# Simulate Data From Seasonal/Nonseasonal ARIMA(p,d,q)*(ps,ds,qs)_s or VARIMA(p,d,q)*(ps,ds,qs)_s Models 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. ```r 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"),...) ``` ## 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. ## 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. ## Returns 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. ## Author(s) Esam Mahdi and A.I. McLeod. ## 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` ## Examples ```r ################################################################################# # 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) ```

varima.sim function