---
title: "Model definition: estimate_truncation()"
output: rmarkdown::html_vignette
bibliography: library.bib
csl: https://raw.githubusercontent.com/citation-style-language/styles/master/apa-numeric-superscript-brackets.csl
vignette: >
  %\VignetteIndexEntry{Model definition: estimate_truncation()}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```
**This is a work in progress. Please consider submitting a PR to improve it.** 

This model deals with the problem of _nowcasting_, or adjusting for right-truncation in reported count data. This occurs when the quantity being observed, for example cases, hospitalisations or deaths, is reported with a delay, resulting in an underestimation of recent counts. The `estimate_truncation()` model attempts to infer parameters of the underlying delay distributions from multiple snapshots of past data. It is designed to be a simple model that can integrate with the other models in the package and therefore may not be ideal for all uses. For a more principled approach to nowcasting please consider using the [epinowcast](https://package.epinowcast.org) package.

Given snapshots $C^{i}_{t}$ reflecting reported counts for time $t$ where $i=1\ldots S$ is in order of recency (earliest snapshots first) and $S$ is the number of past snapshots used for estimation, we try to infer the parameters of a discrete truncation distribution $\zeta(\tau | \mu_{\zeta}, \sigma_{\zeta})$ with corresponding probability mass function $\Zeta(\tau | \mu_{\zeta}$.

The model assumes that final counts $D_{t}$ are related to observed snapshots via the truncation distribution such that

\begin{equation}
C^{i < S)_{t}_\sim \mathcal{NegBinom}\left(\Zeta (T_i - t | \mu_{\Zeta}, \sigma_{\Zeta}) D(t) + \sigma, \varphi\right)
\end{equation}

where $T_i$ is the date of the final observation in snapshot $i$, $\Zeta(\tau)$
is defined to be zero for negative values of $\tau$ and $\sigma$ is an
additional error term.

The final counts $D_{t}$ are estimated from the most recent snapshot as

\begin{equation}
D_t = \frac{C^{S}}{\Zeta (T_\mathrm{S} - t | \mu_{\Zeta}, \sigma_{\Zeta})}
\end{equation}

Relevant priors are:

\begin{align}
\mu_\zeta &\sim \mathrm{Normal}(0, 1)\\
\sigma_\zeta &\sim \mathrm{HalfNormal}(0, 1)\\
\varphi &\sim \mathrm{HalfNormal}(0, 1)\\
\sigma &\sim \mathrm{HalfNormal}(0, 1)
\end{align}

