---
title: "Model checking"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Model checking}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
bibliography: ref.bib
link-citations: true
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.dim = c(10, 6),
  out.width = "80%",
  fig.align = "center",
  fig.path = "fHMM-"
)
library("fHMM")
```

This vignette^[This vignette was built using R `r paste(R.Version()[c("major", "minor")], collapse = ".")` with the `{fHMM}` `r utils::packageVersion("fHMM")` package.] discusses model checking in `{fHMM}`, that means the task of verifying whether the fitted model describes the data well.

## Model checking using pseudo-residuals

Since the observations are explained by different distributions (depending on the active state), model checking cannot be done by analyzing standard residuals. Instead, we consider "pseudo-residuals". To transform all observations on a common scale, we proceed as follows: If $X_t$ has the invertible distribution function $F_{X_t}$, then

\begin{align*}
Z_t=\Phi^{-1}(F_{X_t} (X_t))
\end{align*}

is standard normally distributed, where $\Phi$ denotes the cumulative distribution function of the standard normal distribution. The observations, $(X_t)_t$, are modeled well if the so-called pseudo-residuals, $(Z_t)_t$, are approximately standard normally distributed, which can be visually assessed using quantile-quantile plots or further investigated using statistical tests such as the Jarque-Bera test [@zuc16]. 

For HHMMs, we first decode the coarse-scale state process using the Viterbi algorithm. Subsequently, we assign each coarse-scale observation its distribution function under the fitted model and perform the transformation described above. Using the Viterbi-decoded coarse-scale states, we then treat the fine-scale observations analogously.

## Implementation

In `{fHMM}`, pseudo-residuals can be computed via the `compute_residuals()` function, provided that the states have been decoded beforehand.

We revisit the DAX example:

```{r dax model}
data(dax_model_3t)
```

The following line computes the residuals and saves them into the `model` object:

```{r res}
dax_model_3t <- compute_residuals(dax_model_3t)
```

The residuals can be visualized as follows:

```{r plot-res}
plot(dax_model_3t, plot_type = "pr")
```

For additional normality tests, the residuals can be extracted from the `model` object via the `residuals()` method. The following lines perform a [Jarque–Bera test](https://en.wikipedia.org/wiki/Jarque%E2%80%93Bera_test) as an example [@jar87]:

```{r, message=FALSE}
tseries::jarque.bera.test(residuals(dax_model_3t))
```

## References
