Synthetic time series

For time series the order of the observations carries information: a synthetic series should reproduce the marginal distribution and the autocorrelation (and, for several series, the cross-correlation). fit_synthesizer() has a method for ts objects with four generators.

`method`	What it preserves	How
`"copula_var"` (default)	marginals, auto- and cross-correlation	VAR on normal scores, bootstrapped innovations
`"block"`	short-range dependence, exact values	moving (circular) block bootstrap
`"stationary"`	short-range dependence, exact values	stationary bootstrap (Politis & Romano, 1994)
`"iid"`	marginals and contemporaneous correlation only	cross-sectional copula

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

Several series

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.

Synthetic time series

Copula-VAR

Several series

Bootstraps

Practical notes

References