There are six types of plots for fitted hhh4 models:

• Plot the "fitted" component means (of selected units) along time along with the observed counts.

• Plot the estimated "season"ality of the three components.

• Plot the time-course of the dominant eigenvalue "maxEV".

• If the units of the corresponding multivariate "sts" object represent different regions, maps of the fitted mean components averaged over time ("maps"), or a map of estimated region-specific intercepts ("ri") of a selected model component can be produced.

• Plot the (estimated) neighbourhood weights ("neweights") as a function of neighbourhood order (shortest-path distance between regions), i.e., w_ji ~ o_ji.

Spatio-temporal "hhh4" models and these plots are illustrated in Meyer et al. (2017, Section 5), see vignette("hhh4_spacetime").

## Usage

# S3 method for hhh4
plot(x, type=c("fitted", "season", "maxEV", "maps", "ri", "neweights"), ...)

plotHHH4_fitted(x, units = 1, names = NULL,
col = c("grey85", "blue", "orange"),
pch = 19, pt.cex = 0.6, pt.col = 1,
par.settings = list(),
legend = TRUE, legend.args = list(),
legend.observed = FALSE,
decompose = NULL, total = FALSE, meanHHH = NULL, ...)

plotHHH4_fitted1(x, unit = 1, main = NULL,
col = c("grey85", "blue", "orange"),
pch = 19, pt.cex = 0.6, pt.col = 1, border = col,
start = x$stsObj@start, end = NULL, xaxis = NULL, xlim = NULL, ylim = NULL, xlab = "", ylab = "No. infected", hide0s = FALSE, decompose = NULL, total = FALSE, meanHHH = NULL) plotHHH4_season(..., components = NULL, intercept = FALSE, xlim = NULL, ylim = NULL, xlab = NULL, ylab = "", main = NULL, par.settings = list(), matplot.args = list(), legend = NULL, legend.args = list(), refline.args = list(), unit = 1, period = NULL) getMaxEV_season(x, period = x$stsObj@freq)

plotHHH4_maxEV(...,
matplot.args = list(), refline.args = list(),
legend.args = list())
getMaxEV(x)

plotHHH4_maps(x, which = c("mean", "endemic", "epi.own", "epi.neighbours"),
prop = FALSE, main = which, zmax = NULL, col.regions = NULL,
labels = FALSE, sp.layout = NULL, ...,
map = x$stsObj@map, meanHHH = NULL) plotHHH4_ri(x, component, exp = FALSE, at = list(n = 10), col.regions = cm.colors(100), colorkey = TRUE, labels = FALSE, sp.layout = NULL, gpar.missing = list(col = "darkgrey", lty = 2, lwd = 2), ...) plotHHH4_neweights(x, plotter = boxplot, ..., exclude = 0, maxlag = Inf) ## Arguments x a fitted hhh4 object. type type of plot: either "fitted" component means of selected units along time along with the observed counts, or "season"ality plots of the model components and the epidemic dominant eigenvalue (which may also be plotted along overall time by type="maxEV", especially if the model contains time-varying neighbourhood weights or unit-specific epidemic effects), or "maps" of the fitted mean components averaged over time, or a map of estimated region-specific random intercepts ("ri") of a specific model component. The latter two require x$stsObj to contain a map.

...

For plotHHH4_season and plotHHH4_maxEV, one or more hhh4-fits, or a single list of these. Otherwise further arguments passed on to other functions.
For the plot-method these go to the specific plot type function.
plotHHH4_fitted passes them to plotHHH4_fitted1, which is called sequentially for every unit in units.
plotHHH4_maps and plotHHH4_ri pass additional arguments to spplot, and plotHHH4_neweights to the plotter.

units,unit

integer or character vector specifying a single unit or possibly multiple units to plot. It indexes colnames(x$stsObj). In plotHHH4_fitted, units=NULL plots all units. In the seasonality plot, selection of a unit is only relevant if the model contains unit-specific intercepts or seasonality terms. names,main main title(s) for the selected unit(s) / components. If NULL (default), plotHHH4_fitted1 will use the appropriate element of colnames(x$stsObj), whereas plotHHH4_season uses default titles.

col,border

length 3 vectors specifying the fill and border colors for the endemic, autoregressive, and spatio-temporal component polygons (in this order).

pch,pt.cex,pt.col

style specifications for the dots drawn to represent the observed counts. pch=NA can be used to disable these dots.

par.settings

list of graphical parameters for par. Sensible defaults for mfrow, mar and las will be applied unless overridden or !is.list(par.settings).

legend

Integer vector specifying in which of the length(units) frames the legend should be drawn. If a logical vector is supplied, which(legend) determines the frame selection, i.e., the default is to drawn the legend in the first (upper left) frame only, and legend=FALSE results in no legend being drawn.

legend.args

list of arguments for legend, e.g., to modify the default positioning list(x="topright", inset=0.02).

legend.observed

logical indicating if the legend should contain a line for the dots corresponding to observed counts.

decompose

if TRUE or (a permutation of) colnames(x$stsObj), the fitted mean will be decomposed into the contributions from each single unit and the endemic part instead of the default endemic + AR + neighbours decomposition. total logical indicating if the fitted components should be summed over all units to be compared with the total observed counts at each time point. If total=TRUE, the units/unit argument is ignored. start,end time range to plot specified by vectors of length two in the form c(year,number), see "sts". xaxis if this is a list (of arguments for addFormattedXAxis), the time axis is nicely labelled similar to stsplot_time. Note that in this case or if xaxis = NA, the basic time indexes 1:nrow(x$stsObj) will be used as x coordinates, which is different from the long-standing default (xaxis = NULL) with a real time scale.

xlim

numeric vector of length 2 specifying the x-axis range. The default (NULL) is to plot the complete time range (type="fitted") or period (type="season"), respectively.

ylim

y-axis range. For type="fitted", this defaults to c(0,max(observed(x$stsObj)[,unit])). For type="season", ylim must be a list of length length(components) specifying the range for every component plot, or a named list to customize only a subset of these. If only one ylim is specified, it will be recycled for all components plots. xlab,ylab axis labels. For plotHHH4_season, ylab specifies the y-axis labels for all components in a list (similar to ylim). If NULL or incomplete, default mathematical expressions are used. If a single name is supplied such as the default ylab="" (to omit y-axis labels), it is used for all components. hide0s logical indicating if dots for zero observed counts should be omitted. Especially useful if there are too many. meanHHH (internal) use different component means than those estimated and available from x. components character vector of component names, i.e., a subset of c("ar", "ne", "end"), for which to plot the estimated seasonality. If NULL (the default), only components which appear in any of the models in ... are plotted. A seasonality plot of the epidemic dominant eigenvalue is also available by including "maxEV" in components, but it only supports models without epidemic covariates/offsets. intercept logical indicating whether to include the global intercept. For plotHHH4_season, the default (FALSE) means to plot seasonality as a multiplicative effect on the respective component. Multiplication by the intercept only makes sense if there are no further (non-centered) covariates/offsets in the component. exp logical indicating whether to exp-transform the color-key axis labels to show the multiplicative effect of the region-specific random intercept on the respective component. Axis labels are then computed using log_breaks from package scales (if that is available) or axisTicks (as a fallback) respecting the colorkey$tick.number setting (default: 7). The default is FALSE.

at

a numeric vector of breaks for the color levels (see levelplot), or a list specifying the number of breaks n (default: 10) and their range (default: range of the random effects, extended to be symmetric around 0). In the latter case, breaks are equally spaced (on the original, non-exp scale of the random intercepts). If exp=TRUE, custom breaks (or range) need to be given on the exp-scale.

matplot.args

list of line style specifications passed to matplot, e.g., lty, lwd, col.

refline.args

list of line style specifications (e.g., lty or col) passed to abline when drawing the reference line (h=1) in plots of seasonal effects (if intercept=FALSE) and of the dominant eigenvalue. The reference line is omitted if refline.args is not a list.

period

a numeric value giving the (longest) period of the harmonic terms in the model. This usually coincides with the freq of the data (the default), but needs to be adjusted if the model contains harmonics with a longer periodicity.

which

a character vector specifying the components of the mean for which to produce maps. By default, the overall mean and all three components are shown.

prop

a logical indicating whether the component maps should display proportions of the total mean instead of absolute numbers.

zmax

a numeric vector of length length(which) (recycled as necessary) specifying upper limits for the color keys of the maps, using a lower limit of 0. A missing element (NA) means to use a map-specific color key only covering the range of the values in that map (can be useful for prop = TRUE). The default zmax = NULL means to use the same scale for the component maps and a separate scale for the map showing the overall mean.

col.regions

a vector of colors used to encode the fitted component means (see levelplot). For plotHHH4_maps, the length of this color vector also determines the number of levels, using 10 heat colors by default.

colorkey

a Boolean indicating whether to draw the color key. Alternatively, a list specifying how to draw it, see levelplot.

map

an object inheriting from "SpatialPolygons" with row.names covering colnames(x).

component

component for which to plot the estimated region-specific random intercepts. Must partially match one of colnames(ranef(x, tomatrix=TRUE)).

labels

determines if and how regions are labeled, see layout.labels.

sp.layout

optional list of additional layout items, see spplot.

gpar.missing

list of graphical parameters for sp.polygons, applied to regions with missing random intercepts, i.e., not included in the model. Such extra regions won't be plotted if !is.list(gpar.missing).

plotter

the (name of a) function used to produce the plot of weights (a numeric vector) as a function of neighbourhood order (a factor variable). It is called as plotter(Weight ~ Distance, ...) and defaults to boxplot. A useful alternative is, e.g., stripplot from package lattice.

exclude

vector of neighbourhood orders to be excluded from plotting (passed to factor). By default, the neighbourhood weight for order 0 is not shown, which is usually zero anyway.

maxlag

maximum order of neighbourhood to be assumed when computing the nbOrder matrix. This additional step is necessary iff neighbourhood(x$stsObj) only specifies a binary adjacency matrix. ## Value plotHHH4_fitted1 invisibly returns a matrix of the fitted component means for the selected unit, and plotHHH4_fitted returns these in a list for all units. plotHHH4_season invisibly returns the plotted y-values, i.e. the multiplicative seasonality effect within each of components. Note that this will include the intercept, i.e. the point estimate of $$exp(intercept + seasonality)$$ is plotted and returned. getMaxEV_season returns a list with elements "maxEV.season" (as plotted by plotHHH4_season(..., components="maxEV"), "maxEV.const" and "Lambda.const" (the Lambda matrix and its dominant eigenvalue if time effects are ignored). plotHHH4_maxEV (invisibly) and getMaxEV return the dominant eigenvalue of the $$\Lambda_t$$ matrix for all time points $$t$$ of x$stsObj.

plotHHH4_maps returns a trellis.object if

length(which) == 1 (a single spplot), and otherwise uses grid.arrange from the

gridExtra package to arrange all length(which)

spplots on a single page.

plotHHH4_ri returns the generated spplot, i.e., a trellis.object.

plotHHH4_neweights eventually calls plotter and thus returns whatever is returned by that function.

Sebastian Meyer

## References

Held, L. and Paul, M. (2012): Modeling seasonality in space-time infectious disease surveillance data. Biometrical Journal, 54, 824-843. doi:10.1002/bimj.201200037

Meyer, S., Held, L. and Höhle, M. (2017): Spatio-temporal analysis of epidemic phenomena using the R package surveillance. Journal of Statistical Software, 77 (11), 1-55. doi:10.18637/jss.v077.i11

other methods for hhh4 fits, e.g., summary.hhh4.

## Examples

data("measlesWeserEms")

## fit a simple hhh4 model
measlesModel <- list(
ar = list(f = ~ 1),
end = list(f = addSeason2formula(~0 + ri(type="iid"), S=1, period=52),
offset = population(measlesWeserEms)),
family = "NegBin1"
)
measlesFit <- hhh4(measlesWeserEms, measlesModel)

## fitted values for a single unit
plot(measlesFit, units=2)

## sum fitted components over all units
plot(measlesFit, total=TRUE)

## 'xaxis' option for a nicely formatted time axis
## default tick locations and labels:
plot(measlesFit, total=TRUE, xaxis=list(epochsAsDate=TRUE, line=1))
## an alternative with monthly ticks:
oopts <- surveillance.options(stsTickFactors = c("%m"=0.75, "%Y" = 1.5))
plot(measlesFit, total=TRUE, xaxis=list(epochsAsDate=TRUE,
xaxis.tickFreq=list("%m"=atChange, "%Y"=atChange),
xaxis.labelFreq=list("%Y"=atMedian), xaxis.labelFormat="%Y"))
surveillance.options(oopts)

## plot the multiplicative effect of seasonality
plot(measlesFit, type="season")

## alternative fit with biennial pattern, plotted jointly with original fit
measlesFit2 <- update(measlesFit,
end = list(f = addSeason2formula(~0 + ri(type="iid"), S=2, period=104)))
plotHHH4_season(measlesFit, measlesFit2, components="end", period=104)

## dominant eigenvalue of the Lambda matrix (cf. Held and Paul, 2012)
getMaxEV(measlesFit)  # here simply constant and equal to exp(ar.1)
plot(measlesFit, type="maxEV")  # not very exciting

## fitted mean components/proportions by district, averaged over time
if (requireNamespace("gridExtra")) {
plot(measlesFit, type="maps", labels=list(cex=0.6),
which=c("endemic", "epi.own"), prop=TRUE, zmax=NA,
main=c("endemic proportion", "autoregressive proportion"))
}

## estimated random intercepts of the endemic component
fixef(measlesFit)["end.ri(iid)"]     # global intercept (log-scale)
ranef(measlesFit, tomatrix = TRUE)   # zero-mean deviations
ranef(measlesFit, intercept = TRUE)  # sum of the above
exp(ranef(measlesFit))               # multiplicative effects
plot(measlesFit, type="ri", component="end",
main="deviations around the endemic intercept (log-scale)")
plot(measlesFit, type="ri", component="end", exp=TRUE,
main="multiplicative effects",
labels=list(font=3, labels="GEN"))

## neighbourhood weights as a function of neighbourhood order
plot(measlesFit, type="neweights")  # boring, model has no "ne" component

## fitted values for the 6 regions with most cases and some customization
bigunits <- tail(names(sort(colSums(observed(measlesWeserEms)))), 6)
plot(measlesFit, units=bigunits,
names=measlesWeserEms@map@data[bigunits,"GEN"],
legend=5, legend.args=list(x="top"), xlab="Time (weekly)",
hide0s=TRUE, ylim=c(0,max(observed(measlesWeserEms)[,bigunits])),
start=c(2002,1), end=c(2002,26), par.settings=list(xaxs="i"))