Model definition: estimate_truncation()

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.

Both estimate_truncation() and estimate_dist() return a delay distribution that downstream functions such as estimate_infections(), estimate_secondary(), or a further call to estimate_truncation() can consume. The main difference is the data they expect: estimate_truncation() takes successive snapshots of the same aggregate counts (a reporting triangle), while estimate_dist() takes individual-level (linelist) data with primary and secondary event dates. Because it works from aggregate counts rather than individual records, estimate_truncation() also fits an observation model for the counts on top of the delay, whereas estimate_dist() estimates the delay distribution alone. As a rough decision rule, use estimate_dist() when you have a linelist and estimate_truncation() when you have repeated snapshots of aggregate counts.

Model

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

$$\begin{equation} D_t = \frac{C^{S}_{t}}{Z(T_S - t | \boldsymbol{\theta})} \end{equation}$$

Priors

$$\begin{align} \boldsymbol{\theta} &\sim \text{as specified by } \texttt{trunc\_opts()} \\ \varphi &\sim \text{as specified by } \texttt{obs\_opts()} \quad \text{(negative binomial only)} \\ \sigma &\sim \text{as specified by } \texttt{noise} \end{align}$$