---
title: "Synthetic time series"
output:
  rmarkdown::html_vignette:
    toc: true
vignette: >
  %\VignetteIndexEntry{Synthetic time series}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>", fig.width = 7,
                      fig.height = 4, dpi = 72, out.width = "100%")
```

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.

```{r}
library(SynthesizerPlus)
```

| `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.

```{r}
fit <- fit_synthesizer(ldeaths)
fit
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:

```{r}
long <- generate(fit, n = 240, start = c(1980, 1), seed = 2)
tsp(long)
```

### Several series

```{r}
returns <- diff(log(EuStockMarkets))
fit_mv <- fit_synthesizer(returns)
syn_mv <- generate(fit_mv, seed = 1)
round(cor(returns) - cor(syn_mv), 3)
```

## 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:

```{r}
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.