---
title: "Copula Processes with V-Transforms"
author: "Alexander J. McNeil and Martin Bladt"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Copula Processes with V-Transforms}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  chunk_output_type: console
---

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

All types of copula process can be combined with a v-transform to model volatile time series. 

## 1. VT-ARMA Copula Processes

VT-ARMA processes are created by adding a v-transform to an armacopula process using the command `vtscopula`. The generic commands `sim`, `fit` and `plot` also work for these processes.

### VT-ARMA(1,1) Example

This example uses a ARMA(1,1) copula and an off-centre linear v-transform. We set up the model and generate some data.

```{r arma11setup, fig.width = 6, fig.height = 3, fig.show='hold', dev.args =list(pointsize=9)}
vtarma11 <- vtscopula(armacopula(list(ar = 0.95, ma = -0.85)),
  Vtransform = Vlinear(delta = 0.6))
vtarma11
set.seed(19)
data <- sim(vtarma11, 2000)
ts.plot(data)
```

We now fit the model with a fixed value for the fulcrum parameter $\delta$ and plot the results. (More on fulcrum choice next.)

```{r arma11fit, fig.show='hold', dev.args =list(pointsize=9)}
vtarma11spec <- vtscopula(armacopula(list(ar = 0, ma = 0)), Vtransform = Vlinear(delta = 0.6))
vtarma11fit <- fit(vtarma11spec, data)
vtarma11fit

plot(vtarma11fit)
plot(vtarma11fit, plottype = "vtransform")
plot(vtarma11fit, plottype = "kendall" )
```

Optimization over the fulcrum parameter $\delta$ does not take place. To identify a reasonable value for $\delta$ we can carry a profile likelihood analysis using different fixed values for the fulcrum parameter. 

```{r profileplot, fig.show='hold', dev.args =list(pointsize=9)}
profilefulcrum(data, tscopula = vtarma11spec, locations = seq(from = 0, to = 1, length = 11))
abline(v = 0.6)
```

## 2. VT-D-Vine Copula Processes (type 2)

VT-D-Vine processes are created by adding a v-transform to an `dvinecopula2` object using the command `vtscopula`. The generic commands `sim`, `fit` and `plot` also work for these processes.

### Construction

We add a 2-parameter V-transform.

```{r vtdvinecopula2}
copmod <- dvinecopula2(family = "joe",
                       kpacf = "kpacf_arma",
                       pars = list(ar = 0.9, ma = -0.85),
                       maxlag = 20)
vcopmod <- vtscopula(copmod,
  Vtransform = V2p(delta = 0.6, kappa = 0.8)
)
vcopmod
```

### Simulation

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

### Estimation

```{r fitvtdvinecopula2}
copspec_Joe <- dvinecopula2(family = "joe",
                            pars = list(ar = 0, ma = 0),
                            maxlag = 30)
vcopspec <- vtscopula(copspec_Joe, V2p(delta = 0.6))
vcopfit <- fit(vcopspec, data2, 
               tsoptions = list(hessian = TRUE),
               control = list(maxit = 1000))
vcopfit
coef(vcopfit)
coef(vcopmod)
```

### Plotting

We can plot the estimated v-transform and well as the goodness-of-fit plots for the `dvinecopula2` object based on Kendall rank correlations.

```{r finalplots, fig.show='hold', dev.args =list(pointsize=9)}
plot(vcopfit, plottype = "vtransform")
plot(vcopfit, plottype = "kendall")
plot(vcopfit, plottype = "residual")
```