---
title: "MGARCH"
author: "James Thorson"
output: rmarkdown::html_vignette
#output: rmarkdown::pdf_document
bibliography: 'dsem_vignettes.bib'
vignette: >
  %\VignetteIndexEntry{MGARCH}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE, warning=FALSE, message=FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
start_time = Sys.time()
# Install locally
#  devtools::install_local( R'(C:\Users\James.Thorson\Desktop\Git\dsem)', force=TRUE )
# Build
#  setwd(R'(C:\Users\James.Thorson\Desktop\Git\dsem)'); devtools::build_rmd("vignettes/spatial_diffusion.Rmd")
```

## Multivariate Generalized Autoregressive Conditional Heteroskedasticity (MGARCH) models

`dsem` can be specified to estimate variation over time in a parameter representing the magnitude of exogenous variance (i.e., two-headed arrow).  This essentially allows `dsem` to function as a Multivariate Generalized Autoregressive Conditional Heteroskedasticity (MGARCH) model, while allowing missing values for covariates that drive changes in variance.

To show this, we simulate three time-series of $T=100$ length, with no correlation but a steady increase in the standard deviation over time:

```{r}
library(dsem)
set.seed(123)

# Specify settings
n_times = 100
n_vars = 3

# SD over time
sigF_t = seq( 0.1, 0.3, length = n_times )

# Simulate and apply time-varying SD
eps_tc = matrix( rnorm(n_times*n_vars), ncol = n_vars )
eps_tc = sweep( eps_tc, MARGIN = 1, FUN = "*", STAT = sigF_t )
```

## Exploratory MGARCH

We first fit this without any covariate.  To do so, we specify a latent variable `F` that follows a random walk with unit variance.  This variable the moderates the double-headed arrows (representing the magnitude of exogenous variance) for each time-series:

```{r}
# Define data including latent factor for heteroskedasticity
dat = data.frame(
  setNames( data.frame(eps_tc),letters[seq_len(n_vars)]),
  F = NA
)

# Define SEM using F as latent moderating variable
sem = "
  a <-> a, 0, F
  b <-> b, 0, F
  c <-> c, 0, F
  F <-> F, 0, sdF, 0.1
  F -> F, 1, NA, 1
"

# exploratory fit
fit1 = dsem(
  tsdata = ts(dat),
  sem = sem,
  estimate_mu = colnames(dat),
  control = dsem_control(
    use_REML = FALSE,
    gmrf_parameterization = "full",
    logscale_moderating_variance = TRUE,
    quiet = TRUE
  )
)

# Inspect estimates
summary(fit1)
```

The model has a nonzero estimate of `sdF` representing the variance over time in heteroskedasticity (in log-space), suggesting that the model detects the heteroskedasticity.

## Confirmatory MGARCH

Alternatively, we might specify a covariate that is hypothesized to drive heteroskedasticity.  In this case, we simply specify a trend over time as covariate, and estimate its impact on the latent moderating variable.  To avoid confounding between the random-walk for the latent variable and the trend covariate, we also remove the random-walk from the latent factor.  Finally, we randomly simulate missing data in the covariate, to show that the MGARCH can still accomodate data that are missing at random:

```{r}
# Define data including latent factor for heteroskedasticity and covariate
dat = data.frame(
  setNames( data.frame(eps_tc),letters[seq_len(n_vars)]),
  F = NA,
  slope = scale( seq_len(n_times), center = TRUE, scale = TRUE )
)

# Randomly simulate 10% missing data for covariate
dat$slope[ sample(seq_len(n_times), n_times/2) ] = NA

# Define SEM using F as latent moderating variable
# and slope as covariate for F
sem = "
  a <-> a, 0, F
  b <-> b, 0, F
  c <-> c, 0, F
  F <-> F, 0, sdF, 0.1
  slope <-> slope, 0, sd_slope
  slope -> slope, 1, NA, 1
  slope -> F, 0, beta
"

# confirmatory MGARCH
fit2 = dsem(
  tsdata = ts(dat),
  sem = sem,
  estimate_mu = colnames(dat),
  control = dsem_control(
    use_REML = FALSE,
    gmrf_parameterization = "full",
    logscale_moderating_variance = TRUE,
    quiet = TRUE
  )
)

# Inspect estimates
summary(fit2)
```

The model has a positive estimate of `beta`, indicating that it attributes some portion of heteroskedasticity to the hypothesized covariate.

## Comparison

We can then plot these estimated variances against the true (simulated) value

```{r, fig.width=4, fig.height=4}
# Bundle true and estimated time-series
Y = cbind(
  True = sigF_t,
  exp(predict(fit1)[,4]),
  exp(predict(fit2)[,4])
)

#
matplot( 
  x = seq_len(n_times), y = Y, type = "l", lty = "solid",
  col = c("black","red","blue"), xlab = "Time", 
  ylab = "SD for heteroskedasticity"
)
legend( "topleft", fill = c("black","red","blue"), bty = "n",
        legend = c("True", "Exploratory", "Confirmatory"))
```

As expected, using a covariate improves the estimated heteroskedasticity even in the presence of missing data.  

```{r, include = FALSE, warning=FALSE, message=FALSE}
run_time = Sys.time() - start_time
```
Runtime for this vignette: `r paste( round(unclass(run_time),2), attr(run_time, "units") )`

