Estimate Rt via smoothing splines

This option estimates the effective reproduction number Rt using penalized cubic basis splines.

Usage

R_estimate_splines(
  knot_distance_global = 4 * 7,
  knot_distance_local = 7,
  R_start_prior_mu = 1,
  R_start_prior_sigma = 0.8,
  R_sd_local_prior_mu = 0,
  R_sd_local_prior_sd = 0.05,
  R_sd_global_prior_shape = 1,
  R_sd_global_prior_scale = 0.01,
  R_sd_global_change_distance = knot_distance_global,
  link = "inv_softplus",
  R_max = 6,
  modeldata = modeldata_init()
)

Arguments

knot_distance_global

Distance between spline breakpoints (knots) for the global spline in days. EpiSewer uses an ensemble of two splines to model Rt. The global spline models larger, long-term changes. Default is 4*7, i.e. one knot every four week.

knot_distance_local

Distance between spline breakpoints (knots) for the local spline in days. EpiSewer uses an ensemble of two splines to model Rt. The local spline models smaller, short-term changes. Default is 7, i.e. one knot each week.

R_start_prior_mu

Prior (mean) on the initial reproduction number (intercept).

R_start_prior_sigma

Prior (standard deviation) on the initial reproduction number (intercept).

R_sd_local_prior_mu

Prior (mean) for the variation of the local (i.e. short-term) spline. This controls the standard deviation of a random walk over the coefficients of the local spline. The prior refers to the daily standard deviation and is thus independent of the knot distance.

R_sd_local_prior_sd

Prior (standard deviation) for the variation of the local (i.e. short-term) spline. See R_sd_local_prior_mu for details.

R_sd_global_prior_shape

Exponential-Gamma prior (shape) for the variation of the global (i.e. long-term) spline. This controls the standard deviation of a random walk over the coefficients of the global spline. The prior refers to the daily standard deviation and is thus independent of the knot distance. The exponential-Gamma prior is sparse, i.e. it has a strong peak towards zero and a long tail. Smaller shape parameters will lead to more sparseness, i.e. a longer tail. Note that when adjusting the shape, you will likely also have to adjust the scale. The variation of the global splines follows a change point model to allow for adaptive changes, see details for more explanation.

R_sd_global_prior_scale

Exponential-Gamma prior (scale) for the variation of the global (i.e. long-term) spline. Larger scales will lead to more variability. See R_sd_global_prior_shape and the details for more explanation about this prior.

R_sd_global_change_distance

Distance between changepoints used to model global variation in Rt. EpiSewer uses an adaptive model for the variation of the global spline, to model both time periods with stable and with volatile transmission dynamics. The default change point distance is equal to knot_distance_global. Making this smaller than the global knot distance will not have a large effect, however making this longer reduces the adaptability of the Rt variation. If set to zero, no change points are modeled, meaning zero adaptability.

link

Link function. Currently supported are inv_softplus (default) and scaled_logit. Both of these links are configured to behave approximately like the identity function around R=1, but become increasingly non-linear below (and in the case of scaled_logit also above) R=1.

R_max

If link=scaled_logit is used, a maximum reproduction number must be assumed. This should be higher than any realistic R value for the modeled pathogen. Default is 6.

modeldata

A modeldata object to which the above model specifications should be added. Default is an empty model given by modeldata_init(). Can also be an already partly specified model returned by other EpiSewer modeling functions.

Value

A modeldata object containing data and specifications of the model to be fitted. Can be passed on to other EpiSewer modeling functions to add further data and model specifications.

The modeldata object also includes information about parameter initialization (.init), meta data (.metainfo), and checks to be performed before model fitting (.checks).

Details

EpiSewer uses a combination of two splines (global, for larger/long-term changes and local, for smaller/short-term changes), with a random walk on the coefficients of each spline. This allows a highly adaptive yet regularized Rt model.

• The prior R_start_prior should reflect your expectation of Rt at the beginning of the time series. If your time series starts in the midst of an outbreak, you might want to use a prior with mean larger than 1 for the intercept.

• The prior R_sd_local_prior on the standard deviation of the local spline coefficients should be interpreted in terms of daily additive changes (this is accurate around Rt=1, and becomes less accurate as Rt approaches 0 or its upper bound as defined by the link function). For example, a baseline half-normal prior with sd=0.05 allows a daily standard deviation between 0 and 0.1. A daily standard deviation of 0.1 in turn roughly allows the spline coefficients to change by ±0.2 (using the 2 sigma rule) each day. The daily standard deviation is summed up over the days between two knots to get the actual standard deviation of the coefficients. This way, the prior is independent of the chosen knot_distance. For example, if knot_distance is 7 days, and a constant daily standard deviation of 0.1 is estimated, the coefficients of two adjacent splines can differ by up to 0.2*sqrt(knot_distance), i.e. ±0.5. Note however that this does not directly translate into a change of Rt by ±0.5, as Rt is always the weighted sum of several basis functions at any given point. It will therefore change more gradually, depending on the distances between knots.

• The prior R_sd_global_prior on the standard deviation of the global spline coefficients is interpreted in the same way as the local spline, i.e. it describes daily additive changes. However, we us a sparse prior for the global variation, combined with a changepoint model. This yields a global spline that expects small changes most of the time but allows for occasional large changes. For example, during the height of an epidemic wave, countermeasures may lead to much faster changes in Rt than observable at other times. These differences in variability are accounted for using change points placed at regular intervals, with the daily global standard deviation evolving linearly between the change points. The values at the changepoints are modeled as independently distributed and follow the Exponential-Gamma (EG) prior distribution defined by R_sd_global_prior. The EG distribution is also known as Lomax distribution and corresponds to an exponential distribution with a Gamma distributed rate parameter.

The smoothness of the Rt estimates is influenced by a combination of the global and local knot distances and by the priors on the global and local standard deviation of the random walk on spline coefficients. Placing knots further apart increases the smoothness of Rt estimates and can speed up model fitting. The Rt time series remains surprisingly flexible even at larger knot distances, but placing knots too far apart can lead to inaccurate estimates. Note that the priors for the variation of the global and local random walk also influence the uncertainty of Rt estimates towards the present / date of estimation, when limited data signal is available. Absent sufficient data signal, Rt estimates will tend to stay at the current level (which corresponds to assuming unchanged transmission dynamics).

The priors of this component have the following functional form:

• initial Rt (intercept): Normal

• daily standard deviation of the random walk over local spline coefficients: Half-normal

• daily standard deviation of the random walk over global spline coefficients: Exponential-Gamma

See also

Other Rt models: R_estimate_approx(), R_estimate_changepoint_splines(), R_estimate_ets(), R_estimate_gp(), R_estimate_piecewise(), R_estimate_rw(), R_estimate_smooth_derivative()