---
title: "Models with Margins"
author: "Alexander J. McNeil and Martin Bladt"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Models with Margins}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
library(tscopula)
```

## 1. Gaussian ARMA Process

### Construction

We generate a Gaussian ARMA model.

```{r standard-arma, fig.show='hold', fig.width = 6, fig.height = 3, dev.args =list(pointsize=9)}
set.seed(13)
data1 <- 0.5 + 2*arima.sim(list(ar =0.95, ma =-0.85), 1000)
ts.plot(data1)
```

### Estimation 

A model spec can be fitted to data with the generic command `fit`.

```{r standard-arma-fit}
copspec <- armacopula(pars = list(ar =0.01, ma =0.01))
margspec <- margin("norm")
fullspec <- tscm(copspec, margspec)
modfit <- fit(fullspec, data1, method = "full")
modfit
```

### Plotting

As well as the copula plots we can also plot the marginal fit.

```{r standardarma-plot, fig.show='hold', dev.args =list(pointsize=9)}
plot(modfit, plottype = "residual")
plot(modfit, plottype = "kendall")
plot(modfit, plottype = "margin")
```

## 2. VT-D-Vine Process (ARMA type) with Skewed Laplace Margin 

### Construction

```{r tscm-dvine}
copmod <- sdvinecopula(basefamily = "joe",
                       kpacf = "kpacf_arma",
                       pars = list(ar = 0.9, ma = -0.8),
                       maxlag = 20)
vcopmod <- vtscopula(copmod,
                     Vtransform = V2p(delta = 0.5, kappa = 2))
margmod <- margin("slaplace",
                  pars = c(mu = 1, scale = 2, gamma = 0.7))
tscmmod <- tscm(vcopmod, margmod)
tscmmod
```

### Simulation

```{r tscm-dvine-sim, fig.show='hold', fig.width = 6, fig.height = 3, dev.args =list(pointsize=9)}
set.seed(13)
data2 <- sim(tscmmod, n= 2000)
hist(data2)
ts.plot(data2)
```

### Estimation

First fit a marginal model only. 

```{r margin-fit}
margfit <- fit(margmod, data2)
```

Now fit the time series copula model stepwise.

```{r stepwise-fit}
tscmfit_step <- fit(tscmmod, data2)
tscmfit_step
coef(tscmfit_step)
coef(tscmmod)
```

Final optimization over all parameters.

```{r full-optimization}
tscmfit_full <- fit(tscmfit_step, data2, method = "full")
tscmfit_full
```

Comparison of model.

```{r compare-models}
AIC(margfit, tscmfit_step, tscmfit_full)
```

### Plotting

We can plot the estimated v-transform and well as the goodness-of-fit plots for the `dvinecopula` object based on Kendall rank correlations.

The first plots relate to the fitted copula.

```{r plots1, fig.show='hold', dev.args =list(pointsize=9)}
plot(tscmfit_full)
plot(tscmfit_full, plottype = "kendall")
```

The next plot is the QQplot of the marginal fit.

```{r plot2, fig.show='hold', dev.args =list(pointsize=9)}
plot(tscmfit_full, plottype = "margin")
```

The next two plots are the estimated v-transform and the estimated volatility profile function.

```{r plots3, fig.show='hold', dev.args =list(pointsize=9)}
plot(tscmfit_full, plottype = "vtransform")
plot(tscmfit_full, plottype = "volprofile")
```

The final plot shows aspect of the fit of the v-transform to the data.

```{r plot4, fig.show='hold', dev.args =list(pointsize=9)}
plot(tscmfit_full, plottype = "volproxy")
```