hhh4_W: Power-Law and Nonparametric Neighbourhood Weights for...
In jimhester/surveillance: Temporal and Spatio-Temporal Modeling and Monitoring of Epidemic Phenomena

Description Usage Arguments Value Author(s) References See Also Examples

Set up power-law or nonparametric weights for the neighbourhood component of hhh4-models as proposed by Meyer and Held (2014). Without normalization, power-law weights are w_ji = o_ji^-d, where o_ji is the order of neighbourhood between regions i and j, see nbOrder, and d is to be estimated. In the nonparametric formulation, maxlag-1 order-specific log-weights are to be estimated (the first-order weight is always fixed to 1 for identifiability).

W_powerlaw(maxlag, normalize = TRUE,
           log = FALSE, initial = if (log) 0 else 1)

W_np(maxlag, to0 = TRUE, normalize = TRUE,
     initial = log(zetaweights(2:maxlag)))

`maxlag`	a single integer specifying a limiting order of neighbourhood. If spatial dependence is not to be truncated at some high order, `maxlag` should be set to the maximum neighbourhood order in the network of regions.
`to0`	`W_np` represents order-specific log-weights up to order `maxlag`. Higher orders are by default (`to0=TRUE`) assumed to have 0 weight as for `W_powerlaw`. Alternatively, `to0=FALSE` requests that the weight at order `maxlag` should be carried forward to higher orders.
`normalize`	logical indicating if the weights should be normalized such that the rows of the weight matrix sum to 1 (default). Note that normalization does not work with islands, i.e., regions without neighbours.
`log`	logical indicating if the decay parameter d should be estimated on the log-scale to ensure positivity.
`initial`	initial value of the parameter vector.

a list which can be passed as a specification of parametric neighbourhood weights in the control$ne$weights argument of hhh4.

Sebastian Meyer

Meyer, S. and Held, L. (2014): Power-law models for infectious disease spread. The Annals of Applied Statistics, 8 (3), 1612-1639.
DOI-Link: http://dx.doi.org/10.1214/14-AOAS743

nbOrder to determine the matrix of neighbourhood orders from a binary adjacency matrix.

siaf.powerlaw, and siaf.step for modelling distance decay as power law or step function in twinstim space-time point process models.

data("measlesWeserEms")

## data contains neighbourhood orders as required for parametric weights
neighbourhood(measlesWeserEms)[1:6,1:6]
max(neighbourhood(measlesWeserEms))  # max order is 5

## fit a power-law decay of spatial interaction
## in a hhh4 model with seasonality and random intercepts in the endemic part
measlesModel <- list(
    ar = list(f = ~ 1),
    ne = list(f = ~ 1, weights = W_powerlaw(maxlag=5, normalize=TRUE, log=FALSE)),
    end = list(f = addSeason2formula(~-1 + ri(), S=1, period=52),
               offset = population(measlesWeserEms)),
    family = "NegBin1")

## fit the model
set.seed(1)  # random intercepts are initialized randomly
measlesFit <- hhh4(measlesWeserEms, measlesModel)
summary(measlesFit)  # "neweights.d" is the decay parameter d

## plot the spatio-temporal weights o_ji^-d / sum_k o_jk^-d
## as a function of neighbourhood order
plot(measlesFit, type="neweights")
## Due to normalization, same distance does not necessarily mean same weight.
## There is no evidence for a power law of spatial interaction in this
## small observation region with only 17 districts.
## A possible simpler model is first-order dependence, i.e., using
## 'weights = neighbourhood(measlesWeserEms) == 1' in the 'ne' component.