portes (version 5.0)

varima.sim: Simulate Data From Seasonal/Nonseasonal ARIMA(p,d,q)*(ps,ds,qs)_s or VARIMA(p,d,q)*(ps,ds,qs)_s Models

Description

Simulate time series from AutoRegressive Integrated Moving Average, ARIMA(p,d,q), or Vector Integrated AutoRegressive Moving Average, VARIMA(p,d,q), where d is a nonnegative difference integer in the ARIMA case and it is a vector of \(k\) differenced components \(d_1,...,d_k\) in the VARIMA case. In general, this function can be implemented in simulating univariate or multivariate Seasonal AutoRegressive Integrated Moving Average, SARIMA(p,d,q)*(ps,ds,qs)_s and SVARIMA(p,d,q)*(ps,ds,qs)_s, where ps and qs are the orders of the seasonal univariate/multivariate AutoRegressive and Moving Average components respectively. ds is a nonnegative difference integer in the SARIMA case and it is a vector of \(k\) differenced components \(ds_1,...,ds_k\) in the SVARIMA case, whereas s is the seasonal period. The simulated process may have a deterministic terms, drift constant and time trend, with non-zero mean. The innovations may have finite or infinite variance.

Usage

varima.sim(model=list(ar=NULL,ma=NULL,d=NULL,sar=NULL,sma=NULL,D=NULL,period=NULL),
          n,k=1,constant=NA,trend=NA,demean=NA,innov=NULL, 
          innov.dist=c("Gaussian","t","stable","bootstrap"),...)

Arguments

model

a list with univariate/multivariate component ar and/or ma and/or sar and/or sma giving the univariate/multivariate AR and/or MA and/or SAR and/or SMA coefficients respectively. period specifies the seasonal period. For seasonality, default is NULL indicates that period =12. d and D are integer or vector representing the order of the usual and seasonal difference. An empty list gives an ARIMA(0, 0, 0)*(0,0,0)_null model, that is white noise.

n

length of the series.

k

number of simulated series. For example, k=1 is used for univariate series and k=2 is used for bivariate series.

constant

a numeric vector represents the intercept in the deterministic equation.

trend

a numeric vector represents the slop in the deterministic equation.

demean

a numeric vector represents the mean of the series.

innov

a vector of univariate or multivariate innovation series. This may used as an initial series to genrate innovations with innov.dist = "stable" or innov.dist = "bootstrap". This argument is irrelevant with the other selections of innov.dist.

innov.dist

distribution to generate univariate or multivariate innovation process. This could be Gaussian, t, stable, or bootstrap using resampled errors rather than distributed errors. Default is Gaussian.

...

arguments to be passed to methods, such as dft degrees of freedom needed to generate innovations with univariate/multivariate series with t-distribution innovations. The argument par.stable may passed this function as a four parameters (vector or matrix of dimension k*4) to generate univariate/multivariate series with dimension n*k using stable distribution with infinite variance innovations. The four parameters as described in the function rStable are Alpha, Beta, Scale, and Location. The argument trunc.lag represents the truncation lag that is used to truncate the infinite MA or VMA Process. IF it is not given, then trunc.lag = min(100, \(n/3\)). Optionally sigma is the variance of a Gaussian or t white noise series.

Value

Simulated data from SARIMA(p,d,q) or SVARIMA(p,d,q)*(ps,ds,qs)_s process that may have a drift and deterministic time trend terms.

Details

This function is used to simulate a univariate/multivariate seasonal/nonseasonal SARIMA or SVARIMA model of order \((p,d,q)\times(ps,ds,qs)_s\) $$\phi(B)\Phi(B^s)d(B)D(B^s)(Z_{t})-\mu = a + b \times t + \theta(B)\Theta(B^s)e_{t},$$ where \(a, b,\) and \(\mu\) correspond to the arguments constant, trend, and demean respectively. The univariate or multivariate series \(e_{t}\) represents the innovations series given from the argument innov. If innov = NULL then \(e_{t}\) will be generated from a univariate or multivariate normal distribution, t-distribution, or stable distribution of infinite variance specified from the argument innov.dist. \(\phi(B)\) and \(\theta(B)\) are the VAR and the VMA coefficient matrices respectively and \(B\) is the backshift time operator. \(\Phi(B^s)\) and \(\Theta(B^s)\) are the Vector SAR Vector SMA coefficient matrices respectively. \(d(B)=diag[(1-B)^{d_{1}},\ldots,(1-B)^{d_{k}}]\) and \(D(B^s)=diag[(1-B^s)^{ds_{1}},\ldots,(1-B^s)^{ds_{k}}]\) are diagonal matrices. This states that each individual series \(Z_{i}, i=1,...,k\) is differenced \(d_{i}ds_{i}\) times to reduce to a stationary Vector ARMA(p,0,q)*(ps,0,qs)_s series.

References

Hipel, K.W. and McLeod, A.I. (2005). "Time Series Modelling of Water Resources and Environmental Systems".

Reinsel, G. C. (1997). "Elements of Multivariate Time Series Analysis". Springer-Verlag, 2nd edition.

See Also

arima.sim, vma.sim, ImpulseVMA, InvertQ, fitstable, rStable

Examples

# NOT RUN {
#################################################################################
# Simulate white noise series from a Gaussian distribution                      #
#################################################################################
set.seed(1234)
Z1 <- varima.sim(n=400)       ## a univariate series
plot(Z1)
Z2 <- varima.sim(n=400,k=2)   ## a bivariate series
plot(Z2)
Z3 <- varima.sim(n=400,k=5)   ## a multivariate series of dimension 5
plot(Z3)
#################################################################################
# Simulate MA(1) where innovation series is provided via argument innov         #
#################################################################################
set.seed(1234)
n <- 200
theta <-  0.6
Z<-varima.sim(list(ma=theta),n=n,innov=rnorm(n),innov.dist="bootstrap")
plot(Z)
#################################################################################
# Simulate seasonal ARIMA(2,1,0)*(0,2,1)_12 process with phi=c(1.3,-0.35),      #
# theta.season = 0.8. Gaussian innovations. The series is truncated at lag 50   #
#################################################################################
set.seed(12834)
n <- 100
phi <- c(1.3, -0.35)
theta.season <- 0.8
Z<-varima.sim(list(ar=phi,d=1,sma=theta.season,D=2),n=n,trunc.lag=50)
plot(Z)
acf(Z)
#################################################################################
# Simulate seasonal ARMA(0,0,0)*(2,0,0)_4 process with intercept = 0.8          #
# phi.season = c(0.9,-0.9), period = 4, t5-distribution innovations with df = 3 #
#################################################################################
set.seed(1234)
n <- 200
phi.season <- c(0.9,-0.9)
Z<-varima.sim(list(sar=phi.season,period=4),n=n,constant=0.8,innov.dist="t",dft=3)
plot(Z)
acf(Z)
arima(Z,order=c(0,0,0),seasonal = list(order = c(2,0,0),period=4))
#################################################################################
# Simulate univariate ARMA(2,1) process with length 500,                        #
# phi = c(1.3, -0.35), theta = 0.1. Drift equation is 8 + 0.05*t                #
# Stable innovations with: Alpha = 1.75, Beta = 0, Scale = 1, Location = 0      #
#################################################################################
set.seed(1234)
n <- 500
phi <-  c(1.3, -0.35)
theta <-  0.1
Alpha <- 1.75
Beta <- 0
Scale <- 1
Location <- 0
Stable <- c(Alpha,Beta,Scale,Location)
Z <- varima.sim(list(ar=phi,ma=theta),n=n,constant=8,trend=0.05,demean=0,
     innov.dist="stable",par.stable=Stable)
plot(Z)
#################################################################################
# Simulate a bivariate white noise series from a multivariate t4-distribution   #
# Then use the nonparametric bootstrap method to generate a seasonal SVARIMA    #
# of order (0,d,0)*(0,0,1)_12 with d = c(1, 0), n= 250, k = 2, and              #
# theta.season=array(c(0.5,0.4,0.1,0.3),dim=c(k,k,1))                           #
#################################################################################
set.seed(1234)
Z1 <- varima.sim(n=250,k=2,innov.dist="t",dft=4)
theta.season=array(c(0.5,0.4,0.1,0.3),dim=c(2,2,1)) 
Z2 <- varima.sim(list(sma=theta.season,d=c(1,0)),n=250,k=2,
                 innov=Z1,innov.dist="bootstrap")
plot(Z2)
#################################################################################
# Simulate a bivariate VARIMA(2,d,1) process with length 300, where d=(1,2).    #
# phi = array(c(0.5,0.4,0.1,0.5,0,0.3,0,0),dim=c(k,k,2)),                       #
# theta = array(c(0,0.25,0,0), dim=c(k,k,1)).                                   #
# innovations are generated from multivariate normal                            #
# The process have mean zero and no deterministic terms.                        #
# The variance covariance is sigma = matrix(c(1,0.71,0.71,2),2,2).              #
# The series is truncated at default value: trunc.lag=ceiling(100/3)=34         #
#################################################################################
set.seed(1234)
k <- 2
n <- 300 
phi <-  array(c(0.5,0.4,0.1,0.5,0,0.3,0,0),dim=c(k,k,2))
theta <-  array(c(0,0.25,0,0),dim=c(k,k,1))
d <- c(1,2)
sigma <- matrix(c(1,0.71,0.71,2),k,k)
Z <- varima.sim(list(ma=phi,ma=theta,d=d),n=n,k=2,sigma=sigma)
plot(Z)
#################################################################################
# Simulate a trivariate Vector SVARMA(1,0,0)*(1,0,0)_12 process with length 300 #
# phi = array(c(0.5,0.4,0.1,0.5,0,0.3,0,0,0.1), dim=c(k,k,1)), where k =3       #
# phi.season = array(c(0,0.25,0,0.5,0.1,0.4,0,0.25,0.6), dim=c(k,k,1)).         #
# innovations are generated from multivariate normal distribution               #
# The process have mean c(10, 0, 12),                                           #
# Drift equation a + b * t, where a = c(2,1,5), and b = c(0.01,0.06,0)          #
# The series is truncated at default value: trunc.lag=ceiling(100/3)=34         #
#################################################################################
set.seed(1234)
k <- 3
n <- 300
phi <-  array(c(0.5,0.4,0.1,0.5,0,0.3,0,0,0.1),dim=c(k,k,1))
phi.season <-  array(c(0,0.25,0,0.5,0.1,0.4,0,0.25,0.6),dim=c(k,k,1))
constant <- c(2,1,5)
trend <- c(0.01,0.06,0)
demean <- c(10,0,12)
Z <- varima.sim(list(ar=phi,sar=phi.season),n=n,k=3,constant=constant,
trend=trend,demean=demean)
plot(Z)
acf(Z)
#################################################################################
# Simulate a bivariate VAR(1) process with length 600.                          #
# Stable dist.: Alpha=(1.3,1.6), Beta=(0,0.2), Scale=(1,1), Location=(0,0.2)    #
# The series is truncated at default value: trunc.lag=min(100,200)=100          #
#################################################################################
set.seed(1234)
k <- 2
n <- 600
phi <- array(c(-0.2,-0.6,0.3,1.1),dim=c(k,k,1))
theta <- array(c(1,-0.2,0.71,0.2),dim=c(k,k,1))
Alpha <- c(1.3,1.6)
Beta <- c(0,0.2)
Scale <-c(1,1)
Location <-c(0,0.2)
Stable <- c(Alpha,Beta,Scale,Location)
Z<-varima.sim(list(ar=phi,ma=theta),n=n,k=2,innov.dist="stable",par.stable=Stable)
plot(Z)
# }