Discretised distributions

Sam Abbott, Adrian Lison, Felix Günther

This vignette describes the parametric delay distributions that are currently available in epinowcast and explains how they are internally discretised.

Available distributions

The currently available parametric delay distributions are continuous probability distributions with (up to) two parameters \(\mu_{g,t}\) and \(\upsilon_{g,t}\). The table below provides a link to the definition of each distribution, specifies how the parameters \(\mu_{g,t}\) and \(\upsilon_{g,t}\) are mapped to the parameters of the distribution (according to the referenced definition), and states the resulting mean of the distribution (before discretization and adjustment for the assumed maximum delay).

Distribution	Parametrization	Mean
Log-normal	\(\mu=\mu_{g,t}\), \(\sigma = \upsilon_{g,t}\)	\(\exp(\mu_{g,t}+\frac{\upsilon_{g,t}^2}{2})\)
Exponential	\(\beta = \exp(-\mu_{g,t})\)	\(\exp(\mu_{g,t})\)
Gamma	\(\alpha = \exp(\mu_{g,t})\), \(\beta = \upsilon_{g,t}\)	\(\exp(\mu_{g,t})/\upsilon_{g,t}\)
Log-logistic	\(\alpha = \exp(\mu_{g,t})\), \(\beta = \upsilon_{g,t}\)	\(\frac{\exp(\mu_{g,t})\,\pi/\upsilon_{g,t}}{\sin(\pi/\upsilon_{g,t})}\)

Discretisation and adjustment for maximum delay

In epinowcast, delays are modeled in discrete time and with an assumed maximum delay (specified via the max_delay argument). Therefore, the continuous delay distributions must be discretised and adjusted for the maximum delay.

To discretise the continuous probability distribution into discrete probabilities \(p^{\prime}_{g,t,d}\), we use its cumulative distribution function \(F^{\mu_{g,t}, \upsilon_{g,t}}\) to compute \[p^{\prime}_{g,t,d} = \frac{F^{\mu_{g,t}, \upsilon_{g,t}}(d+1) - F^{\mu_{g,t}, \upsilon_{g,t}}(d)}{F^{\mu_{g,t}, \upsilon_{g,t}}(D)}.\] In words, the discrete probability mass function (pmf) is obtained from the continuous distribution by dividing it into \(D\) different bins \[(0,1],\, (1,2],\, \ldots,\, (D-1,D],\] where \(D\) is the maximum delay covered by the model. Importantly, the bins are normalized by \(F^{\mu_{g,t}, \upsilon_{g,t}}(D)\), which ensures that the \(p^{\prime}_{g,t,d}\) sum to 1. Since \(F^{\mu_{g,t}, \upsilon_{g,t}}(D)\) is the probability of reporting before the maximum delay, this can also be interpreted as conditioning our distribution on the maximum delay.

Note that because of the discretisation and normalization, the discrete random variable we obtain generally does not have exactly the same moments as the original, continuous distribution.