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 $$w_{ji} = o_{ji}^{-d}$$ (if $$o_{ji} > 0$$, otherwise $$w_{ji} = 0$$), where $$o_{ji}$$ ($$=o_{ij}$$) is the adjacency order between regions $$i$$ and $$j$$, and the decay parameter $$d$$ is to be estimated. In the nonparametric formulation, unconstrained log-weights will be estimated for each of the adjacency orders 2:maxlag (the first-order weight is fixed to 1 for identifiability). Both weight functions can be modified to include a 0-distance weight, which enables hhh4 models without a separate autoregressive component.

W_powerlaw(maxlag, normalize = TRUE, log = FALSE,
initial = if (log) 0 else 1, from0 = FALSE)

W_np(maxlag, truncate = TRUE, normalize = TRUE,
initial = log(zetaweights(2:(maxlag+from0))),
from0 = FALSE, to0 = truncate)

## Arguments

maxlag

a single integer specifying a limiting order of adjacency. If spatial dependence is not to be truncated at some high order, maxlag should be set to the maximum adjacency order in the network of regions. The smallest possible value for maxlag is 2 if from0=FALSE and 1 otherwise.

truncate,to0

W_np represents order-specific log-weights up to order maxlag. Higher orders are by default (truncate=TRUE) assumed to have zero weight (similar to W_powerlaw). Alternatively, truncate=FALSE requests that the weight at order maxlag should be carried forward to higher orders. truncate has previously been called to0 (deprecated).

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.

from0

logical indicating if these parametric weights should include the 0-distance (autoregressive) case. In the default setting (from0 = FALSE), adjacency order 0 has zero weight, which is suitable for hhh4 models with a separate autoregressive component. With from0 = TRUE (Meyer and Held, 2017), the power law is based on $$(o_{ji} + 1)$$, and nonparametric weights are estimated for adjacency orders 1:maxlag, respectively, where the 0-distance weight is $$w_{jj} = 1$$ (without normalization). Note that the corresponding hhh4 model should then exclude a separate autoregressive component (control$ar$f = ~ -1).

## Value

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

## Details

hhh4 will take adjacency orders from the neighbourhood slot of the "sts" object, so these must be prepared before fitting a model with parametric neighbourhood weights. The function nbOrder can be used to derive adjacency orders from a binary adjacency matrix.

## References

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

Meyer, S. and Held, L. (2017): Incorporating social contact data in spatio-temporal models for infectious disease spread. Biostatistics, 18 (2), 338-351. doi: 10.1093/biostatistics/kxw051

## Author

Sebastian Meyer

nbOrder to determine adjacency orders from a binary adjacency matrix.

getNEweights and coefW to extract the estimated neighbourhood weight matrix and coefficients from an hhh4 model.

## Examples

data("measlesWeserEms")

## data contains adjaceny orders as required for parametric weights
plot(measlesWeserEms, type = observed ~ unit, labels = TRUE)
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)),
end = list(f = addSeason2formula(~-1 + ri(), S=1, period=52)),
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
coefW(measlesFit)

## plot the spatio-temporal weights o_ji^-d / sum_k o_jk^-d
## as a function of adjacency order
plot(measlesFit, type = "neweights", xlab = "adjacency order")
## normalization => same distance does not necessarily mean same weight.
## to extract the whole weight matrix W: getNEweights(measlesFit)

## visualize contributions of the three model components
## to the overall number of infections (aggregated over all districts)
plot(measlesFit, total = TRUE)
## little contribution from neighbouring districts

if (surveillance.options("allExamples")) {
## simpler model with autoregressive effects captured by the ne component
measlesModel2 <- list(
ne = list(f = ~ 1, weights = W_powerlaw(maxlag=5, from0=TRUE)),
end = list(f = addSeason2formula(~-1 + ri(), S=1, period=52)),
family = "NegBin1")
measlesFit2 <- hhh4(measlesWeserEms, measlesModel2)
## omitting the separate AR component simplifies model extensions/selection
## and interpretation of covariate effects (only two predictors left)

plot(measlesFit2, type = "neweights", exclude = NULL, xlab = "adjacency order")
## strong decay, again mostly within-district transmission
## (one could also try a purely autoregressive model)
plot(measlesFit2, total = TRUE,
legend.args = list(legend = c("epidemic", "endemic")))
## almost the same RMSE as with separate AR and NE effects
c(rmse1 = sqrt(mean(residuals(measlesFit, "response")^2)),
rmse2 = sqrt(mean(residuals(measlesFit2, "response")^2)))
}