---
title: "Comparing Copula VAR Models"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Comparing Copula VAR Models}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
has_stan <- requireNamespace("rstan", quietly = TRUE)
eval_fits <- identical(Sys.getenv("DCVAR_EVAL_VIGNETTES"), "true") && has_stan
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  eval = eval_fits
)
```

## Overview

The dcvar package provides three copula VAR models:

1. **DC-VAR** (`dcvar()`): Time-varying correlation via a random walk on the Fisher-z scale. Best for smooth, gradual changes.

2. **HMM Copula** (`dcvar_hmm()`): Discrete regime-switching with K hidden states. Best for abrupt changes between distinct coupling regimes.

3. **Constant Copula** (`dcvar_constant()`): Single time-invariant correlation. Serves as a null/baseline model.

## Mathematical Background

For normal-margin fits, all three models share a common decomposition of the
bivariate likelihood into marginal and copula components. The package also
supports exponential, gamma, and skew-normal margins for these three models,
but the copula structure below is the common part. All three single-level
models additionally support *per-variable* margins, where each variable can use
a different marginal family (see the mixed-margins section below).

### VAR(1) Marginals

Each variable follows a VAR(1) process with innovations:

$$y_t = \mu + \Phi \, (y_{t-1} - \mu) + \varepsilon_t, \qquad \varepsilon_t \sim N(0, \text{diag}(\sigma_{\varepsilon}^2))$$

The normal marginal densities provide the per-variable likelihoods in this
special case. For non-normal margins, dcvar uses the same VAR recursion and
Gaussian copula but swaps in the corresponding marginal likelihood and
transform.

### Gaussian Copula

The dependence between variables is captured by a Gaussian copula with
time-varying correlation $\rho_t$. The log-likelihood contribution of the
copula at time $t$ is:

$$\log c(u_{1t}, u_{2t}; \rho_t) = -\frac{1}{2}\log(1 - \rho_t^2) - \frac{\rho_t^2 \, (z_{1t}^2 + z_{2t}^2) - 2\rho_t \, z_{1t} z_{2t}}{2(1 - \rho_t^2)}$$

where $z_{it} = \Phi^{-1}(u_{it})$ and $u_{it} = \Phi(\varepsilon_{it} / \sigma_{\varepsilon,i})$.

### How the Models Differ

- **DC-VAR**: The correlation evolves as a random walk on the Fisher-z scale:
  $z_t = z_{t-1} + \omega_t$ with $\omega_t \sim N(0, \sigma_\omega^2)$, and
  $\rho_t = \tanh(z_t)$.

- **HMM Copula**: The correlation is state-dependent: $\rho_t = \rho_{s_t}$,
  where $s_t \in \{1, \ldots, K\}$ follows a Markov chain with transition
  matrix $A$. State-specific correlations use an ordered constraint
  ($\rho_1 < \rho_2 < \ldots$) to prevent label switching.

- **Constant Copula**: $\rho_t = \rho$ for all $t$. A special case of both
  the DC-VAR ($\sigma_\omega = 0$) and HMM ($K = 1$).

## Per-variable (mixed) margins

A copula separates the marginal distributions from the dependence structure, so
each variable is free to follow its own marginal family. `dcvar_constant()`,
`dcvar()`, and `dcvar_hmm()` expose this by accepting a length-2 `margins`
vector. For example, a roughly-symmetric first series paired with a
positively-skewed second series can be modelled as `normal` and `exponential`
margins under a single Gaussian copula:

```{r mixed-margins}
library(dcvar)

sim_mix <- simulate_dcvar(
  n_time = 200,
  rho_trajectory = rho_constant(200, rho = 0.6),
  margins = c("normal", "exponential"),
  skew_direction = c(1, 1),
  seed = 11
)

fit_mix <- dcvar_constant(
  sim_mix$Y_df,
  vars = c("y1", "y2"),
  margins = c("normal", "exponential"),
  skew_direction = c(1, 1),
  seed = 11
)

# Each dimension is reported under its own family: sigma_eps for the normal
# variable, sigma_exp for the exponential variable.
coef(fit_mix)[c("sigma_eps", "sigma_exp")]
```

`skew_direction` is required whenever any dimension uses an exponential or gamma
margin, and only those dimensions consult it. A single `margins` string still
applies the same family to both variables, and an all-identical vector such as
`c("normal", "normal")` is treated exactly like the scalar form (it reuses the
specialised single-family model), so existing code is unaffected. Mixed margins
currently require the Gaussian copula and are available for the three
single-level models (`dcvar_constant()`, `dcvar()`, `dcvar_hmm()`).

## Time-varying coefficients and scales

In `dcvar()` the copula correlation always evolves over time. The
autoregressive dynamics and the residual scales can be allowed to evolve as
well, each as a random walk around a constant baseline. Two flags turn these on:

- `tv_phi` lets the VAR(1) coefficients drift. `tv_phi = TRUE` varies all four
  (the autoregressive coefficients phi11, phi22 and the cross-lagged
  coefficients phi12, phi21); a selector varies only a subset. Use `"ar"` to
  let only the autoregressive coefficients drift (for example to capture
  changing inertia), `"cross"` for only the cross-lagged coefficients, or pass
  specific names such as `c("phi11", "phi22")`.
- `tv_sigma` lets the residual scales drift. Normal and skew-normal dimensions
  scale multiplicatively on the log scale. Exponential and gamma dimensions use
  a soft-barrier transform so their scale can vary even though the shifted
  margin has a hard support boundary; `tv_sigma_k` controls how closely it
  approximates the exact margin.

With both flags off, `dcvar()` is the usual dynamic copula model with constant
coefficients and scales. Smaller models are nested in larger ones, which
suggests fitting a short ladder and comparing the fits.

```{r tv-fit}
library(dcvar)

# A series with declining coupling to illustrate the time-varying fits
sim_tv <- simulate_dcvar(
  n_time = 150,
  rho_trajectory = rho_decreasing(150, rho_start = 0.7, rho_end = 0.3),
  seed = 3
)

# Only the autoregressive coefficients evolve over time
fit_ar <- dcvar(
  sim_tv$Y_df,
  vars = c("y1", "y2"),
  tv_phi = "ar",
  seed = 4
)

# Time-varying coefficients and time-varying scales together, with a
# right-skewed second margin (the exponential dimension uses the soft barrier)
fit_full <- dcvar(
  sim_tv$Y_df,
  vars = c("y1", "y2"),
  margins = c("normal", "exponential"),
  skew_direction = c(1, 1),
  tv_phi = TRUE,
  tv_sigma = TRUE,
  seed = 5
)
```

The coefficient and scale paths come back as posterior summaries, in the same
shape as the correlation trajectory, and have their own plotting helpers.

```{r tv-extract}
# Posterior summaries of phi11(t), phi12(t), phi21(t), phi22(t)
head(phi_trajectory(fit_ar))

# Per-variable residual-scale paths
head(sigma_trajectory(fit_full))

# Facetted trajectory plots
plot_phi_trajectory(fit_ar)
plot_sigma_trajectory(fit_full)
```

## Fitting All Three Models

```{r fit-all}
library(dcvar)

# Simulate data with a step-function trajectory
sim <- simulate_dcvar(
  n_time = 200,
  rho_trajectory = rho_step(200, rho_before = 0.7, rho_after = 0.3),
  seed = 42
)

# Fit all three models on the same data
fit_dc  <- dcvar(sim$Y_df, vars = c("y1", "y2"), seed = 1)
fit_hmm <- dcvar_hmm(sim$Y_df, vars = c("y1", "y2"), K = 2, seed = 2)
fit_con <- dcvar_constant(sim$Y_df, vars = c("y1", "y2"), seed = 3)
```

## LOO-CV Comparison

```{r compare}
dcvar_compare(dcvar = fit_dc, hmm = fit_hmm, constant = fit_con)
```

## Extracting Tidy Summaries

All fit objects support `as.data.frame()` for exporting tidy parameter
summaries. All five fit families also support `fitted()`/`predict()` for
one-step-ahead values (for SEM and multilevel fits as well). Prediction
intervals are restricted to normal margins for single-level, multilevel, and
naive-SEM fits; indicator-SEM predictions are available for all supported SEM
margins:

```{r tidy}
# Full parameter summary as a data frame
param_df <- as.data.frame(fit_dc)
head(param_df)

# Predictions with marginal intervals
pred_df <- predict(fit_hmm)
head(pred_df)
```

## HMM-Specific Outputs

The HMM model provides additional outputs related to state inference:

```{r hmm}
# State posteriors, Viterbi path, transition matrix
states <- hmm_states(fit_hmm)

# State-specific rho values
states$rho_state

# Transition matrix
states$A

# Visualise state posteriors
plot(fit_hmm, type = "states")

# Transition matrix heatmap
plot(fit_hmm, type = "transition")
```

## Clinical Interpretation

`interpret_rho_trajectory()` provides a model-aware narrative:

```{r interpret}
interpret_rho_trajectory(fit_dc)
interpret_rho_trajectory(fit_hmm)
interpret_rho_trajectory(fit_con)
```

## When to Use Which Model

- **Smooth dynamics** (therapy effects, developmental trends): Use DC-VAR
- **Abrupt regime switches** (treatment onset, relapse): Use HMM Copula
- **No time variation expected**: Use Constant Copula as baseline
- **Uncertain**: Fit all three and compare with LOO-CV
