# 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 \(\Zeta(\tau | \mu_{\zeta}\).

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:

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