472.

Examples
#### ==> Warning: running the examples may take a few minutes! <== ####
## Not run:
set.seed(4521222)
### Univariate densities
## Unif(-r,r)
y <- arms(runif(1,-1,1), function(x,r) 1, function(x,r) (x>-r)*(x<r), 5000, r=2)
summary(y); hist(y,prob=TRUE,main="Unif(-r,r); r=2")
## Normal(mean,1)
norldens <- function(x,mean) -(x-mean)^2/2
y <- arms(runif(1,3,17), norldens, function(x,mean) ((x-mean)>-7)*((x-mean)<7),
5000, mean=10)
summary(y); hist(y,prob=TRUE,main="Gaussian(m,1); m=10")
curve(dnorm(x,mean=10),3,17,add=TRUE)
## Exponential(1)
y <- arms(5, function(x) -x, function(x) (x>0)*(x<70), 5000)
summary(y); hist(y,prob=TRUE,main="Exponential(1)")
curve(exp(-x),0,8,add=TRUE)
## Gamma(4.5,1)
y <- arms(runif(1,1e-4,20), function(x) 3.5*log(x)-x,
function(x) (x>1e-4)*(x<20), 5000)
summary(y); hist(y,prob=TRUE,main="Gamma(4.5,1)")
curve(dgamma(x,shape=4.5,scale=1),1e-4,20,add=TRUE)
## Gamma(0.5,1) (this one is not log-concave)
y <- arms(runif(1,1e-8,10), function(x) -0.5*log(x)-x,
function(x) (x>1e-8)*(x<10), 5000)
summary(y); hist(y,prob=TRUE,main="Gamma(0.5,1)")
curve(dgamma(x,shape=0.5,scale=1),1e-8,10,add=TRUE)
## Beta(.2,.2) (this one neither)
y <- arms(runif(1), function(x) (0.2-1)*log(x)+(0.2-1)*log(1-x),
function(x) (x>1e-5)*(x<1-1e-5), 5000)
summary(y); hist(y,prob=TRUE,main="Beta(0.2,0.2)")
curve(dbeta(x,0.2,0.2),1e-5,1-1e-5,add=TRUE)
## Triangular
y <- arms(runif(1,-1,1), function(x) log(1-abs(x)), function(x) abs(x)<1, 5000)
4 arms

summary(y); hist(y,prob=TRUE,ylim=c(0,1),main="Triangular")
curve(1-abs(x),-1,1,add=TRUE)
## Multimodal examples (Mixture of normals)
lmixnorm <- function(x,weights,means,sds) {
log(crossprod(weights, exp(-0.5*((x-means)/sds)^2 - log(sds))))
}
y <- arms(0, lmixnorm, function(x,...) (x>(-100))*(x<100), 5000, weights=c(1,3,2),
means=c(-10,0,10), sds=c(1.5,3,1.5))
summary(y); hist(y,prob=TRUE,main="Mixture of Normals")
curve(colSums(c(1,3,2)/6*dnorm(matrix(x,3,length(x),byrow=TRUE),c(-10,0,10),c(1.5,3,1.5))),
par("usr")[1], par("usr")[2], add=TRUE)

### Bivariate densities

## Bivariate standard normal
y <- arms(c(0,2), function(x) -crossprod(x)/2,
function(x) (min(x)>-5)*(max(x)<5), 500)
plot(y, main="Bivariate standard normal", asp=1)
## Uniform in the unit square
y <- arms(c(0.2,.6), function(x) 1,
function(x) (min(x)>0)*(max(x)<1), 500)
plot(y, main="Uniform in the unit square", asp=1)
polygon(c(0,1,1,0),c(0,0,1,1))
## Uniform in the circle of radius r
y <- arms(c(0.2,0), function(x,...) 1,
function(x,r2) sum(x^2)<r2, 500, r2=2^2)
plot(y, main="Uniform in the circle of radius r; r=2", asp=1)
curve(-sqrt(4-x^2), -2, 2, add=TRUE)
curve(sqrt(4-x^2), -2, 2, add=TRUE)
## Uniform on the simplex
simp <- function(x) if ( any(x<0) || (sum(x)>1) ) 0 else 1
y <- arms(c(0.2,0.2), function(x) 1, simp, 500)
plot(y, xlim=c(0,1), ylim=c(0,1), main="Uniform in the simplex", asp=1)
polygon(c(0,1,0), c(0,0,1))
## A bimodal distribution (mixture of normals)
bimodal <- function(x) { log(prod(dnorm(x,mean=3))+prod(dnorm(x,mean=-3))) }
y <- arms(c(-2,2), bimodal, function(x) all(x>(-10))*all(x<(10)), 500)
plot(y, main="Mixture of bivariate Normals", asp=1)

## A bivariate distribution with non-convex support

support <- function(x) {
return(as.numeric( -1 < x[2] && x[2] < 1 &&
-2 < x[1] &&
( x[1] < 1 || crossprod(x-c(1,0)) < 1 ) ) )
}
Min.log <- log(.Machine$double.xmin) + 10
logf <- function(x) {
if ( x[1] < 0 ) return(log(1/4))
else
if (crossprod(x-c(1,0)) < 1 ) return(log(1/pi))
return(Min.log)
}
x <- as.matrix(expand.grid(seq(-2.2,2.2,length=40),seq(-1.1,1.1,length=40)))
y <- sapply(1:nrow(x), function(i) support(x[i,]))
ARtransPars 5

plot(x,type='n',asp=1)
points(x[y==1,],pch=1,cex=1,col='green')
z <- arms(c(0,0), logf, support, 1000)
points(z,pch=20,cex=0.5,col='blue')
polygon(c(-2,0,0,-2),c(-1,-1,1,1))
curve(-sqrt(1-(x-1)^2),0,2,add=TRUE)
curve(sqrt(1-(x-1)^2),0,2,add=TRUE)
sum( z[,1] < 0 ) # sampled points in the square
sum( apply(t(z)-c(1,0),2,crossprod) < 1 ) # sampled points in the circle

## End(Not run)

ARtransPars Function to parametrize a stationary AR process

Description
The function maps a vector of length p to the vector of autoregressive coefficients of a stationary
AR(p) process. It can be used to parametrize a stationary AR(p) process

Usage
ARtransPars(raw)

Arguments
raw a vector of length p

Details
The function first maps each element of raw to (0,1) using tanh. The numbers obtained are treated
as the first partial autocorrelations of a stationary AR(p) process and the vector of the corresponding
autoregressive coefficients is computed and returned.

Value
The vector of autoregressive coefficients of a stationary AR(p) process corresponding to the param-
eters in raw.

Author(s)
Giovanni Petris, <GPetris@uark.edu>

References
Jones, 1987. Randomly choosing parameters from the stationarity and invertibility region of autoregressive-
moving average models. Applied Statistics, 36.
6 convex.bounds

Examples
(ar <- ARtransPars(rnorm(5)))
all( Mod(polyroot(c(1,-ar))) > 1 ) # TRUE

bdiag Build a block diagonal matrix

Description
The function builds a block diagonal matrix.

Usage
bdiag(...)

Arguments
... individual matrices, or a list of matrices.

Value
A matrix obtained by combining the arguments.

Author(s)
Giovanni Petris <GPetris@uark.edu>

Examples
bdiag(matrix(1:4,2,2),diag(3))
bdiag(matrix(1:6,3,2),matrix(11:16,2,3))

convex.bounds Find the boundaries of a convex set

Description
Finds the boundaries of a bounded convex set along a specified straight line, using a bisection
approach. It is mainly intended for use within arms.

Usage
convex.bounds(x, dir, indFunc, ..., tol=1e-07)
dlm 7

Arguments
x a point within the set
dir a vector specifying a direction
indFunc indicator function of the set
... parameters passed to indFunc
tol tolerance

Details
Uses a bisection algorithm along a line having parametric representation x + t * dir.

Value
A vector ans of length two. The boundaries of the set are x + ans[1] * dir and x + ans[2] * dir.

Author(s)
Giovanni Petris <GPetris@uark.edu>

Examples
## boundaries of a unit circle
convex.bounds(c(0,0), c(1,1), indFunc=function(x) crossprod(x)<1)

dlm dlm objects

Description
The function dlm is used to create Dynamic Linear Model objects. as.dlm and is.dlm coerce an
object to a Dynamic Linear Model object and test whether an object is a Dynamic Linear Model.

Usage
dlm(...)
as.dlm(obj)
is.dlm(obj)

Arguments
... list with named elements m0, C0, FF, V, GG, W and, optionally, JFF, JV, JGG, JW,
and X. The first six are the usual vector and matrices that define a time-invariant
DLM. The remaining elements are used for time-varying DLM. X, if present,
should be a matrix. If JFF is not NULL, then it must be a matrix of the same
dimension of FF, with the (i, j) element being zero if FF[i,j] is time-invariant,
and a positive integer k otherwise. In this case the (i, j) element of FF at time
8 dlm

t will be X[t,k]. A similar interpretation holds for JV, JGG, and JW. ... may
have additional components, that are not used by dlm. The named components
may also be passed to the function as individual arguments.
obj an arbitrary R object.

Details

The function dlm is used to create Dynamic Linear Model objects. These are lists with the named
elements described above and with class of "dlm".
Class "dlm" has a number of methods. In particular, consistent DLM can be added together to
produce another DLM.

Value

For dlm, an object of class "dlm".

Author(s)

Giovanni Petris <GPetris@uark.edu>

References

Giovanni Petris (2010), An R Package for Dynamic Linear Models. Journal of Statistical Software,
36(12), 1-16. http://www.jstatsoft.org/v36/i12/.
Petris, Petrone, and Campagnoli, Dynamic Linear Models with R, Springer (2009).
West and Harrison, Bayesian forecasting and dynamic models (2nd ed.), Springer (1997).

See Also

dlmModReg, dlmModPoly, dlmModARMA, dlmModSeas, to create particular objects of class "dlm".

Examples

## Linear regression as a DLM

x <- matrix(rnorm(10),nc=2)
mod <- dlmModReg(x)
is.dlm(mod)

## Adding dlm's
dlmModPoly() + dlmModSeas(4) # linear trend plus quarterly seasonal component
dlmBSample 9

dlmBSample Draw from the posterior distribution of the state vectors

Description
The function simulates one draw from the posterior distribution of the state vectors.

Usage
dlmBSample(modFilt)

Arguments
modFilt a list, typically the ouptut from dlmFilter, with elements m, U.C, D.C, a, U.R,
D.R (see the value returned by dlmFilter), and mod The latter is an object of
class "dlm" or a list with elements GG, W and, optionally, JGG, JW, and X

Details
The calculations are based on singular value decomposition.

Value
The function returns a draw from the posterior distribution of the state vectors. If m is a time series
then the returned value is a time series with the same tsp, otherwise it is a matrix or vector.

Author(s)
Giovanni Petris <GPetris@uark.edu>

References
Giovanni Petris (2010), An R Package for Dynamic Linear Models. Journal of Statistical Software,
36(12), 1-16. http://www.jstatsoft.org/v36/i12/.
Petris, Petrone, and Campagnoli, Dynamic Linear Models with R, Springer (2009).
West and Harrison, Bayesian forecasting and dynamic models (2nd ed.), Springer (1997).

See Also
See also dlmFilter

Examples
nileMod <- dlmModPoly(1, dV = 15099.8, dW = 1468.4)
nileFilt <- dlmFilter(Nile, nileMod)
nileSmooth <- dlmSmooth(nileFilt) # estimated "true" level
plot(cbind(Nile, nileSmooth$s[-1]), plot.type = "s",
col = c("black", "red"), ylab = "Level",
main = "Nile river", lwd = c(2, 2))
10 dlmFilter

for (i in 1:10) # 10 simulated "true" levels

lines(dlmBSample(nileFilt[-1]), lty=2)

dlmFilter DLM filtering

Description
The functions applies Kalman filter to compute filtered values of the state vectors, together with their
variance/covariance matrices. By default the function returns an object of class "dlmFiltered".
Methods for residuals and tsdiag for objects of class "dlmFiltered" exist.

Usage
dlmFilter(y, mod, debug = FALSE, simplify = FALSE)

Arguments
y the data. y can be a vector, a matrix, a univariate or multivariate time series.
mod an object of class dlm, or a list with components m0, C0, FF, V, GG, W, and option-
ally JFF, JV, JGG, JW, and X, defining the model and the parameters of the prior
distribution.
debug if FALSE, faster C code will be used, otherwise all the computations will be
performed in R.
simplify should the data be included in the output?

Details
The calculations are based on the singular value decomposition (SVD) of the relevant matrices.
Variance matrices are returned in terms of their SVD.
Missing values are allowed in y.

Value
A list with the components described below. If simplify is FALSE, the returned list has class
"dlmFiltered".

y The input data, coerced to a matrix. This is present only if simplify is FALSE.
mod The argument mod (possibly simplified).
m Time series (or matrix) of filtered values of the state vectors. The series starts
one time unit before the first observation.
U.C See below.
D.C Together with U.C, it gives the SVD of the variances of the estimation errors.
The variance of m[t, ]−theta[t, ] is given by U.C[[t]] %*% diag(D.C[t,]^2) %*% t(U.C[[t]]).
dlmFilter 11

a Time series (or matrix) of predicted values of the state vectors given the obser-
vations up and including the previous time unit.
U.R See below.
D.R Together with U.R, it gives the SVD of the variances of the prediction errors. The
variance of a[t, ]−theta[t, ] is given by U.R[[t]] %*% diag(D.R[t,]^2) %*% t(U.R[[t]]).
f Time series (or matrix) of one-step-ahead forecast of the observations.

Warning

The observation variance V in mod must be nonsingular.

Author(s)

Giovanni Petris <GPetris@uark.edu>

References

Zhang, Y. and Li, X.R., Fixed-interval smoothing algorithm based on singular value decomposition,
Proceedings of the 1996 IEEE International Conference on Control Applications.
Giovanni Petris (2010), An R Package for Dynamic Linear Models. Journal of Statistical Software,
36(12), 1-16. http://www.jstatsoft.org/v36/i12/.
Petris, Petrone, and Campagnoli, Dynamic Linear Models with R, Springer (2009).

See Also

See dlm for a description of dlm objects, dlmSvd2var to obtain a variance matrix from its SVD,
dlmMLE for maximum likelihood estimation, dlmSmooth for Kalman smoothing, and dlmBSample
for drawing from the posterior distribution of the state vectors.

Examples

nileBuild <- function(par) {

dlmModPoly(1, dV = exp(par[1]), dW = exp(par[2]))
}
nileMLE <- dlmMLE(Nile, rep(0,2), nileBuild); nileMLE$conv
nileMod <- nileBuild(nileMLE$par)
V(nileMod)
W(nileMod)
nileFilt <- dlmFilter(Nile, nileMod)
nileSmooth <- dlmSmooth(nileFilt)
plot(cbind(Nile, nileFilt$m[-1], nileSmooth$s[-1]), plot.type='s',
col=c("black","red","blue"), ylab="Level", main="Nile river", lwd=c(1,2,2))
12 dlmForecast

dlmForecast Prediction and simulation of future observations

Description
The function evaluates the expected value and variance of future observations and system states. It
can also generate a sample from the distribution of future observations and system states.

Usage
dlmForecast(mod, nAhead = 1, method = c("plain", "svd"), sampleNew = FALSE)

Arguments
mod an object of class "dlm", or a list with components m0, C0, FF, V, GG, and W,
defining the model and the parameters of the prior distribution. mod can also be
an object of class "dlmFiltered", such as the output from dlmFilter.
nAhead number of steps ahead for which a forecast is requested.
method method="svd" uses singular value decomposition for the calculations. Cur-
rently, only method="plain" is implemented.
sampleNew if sampleNew=n for an integer n, them a sample of size n from the forecast
distribution of states and observables will be returned.

Value
A list with components

a matrix of expected values of future states

R list of variances of future states
f matrix of expected values of future observations
Q list of variances of future observations
newStates list of matrices containing the simulated future values
of the states. Each component of the list corresponds
to one simulation.
newObs same as newStates, but for the observations.

The last two components are not present if sampleNew=FALSE.

Note
The function is currently entirely written in R and is not particularly fast. Currently, only constant
models are allowed.

Author(s)
Giovanni Petris <GPetris@uark.edu>
dlmGibbsDIG 13

Examples
## Comparing theoretical prediction intervals with sample quantiles
set.seed(353)
n <- 20; m <- 1; p <- 5
mod <- dlmModPoly() + dlmModSeas(4, dV=0)
W(mod) <- rwishart(2*p,p) * 1e-1
m0(mod) <- rnorm(p, sd=5)
C0(mod) <- diag(p) * 1e-1
new <- 100
fore <- dlmForecast(mod, nAhead=n, sampleNew=new)
ciTheory <- (outer(sapply(fore$Q, FUN=function(x) sqrt(diag(x))), qnorm(c(0.1,0.9))) +
as.vector(t(fore$f)))
ciSample <- t(apply(array(unlist(fore$newObs), dim=c(n,m,new))[,1,], 1,
FUN=function(x) quantile(x, c(0.1,0.9))))
plot.ts(cbind(ciTheory,fore$f[,1]),plot.type="s", col=c("red","red","green"),ylab="y")
for (j in 1:2) lines(ciSample[,j], col="blue")
legend(2,-40,legend=c("forecast mean", "theoretical bounds", "Monte Carlo bounds"),
col=c("green","red","blue"), lty=1, bty="n")

dlmGibbsDIG Gibbs sampling for d-inverse-gamma model

Description
The function implements a Gibbs sampler for a univariate DLM having one or more unknown
variances in its specification.

Usage
dlmGibbsDIG(y, mod, a.y, b.y, a.theta, b.theta, shape.y, rate.y,
shape.theta, rate.theta, n.sample = 1,
thin = 0, ind, save.states = TRUE,
progressBar = interactive())

Arguments
y data vector or univariate time series
mod a dlm for univariate observations
a.y prior mean of observation precision
b.y prior variance of observation precision
a.theta prior mean of system precisions (recycled, if needed)
b.theta prior variance of system precisions (recycled, if needed)
shape.y shape parameter of the prior of observation precision
rate.y rate parameter of the prior of observation precision
shape.theta shape parameter of the prior of system precisions (recycled, if needed)
14 dlmGibbsDIG

rate.theta rate parameter of the prior of system precisions (recycled, if needed)

n.sample requested number of Gibbs iterations
thin discard thin iterations for every saved iteration
ind indicator of the system variances that need to be estimated
save.states should the simulated states be included in the output?
progressBar should a text progress bar be displayed during execution?

Details

The d-inverse-gamma model is a constant univariate DLM with unknown observation variance, di-
agonal system variance with unknown diagonal entries. Some of these entries may be known, in
which case they are typically zero. Independent inverse gamma priors are assumed for the unknown
variances. These can be specified be mean and variance or, alternatively, by shape and rate. Re-
cycling is applied for the prior parameters of unknown system variances. The argument ind can
be used to specify the index of the unknown system variances, in case some of the diagonal ele-
ments of W are known. The unobservable states are generated in the Gibbs sampler and are returned
if save.states = TRUE. For more details on the model and usage examples, see the package
vignette.

Value

The function returns a list of simulated values.

dV simulated values of the observation variance.

dW simulated values of the unknown diagonal elements of the system variance.
theta simulated values of the state vectors.

Author(s)

Giovanni Petris <GPetris@uark.edu>

References

Giovanni Petris (2010), An R Package for Dynamic Linear Models. Journal of Statistical Software,
36(12), 1-16. http://www.jstatsoft.org/v36/i12/.
Petris, Petrone, and Campagnoli, Dynamic Linear Models with R, Springer (2009).

Examples
## See the package vignette for an example
dlmLL 15

dlmLL Log likelihood evaluation for a state space model

Description
Function that computes the log likelihood of a state space model.

Usage
dlmLL(y, mod, debug=FALSE)

Arguments
y a vector, matrix, or time series of data.
mod an object of class "dlm", or a list with components m0, C0, FF, V, GG, W defining
the model and the parameters of the prior distribution.
debug if debug=TRUE, the function uses R code, otherwise it uses faster C code.

Details
The calculations are based on singular value decomposition. Missing values are allowed in y.

Value
The function returns the negative of the loglikelihood.

Warning
The observation variance V in mod must be nonsingular.

Author(s)
Giovanni Petris <GPetris@uark.edu>

References
Durbin and Koopman, Time series analysis by state space methods, Oxford University Press, 2001.

See Also
dlmMLE, dlmFilter for the definition of the equations of the model.

Examples
##---- See the examples for dlmMLE ----
16 dlmMLE

dlmMLE Parameter estimation by maximum likelihood

Description
The function returns the MLE of unknown parameters in the specification of a state space model.

Usage
dlmMLE(y, parm, build, method = "L-BFGS-B", ..., debug = FALSE)

Arguments
y a vector, matrix, or time series of data.
parm vector of initial values - for the optimization routine - of the unknown parame-
ters.
build a function that takes a vector of the same length as parm and returns an object of
class dlm, or a list that may be interpreted as such.
method passed to optim.
... additional arguments passed to optim and build.
debug if debug=TRUE, the likelihood calculations are done entirely in R, otherwise C
functions are used.

Details
The evaluation of the loglikelihood is done by dlmLL. For the optimization, optim is called. It
is possible for the model to depend on additional parameters, other than those in parm, passed to
build via the ... argument.

Value
The function dlmMLE returns the value returned by optim.

Warning
The build argument must return a dlm with nonsingular observation variance V.

Author(s)
Giovanni Petris <GPetris@uark.edu>

References
Giovanni Petris (2010), An R Package for Dynamic Linear Models. Journal of Statistical Software,
36(12), 1-16. http://www.jstatsoft.org/v36/i12/.
Petris, Petrone, and Campagnoli, Dynamic Linear Models with R, Springer (2009).
dlmModARMA 17

See Also
dlmLL, dlm.

Examples
data(NelPlo)
### multivariate local level -- seemingly unrelated time series
buildSu <- function(x) {
Vsd <- exp(x[1:2])
Vcorr <- tanh(x[3])
V <- Vsd %o% Vsd
V[1,2] <- V[2,1] <- V[1,2] * Vcorr
Wsd <- exp(x[4:5])
Wcorr <- tanh(x[6])
W <- Wsd %o% Wsd
W[1,2] <- W[2,1] <- W[1,2] * Wcorr
return(list(
m0 = rep(0,2),
C0 = 1e7 * diag(2),
FF = diag(2),
GG = diag(2),
V = V,
W = W))
}

suMLE <- dlmMLE(NelPlo, rep(0,6), buildSu); suMLE

buildSu(suMLE$par)[c("V","W")]
StructTS(NelPlo[,1], type="level") ## compare with W[1,1] and V[1,1]
StructTS(NelPlo[,2], type="level") ## compare with W[2,2] and V[2,2]

## multivariate local level model with homogeneity restriction

buildHo <- function(x) {
Vsd <- exp(x[1:2])
Vcorr <- tanh(x[3])
V <- Vsd %o% Vsd
V[1,2] <- V[2,1] <- V[1,2] * Vcorr
return(list(
m0 = rep(0,2),
C0 = 1e7 * diag(2),
FF = diag(2),
GG = diag(2),
V = V,
W = x[4]^2 * V))
}

hoMLE <- dlmMLE(NelPlo, rep(0,4), buildHo); hoMLE

buildHo(hoMLE$par)[c("V","W")]

dlmModARMA Create a DLM representation of an ARMA process

18 dlmModARMA

Description

The function creates an object of class dlm representing a specified univariate or multivariate
ARMA process

Usage

dlmModARMA(ar = NULL, ma = NULL, sigma2 = 1, dV, m0, C0)

Arguments

ar a vector or a list of matrices (in the multivariate case) containing the autoregres-
sive coefficients.
ma a vector or a list of matrices (in the multivariate case) containing the moving
average coefficients.
sigma2 the variance (or variance matrix) of the innovations.
dV the variance, or the diagonal elements of the variance matrix in the multivariate
case, of the observation noise. V is assumed to be diagonal and it defaults to
zero.
m0 m0 , the expected value of the pre-sample state vector.
C0 C0 , the variance matrix of the pre-sample state vector.

Details

The returned DLM only gives one of the many possible representations of an ARMA process.

Value

The function returns an object of class dlm representing the ARMA model specified by ar, ma, and
sigma2.

Author(s)

Giovanni Petris <GPetris@uark.edu>

References

Giovanni Petris (2010), An R Package for Dynamic Linear Models. Journal of Statistical Software,
36(12), 1-16. http://www.jstatsoft.org/v36/i12/.
Petris, Petrone, and Campagnoli, Dynamic Linear Models with R, Springer (2009).
Durbin and Koopman, Time series analysis by state space methods, Oxford University Press, 2001.

See Also

dlmModPoly, dlmModSeas, dlmModReg

dlmModPoly 19

Examples
## ARMA(2,3)
dlmModARMA(ar = c(.5,.1), ma = c(.4,2,.3), sigma2=1)
## Bivariate ARMA(2,1)
dlmModARMA(ar = list(matrix(1:4,2,2), matrix(101:104,2,2)),
ma = list(matrix(-4:-1,2,2)), sigma2 = diag(2))

dlmModPoly Create an n-th order polynomial DLM

Description
The function creates an nth order polynomial DLM.

Usage
dlmModPoly(order = 2, dV = 1, dW = c(rep(0, order - 1), 1),
m0 = rep(0, order), C0 = 1e+07 * diag(nrow = order))

Arguments
order order of the polynomial model. The default corresponds to a stochastic linear
trend.
dV variance of the observation noise.
dW diagonal elements of the variance matrix of the system noise.
m0 m0 , the expected value of the pre-sample state vector.
C0 C0 , the variance matrix of the pre-sample state vector.

Value
An object of class dlm representing the required n-th order polynomial model.

Author(s)
Giovanni Petris <GPetris@uark.edu>

References
Giovanni Petris (2010), An R Package for Dynamic Linear Models. Journal of Statistical Software,
36(12), 1-16. http://www.jstatsoft.org/v36/i12/.
Petris, Petrone, and Campagnoli, Dynamic Linear Models with R, Springer (2009).
West and Harrison, Bayesian forecasting and dynamic models (2nd ed.), Springer, 1997.

See Also
dlmModARMA, dlmModReg, dlmModSeas
20 dlmModReg

Examples
## the default
dlmModPoly()
## random walk plus noise
dlmModPoly(1, dV = .3, dW = .01)

dlmModReg Create a DLM representation of a regression model

Description
The function creates a dlm representation of a linear regression model.

Usage
dlmModReg(X, addInt = TRUE, dV = 1, dW = rep(0, NCOL(X) + addInt),
m0 = rep(0, length(dW)),
C0 = 1e+07 * diag(nrow = length(dW)))

Arguments
X the design matrix
addInt logical: should an intercept be added?
dV variance of the observation noise.
dW diagonal elements of the variance matrix of the system noise.
m0 m0 , the expected value of the pre-sample state vector.
C0 C0 , the variance matrix of the pre-sample state vector.

Details
By setting dW equal to a nonzero vector one obtains a DLM representation of a dynamic regression
model. The default value zero of dW corresponds to standard linear regression. Only univariate
regression is currently covered.

Value
An object of class dlm representing the specified regression model.

Author(s)
Giovanni Petris <GPetris@uark.edu>

References
Giovanni Petris (2010), An R Package for Dynamic Linear Models. Journal of Statistical Software,
36(12), 1-16. http://www.jstatsoft.org/v36/i12/.
Petris, Petrone, and Campagnoli, Dynamic Linear Models with R, Springer (2009).
West and Harrison, Bayesian forecasting and dynamic models (2nd ed.), Springer, 1997.
dlmModSeas 21

See Also
dlmModARMA, dlmModPoly, dlmModSeas

Examples
x <- matrix(runif(6,4,10), nc = 2); x
dlmModReg(x)
dlmModReg(x, addInt = FALSE)

dlmModSeas Create a DLM for seasonal factors

Description
The function creates a DLM representation of seasonal component.

Usage
dlmModSeas(frequency, dV = 1, dW = c(1, rep(0, frequency - 2)),
m0 = rep(0, frequency - 1),
C0 = 1e+07 * diag(nrow = frequency - 1))

Arguments
frequency how many seasons?
dV variance of the observation noise.
dW diagonal elements of the variance matrix of the system noise.
m0 m0 , the expected value of the pre-sample state vector.
C0 C0 , the variance matrix of the pre-sample state vector.

Value
An object of class dlm representing a seasonal factor for a process with frequency seasons.

Author(s)
Giovanni Petris <GPetris@uark.edu>

References
Giovanni Petris (2010), An R Package for Dynamic Linear Models. Journal of Statistical Software,
36(12), 1-16. http://www.jstatsoft.org/v36/i12/.
Petris, Petrone, and Campagnoli, Dynamic Linear Models with R, Springer (2009).
Harvey, Forecasting, structural time series models and the Kalman filter, Cambridge University
Press, 1989.
22 dlmModTrig

See Also
dlmModARMA, dlmModPoly, dlmModReg, and dlmModTrig for the Fourier representation of a seasonal
component.

Examples
## seasonal component for quarterly data
dlmModSeas(4, dV = 3.2)

dlmModTrig Create Fourier representation of a periodic DLM component

Description
The function creates a dlm representing a specified periodic component.

Usage
dlmModTrig(s, q, om, tau, dV = 1, dW = 0, m0, C0)

Arguments
s the period, if integer.
q number of harmonics in the DLM.
om the frequency.
tau the period, if not an integer.
dV variance of the observation noise.
dW a single number expressing the variance of the system noise.
m0 m0 , the expected value of the pre-sample state vector.
C0 C0 , the variance matrix of the pre-sample state vector.

Details
The periodic component is specified by one and only one of s, om, and tau. When s is given, the
function assumes that the period is an integer, while a period specified by tau is assumed to be
noninteger. Instead of tau, the frequency om can be specified. The argument q specifies the number
of harmonics to include in the model. When tau or omega is given, then q is required as well, since
in this case the implied Fourier representation has infinitely many harmonics. On the other hand, if
s is given, q defaults to all the harmonics in the Fourier representation, that is floor(s/2).
The system variance of the resulting dlm is dW times the identity matrix of the appropriate dimen-
sion.

Value
An object of class dlm, representing a periodic component.
dlmRandom 23

Author(s)
Giovanni Petris <GPetris@uark.edu>

References
Giovanni Petris (2010), An R Package for Dynamic Linear Models. Journal of Statistical Software,
36(12), 1-16. http://www.jstatsoft.org/v36/i12/.
Petris, Petrone, and Campagnoli, Dynamic Linear Models with R, Springer (2009).
West and Harrison, Bayesian forecasting and dynamic models (2nd ed.), Springer (1997).

See Also
dlmModSeas, dlmModARMA, dlmModPoly, dlmModReg

Examples
dlmModTrig(s = 3)
dlmModTrig(tau = 3, q = 1) # same thing
dlmModTrig(s = 4) # for quarterly data
dlmModTrig(s = 4, q = 1)
dlmModTrig(tau = 4, q = 2) # a bad idea!
m1 <- dlmModTrig(tau = 6.3, q = 2); m1
m2 <- dlmModTrig(om = 2 * pi / 6.3, q = 2)
all.equal(unlist(m1), unlist(m2))

dlmRandom Random DLM

Description
Generate a random (constant or time-varying) object of class "dlm", along with states and observa-
tions from it.

Usage
dlmRandom(m, p, nobs = 0, JFF, JV, JGG, JW)

Arguments
m dimension of the observation vector.
p dimension of the state vector.
nobs number of states and observations to simulate from the model.
JFF should the model have a time-varying FF component?
JV should the model have a time-varying V component?
JGG should the model have a time-varying GG component?
JW should the model have a time-varying W component?
24 dlmSmooth

Details
The function generates randomly the system and observation matrices and the variances of a DLM
having the specified state and observation dimension. The system matrix GG is guaranteed to have
eigenvalues strictly less than one, which implies that a constant DLM is asymptotically stationary.
The default behavior is to generate a constant DLM. If JFF is TRUE then a model for nobs obser-
vations in which all the elements of FF are time-varying is generated. Similarly with JV, JGG, and
JW.

Value
The function returns a list with the following components.
mod An object of class "dlm".
theta Matrix of simulated state vectors from the model.
y Matrix of simulated observations from the model.
If nobs is zero, only the mod component is returned.

Author(s)
Giovanni Petris <GPetris@uark.edu>

References
Anderson and Moore, Optimal filtering, Prentice-Hall (1979)

See Also
dlm

Examples
dlmRandom(1, 3, 5)

dlmSmooth DLM smoothing

Description
The function apply Kalman smoother to compute smoothed values of the state vectors, together
with their variance/covariance matrices.

Usage
dlmSmooth(y, ...)
## Default S3 method:
dlmSmooth(y, mod, ...)
## S3 method for class 'dlmFiltered'
dlmSmooth(y, ..., debug = FALSE)
dlmSmooth 25

Arguments
y an object used to select a method.
... futher arguments passed to or from other methods.
mod an object of class "dlm".
debug if debug=FALSE, faster C code will be used, otherwise all the computations will
be performed in R.

Details
The default method returns means and variances of the smoothing distribution for a data vector (or
matrix) y and a model mod.
dlmSmooth.dlmFiltered produces the same output based on a dlmFiltered object, typically one
produced by a call to dlmFilter.
The calculations are based on the singular value decomposition (SVD) of the relevant matrices.
Variance matrices are returned in terms of their SVD.

Value
A list with components

s Time series (or matrix) of smoothed values of the state vectors. The series starts
one time unit before the first observation.
U.S See below.
D.S Together with U.S, it gives the SVD of the variances of the smoothing errors.

Warning
The observation variance V in mod must be nonsingular.

Author(s)
Giovanni Petris <GPetris@uark.edu>

References
Zhang, Y. and Li, X.R., Fixed-interval smoothing algorithm based on singular value decomposition,
Proceedings of the 1996 IEEE International Conference on Control Applications.
Giovanni Petris (2010), An R Package for Dynamic Linear Models. Journal of Statistical Software,
36(12), 1-16. http://www.jstatsoft.org/v36/i12/.
Petris, Petrone, and Campagnoli, Dynamic Linear Models with R, Springer (2009).

See Also
See dlm for a description of dlm objects, dlmSvd2var to obtain a variance matrix from its SVD,
dlmFilter for Kalman filtering, dlmMLE for maximum likelihood estimation, and dlmBSample for
drawing from the posterior distribution of the state vectors.
26 dlmSum

Examples

s <- dlmSmooth(Nile, dlmModPoly(1, dV = 15100, dW = 1470))

plot(Nile, type ='o')
lines(dropFirst(s$s), col = "red")

## Multivariate
set.seed(2)
tmp <- dlmRandom(3, 5, 20)
obs <- tmp$y
m <- tmp$mod
rm(tmp)

f <- dlmFilter(obs, m)
s <- dlmSmooth(f)
all.equal(s, dlmSmooth(obs, m))

dlmSum Outer sum of Dynamic Linear Models

Description
dlmSum creates a unique DLM out of two or more independent DLMs. %+% is an alias for dlmSum.

Usage
dlmSum(...)
x %+% y

Arguments
... any number of objects of class dlm, or a list of such objects.
x, y objects of class dlm.

Value
An object of class dlm, representing the outer sum of the arguments.

Author(s)
Giovanni Petris <GPetris@uark.edu>

References
Giovanni Petris (2010), An R Package for Dynamic Linear Models. Journal of Statistical Software,
36(12), 1-16. http://www.jstatsoft.org/v36/i12/.
Petris, Petrone, and Campagnoli, Dynamic Linear Models with R, Springer (2009).
dlmSvd2var 27

Examples
m1 <- dlmModPoly(2)
m2 <- dlmModPoly(1)
dlmSum(m1, m2)
m1 %+% m2 # same thing

dlmSvd2var Compute a nonnegative definite matrix from its Singular Value Decom-
position

Description

The function computes a nonnegative definite matrix from its Singular Value Decomposition.

Usage

dlmSvd2var(u, d)

Arguments

u a square matrix, or a list of square matrices for a vectorized usage.

d a vector, or a matrix for a vectorized usage.

Details

The SVD of a nonnegative definite n by n square matrix x can be written as ud2 u0 , where u is an
n by n orthogonal matrix and d is a diagonal matrix. For a single matrix, the function returns just
ud2 u0 . Note that the argument d is a vector containing the diagonal elements of d. For a vectorized
usage, u is a list of square matrices, and d is a matrix. The returned value in this case is a list of
matrices, with the element i being u[[i]] %*% diag(d[i,]^2) %*% t(u[[i]]).

Value

The function returns a nonnegative definite matrix, reconstructed from its SVD, or a list of such
matrices (see details above).

Author(s)

Giovanni Petris <GPetris@uark.edu>

References

Horn and Johnson, Matrix analysis, Cambridge University Press (1985)

28 dropFirst

Examples
x <- matrix(rnorm(16),4,4)
x <- crossprod(x)
tmp <- La.svd(x)
all.equal(dlmSvd2var(tmp$u, sqrt(tmp$d)), x)
## Vectorized usage
x <- dlmFilter(Nile, dlmModPoly(1, dV=15099, dW=1469))
x$se <- sqrt(unlist(dlmSvd2var(x$U.C, x$D.C)))
## Level with 50% probability interval
plot(Nile, lty=2)
lines(dropFirst(x$m), col="blue")
lines(dropFirst(x$m - .67*x$se), lty=3, col="blue")
lines(dropFirst(x$m + .67*x$se), lty=3, col="blue")

dropFirst Drop the first element of a vector or matrix

Description
A utility function, dropFirst drops the first element of a vector or matrix, retaining the correct
time series attributes, in case the argument is a time series object.

Usage
dropFirst(x)

Arguments
x a vector or matrix.

Value
The function returns x[-1] or x[-1,], if the argument is a matrix. For an argument of class ts the
class is preserved, together with the correct tsp attribute.

Author(s)
Giovanni Petris <GPetris@uark.edu>

Examples
(pres <- dropFirst(presidents))
start(presidents)
start(pres)
FF 29

FF Components of a dlm object

Description
Functions to get or set specific components of an object of class dlm

Usage
## S3 method for class 'dlm'
FF(x)
## S3 replacement method for class 'dlm'
FF(x) <- value
## S3 method for class 'dlm'
V(x)
## S3 replacement method for class 'dlm'
V(x) <- value
## S3 method for class 'dlm'
GG(x)
## S3 replacement method for class 'dlm'
GG(x) <- value
## S3 method for class 'dlm'
W(x)
## S3 replacement method for class 'dlm'
W(x) <- value
## S3 method for class 'dlm'
m0(x)
## S3 replacement method for class 'dlm'
m0(x) <- value
## S3 method for class 'dlm'
C0(x)
## S3 replacement method for class 'dlm'
C0(x) <- value
## S3 method for class 'dlm'
JFF(x)
## S3 replacement method for class 'dlm'
JFF(x) <- value
## S3 method for class 'dlm'
JV(x)
## S3 replacement method for class 'dlm'
JV(x) <- value
## S3 method for class 'dlm'
JGG(x)
## S3 replacement method for class 'dlm'
JGG(x) <- value
## S3 method for class 'dlm'
JW(x)
30 FF

## S3 replacement method for class 'dlm'

JW(x) <- value
## S3 method for class 'dlm'
X(x)
## S3 replacement method for class 'dlm'
X(x) <- value

Arguments
x an object of class dlm.
value a numeric matrix (or vector for m0).

Details
Missing or infinite values are not allowed in value. The dimension of value must match the
dimension of the current value of the specific component in x

Value
For the assignment forms, the updated dlm object.
For the other forms, the specific component of x.

Author(s)
Giovanni Petris <GPetris@uark.edu>

See Also
dlm

Examples
set.seed(222)
mod <- dlmRandom(5, 6)
all.equal( FF(mod), mod$FF )
all.equal( V(mod), mod$V )
all.equal( GG(mod), mod$GG )
all.equal( W(mod), mod$W )
all.equal( m0(mod), mod$m0 )
all.equal( C0(mod), mod$C0)
m0(mod)
m0(mod) <- rnorm(6)
C0(mod)
C0(mod) <- rwishart(10, 6)
### A time-varying model
mod <- dlmModReg(matrix(rnorm(10), 5, 2))
JFF(mod)
X(mod)
mcmc 31

mcmc Utility functions for MCMC output analysis

Description
Returns the mean, the standard deviation of the mean, and a sequence of partial means of the input
vector or matrix.

Usage
mcmcMean(x, sd = TRUE)
mcmcMeans(x, sd = TRUE)
mcmcSD(x)
ergMean(x, m = 1)

Arguments
x vector or matrix containing the output of a Markov chain Monte Carlo simula-
tion.
sd logical: should an estimate of the Monte Carlo standard deviation be reported?
m ergodic means are computed for i in m:NROW(x)

Details
The argument x is typically the output from a simulation. If a matrix, rows are considered consec-
utive simulations of a target vector. In this case means, standard deviations, and ergodic means are
returned for each column. The standard deviation of the mean is estimated using Sokal’s method
(see the reference). mcmcMeans is an alias for mcmcMean.

Author(s)
Giovanni Petris <GPetris@uark.edu>

References
P. Green (2001). A Primer on Markov Chain Monte Carlo. In Complex Stochastic Systems,
(Barndorff-Nielsen, Cox and Kl\"uppelberg, eds.). Chapman and Hall/CRC.

Examples
x <- matrix(rexp(1000), nc=4)
dimnames(x) <- list(NULL, LETTERS[1:NCOL(x)])
mcmcSD(x)
mcmcMean(x)
em <- ergMean(x, m = 51)
plot(ts(em, start=51), xlab="Iteration", main="Ergodic means")
32 residuals.dlmFiltered

NelPlo Nelson-Plosser macroeconomic time series

Description
A subset of Nelson-Plosser data.

Usage
data(NelPlo)

Format
The format is: mts [1:43, 1:2] -4.39 3.12 1.08 -1.50 3.91 ... - attr(*, "tsp")= num [1:3] 1946 1988 1
- attr(*, "class")= chr [1:2] "mts" "ts" - attr(*, "dimnames")=List of 2 ..$ : NULL ..$ : chr [1:2] "ip"
"stock.prices"

Details
The series are 100*diff(log()) of industrial production and stock prices (S&P500) from 1946 to
1988.

Source
The complete data set is available in package tseries.

Examples
data(NelPlo)
plot(NelPlo)

residuals.dlmFiltered One-step forecast errors

Description
The function computes one-step forecast errors for a filtered dynamic linear model.

Usage
## S3 method for class 'dlmFiltered'
residuals(object, ..., type = c("standardized", "raw"), sd = TRUE)
residuals.dlmFiltered 33

Arguments

object an object of class "dlmFiltered", such as the output from dlmFilter

... unused additional arguments.
type should standardized or raw forecast errors be produced?
sd when sd = TRUE, standard deviations are returned as well.

Value

A vector or matrix (in the multivariate case) of one-step forecast errors, standardized if type = "standardized".
Time series attributes of the original observation vector (matrix) are retained by the one-step fore-
cast errors.
If sd = TRUE then the returned value is a list with the one-step forecast errors in component res
and the corresponding standard deviations in component sd.

Note

The object argument must include a component y containing the data. This component will not be
present if object was obtained by calling dlmFilter with simplify = TRUE.

Author(s)

Giovanni Petris <GPetris@uark.edu>

References

Giovanni Petris (2010), An R Package for Dynamic Linear Models. Journal of Statistical Software,
36(12), 1-16. http://www.jstatsoft.org/v36/i12/.
Petris, Petrone, and Campagnoli, Dynamic Linear Models with R, Springer (2009).
West and Harrison, Bayesian forecasting and dynamic models (2nd ed.), Springer (1997).

See Also

dlmFilter

Examples
## diagnostic plots
nileMod <- dlmModPoly(1, dV = 15100, dW = 1468)
nileFilt <- dlmFilter(Nile, nileMod)
res <- residuals(nileFilt, sd=FALSE)
qqnorm(res)
tsdiag(nileFilt)
34 rwishart

rwishart Random Wishart matrix

Description
Generate a draw from a Wishart distribution.

Usage
rwishart(df, p = nrow(SqrtSigma), Sigma, SqrtSigma = diag(p))

Arguments
df degrees of freedom. It has to be integer.
p dimension of the matrix to simulate.
Sigma the matrix parameter Sigma of the Wishart distribution.
SqrtSigma a square root of the matrix parameter Sigma of the Wishart distribution. Sigma
must be equal to crossprod(SqrtSigma).

Details
The Wishart is a distribution on the set of nonnegative definite symmetric matrices. Its density is

c|W |(n−p−1)/2
 
1 −1
p(W ) = exp − tr(Σ W )
|Σ|n/2 2

where n is the degrees of freedom parameter df and c is a normalizing constant. The mean of the
Wishart distribution is nΣ and the variance of an entry is

Var(Wij ) = n(Σ2ij + Σii Σjj )

The matrix parameter, which should be a positive definite symmetric matrix, can be specified via
either the argument Sigma or SqrtSigma. If Sigma is specified, then SqrtSigma is ignored. No
checks are made for symmetry and positive definiteness of Sigma.

Value
The function returns one draw from the Wishart distribution with df degrees of freedom and matrix
parameter Sigma or crossprod(SqrtSigma)

Warning
The function only works for an integer number of degrees of freedom.

Note
From a suggestion by B.Venables, posted on S-news
USecon 35

Author(s)
Giovanni Petris <GPetris@uark.edu>

References
Press (1982). Applied multivariate analysis.

Examples
rwishart(25, p = 3)
a <- matrix(rnorm(9), 3)
rwishart(30, SqrtSigma = a)
b <- crossprod(a)
rwishart(30, Sigma = b)

USecon US macroeconomic time series

Description
US macroeconomic data.

Usage
data(USecon)

Format
The format is: mts [1:40, 1:2] 0.1364 0.0778 -0.3117 -0.5478 -1.2636 ... - attr(*, "dimnames")=List
of 2 ..$ : NULL ..$ : chr [1:2] "M1" "GNP" - attr(*, "tsp")= num [1:3] 1978 1988 4 - attr(*, "class")=
chr [1:2] "mts" "ts"

Details
The series are 100*diff(log()) of seasonally adjusted real U.S. money ’M1’ and GNP from 1978
to 1987.

Source
The complete data set is available in package tseries.

Examples
data(USecon)
plot(USecon)
Index

∗Topic array dlmFilter, 10

dlmSvd2var, 27 dlmSmooth, 24
∗Topic datagen dlmSum, 26
dlmRandom, 23 dropFirst, 28
∗Topic datasets FF, 29
NelPlo, 32 %+% (dlmSum), 26
USecon, 35
∗Topic distribution arms, 2, 6
arms, 2 ARtransPars, 5
rwishart, 34 as.dlm (dlm), 7
∗Topic misc
bdiag, 6
arms, 2
ARtransPars, 5 C0 (FF), 29
bdiag, 6 C0<- (FF), 29
convex.bounds, 6 convex.bounds, 6
dlm, 7
dlmBSample, 9 dlm, 7, 11, 17, 24, 25, 30
dlmFilter, 10 dlmBSample, 9, 11, 25
dlmForecast, 12 dlmFilter, 9, 10, 15, 25, 33
dlmGibbsDIG, 13 dlmForecast, 12
dlmLL, 15 dlmGibbsDIG, 13
dlmMLE, 16 dlmLL, 15, 17
dlmModARMA, 17 dlmMLE, 11, 15, 16, 25
dlmModPoly, 19 dlmModARMA, 8, 17, 19, 21–23
dlmModReg, 20 dlmModPoly, 8, 18, 19, 21–23
dlmModSeas, 21 dlmModReg, 8, 18, 19, 20, 22, 23
dlmModTrig, 22 dlmModSeas, 8, 18, 19, 21, 21, 23
dlmRandom, 23 dlmModTrig, 22, 22
dlmSmooth, 24 dlmRandom, 23
dlmSum, 26 dlmSmooth, 11, 24
dlmSvd2var, 27 dlmSum, 26
dropFirst, 28 dlmSvd2var, 11, 25, 27
mcmc, 31 dropFirst, 28
residuals.dlmFiltered, 32
ergMean (mcmc), 31
∗Topic multivariate
arms, 2 FF, 29
∗Topic smooth FF<- (FF), 29
dlmSmooth, 24
∗Topic ts GG (FF), 29

36
INDEX 37

GG<- (FF), 29

is.dlm (dlm), 7

JFF (FF), 29
JFF<- (FF), 29
JGG (FF), 29
JGG<- (FF), 29
JV (FF), 29
JV<- (FF), 29
JW (FF), 29
JW<- (FF), 29

m0 (FF), 29
m0<- (FF), 29
mcmc, 31
mcmcMean (mcmc), 31
mcmcMeans (mcmc), 31
mcmcSD (mcmc), 31

NelPlo, 32

residuals.dlmFiltered, 32
rwishart, 34

USecon, 35

V (FF), 29
V<- (FF), 29

W (FF), 29
W<- (FF), 29

X (FF), 29
X<- (FF), 29