---
title: "Forecasting with fastTS"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Forecasting with fastTS}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  chunk_output_type: console
---

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

```{r setup}
library(fastTS)
library(tibble)
set.seed(123)
```

# Lake Huron data set

```{r sunspot}

data("LakeHuron")
years <- time(LakeHuron)
fit <- fastTS(LakeHuron, n_lags_max = 3)
fit
```

## What does `predict` do? 

Let $y_t$ refer to our outcome series, and $\hat y_t^{(k)}$ refer to the $k$-step-ahead prediction for $y_t$. 

The predicted value returned at any time point $t$ is the model's prediction for that point $\hat y_t$, given the model and all data up to $t -$ `n_ahead`. This means that

- The 1-step prediction $\hat y_t^{(1)}$ is computed by using lags of $y_t$ deemed important by the fitting process. 

- The 2-step prediction $\hat y_t^{(2)}$ is computed by using important lags of $y_t$, but replacing the first lag $y_{t-1}$ with $\hat y_{t-1}^{(1)}$. 

- The 3-step prediction $\hat y_t^{(3)}$ is computed by replacing the first lag $y_{t-1}$ with $\hat y_{t-1}^{(2)}$ and the second lag $y_{t-2}$ with $\hat y_{t-2}^{(1)}$. 

- And so on until the $k$-step prediction $\hat y_t^{(k)}$ is similarly computed by replacing lags of $y_t$ with predicted values as necessary.   

Here is an example with the `LakeHuron` data set. 

```{r predict}
p1 <- predict(fit, n_ahead = 1)
p7 <- predict(fit, n_ahead = 7)
predictions <- tibble(years, LakeHuron, p1, p7)
head(predictions, 10)
tail(predictions)
```

- The `predict` function returns missing values for the first `n_lags_max` observations for 1-step ahead predictions. The prediction process back-fill real values when necessary for early predictions, but resets to NA before returning predictions.
- In 1884, the model's 1-step prediction, the one that would be made in 1883, is `r p1[10]`. 
- The 7-step prediction for 1884, the one "made" in 1877, is `r p7[10]`.

Note: there is a "burn-in" component to `fastTS` objects that means the first `n_lags_max` observations are back-filled in.

## Forecasting 

By default, the `predict` function does **not** produce forecasts. In order to get forecasts, we need to set `forecast_ahead = TRUE`, which will return forecasted values at the tail end of the returned vector. 

```{r forecasting}
p1 <- predict(fit, n_ahead = 1, forecast_ahead = TRUE) 
predictions <- tibble(time = c(1973), p1)


# For 7-step ahead forecasts
p7 <- predict(fit, n_ahead = 7, forecast_ahead = TRUE)
predictions <- tibble(time = c(1973:1979), p7)
predictions
```

Finally, the `return_intermediate` option allows users to collect all of the step-ahead predictions up to $k$: 

```{r}
p1_p7 <- predict(fit, n_ahead = 7, return_intermediate = TRUE)

predictions <- tibble(years, LakeHuron, p1_p7)
tail(predictions)
```

