---
title: "Bitcoin Analysis"
author: "Alexander J. McNeil and Martin Bladt"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Bitcoin Analysis}
  %\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)
tsoptions <- list(hessian = TRUE, method = "Nelder-Mead", avoidzero= FALSE)

```

## The Bitcoin Log-Return Data

We first load the data and calculate log-returns.

```{r define-data, fig.show='hold', fig.width = 6, fig.height = 3, dev.args =list(pointsize=9)}
data(bitcoin)
X <- (diff(log(bitcoin))[-1]) * 100 # log-returns (as percentages)
length(X)
plot(X, type = "h")
```

## Copula Processes

### ARMA Copulas

We start with the fulcrum profile-likelihood plot which indicates that a value $\delta = 0.45$ is about right.

```{r arma-copula-profile, fig.show='hold', fig.width = 6, fig.height = 3, dev.args =list(pointsize=9)}
U <- strank(X)

copmod_Gauss <- armacopula(pars = list(ar = 0.95, ma = -0.85))
pts <- seq(from=0.01, to=0.99, length=51) + 0.005
profilefulcrum(U, copmod_Gauss, locations = pts)
abline(v = 0.45)
```

Our baseline model is a model with ARMA(1,1) copula and v-transform with fulcrum at $\delta=0.45$.

```{r arma-copula-fit}
mod_Gauss <- vtscopula(copmod_Gauss, Vlinear(0.45))
fit_Gauss <- fit(mod_Gauss, U, tsoptions = tsoptions)
fit_Gauss
```

### D-Vine Copula (second kind)

In this next section we try a d-vine copula model of the second kind with a finite value for `maxlag` and a linear v-transform. We continue to use the Frank copula. The resulting model is superior to the ARMA copula model and has the same number of parameters. Note that an infinite value for `maxlag` makes next to no difference in the fit.

```{r dvinecopula2}
copmod_Frank <- dvinecopula2(family = "frank",
                             pars = list(ar = 0.95, ma = -0.85),
                             maxlag = 30)
mod_Frank <- vtscopula(copmod_Frank, Vlinear(0.45))
fit_Frank <- fit(mod_Frank, U, tsoptions = tsoptions)

AIC(fit_Gauss, fit_Frank)
```

We provide plots for the model with Frank copula to show aspects of the fit.

```{r dvinecopula2-plots, fig.show='hold', fig.width = 6, fig.height = 3, dev.args =list(pointsize=9)}
plot(fit_Frank, plottype = "residual")
plot(fit_Frank, plottype = "kendall")
```

## Marginal Models

We fit 6 marginal distributions (3 symmetric and 3 skewed) of which the double Weibull gives the lowest AIC value.

```{r margmods}
marg_st <- fit(margin("st"), X)
marg_sst <- fit(margin("sst"), X)
marg_lp <- fit(margin("laplace", 
                      pars = c(mu = 0.2, scale = 2.7)), X)
marg_slp <- fit(margin("slaplace", 
                pars = c(mu = 0.2, scale = 2.7, gamma = 0.9)), X)
marg_dw <- fit(margin("doubleweibull", 
                      pars = c(mu = 0.2, shape = 0.8, scale = 2.7)), X)
marg_sdw <- fit(margin("sdoubleweibull", 
                pars = c(mu = 0.2, shape = 0.8, scale = 2.7, gamma = 0.9)), X)
AIC(marg_st, marg_sst, marg_slp, marg_lp, marg_dw, marg_sdw)
```

## Full Models

### D-Vine Copula

We fit a full model combining the double Weibull margin with the VT-d-vine model of the second kind.

```{r fullmod-fit,fig.show='hold', dev.args =list(pointsize=9)}
fullmod <- tscm(fit_Frank, margin = marg_dw)
fullmod <- fit(fullmod, as.numeric(X), 
               method = "full", tsoptions = tsoptions)
fullmod
AIC(marg_dw, fullmod)
```

We show all the possible plots for the full model.

```{r fullmod-plots1,fig.show='hold', dev.args =list(pointsize=9)}
plot(fullmod, plottype = "residual")
plot(fullmod, plottype = "kendall")
plot(fullmod, plottype = "margin")
plot(fullmod, plottype = "vtransform")
plot(fullmod, plottype = "volprofile")
plot(fullmod, plottype = "volproxy")
```

```{r fullmod-plots2, fig.show='hold', fig.width = 6, fig.height = 6, dev.args =list(pointsize=9)}
plot(fullmod, plottype = "glag")
```