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","bootstrap"),...)

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.

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 = "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, 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 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.

Author

Esam Mahdi and A.I. McLeod.

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 or t-distribution. $\phi(B)$ 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.

Examples

Run this code

#################################################################################
# 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
ma <-  0.6
Z<-varima.sim(list(ma=ma),n=n,innov=rnorm(n),innov.dist="bootstrap")
plot(Z)
#################################################################################
# Simulate seasonal ARIMA(2,1,0)*(0,2,1)_12 process with ar=c(1.3,-0.35),      #
# ma.season = 0.8. Gaussian innovations. The series is truncated at lag 50   #
#################################################################################
set.seed(12834)
n <- 100
ar <- c(1.3, -0.35)
ma.season <- 0.8
Z<-varima.sim(list(ar=ar,d=1,sma=ma.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          #
# ar.season = c(0.9,-0.9), period = 4, t5-distribution innovations with df = 3 #
#################################################################################
set.seed(1234)
n <- 200
ar.season <- c(0.9,-0.9)
Z<-varima.sim(list(sar=ar.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 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              #
# ma.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)
ma.season=array(c(0.5,0.4,0.1,0.3),dim=c(2,2,1)) 
Z2 <- varima.sim(list(sma=ma.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).    #
# ar = array(c(0.5,0.4,0.1,0.5,0,0.3,0,0),dim=c(k,k,2)),                       #
# ma = 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 
ar <-  array(c(0.5,0.4,0.1,0.5,0,0.3,0,0),dim=c(k,k,2))
ma <-  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=ar,ma=ma,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 #
# ar = 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       #
# ar.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
ar <-  array(c(0.5,0.4,0.1,0.5,0,0.3,0,0,0.1),dim=c(k,k,1))
ar.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=ar,sar=ar.season),n=n,k=3,constant=constant,
trend=trend,demean=demean)
plot(Z)
acf(Z)

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