Synthetic time series

Copula-VAR

Each series is transformed to normal scores through its empirical distribution, a vector autoregression is fitted (order chosen by AIC up to max_lag), new paths are simulated with innovations resampled from the fitted residuals, and the paths are mapped back through the empirical quantile functions.

fit <- fit_synthesizer(ldeaths)
fit
#> <sp_ts_synthesizer>
#>   trained on : 72 time points, 1 series (frequency 12)
#>   method     : copula_var (VAR order 5)
syn <- generate(fit, seed = 1)
plot_ts(ldeaths, syn)

plot_acf(ldeaths, syn, lag_max = 24)

Synthetic series can be of any length and start anywhere:

long <- generate(fit, n = 240, start = c(1980, 1), seed = 2)
tsp(long)
#> [1] 1980.000 1999.917   12.000

returns <- diff(log(EuStockMarkets))
fit_mv <- fit_synthesizer(returns)
syn_mv <- generate(fit_mv, seed = 1)
round(cor(returns) - cor(syn_mv), 3)
#>         DAX    SMI    CAC   FTSE
#> DAX   0.000  0.008  0.000 -0.028
#> SMI   0.008  0.000 -0.007 -0.012
#> CAC   0.000 -0.007  0.000 -0.022
#> FTSE -0.028 -0.012 -0.022  0.000

Bootstraps

Block bootstraps resample stretches of the observed series, so they keep short-range dependence and the exact values (and non-linear features within blocks), at the price of repeating observed values:

fit_b <- fit_synthesizer(lh, method = "stationary", block_length = 8)
syn_b <- generate(fit_b, n = 96, seed = 1)
plot_ts(lh, syn_b)

Practical notes

• All four methods assume (approximately) stationary series: the marginal distribution and the dependence structure are taken to be constant over time. A trend, a level shift, changing variance or strong deterministic seasonality is reproduced only imperfectly. Model the deterministic part yourself (difference, detrend, deseasonalise or use growth rates), synthesise the stationary remainder and add the deterministic part back. The order of the copula-VAR should cover the seasonal period (e.g. order = 12 for monthly data, see vignette("augmentation")).
• Missing values are linearly interpolated before fitting, with a warning.
• simulate(fit, nsim = 10) returns ten independent synthetic series.

References

Politis, D. N. and Romano, J. P. (1994). The stationary bootstrap. Journal of the American Statistical Association, 89(428), 1303–1313.