Set up power-law or nonparametric weights for the neighbourhood
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,
nbOrder, and d is to be estimated.
In the nonparametric formulation,
log-weights are to be estimated (the first-order weight is always
fixed to 1 for identifiability).
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a single integer specifying a limiting order of
neighbourhood. If spatial dependence is not to be truncated at some
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.
logical indicating if the decay parameter d should be estimated on the log-scale to ensure positivity.
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
Meyer, S. and Held, L. (2014): Power-law models for infectious disease spread. The Annals of Applied Statistics, 8 (3), 1612-1639. doi: 10.1214/14-AOAS743
nbOrder to determine the matrix of neighbourhood orders
from a binary adjacency matrix.
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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.
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