---
title: "Gaussian and Student t Copula Time Series Models for Count Data"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Gaussian and Student t Copula Time Series Models for Count Data}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---


```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.align = "center",
  fig.width = 6,
  fig.height = 4
)
library(gctsc)
set.seed(1)
```


The `gctsc` package provides fast and scalable tools for fitting
**Gaussian and Student t copula models for count time series**, supporting a wide range of discrete marginals:

- Poisson  
- Negative Binomial  
- Binomial  
- Beta Binomial
- Zero-Inflated Poisson (ZIP)  
- Zero-Inflated Binomial (ZIB)  
- Zero-Inflated Beta-Binomial (ZIBB)

These models are constructed from a latent **ARMA process**
combined with a discrete marginal distribution. Likelihood evaluation requires computing multivariate
probabilities under Gaussian or Student–t dependence structures.


The package provides several likelihood approximation methods:

- **TMET** — Time Series Minimax Exponential Tilting  
- **GHK** — Geweke–Hajivassiliou–Keane simulator  
- **CE** — Continuous Extension approximation  

In addition, the package offers:

- Simulation of count time series with ARMA latent structure  
- Maximum likelihood estimation using the above likelihoods  
- Residual diagnostic tools  
- One-step-ahead prediction  

A key feature of the package is the TMET method, which exploits the
conditional structure of ARMA processes to compute likelihood
approximations efficiently even for long time series. The package
also includes the classical GHK simulator and the Continuous Extension
approximation for comparison.

This vignette illustrates the main functionality of the `gctsc` package,
including simulation, likelihood approximation, model fitting, diagnostics,
and prediction for copula-based count time series models.


## Installation

The package can be installed from CRAN using

```r
install.packages("gctsc")
```

The development version is available at

```r
remotes::install_github("QNNHU/gctsc")
```
## Package overview

The package provides tools for simulation, likelihood evaluation,
estimation, diagnostics, and prediction for copula-based count time series
models with Gaussian or Student–t copula dependence.

The package implements several likelihood approximation methods, including
Time Series Minimax Exponential Tilting (TMET), the classical
Geweke–Hajivassiliou–Keane (GHK) simulator, and the Continuous Extension (CE)
approximation.

### Main functions

| Function | Description |
|--------|-------------|
| `sim_*()` | Simulate copula count time series with latent ARMA dependence |
| `pmvn_*()` / `pmvt_*()` | Compute multivariate rectangle probabilities for Gaussian and Student–t copulas |
| `gctsc()` | Fit Gaussian or Student–t copula count time series models |
| `predict()` | Compute one-step-ahead predictive distributions and moments |
| `plot()` | Produce residual diagnostic plots |
| `summary()` | Display parameter estimates and model fit statistics |

The following sections illustrate the main functionality of the package,
including simulation, likelihood approximation, model estimation,
diagnostics, and prediction.


---
### Example 1: Simulating Poisson AR(1) counts under a Gaussian copula
```{r}
n  <- 300
mu <- 10
phi <- 0.2
arma_order <- c(1, 0)
tau <- c(phi)

sim_data <- sim_poisson(
  mu = mu,
  tau = tau,
  arma_order = arma_order,
  nsim = n,
  family = "gaussian",
  seed = 7
)

y <- sim_data$y
plot(y, type = "l", main = "Simulated Poisson AR(1) Counts: Gaussian Copula")
```


### Example 2: Simulating Poisson AR(1) counts under a Student–t copula
```{r}
n  <- 300
mu <- 10
phi <- 0.2
arma_order <- c(1, 0)
tau <- c(phi)

sim_data <- sim_poisson(
  mu = mu,
  tau = tau,
  arma_order = arma_order,
  nsim = n,
  family  = "t",
  df = 5,
  seed = 7
)

y <- sim_data$y
plot(y, type = "l", main = "Simulated Poisson AR(1) Counts: t Copula")
```


### Example 3: Simulating Zero–Inflated Beta–Binomial (ZIBB) AR(1) Count Time Series
```{r}
n <- 300
phi <- 0.5
tau <- c(phi)

beta_0 <- 1.2
prob   <- plogis(beta_0)
rho    <- 0.1
pi0    <- 0.8
size   <- 24

set.seed(7)
sim_data <- sim_zibb(prob = prob * rep(1,n),
                     rho = rho, pi0 = pi0,
                     size = size, tau = tau,
                     arma_order = c(1,0),
                     family = "gaussian",
                     nsim = n)
y <- sim_data$y

plot(y, type = "l", main = "Simulated ZIBB AR(1) Counts: gaussian Copula")


```

### Example 4: Likelihood approximation (TMET vs GHK vs CE)
```{r}
n <- 300
mu <- 10
phi <- 0.5
theta <- 0.2
arma_order <- c(1,1)
tau <- c(phi, theta)

# Simulate Poisson count data
sim_data <- sim_poisson(mu, tau, arma_order, nsim = n, seed = 1)
y <- sim_data$y

# Compute bounds for truncated MVN using marginal object
marg <- poisson.marg()
lower <- qnorm(ppois(y - 1, lambda = mu))
upper <- qnorm(ppois(y, lambda = mu))

# Likelihood approximation
llk_tmet <- pmvn_tmet(lower, upper, tau, od = arma_order, pm = 30, QMC = TRUE)
llk_ghk  <- pmvn_ghk(lower, upper, tau, od = arma_order, QMC = TRUE)
llk_ce  <- pmvn_ce(lower, upper, tau, od = arma_order)

c(TMET = llk_tmet, GHK = llk_ghk, CE = llk_ce)

```

### Example 4: Fitting a Gaussian Copula Count Model

```{r}

n <- 300
mu <- 10
phi <- 0.5
theta <- 0.2
arma_order <- c(1,1)
tau <- c(phi, theta)

# Simulate Poisson count data
sim_data <- sim_poisson(mu, tau, arma_order, nsim = n, seed = 1, family ="gaussian")
y <- sim_data$y

fit <- gctsc(
  formula  = y ~ 1,
  marginal = poisson.marg(),
  cormat   = arma.cormat(p = 1, q = 1),
  method   = "CE", family ="gaussian"
)

summary(fit)


```

### Example 5: Residual diagnostics
```{r}
oldpar <- par(no.readonly = TRUE)

par(mfrow = c(2,3))
plot(fit)

par(oldpar)
```


### Example 6: One-step-ahead prediction
```{r}
prediction <- predict(
  fit,
  y_obs   = 10
)

prediction
```

### Example 7: Fitting a t Copula Count Model
```{r}
n <- 100
mu <- 10
phi <- 0.5
theta <- 0.2
arma_order <- c(1,1)
tau <- c(phi, theta)

# Simulate Poisson count data
sim_data <- sim_poisson(mu, tau, arma_order, nsim = n, family = "t", df= 15, seed = 1)
y <- sim_data$y

fit <- gctsc(
  formula  = y ~ 1,
  marginal = poisson.marg(),
  cormat   = arma.cormat(p = 1, q = 1),
  method   = "CE", family ="t", df= 15
)

summary(fit)

```


### Example 7: Real Data Analysis — Campylobacter Time Series

```{r}
## Load weekly Campylobacter incidence data
data("campyl", package = "gctsc")
y <- campyl[,"X1"]
n <- length(y)

## Plot the time series
ts_y <- ts(y, start = c(2001, 1), frequency = 52)
plot(ts_y, main = "Weekly Campylobacter Incidence",
     ylab = "Cases", xlab = "Year")

## ---------------------------------------------------------------
## Construct seasonal covariates
## ---------------------------------------------------------------
## Seasonal structure appears to have yearly periodicity,
## so we include sine/cosine terms with period = 52 weeks.

time <- 1:n
X <- data.frame(
  intercept = 1,
  sin52 = sin(2 * pi * time / 52),
  cos52 = cos(2 * pi * time / 52)
)

## Combine response and covariates
data_df <- data.frame(Y = y, X)

## Use the first 800 observations for model fitting
train_end <- 1000
data_train <- data_df[1:train_end, ]

## ---------------------------------------------------------------
## Fit a Negative Binomial Gaussian Copula model
## ---------------------------------------------------------------
## We use:
##   - Negative Binomial marginal (log link)
##   - ARMA(1,1) latent correlation structure
##   - GHK likelihood approximation
##
## The model is:
##     E[Y_t] = exp(β0 + β1 sin + β2 cos)
##
## Covariates enter only through the marginal mean.

fit <- gctsc(
  formula  = Y ~ sin52 + cos52,
  data     = data_train,
  marginal = negbin.marg(link = "log"),
  cormat   = arma.cormat(p = 1, q = 1),
  method   = "CE", family ="gaussian",
  options  = gctsc.opts(seed = 1)   # fixed seed for reproducibility
)

summary(fit)

## ---------------------------------------------------------------
## Residual diagnostics
## ---------------------------------------------------------------
oldpar <- par(no.readonly = TRUE)
on.exit(par(oldpar))   # restore user settings

par(mfrow=c(2,3))
plot(fit)


## ---------------------------------------------------------------
## One-step-ahead prediction
## ---------------------------------------------------------------
## Predict Y_{801} using fitted model
new_obs_index <- train_end + 1
pred <- predict(
  fit,
  X_test = data_df[new_obs_index, ],
  y_obs  = data_df[new_obs_index, "Y"]
)

pred

```

*Further examples (Negative Binomial, Binomial,Beta-Binomial, ZIP, ZIB, ZIBB, including cases with covariates) and another real data analysis are available in the `inst/examples/` directory of the package source.*