# Model definition: estimate_truncation()

Source:`vignettes/estimate_truncation.Rmd`

`estimate_truncation.Rmd`

**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 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 $Z(\tau | \mu_{\zeta}$).

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

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

where $T_i$ is the date of the final observation in snapshot $i$, $Z(\tau)$ 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

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

Relevant priors are:

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