---
title: "Forecasting"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Forecasting}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
  \newcommand{\R}{\mathbb{R}}
  \newcommand{\B}{\boldsymbol\beta}
  \newcommand{\hb}{\boldsymbol{\hat\beta}}
  \newcommand{\E}{\boldsymbol\epsilon}
  \DeclareMathOperator*{\argmin}{argmin}
  \DeclareMathOperator*{\argmax}{argmax}
  \newcommand{\defn}{\mathpunct{:}=}
  \newcommand{\X}{\mathbf{X}}
  \newcommand{\Y}{\mathbf{Y}}
  \newcommand{\by}{\mathbf{y}}
  \newcommand{\bz}{\mathbf{Z}}
  \newcommand{\ba}{\boldsymbol{\alpha}}
  \newcommand{\bc}{\mathbf{c}}
  \newcommand{\bu}{\mathbf{u}}
  \def\Cov{\mathrm{Cov}}
  \def\Var{\mathrm{Var}}
  \def\Corr{\mathrm{Corr}}
  \def\vec{\mathrm{vec}}
---

```{r rmdsetup, include = FALSE}
knitr::opts_chunk$set(
  comment = "#>",
  collapse = TRUE,
  out.width = "70%",
  fig.align = "center",
  fig.width = 6,
  fig.asp = .618
)
orig_opts <- options("digits")
options(digits = 3)
set.seed(1)
```

```{r setup}
library(bvhar)
```

# Simulation

Given VAR coefficient and VHAR coefficient each,

- `sim_var(num_sim, num_burn, var_coef, var_lag, sig_error, init)` generates VAR process
- `sim_vhar(num_sim, num_burn, vhar_coef, sig_error, init)` generates VHAR process

We use coefficient matrix estimated by VAR(5) in introduction vignette.

```{r evalcoef, echo=FALSE}
etf_eval <- 
  etf_vix %>% 
  dplyr::select(GVZCLS, OVXCLS, EVZCLS, VXFXICLS) %>% 
  divide_ts(20)
etf_train <- etf_eval$train
etf_test <- etf_eval$test
ex_fit <- var_lm(etf_train, p = 5)
```

Consider

```{r whatcoef}
coef(ex_fit)
ex_fit$covmat
```

Then

```{r simvar}
m <- ncol(ex_fit$coefficients)
# generate VAR(5)-----------------
y <- sim_var(
  num_sim = 1500, 
  num_burn = 100, 
  var_coef = coef(ex_fit), 
  var_lag = 5L, 
  sig_error = ex_fit$covmat, 
  init = matrix(0L, nrow = 5L, ncol = m)
)
# colname: y1, y2, ...------------
colnames(y) <- paste0("y", 1:m)
head(y)
```

```{r outofsample}
h <- 20
y_eval <- divide_ts(y, h)
y_train <- y_eval$train # train
y_test <- y_eval$test # test
```


# Fitting Models

## VAR(5) and VHAR

```{r fitvar}
# VAR(5)
model_var <- var_lm(y_train, 5)
# VHAR
model_vhar <- vhar_lm(y_train)
```

## BVAR(5)

Minnesota prior

```{r fitbvar}
# hyper parameters---------------------------
y_sig <- apply(y_train, 2, sd) # sigma vector
y_lam <- .2 # lambda
y_delta <- rep(.2, m) # delta vector (0 vector since RV stationary)
eps <- 1e-04 # very small number
spec_bvar <- set_bvar(y_sig, y_lam, y_delta, eps)
# fit---------------------------------------
model_bvar <- bvar_minnesota(y_train, p = 5, bayes_spec = spec_bvar)
```

## BVHAR

BVHAR-S

```{r fitbvhars}
spec_bvhar_v1 <- set_bvhar(y_sig, y_lam, y_delta, eps)
# fit---------------------------------------
model_bvhar_v1 <- bvhar_minnesota(y_train, bayes_spec = spec_bvhar_v1)
```

BVHAR-L

```{r fitbvharl}
# weights----------------------------------
y_day <- rep(.1, m)
y_week <- rep(.01, m)
y_month <- rep(.01, m)
# spec-------------------------------------
spec_bvhar_v2 <- set_weight_bvhar(
  y_sig,
  y_lam,
  eps,
  y_day,
  y_week,
  y_month
)
# fit--------------------------------------
model_bvhar_v2 <- bvhar_minnesota(y_train, bayes_spec = spec_bvhar_v2)
```

# Splitting

You can forecast using `predict()` method with above objects.
You should set the step of the forecasting using `n_ahead` argument.

In addition, the result of this forecast will return another class called `predbvhar` to use some methods,

- Plot: `autoplot.predbvhar()`
- Evaluation: `mse.predbvhar()`, `mae.predbvhar()`, `mape.predbvhar()`, `mase.predbvhar()`, `mrae.predbvhar()`, `relmae.predbvhar()`
- Relative error: `rmape.predbvhar()`, `rmase.predbvhar()`, `rmase.predbvhar()`, `rmsfe.predbvhar()`, `rmafe.predbvhar()`

## VAR

```{r predvar}
(pred_var <- predict(model_var, n_ahead = h))
```

```{r varpredlist}
class(pred_var)
names(pred_var)
```

The package provides the evaluation function

- `mse(predbvhar, test)`: MSE
- `mape(predbvhar, test)`: MAPE

```{r msevar}
(mse_var <- mse(pred_var, y_test))
```

## VHAR

```{r predvhar}
(pred_vhar <- predict(model_vhar, n_ahead = h))
```

MSE:

```{r msevhar}
(mse_vhar <- mse(pred_vhar, y_test))
```

## BVAR

```{r predbvar}
(pred_bvar <- predict(model_bvar, n_ahead = h))
```

MSE:

```{r msebvar}
(mse_bvar <- mse(pred_bvar, y_test))
```

## BVHAR

### VAR-type Minnesota

```{r predbvharvar}
(pred_bvhar_v1 <- predict(model_bvhar_v1, n_ahead = h))
```

MSE:

```{r msebvharvar}
(mse_bvhar_v1 <- mse(pred_bvhar_v1, y_test))
```

### VHAR-type Minnesota

```{r predbvharvhar}
(pred_bvhar_v2 <- predict(model_bvhar_v2, n_ahead = h))
```

MSE:

```{r msebvharvhar}
(mse_bvhar_v2 <- mse(pred_bvhar_v2, y_test))
```

## Compare

### Region

`autoplot(predbvhar)` and `autolayer(predbvhar)` draws the results of the forecasting.

```{r predplot}
autoplot(pred_var, x_cut = 1470, ci_alpha = .7, type = "wrap") +
  autolayer(pred_vhar, ci_alpha = .5) +
  autolayer(pred_bvar, ci_alpha = .4) +
  autolayer(pred_bvhar_v1, ci_alpha = .2) +
  autolayer(pred_bvhar_v2, ci_alpha = .1) +
  geom_eval(y_test, colour = "#000000", alpha = .5)
```

### Error

Mean of MSE

```{r msevalues}
list(
  VAR = mse_var,
  VHAR = mse_vhar,
  BVAR = mse_bvar,
  BVHAR1 = mse_bvhar_v1,
  BVHAR2 = mse_bvhar_v2
) %>% 
  lapply(mean) %>% 
  unlist() %>% 
  sort()
```

For each variable, we can see the error with plot.

```{r evalplot}
list(
  pred_var,
  pred_vhar,
  pred_bvar,
  pred_bvhar_v1,
  pred_bvhar_v2
) %>% 
  gg_loss(y = y_test, "mse")
```

Relative MAPE (MAPE), benchmark model: VAR

```{r relmape}
list(
  VAR = pred_var,
  VHAR = pred_vhar,
  BVAR = pred_bvar,
  BVHAR1 = pred_bvhar_v1,
  BVHAR2 = pred_bvhar_v2
) %>% 
  lapply(rmape, pred_bench = pred_var, y = y_test) %>% 
  unlist()
```


# Out-of-Sample Forecasting

In time series research, out-of-sample forecasting plays a key role.
So, we provide out-of-sample forecasting function based on

- Rolling window: `forecast_roll(object, n_ahead, y_test)`
- Expanding window: `forecast_expand(object, n_ahead, y_test)`

## Rolling windows

`forecast_roll(object, n_ahead, y_test)` conducts h >= 1 step rolling windows forecasting.

It fixes window size and moves the window. The window is the training set.
In this package, we set *window size = original input data*.

Iterating the step

1. The model is fitted in the training set.
2. With the fitted model, researcher should forecast the next h >= 1 step ahead. The longest forecast horizon is `num_test - h + 1`.
3. After this window, move the window and do the same process.
4. Get forecasted values until possible (longest forecast horizon).

5-step out-of-sample:

```{r rollvar}
(var_roll <- forecast_roll(model_var, 5, y_test))
```

Denote that the nrow is longest forecast horizon.

```{r rollvarlist}
class(var_roll)
names(var_roll)
```

To apply the same evaluation methods, a class named `bvharcv` has been defined. You can use the functions above.

```{r otherroll}
vhar_roll <- forecast_roll(model_vhar, 5, y_test)
bvar_roll <- forecast_roll(model_bvar, 5, y_test)
bvhar_roll_v1 <- forecast_roll(model_bvhar_v1, 5, y_test)
bvhar_roll_v2 <- forecast_roll(model_bvhar_v2, 5, y_test)
```

Relative MAPE, benchmark model: VAR 

```{r relroll}
list(
  VAR = var_roll,
  VHAR = vhar_roll,
  BVAR = bvar_roll,
  BVHAR1 = bvhar_roll_v1,
  BVHAR2 = bvhar_roll_v2
) %>% 
  lapply(rmape, pred_bench = var_roll, y = y_test) %>% 
  unlist()
```

## Expanding Windows

`forecast_expand(object, n_ahead, y_test)` conducts h >= 1 step expanding window forecasting.

Different with rolling windows, expanding windows method fixes the starting point. The other is same.

```{r expandvar}
(var_expand <- forecast_expand(model_var, 5, y_test))
```

The class is `bvharcv`.

```{r expandvarlist}
class(var_expand)
names(var_expand)
```

```{r otherexpand}
vhar_expand <- forecast_expand(model_vhar, 5, y_test)
bvar_expand <- forecast_expand(model_bvar, 5, y_test)
bvhar_expand_v1 <- forecast_expand(model_bvhar_v1, 5, y_test)
bvhar_expand_v2 <- forecast_expand(model_bvhar_v2, 5, y_test)
```

Relative MAPE, benchmark model: VAR

```{r relexpand}
list(
  VAR = var_expand,
  VHAR = vhar_expand,
  BVAR = bvar_expand,
  BVHAR1 = bvhar_expand_v1,
  BVHAR2 = bvhar_expand_v2
) %>% 
  lapply(rmape, pred_bench = var_expand, y = y_test) %>% 
  unlist()
```

```{r resetopts, include=FALSE}
options(orig_opts)
```
