---
title: "Introduction to ZIM"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{ZIM}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r}
library(ZIM)
```

```{r}
data(syph)
count <- syph$a33
count.lag1 <- bshift(count > 0)
trend <- 1:length(count) / 1000
```

```{r}
plot(ts(count), xlab = "Week", ylab = "Number of Syphilis Cases", 
     main = "Maryland (2007 - 2010)", type = "o", pch = 20, las = 1)
table(count)
```

# Observation-Driven Models

## ZIP Autoregression

We first fit a ZIP autoregression with an AR(1) correlation structure. The linear trend is included in both the log-linear and logistic parts of the model. 

```{r}
m1 <- zim(count ~ count.lag1 + trend | trend)
m1
```

The EM-NR algorithm is used as the default algorithm in the zim function. The score test for overdispersion suggests that the ZINB model could provide a better fit to the syphilis data (p = `r round(m1$p.value, 4)`).

## ZINB Autoregression

As suggested by the score test, we next fit a ZINB autoregression, with all the other components remaining the same as in the ZIP autoregression.

```{r}
m2 <- zim(count ~ count.lag1 + trend | trend, dist = "zinb")
m2
```

The AIC and TIC suggest a marginal improvement when the ZINB autoregression is used. However, the BIC values for the ZIP and ZINB autoregressions are not distinguishable. This should not be surprising as BIC tends to penalize more for complexity.

# Parameter-Driven Models

## Dynamic ZIP Model

We now fit a dynamic ZIP model to the syphilis data. The trend is included as a deterministic covariate in the log-linear model. The zero-inflation parameter is assumed to be constant over time.

```{r dzim-zip, cache=TRUE}
set.seed(123)
system.time(m3 <- dzim(count ~ trend, dist = "zip", N = 200, R = 200, niter = 100))
m3
```

```{r}
dzim.plot(m3)
```

## Dynamic ZINB Model

We next fit a dynamic ZINB model to see whether a need remains for the NB dispersion parameter.

```{r dzim-zinb, cache=TRUE}
set.seed(123)
system.time(m4 <- dzim(count ~ trend, dist = "zinb", N = 200, R = 200, niter = 100))
m4
```

```{r}
dzim.plot(m4)
```