Simulate fractional ARIMA Time Series

Generates simulated long-memory time series data from the fractional ARIMA(p,d,q) model. This is a test problem generator for fracdiff.

Note that the MA coefficients have inverted signs compared to other parametrizations, see the details in fracdiff.

Usage

fracdiff.sim(n, ar = NULL, ma = NULL, d,
             rand.gen = rnorm, innov = rand.gen(n+q, ...),
             n.start = NA, backComp = TRUE, allow.0.nstart = FALSE,
             start.innov = rand.gen(n.start, ...),
             ..., mu = 0)

Arguments

n: length of the time series.
ar: vector of autoregressive parameters; empty by default.
ma: vector of moving average parameters; empty by default.
d: fractional differencing parameter.
rand.gen: a function to generate the innovations; the default, rnorm generates white N(0,1) noise.
innov: an optional times series of innovations. If not provided, rand.gen() is used.
n.start: length of “burn-in” period. If NA, the default, the same value as in arima.sim is computed.
backComp: logical indicating if back compatibility with older versions of fracdiff.sim is desired. Otherwise, for d = 0, compatibility with R's arima.sim is achieved.
allow.0.nstart: logical indicating if n.start = 0 should be allowed even when \(p + q > 0\). This not recommended unless for producing the same series as with older versions of fracdiff.sim.
start.innov: an optional vector of innovations to be used for the burn-in period. If supplied there must be at least n.start values.
...: additional arguments for rand.gen(). Most usefully, the standard deviation of the innovations generated by rnorm can be specified by sd.
mu: time series mean (added at the end).

Value

a list containing the following elements :

series: time series
ar, ma, d, mu, n.start: same as input

Examples

## Pretty (too) short to "see" the long memory
fracdiff.sim(100, ar = .2, ma = .4, d = .3)
#> $series
#>   [1]  1.638800218 -1.603146117 -0.054717460  1.403075920 -0.659829202
#>   [6]  0.013055359 -0.972348581  0.536532022  0.440775043 -0.873033650
#>  [11] -0.432069381  0.094774774 -1.648142812 -0.210777100  0.617922101
#>  [16]  2.764655474  0.753441848  0.515446273 -0.014856082  2.121160874
#>  [21]  1.445149215  1.289507605 -0.707403820  0.879238788  2.614860696
#>  [26]  1.172039268  0.721397889  2.188541912  1.006855452  0.925061611
#>  [31] -0.459106839  1.847786874  1.402210782  0.427446021  2.097796988
#>  [36] -0.673488716 -0.230542183  0.568655111  2.609624653  1.771011342
#>  [41]  0.516286980  1.352795573  1.562642739  1.934142563  1.124433135
#>  [46]  1.270348383  1.361166793  0.175551401  0.755878082  0.849239877
#>  [51]  0.477828428  0.473281788  0.169620245  0.526362814  1.349228496
#>  [56]  1.239364747  1.193070841  1.204387420  1.641551435  1.041120684
#>  [61]  1.626609504  1.724350559  1.248292526  1.940142594  0.081973195
#>  [66] -0.444548735 -0.796570297 -0.546593677  0.717776767  0.040881270
#>  [71]  1.793192136  0.159579097  0.015848061  0.964101534  0.448662554
#>  [76] -1.724601439 -1.407805075 -0.899407175 -0.181081262 -0.224475580
#>  [81]  1.507365541  0.974737047 -1.220979546 -0.350581962  0.009325098
#>  [86]  0.253554447  0.629093465 -0.897742019  0.014869018  0.821637422
#>  [91] -1.163497727 -0.400604240  1.110154286 -0.997538749  0.870989573
#>  [96] -0.133547186 -0.744269960  1.041665315  0.846093565  0.482715675
#> 
#> $ar
#> [1] 0.2
#> 
#> $ma
#> [1] 0.4
#> 
#> $d
#> [1] 0.3
#> 
#> $mu
#> [1] 0
#> 
#> $n.start
#> [1] 6
#> 

## longer with "extreme" ar:
r <- fracdiff.sim(n=1500, ar=-0.9, d= 0.3)
plot(as.ts(r$series))


## Show that  MA  coefficients meaning is inverted
## compared to   stats :: arima :

AR <- 0.7
MA <- -0.5
n.st <- 2

AR <- c(0.7, -0.1)
MA <- c(-0.5, 0.4)
n <- 512 ; sd <- 0.1
n.st <- 10

set.seed(101)
Y1 <- arima.sim(list(ar = AR, ma = MA), n = n, n.start = n.st, sd = sd)
plot(Y1)

# For our fracdiff, reverse the MA sign:
set.seed(101)
Y2 <- fracdiff.sim(n = n, ar = AR, ma = - MA, d = 0,
                   n.start = n.st, sd = sd)$series
lines(Y2, col=adjustcolor("red", 0.5))

## .. no, you don't need glasses ;-)  Y2 is Y1 shifted slightly

##' rotate left by k (k < 0: rotate right)
rot <- function(x, k) {
  stopifnot(k == round(k))
  n <- length(x)
  if(k <- k %% n) x[c((k+1):n, 1:k)] else x
}
k <- n.st - 2
Y2.s <- rot(Y2, k)
head.matrix(cbind(Y1, Y2.s))
#>                Y1         Y2.s
#> [1,]  0.105653840  0.105986817
#> [2,] -0.043723547 -0.043557047
#> [3,]  0.162403833  0.162487085
#> [4,] -0.131808486 -0.131766860
#> [5,] -0.001723455 -0.001702642
#> [6,] -0.054197985 -0.054187579
plot(Y1, Y2.s); i <- (n-k+1):n
text(Y1[i], Y2.s[i], i, adj = c(0,0)-.1, col=2)


## With  backComp = FALSE,  get *the same* as arima.sim():
set.seed(101)
Y2. <- fracdiff.sim(n = n, ar = AR, ma = - MA, d = 0,
                    n.start = n.st, sd = sd, backComp = FALSE)$series
stopifnot( all.equal( c(Y1), Y2., tolerance= 1e-15))

Usage

Arguments

Value

See also

Examples