Update estimate_truncation.Rmd

vignettes/estimate_truncation.Rmd

@@ -19,22 +19,22 @@ knitr::opts_chunk$set(
 
 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](https://package.epinowcast.org) 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 $\Zeta(\tau | \mu_{\zeta}$.
+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(\Zeta (T_i - t | \mu_{\Zeta}, \sigma_{\Zeta}) D(t) + \sigma, \varphi\right)
+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 $i$, $\Zeta(\tau)$
+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}}{\Zeta (T_\mathrm{S} - t | \mu_{\Zeta}, \sigma_{\Zeta})}
+D_t = \frac{C^{S}}{Z (T_\mathrm{S} - t | \mu_{Z}, \sigma_{Z})}
 \end{equation}
 
 Relevant priors are: