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 infers parameters of the
underlying delay distribution from multiple snapshots of past data. This
can be thought of as a Bayesian form of the chain-ladder nowcasting
approach implemented in the baselinenowcast
package, with the added benefit of joint uncertainty quantification and
delay estimation. For settings requiring time-varying delays or more
detailed reporting structure, see 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 infer the
parameters \(\boldsymbol{\theta}\) of a
discrete truncation distribution with cumulative mass function \(Z(\tau | \boldsymbol{\theta})\). The
truncation distribution can be any family supported by
dist_spec (e.g. log-normal, gamma).
The model assumes that final counts \(D_{t}\) are related to observed snapshots via the truncation distribution such that
\[\begin{equation} C^{i < S}_{t} \sim F\left(Z(T_i - t | \boldsymbol{\theta}) \cdot D(t) + \sigma\right) \end{equation}\]
where \(T_i\) is the date of the
final observation in snapshot \(i\),
\(Z(\tau)\) is defined to be zero for
negative values of \(\tau\), \(\sigma\) is an additive noise term
(controlled via the noise argument), and \(F\) is the observation model (Poisson or
negative binomial, controlled via obs_opts()).
The final counts \(D_{t}\) are estimated from the most recent snapshot as
\[\begin{equation} D_t = \frac{C^{S}_{t}}{Z(T_S - t | \boldsymbol{\theta})} \end{equation}\]
\[\begin{align} \boldsymbol{\theta} &\sim \text{as specified by } \texttt{trunc\_opts()} \\ \varphi &\sim \text{as specified by } \texttt{obs\_opts()} \quad \text{(negative binomial only)} \\ \sigma &\sim \text{as specified by } \texttt{noise} \end{align}\]