# Model definition: estimate_truncation()

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:

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