%\VignetteIndexEntry{twinstim: An endemic-epidemic modeling framework for spatio-temporal point patterns}
%\VignetteEngine{knitr::knitr}
%% additional dependencies beyond what is required for surveillance anyway:
%\VignetteDepends{surveillance, xtable, grid, sf, polyclip, memoise, colorspace}
%% 'sf' is not strictly required: sp:::is.projectedCRS() still has a fallback

<<setup, cache = FALSE, include = FALSE, purl = FALSE>>=
## purl=FALSE => not included in the tangle'd R script
knitr::opts_chunk$set(echo = TRUE, tidy = FALSE, results = 'markup',
                      fig.path='plots/twinstim-', fig.width = 8, fig.height = 4,
                      fig.align = "center", fig.scap = NA, out.width = NULL,
                      cache = FALSE, error = FALSE, warning = FALSE, message = FALSE)
knitr::render_sweave()             # use Sweave environments
knitr::set_header(highlight = '')  # no \usepackage{Sweave} (part of jss class)
## add a chunk option "strip.white.output" to remove leading and trailing white
## space (empty lines) from output chunks ('strip.white' has no effect)
local({
    default_output_hook <- knitr::knit_hooks$get("output")
    knitr::knit_hooks$set(output = function (x, options) {
        if (isTRUE(options[["strip.white.output"]])) {
            x <- sub("[[:space:]]+$", "\n", # set a single trailing \n
                     sub("^[[:space:]]+", "", x)) # remove leading space
        }
        default_output_hook(x, options)
    })
})
## R settings
options(prompt = "R> ", continue = "+  ", useFancyQuotes = FALSE)  # JSS
options(width = 85, digits = 4)
options(scipen = 1)  # so that 1e-4 gets printed as 0.0001
## xtable settings
library("xtable")
options(xtable.booktabs = TRUE, xtable.size = "small",
        xtable.sanitize.text.function = identity,
        xtable.comment = FALSE)
@


\documentclass[nojss,nofooter,article]{jss}
\usepackage{fixjss}
\usepackage[utf8]{inputenc} % Rnw is ASCII, but auto-generated bib file isn't
                            % (specification is redundant in LaTeX >= 2018-04-01)

\title{%
  \vspace{-1.5cm}
  \fbox{\vbox{\normalfont\footnotesize
      This introduction to the \code{twinstim} modeling framework of the
      \proglang{R}~package \pkg{surveillance} is based on a publication in the
      \textit{Journal of Statistical Software} --
      \citet[Section~3]{meyer.etal2017} -- which is the suggested reference
      if you use the \code{twinstim} implementation in your own work.}}\\[1cm]
  \code{twinstim}: An endemic-epidemic modeling framework for spatio-temporal point patterns}
\Plaintitle{twinstim: An endemic-epidemic modeling framework for spatio-temporal point patterns}
\Shorttitle{Endemic-epidemic modeling of spatio-temporal point patterns}

\author{Sebastian Meyer\thanks{Author of correspondence: \email{seb.meyer@fau.de}}\\Friedrich-Alexander-Universit{\"a}t\\Erlangen-N{\"u}rnberg \And
        Leonhard Held\\University of Zurich \And
        Michael H\"ohle\\Stockholm University}
\Plainauthor{Sebastian Meyer, Leonhard Held, Michael H\"ohle}

%% Basic packages
\usepackage{lmodern}          % successor of CM -> searchable Umlauts (1 char)
\usepackage[english]{babel}   % language of the manuscript is American English

%% Math packages
\usepackage{amsmath,amsfonts} % amsfonts defines \mathbb
\usepackage{bm}               % \bm: alternative to \boldsymbol from amsfonts

%% Packages for figures and tables
\usepackage{booktabs}         % make tables look nicer
\usepackage{subcaption}       % successor of subfig, which supersedes subfigure
%% knitr uses \subfloat, which subcaption only provides since v1.3 (2019/08/31)
\providecommand{\subfloat}[2][need a sub-caption]{\subcaptionbox{#1}{#2}}

%% Handy math commands
\newcommand{\abs}[1]{\lvert#1\rvert}
\newcommand{\norm}[1]{\lVert#1\rVert}
\newcommand{\given}{\,\vert\,}
\newcommand{\dif}{\,\mathrm{d}}
\newcommand{\IR}{\mathbb{R}}
\newcommand{\IN}{\mathbb{N}}
\newcommand{\ind}{\mathbb{I}}
\DeclareMathOperator{\Po}{Po}
\DeclareMathOperator{\NegBin}{NegBin}
\DeclareMathOperator{\N}{N}

%% Additional commands
\newcommand{\class}[1]{\code{#1}}  % could use quotes (JSS does not like them)
\newcommand{\CRANpkg}[1]{\href{https://CRAN.R-project.org/package=#1}{\pkg{#1}}}

%% Reduce the font size of code input and output
\DefineVerbatimEnvironment{Sinput}{Verbatim}{fontshape=sl, fontsize=\small}
\DefineVerbatimEnvironment{Soutput}{Verbatim}{fontsize=\small}

%% Abstract
\Abstract{
The availability of geocoded health data and the inherent temporal
structure of communicable diseases have led to
an increased interest in statistical models and software for
spatio-temporal data with epidemic features.
The \proglang{R}~package \pkg{surveillance} can handle
various levels of aggregation at which infective events have been recorded.
This vignette illustrates the analysis of \emph{point-referenced}
surveillance data using the endemic-epidemic point process model
``\code{twinstim}'' proposed by \citet{meyer.etal2012} and extended in
\citet{meyer.held2014}.
%% (For other types of surveillance data, see
%% \code{vignette("twinSIR")} and \code{vignette("hhh4\_spacetime")}.)
We first describe the general modeling approach and then exemplify
data handling, model fitting, visualization, and simulation methods
for time-stamped geo-referenced case reports of invasive meningococcal disease
(IMD) caused by the two most common bacterial finetypes of meningococci in
Germany, 2002--2008.
}

\Keywords{%
spatio-temporal point pattern,
endemic-epidemic modeling,
infectious disease epidemiology,
self-exciting point process,
spatial interaction function,
branching process with immigration}



\begin{document}

<<include = FALSE>>=
## load the "cool" package
library("surveillance")

## Compute everything or fetch cached results?
message("Doing computations: ",
        COMPUTE <- !file.exists("twinstim-cache.RData"))
if (!COMPUTE) load("twinstim-cache.RData", verbose = TRUE)
@


\section[Model class]{Model class: \code{twinstim}} \label{sec:twinstim:methods}


Infective events occur at specific points in continuous space and
time, which gives rise to a spatio-temporal point pattern
$\{(\bm{s}_i,t_i): i = 1,\dotsc,n\}$
from a region~$\bm{W}$ observed during a period~$(0,T]$.
The locations~$\bm{s}_i$ and time points~$t_i$ of the $n$~events can be regarded
as a realization of a self-exciting spatio-temporal point process,
which can be characterized by its
conditional intensity function (CIF, also termed intensity process)
$\lambda(\bm{s},t)$.
It represents the instantaneous event rate at location~$\bm{s}$ at time point~$t$
given all past events, and is often more verbosely denoted by~$\lambda^*$
or by explicit conditioning on the ``history''~$\mathcal{H}_t$ of the process.
\citet[Chapter~7]{Daley.Vere-Jones2003} provide a rigorous mathematical
definition of this concept, which is key to likelihood analysis and simulation
of ``evolutionary'' point processes.

\citet{meyer.etal2012} formulated the model class ``\code{twinstim}'' --
a \emph{two}-component \emph{s}patio-\emph{t}emporal \emph{i}ntensity \emph{m}odel --
by a superposition of an endemic and an epidemic component:

\begin{equation} \label{eqn:twinstim}
  \lambda(\bm{s},t) = \nu_{[\bm{s}][t]} +
  \sum_{j \in I(\bm{s},t)} \eta_j \, f(\norm{\bm{s}-\bm{s}_j}) \, g(t-t_j) \:.
\end{equation}

This model constitutes a branching process with immigration.
Part of the event rate is due to the first, endemic component, which
reflects sporadic events caused by unobserved sources of infection.
This background rate of new events
is modeled by a log-linear predictor
$\nu_{[\bm{s}][t]}$ incorporating regional and/or time-varying characteristics.
Here, the space-time index $[\bm{s}][t]$ refers to the region covering $\bm{s}$
during the period containing $t$ and thus spans a whole spatio-temporal grid on
which the involved covariates are measured, e.g., district $\times$ month.
We will later see that the endemic component therefore simply equals an
inhomogeneous Poisson process for the event counts by cell of that grid.

The second, observation-driven epidemic component adds ``infection pressure''
from the set
\begin{equation*}
  I(\bm{s},t) = \big\{ j : t_j < t \:\wedge\: t-t_j \le \tau_j
  \:\wedge\: \norm{\bm{s}-\bm{s}_j} \le \delta_j \big\}
\end{equation*}
of past events and hence makes the process ``self-exciting''.
During its infectious period of length~$\tau_j$
and within its spatial interaction radius~$\delta_j$,
the model assumes each event~$j$ to trigger further events, which are
called offspring, secondary cases, or aftershocks, depending on the application.
The triggering rate (or force of infection) is proportional
to a log-linear predictor~$\eta_j$ associated with event-specific
characteristics (``marks'') $\bm{m}_j$, which are usually
attached to the point pattern of events.
The decay of infection pressure with increasing spatial and temporal
distance from the infective event is modeled by parametric interaction
functions~$f$ and~$g$, respectively.
A simple assumption for the time course of infectivity is $g(t) = 1$.
Alternatives include exponential decay, a step function,
or empirically derived functions such as Omori's law for aftershock intervals.
With regard to spatial interaction, a
Gaussian kernel $f(x) = \exp\left\{-x^2/(2 \sigma^2)\right\}$ could be chosen.
However, in modeling the spread of human infectious diseases on larger scales,
a heavy-tailed power-law kernel $f(x) = (x+\sigma)^{-d}$ was found to
perform better \citep{meyer.held2014}.
The (possibly infinite) upper bounds~$\tau_j$ and~$\delta_j$ provide a way of
modeling event-specific interaction ranges. However, since these need to be
pre-specified, a common assumption is $\tau_j \equiv \tau$ and
$\delta_j \equiv \delta$, where the infectious period~$\tau$ and the spatial
interaction radius~$\delta$ are determined by subject-matter considerations.


\subsection{Model-based effective reproduction numbers}

Similar to the simple SIR model
\citep[see, e.g.,][Section 2.1]{Keeling.Rohani2008}, the above point process
model~\eqref{eqn:twinstim} features a reproduction number derived from its
branching process interpretation.
As soon as an event occurs (individual becomes infected), it triggers offspring
(secondary cases) around its origin $(\bm{s}_j, t_j)$
according to an inhomogeneous Poisson process with rate
$\eta_j \, f(\norm{\bm{s}-\bm{s}_j}) \, g(t-t_j)$.
Since this triggering process is independent of the event's parentage and of other
events, the expected number $\mu_j$ of events triggered by event $j$
can be obtained by integrating the triggering rate over
the observed interaction domain:
\begin{equation} \label{eqn:R0:twinstim}
  \mu_j = \eta_j \cdot
  \left[ \int_0^{\min(T-t_j,\tau_j)} g(t) \,dt \right] \cdot
  \left[ \int_{\bm{R}_j} f(\norm{\bm{s}}) \,d\bm{s} \right] \:,
\end{equation}
where
\begin{equation} \label{eqn:twinstim:IR}
  \bm{R}_j = (b(\bm{s}_j,\delta_j) \cap \bm{W}) - \bm{s}_j
\end{equation}
is event $j$'s influence region centered at $\bm{s}_j$, and
$b(\bm{s}_j, \delta_j)$ denotes the disc centered at $\bm{s}_j$ with radius $\delta_j$.
Note that the above model-based reproduction number $\mu_j$ is event-specific
since it depends on event marks through $\eta_j$,
on the interaction ranges $\delta_j$ and $\tau_j$,
as well as on the event location $\bm{s}_j$ and time point $t_j$.
If the model assumes unique interaction ranges $\delta$ and $\tau$,
a single reference number of secondary cases can be extrapolated from
Equation~\ref{eqn:R0:twinstim} by imputing an unbounded domain
$\bm{W} = \IR^2$ and $T = \infty$ \citep{meyer.etal2016}.

Equation~\ref{eqn:R0:twinstim} can also be motivated by looking at
a spatio-temporal version of the simple SIR model
wrapped into the \class{twinstim} class~\eqref{eqn:twinstim}. This means:
no endemic component,
homogeneous force of infection ($\eta_j \equiv \beta$),
homogeneous mixing in space ($f(x) = 1$, $\delta_j \equiv \infty$),
and exponential decay of infectivity over time ($g(t) = e^{-\alpha t}$, $\tau_j
\equiv \infty$).
Then, for $T \rightarrow \infty$,
\begin{equation*}
  \mu = \beta \cdot
  \left[ \int_0^\infty e^{-\alpha t} \,dt \right] \cdot
  \left[ \int_{\bm{W}-\bm{s}_j} 1 \,d\bm{s} \right] =
  \beta \cdot \abs{\bm{W}} / \alpha \:,
\end{equation*}
which corresponds to the basic reproduction number
known from the simple SIR model by interpreting $\abs{\bm{W}}$ as the population
size, $\beta$ as the transmission rate and $\alpha$ as the removal rate.
If $\mu < 1$, the process is sub-critical, i.e.,
its eventual extinction is almost sure.

However, it is crucial to understand that in a full model with an
endemic component, new infections may always occur via ``immigration''.
Hence, reproduction numbers in \class{twinstim} are adjusted for
infections occurring independently of previous infections.
This also means that a misspecified endemic component may distort model-based
reproduction numbers \citep{meyer.etal2016}.
Furthermore, under-reporting and implemented control measures imply that the
estimates are to be thought of as \emph{effective} reproduction numbers.


\subsection{Likelihood inference}

The log-likelihood of the point process model~\eqref{eqn:twinstim} is a function
of all parameters in the log-linear predictors $\nu_{[\bm{s}][t]}$ and $\eta_j$ and in
the interaction functions $f$ and $g$. It has the form
%% \begin{equation} \label{eqn:twinstim:marked:loglik}
%%   l(\bm{\theta}) = \left[ \sum_{i=1}^{n} \log\lambda(\bm{s}_i,t_i,k_i) \right] -
%%   \sum_{k\in\mathcal{K}} \int_0^T \int_{\bm{W}} \lambda(\bm{s},t,k) \dif\bm{s}
%%   \dif t \:,
%% \end{equation}
\begin{equation} \label{eqn:twinstim:loglik}
  \left[ \sum_{i=1}^{n} \log\lambda(\bm{s}_i,t_i) \right] -
  \int_0^T \int_{\bm{W}} \lambda(\bm{s},t) \dif\bm{s} \dif t \:.
\end{equation}
%\citep[Proposition~7.3.III]{Daley.Vere-Jones2003}
To estimate the model parameters, we maximize the above log-likelihood
numerically using the quasi-Newton algorithm available through the
\proglang{R}~function \code{nlminb}. We thereby employ the
analytical score function and an approximation of the expected Fisher information
worked out by \citet[Web Appendices A and B]{meyer.etal2012}.

The space-time integral in the log-likelihood \eqref{eqn:twinstim:loglik} poses no difficulties for
the endemic component of $\lambda(\bm{s},t)$, since $\nu_{[\bm{s}][t]}$ is
defined on a spatio-temporal grid.
However, integration of the epidemic component
involves two-dimensional integrals
$\int_{\bm{R}_i} f(\norm{\bm{s}}) \dif\bm{s}$
over the influence regions~$\bm{R}_i$,
which are represented by polygons (as is~$\bm{W}$).
Similar integrals appear in the score function, where $f(\norm{\bm{s}})$ is
replaced by partial derivatives with respect to kernel parameters.
Calculation of these integrals is trivial for (piecewise) constant~$f$, but otherwise requires
numerical integration. The \proglang{R}~package
\CRANpkg{polyCub} \citep{meyer2019} offers various cubature methods for polygonal
domains.
% For Gaussian~$f$, we apply a midpoint rule with $\sigma$-adaptive bandwidth
% %% combined with an analytical formula via the $\chi^2$ distribution
% %% if the $6\sigma$-circle around $\bm{s}_i$ is contained in $\bm{R}_i$.
% and use product Gauss cubature \citep{sommariva.vianello2007}
% to approximate the integrals in the score function.
% For the recently implemented power-law kernels,
Of particular relevance for \code{twinstim} is the \code{polyCub.iso} method, which takes
advantage of the assumed isotropy of spatial interaction
such that numerical integration remains in only one dimension
\citep[Supplement~B, Section~2]{meyer.held2014}.
We \CRANpkg{memoise} \citep{R:memoise} the cubature function during
log-likelihood maximization to avoid integration
for unchanged parameters of~$f$.


\subsection{Special cases: Single-component models}

If the \emph{epidemic} component is omitted in Equation~\ref{eqn:twinstim},
the point process model
becomes equivalent to a Poisson regression model for aggregated counts.
This provides a link to
ecological regression approaches in general
and to the count data model \code{hhh4} illustrated in
\code{vignette("hhh4")} and \code{vignette("hhh4\_spacetime")}.
To see this, recall that the endemic component $\nu_{[\bm{s}][t]}$
is piecewise constant on the spatio-temporal grid with cells $([\bm{s}],[t])$.
Hence the log-likelihood~\eqref{eqn:twinstim:loglik} of an endemic-only
\code{twinstim} simplifies to a sum over all these cells,
\begin{equation*}
\sum_{[\bm{s}],[t]} \left\{
  Y_{[\bm{s}][t]} \log\nu_{[\bm{s}][t]} -
  \abs{[\bm{s}]} \, \abs{[t]} \, \nu_{[\bm{s}][t]}
\right\} \:,
\end{equation*}
where $Y_{[\bm{s}][t]}$ is the aggregated number of events observed in cell
$([\bm{s}],[t])$, and $\abs{[\bm{s}]}$ and $\abs{[t]}$ denote cell area and length,
respectively.
Except for an additive constant, the above log-likelihood is equivalently
obtained from the Poisson model
$Y_{[\bm{s}][t]} \sim \Po( \abs{[\bm{s}]} \, \abs{[t]} \, \nu_{[\bm{s}][t]})$.
This relation offers a means of code validation using the established
\code{glm} function to fit an endemic-only \code{twinstim} model --
see the examples in \code{help("glm_epidataCS")}.
%% The \code{help("glm_epidataCS")} also shows how to fit
%% an equivalent endemic-only \code{hhh4} model.

If, in contrast, the \emph{endemic} component is omitted,
all events are necessarily triggered by other observed events.
For such a model to be identifiable,
a prehistory of events must exist to trigger the first event,
and interaction typically needs to be unbounded
such that each event can actually be linked to potential source events.


\subsection[Extension: Event types]{Extension: \code{twinstim} with event types}

To model the example data on invasive meningococcal disease in the remainder of this section,
we actually need to use an extended version $\lambda(\bm{s},t,k)$ of
Equation~\ref{eqn:twinstim}, which accounts for different event types~$k$ with
own transmission dynamics.
This introduces a further dimension in the point process, and the second
log-likelihood component in Equation~\ref{eqn:twinstim:loglik}
accordingly splits into a sum over all event types.
We refer to \citet[Sections~2.4 and~3]{meyer.etal2012} for the technical details
of this type-specific \code{twinstim} class.
The basic idea is that the meningococcal finetypes share the same endemic pattern
(e.g., seasonality), while infections of different finetypes are not associated
via transmission. This means that the force of infection is restricted to
previously infected individuals with the same bacterial finetype~$k$,
i.e., the epidemic sum in Equation~\ref{eqn:twinstim} is over the set
$I(\bm{s},t,k) = I(\bm{s},t) \cap \{j: k_j = k\}$.
The implementation has limited support for type-dependent interaction
functions $f_{k_j}$ and $g_{k_j}$ (not further considered here).



\section[Data structure]{Data structure: \class{epidataCS}} \label{sec:twinstim:data}


<<imdepi_components, echo=FALSE>>=
## extract components from imdepi to reconstruct
data("imdepi")
events <- SpatialPointsDataFrame(
    coords = coordinates(imdepi$events),
    data = marks(imdepi, coords=FALSE),
    proj4string = imdepi$events@proj4string  # ETRS89 projection (+units=km)
    )
stgrid <- imdepi$stgrid[,-1]
@
<<load_districtsD, echo=FALSE>>=
load(system.file("shapes", "districtsD.RData", package = "surveillance"))
@

The first step toward fitting a \code{twinstim} is to turn the relevant
data into an object of the dedicated class \class{epidataCS}.\footnote{
  The suffix ``CS'' indicates that the data-generating point process is indexed
  in continuous space.
}
The primary ingredients of this class are a spatio-temporal point pattern
(\code{events}) and its underlying observation region (\code{W}).
An additional spatio-temporal grid (\code{stgrid}) holds (time-varying)
area-level covariates for the endemic regression part.
We exemplify this data class by the \class{epidataCS} object for the
\Sexpr{nobs(imdepi)} cases of invasive meningococcal disease in Germany
originally analyzed by \citet{meyer.etal2012}.
It is already contained in the \pkg{surveillance} package as
\code{data("imdepi")} and has been constructed as follows:
<<imdepi_construct, results="hide", eval=FALSE>>=
imdepi <- as.epidataCS(events = events, W = stateD, stgrid = stgrid,
  qmatrix = diag(2), nCircle2Poly = 16)
@
The function \code{as.epidataCS} checks the consistency of the
three data ingredients described in detail below.
It also pre-computes auxiliary variables for model fitting, e.g.,
the individual influence regions~\eqref{eqn:twinstim:IR},
which are intersections of the observation region with discs
%of radius \code{eps.s} centered at the event location
approximated by polygons with \code{nCircle2Poly = 16} edges.
The intersections are computed using
functionality of the package \CRANpkg{polyclip} \citep{R:polyclip}.
For multitype epidemics as in our example, the additional indicator matrix
\code{qmatrix} specifies transmissibility across event types.
An identity matrix corresponds to an independent spread of the event
types, i.e., cases of one type can not produce cases of another type.


\subsection{Data ingredients}

The core \code{events} data must be provided in the form of a
\class{SpatialPointsDataFrame} as defined by the package \CRANpkg{sp}
\citep{R:sp}:
<<imdepi_events_echo, results="hide">>=
summary(events)
@
<<imdepi_events, echo=FALSE, purl=FALSE>>=
oopt <- options(width=100)
## hack to reduce the 'print.gap' in the data summary but not for the bbox
## Note: sp >= 2.0-0 loads 'sf' for summary(events), but has a fallback,
##       see https://github.com/r-spatial/evolution/issues/10
local({
    print.summary.Spatial <- sp:::print.summary.Spatial
    environment(print.summary.Spatial) <- environment()
    print.table <- function (x, ..., print.gap = 0) {
        base::print.table(x, ..., print.gap = print.gap)
    }
    print.summary.Spatial(summary(events))
})
options(oopt)
@
The associated event coordinates are residence postcode centroids, projected in
the \emph{European Terrestrial Reference System 1989} (in kilometer units)
to enable Euclidean geometry.
See the \code{spTransform}-methods
for how to project latitude and longitude coordinates
into a planar coordinate reference system (CRS).
The data frame associated with these spatial coordinates ($\bm{s}_i$) contains a
number of required variables and additional event marks
(in the notation of Section~\ref{sec:twinstim:methods}:
$\{(t_i,[\bm{s}_i],k_i,\tau_i,\delta_i,\bm{m}_i): i = 1,\dotsc,n\}$).
For the IMD data, the event
\code{time} is measured in days since the beginning of the observation period
2002--2008 and is subject to a tie-breaking procedure (described later).
The \code{tile} column refers to the region of the spatio-temporal grid where
the event occurred and here contains the official key of the administrative
district of the patient's residence.
There are two \code{type}s of events labeled as \code{"B"} and \code{"C"}, which refer
to the serogroups of the two meningococcal finetypes \emph{B:P1.7-2,4:F1-5} and
\emph{C:P1.5,2:F3-3} contained in the data.
The \code{eps.t} and \code{eps.s} columns specify upper limits
for temporal and spatial interaction, respectively.
Here, the infectious period is assumed to last a maximum of 30 days and
spatial interaction is limited to a 200 km radius for all cases.
The latter has numerical advantages for a Gaussian interaction function $f$ with a
relatively small standard deviation. For a power-law kernel, however, this
restriction will be dropped to enable occasional long-range transmission.
The last two data attributes displayed in the above \code{event} summary are
covariates from the case reports: the gender and age group of the patient.

For the observation region \code{W}, we use a polygon representation
of Germany's boundary.
Since the observation region defines the integration domain
in the point process log-likelihood~\eqref{eqn:twinstim:loglik},
the more detailed the polygons of \code{W} are
the longer it will take to fit a \code{twinstim}.
It is thus advisable to sacrifice some shape details for speed by reducing
the polygon complexity. In \proglang{R} this can be achieved via, e.g.,
\code{ms_simplify} from the \CRANpkg{rmapshaper} package \citep{R:rmapshaper},
or \code{simplify.owin} from \CRANpkg{spatstat.geom} \citep{R:spatstat}.
% \code{thinnedSpatialPoly} in package \CRANpkg{maptools},  % will retire
% which implements the Douglas-Peucker reduction method.
The \pkg{surveillance} package already contains a simplified representation of
Germany's boundaries:
<<imdepi_W_echo, echo=TRUE, eval=FALSE, purl=FALSE>>=
<<load_districtsD>>
@
This file contains both the \class{SpatialPolygonsDataFrame} \code{districtsD}
of Germany's \Sexpr{length(districtsD)} administrative districts as at
January 1, 2009,
as well as their union \code{stateD}.
%obtained by the call \code{rgeos::gUnaryUnion(districtsD)} \citep{R:rgeos}.
These boundaries are projected in the same CRS as the \code{events} data.

The \code{stgrid} input for the endemic model component
is a data frame with (time-varying) area-level covariates,
e.g., socio-economic or ecological characteristics.
In our example:
<<imdepi_stgrid, echo=FALSE>>=
.stgrid.excerpt <- format(rbind(head(stgrid, 3), tail(stgrid, 3)), digits=3)
rbind(.stgrid.excerpt[1:3,], "..."="...", .stgrid.excerpt[4:6,])
@
Numeric (\code{start},\code{stop}] columns index the time periods and
the factor variable \code{tile} identifies the regions of the grid.
Note that the given time intervals (here: months) also define the resolution of
possible time trends and seasonality of the piecewise constant endemic intensity.
We choose monthly intervals to reduce package size and computational cost
compared to the weekly resolution originally used by \citet{meyer.etal2012}
and \citet{meyer.held2014}.
The above \code{stgrid} data frame thus consists of 7 (years) times 12 (months)
blocks of \Sexpr{nlevels(stgrid[["tile"]])} (districts) rows each.
The \code{area} column gives the area of the respective \code{tile} in square
kilometers (compatible with the CRS used for \code{events} and \code{W}).
A geographic representation of the regions in \code{stgrid} is not required for
model estimation, and is thus not part of the \class{epidataCS} class.
%It is, however, necessary for plots of the fitted intensity and for
%simulation from the estimated model.
In our example, the area-level data only consists of the population density
\code{popdensity}, whereas \citet{meyer.etal2012} additionally incorporated
(lagged) weekly influenza counts by district as a time-dependent covariate.
%% In another application, \citet{meyer.etal2016} used a large number of socio-economic
%% characteristics to model psychiatric hospital admissions.


\pagebreak
\subsection{Data handling and visualization}

The generated \class{epidataCS} object \code{imdepi} is a simple list of the
checked ingredients
<<echo = FALSE, results = "asis", purl=FALSE>>=
cat(paste0('\\code{', names(imdepi), '}', collapse = ", "), ".", sep = "")
@
Several methods for data handling and visualization are available for
such objects as listed in Table~\ref{tab:methods:epidataCS}
and briefly presented in the remainder of this section.

<<epidataCS-methods, echo=FALSE, results="asis", purl=FALSE>>=
print(xtable(
    surveillance:::functionTable(
        class = "epidataCS",
        functions = list(
            Convert = c("epidataCS2sts"),
            Extract = c("getSourceDists"))),
    caption="Generic and \\textit{non-generic} functions applicable to \\class{epidataCS} objects.",
    label="tab:methods:epidataCS"
    ), include.rownames = FALSE)
@

Printing an \class{epidataCS} object presents some
metadata and the first \Sexpr{formals(surveillance:::print.epidataCS)[["n"]]}
events by default:
<<imdepi_print>>=
imdepi
@
During conversion to \class{epidataCS}, the last three columns \code{BLOCK}
(time interval index), \code{start} and \code{popdensity} have been merged from
the checked \code{stgrid} to the \code{events} data frame.
The event marks including time and location can be extracted in a standard
data frame by \code{marks(imdepi)} -- inspired by package \CRANpkg{spatstat} --
and this is summarized by \code{summary(imdepi)}.
<<imdepi_summary, include = FALSE>>=
(simdepi <- summary(imdepi))
@
The number of potential sources of infection per event (denoted
\texttt{|.sources|} in the above output) is additionally summarized.
It is determined by the events' maximum ranges of interaction \code{eps.t} and
\code{eps.s}. The event-specific set of potential sources is stored in the
(hidden) list \code{imdepi$events$.sources} (events are referenced by row
index), and the event-specific numbers of potential sources are stored in the
summarized object as \code{simdepi$nSources}.

A simple plot of the number of infectives as a function of time
(Figure~\ref{fig:imdepi_stepfun})
%determined by the event times and infectious periods
can be obtained by the step function converter:
<<imdepi_stepfun, echo=2, fig.cap="Time course of the number of infectives assuming infectious periods of 30 days.">>=
par(mar = c(5, 5, 1, 1), las = 1)
plot(as.stepfun(imdepi), xlim = summary(imdepi)$timeRange, xaxs = "i",
  xlab = "Time [days]", ylab = "Current number of infectives", main = "")
#axis(1, at = 2557, labels = "T", font = 2, tcl = -0.3, mgp = c(3, 0.3, 0))
@

\pagebreak[1]
The \code{plot}-method for \class{epidataCS} offers aggregation of the events over time or space:
<<imdepi_plot, fig.cap="Occurrence of the two finetypes viewed in the temporal and spatial dimensions.", fig.subcap=c("Temporal pattern.","Spatial pattern."), fig.width=5, fig.height=6, echo=c(2,4,5), out.width="0.5\\linewidth", fig.pos="!htb">>=
par(las = 1)
plot(imdepi, "time", col = c("indianred", "darkblue"), ylim = c(0, 20))
par(mar = c(0, 0, 0, 0))
plot(imdepi, "space", lwd = 2,
  points.args = list(pch = c(1, 19), col = c("indianred", "darkblue")))
layout.scalebar(imdepi$W, scale = 100, labels = c("0", "100 km"), plot = TRUE)
@

\pagebreak[1]
The time-series plot (Figure~\ref{fig:imdepi_plot-1}) shows the monthly aggregated number of cases by finetype in
a stacked histogram as well as each type's cumulative number over time.
The spatial plot (Figure~\ref{fig:imdepi_plot-2}) shows the observation window \code{W} with the locations of all
cases (by type), where the areas of the points are proportional to the number of
cases at the respective location.
Additional shading by the population is possible and exemplified in
\code{help("plot.epidataCS")}.
%The above static plots do not capture the space-time dynamics of epidemic spread.
An animation may provide additional insight and can be produced by the
corresponding \code{animate}-method. For instance, to look at the first year of
the B-type in a weekly sequence of snapshots in a web browser (using facilities of the
\CRANpkg{animation} package of \citealp{R:animation}):
<<imdepi_animate_saveHTML, eval=FALSE>>=
animation::saveHTML(
  animate(subset(imdepi, type == "B"), interval = c(0, 365), time.spacing = 7),
  nmax = Inf, interval = 0.2, loop = FALSE, title = "First year of type B")
@
Selecting events from \class{epidataCS} as for the animation above is enabled
by the \code{[}- and \code{subset}-methods, which return a new
\class{epidataCS} object containing only the selected \code{events}.

A limited data sampling resolution may lead to tied event times or locations,
which are in conflict with a continuous spatio-temporal point process model.
For instance, a temporal residual analysis
would suggest model deficiencies \citep[Figure 4]{meyer.etal2012},
and a power-law kernel for spatial interaction may diverge if
there are events with zero distance to potential source events
\citep{meyer.held2014}. The function \code{untie} breaks ties by
random shifts. This has already been applied to the event \emph{times} in
the provided \code{imdepi} data by subtracting a U$(0,1)$-distributed random
number from the original dates.
The event \emph{coordinates} in the IMD data are subject to interval censoring
at the level of Germany's postcode regions. A possible replacement for the given
centroids would thus be a random location within the corresponding postcode area.
Lacking a suitable shapefile,
\citet{meyer.held2014} shifted all locations by a random
vector with length up to half the observed minimum spatial separation:
<<imdepi_untied>>=
eventDists <- dist(coordinates(imdepi$events))
minsep <- min(eventDists[eventDists > 0])
set.seed(321)
imdepi_untied <- untie(imdepi, amount = list(s = minsep / 2))
@
Note that random tie-breaking requires sensitivity analyses as
discussed by \citet{meyer.held2014}, but these are skipped here for the sake of brevity.

The \code{update}-method is useful to change the values of the maximum
interaction ranges \code{eps.t} and \code{eps.s}, since it takes care of the
necessary updates of the hidden auxiliary variables in an \class{epidataCS}
object. For unbounded spatial interaction:
<<imdepi_untied_infeps>>=
imdepi_untied_infeps <- update(imdepi_untied, eps.s = Inf)
@

Last but not least, \class{epidataCS} can be aggregated to
\class{epidata} (from \code{vignette("twinSIR")})
or \class{sts} (from \code{vignette("hhh4_spacetime")}).
The method \code{as.epidata.epidataCS} aggregates events by region (\code{tile}),
and the function \code{epidataCS2sts} yields counts by region and time interval.
The latter could be analyzed by an
areal time-series model such as \code{hhh4} (see \code{vignette("hhh4\_spacetime")}).
We can also use \class{sts} visualizations, e.g.\ (Figure~\ref{fig:imdsts_plot}):
<<imdsts_plot, fig.cap="IMD cases (joint types) aggregated as an \\class{sts} object by month and district.", fig.subcap=c("Time series of monthly counts.", "Disease incidence (per 100\\,000 inhabitants)."), fig.width=5, fig.height=5, out.width="0.5\\linewidth", fig.pos="ht", echo=-2>>=
imdsts <- epidataCS2sts(imdepi, freq = 12, start = c(2002, 1), tiles = districtsD,
                        neighbourhood = NULL) # skip adjacency matrix (needs spdep)
par(las = 1, lab = c(7,7,7), mar = c(5,5,1,1))
plot(imdsts, ~time)
plot(imdsts, ~unit, population = districtsD$POPULATION / 100000)
@



\section{Modeling and inference} \label{sec:twinstim:fit}


Having prepared the data as an object of class \class{epidataCS},
the function \code{twinstim} can be used to perform likelihood inference
for conditional intensity models of the form~\eqref{eqn:twinstim}.
The main arguments for \code{twinstim}
are the formulae of the \code{endemic} and \code{epidemic} linear predictors
($\nu_{[\bm{s}][t]} = \exp$(\code{endemic}) and $\eta_j = \exp$(\code{epidemic})), and
the spatial and temporal interaction functions \code{siaf} ($f$) and \code{tiaf} ($g$), respectively.
Both formulae are parsed internally using the standard \code{model.frame}
toolbox from package \pkg{stats} and thus can handle factor variables and
interaction terms.
While the \code{endemic} linear predictor incorporates
%time-dependent and/or area-level
covariates from \code{stgrid},
%% and in the disease mapping context usually contains at least the population density as a multiplicative offset, i.e.,
%% \code{endemic = ~offset(log(popdensity))}. There can be additional effects of time,
%% which are functions of the variable \code{start} from \code{stgrid},
%% or effects of, e.g., socio-demographic and ecological variables.
the \code{epidemic} formula may use both \code{stgrid} variables and event marks
to be associated with the force of infection.
%% For instance, \code{epidemic = ~log(popdensity) + type} corresponds to
%% $\eta_j = \rho_{[\bm{s}_j]}^{\gamma_{\rho}} \exp(\gamma_0 + \gamma_C \ind(k_j=C))$,
%% which models different infectivity of the event types, and scales
%% with population density (a grid-based covariate) to reflect higher
%% contact rates and thus infectivity in more densely populated regions.
For the interaction functions, several alternatives are predefined as listed in
Table~\ref{tab:iafs}. They are applicable out-of-the-box and
illustrated as part of the following modeling exercise for the IMD data.
Own interaction functions can also be implemented following the structure
described in \code{help("siaf")} and \code{help("tiaf")}, respectively.

<<tab_iafs, echo=FALSE, results="asis", purl=FALSE>>=
twinstim_iafs <- suppressWarnings(
    cbind("Spatial (\\code{siaf.*})" = ls(pattern="^siaf\\.", pos="package:surveillance"),
          "Temporal (\\code{tiaf.*})" = ls(pattern="^tiaf\\.", pos="package:surveillance"))
    )
twinstim_iafs <- apply(twinstim_iafs, 2, function (x) {
    is.na(x) <- duplicated(x)
    x
})
print(xtable(substring(twinstim_iafs, 6), label="tab:iafs",
             caption="Predefined spatial and temporal interaction functions."),
      include.rownames=FALSE,
      sanitize.text.function=function(x) paste0("\\code{", x, "}"),
      sanitize.colnames.function=identity, sanitize.rownames.function=identity)
@


\subsection{Basic example}

To illustrate statistical inference with \code{twinstim},
we will estimate several models for the simplified and ``untied'' IMD data
presented in Section~\ref{sec:twinstim:data}.
In the endemic component, we include the district-specific population density as a multiplicative
offset, a (centered) time trend, and
a sinusoidal wave of frequency $2\pi/365$ to capture seasonality,
where the \code{start} variable from \code{stgrid} measures time:
<<endemic_formula>>=
(endemic <- addSeason2formula(~offset(log(popdensity)) + I(start / 365 - 3.5),
  period = 365, timevar = "start"))
@
See \citet[Section~2.2]{held.paul2012} for how such sine/cosine terms
reflect seasonality.
Because of the aforementioned integrations in the log-likelihood~\eqref{eqn:twinstim:loglik}, it
is advisable to first fit an endemic-only model to obtain reasonable
start values for more complex epidemic models:
<<imdfit_endemic, results="hide">>=
imdfit_endemic <- twinstim(endemic = endemic, epidemic = ~0,
  data = imdepi_untied, subset = !is.na(agegrp))
@
We exclude the single case with unknown age group from this analysis since we
will later estimate an effect of the age group on the force of infection.

Many of the standard functions to access model fits in \proglang{R} are also
implemented for \class{twinstim} fits (see Table~\ref{tab:methods:twinstim}).
For example, we can produce the usual model summary:
<<strip.white.output=TRUE>>=
summary(imdfit_endemic)
@
Because of the aforementioned equivalence of the endemic component with a
Poisson regression model, the coefficients can be interpreted as
log rate ratios in the usual way. For instance, the endemic rate is estimated to
decrease by \code{1 - exp(coef(imdfit_endemic)[2])} $=$
\Sexpr{round(100*(1-exp(coef(imdfit_endemic)[2])),1)}\% per year.
Coefficient correlations can be retrieved via the argument
\code{correlation = TRUE} in the \code{summary} call just like for
\code{summary.glm}, or via
\code{cov2cor(vcov(imdfit_endemic))}.

<<twinstim-methods, echo=FALSE, results="asis", purl=FALSE>>=
print(xtable(
    surveillance:::functionTable(
        class = "twinstim",
        functions = list(
            Display = c("iafplot", "checkResidualProcess"),
            Extract = c("intensity.twinstim", "simpleR0"),
            Modify = c("stepComponent"),
            Other = c("epitest"))),
    caption="Generic and \\textit{non-generic} functions applicable to \\class{twinstim} objects. Note that there is no need for specific \\code{coef}, \\code{confint}, \\code{AIC} or \\code{BIC} methods, since the respective default methods from package \\pkg{stats} apply outright.",
    label="tab:methods:twinstim"
    ), include.rownames = FALSE)
@

We now update the endemic model to take additional spatio-temporal dependence between
events into account. Infectivity shall depend on the meningococcal finetype and
the age group of the patient, and is assumed to be constant over time (default),
$g(t)=\ind_{(0,30]}(t)$, with a Gaussian distance-decay
$f(x) = \exp\left\{-x^2/(2 \sigma^2)\right\}$.
This model was originally selected by \citet{meyer.etal2012} and can be fitted as follows:
<<imdfit_Gaussian, results="hide", eval=COMPUTE>>=
imdfit_Gaussian <- update(imdfit_endemic, epidemic = ~type + agegrp,
  siaf = siaf.gaussian(), cores = 2 * (.Platform$OS.type == "unix"))
@
On Unix-alikes, the numerical integrations
of $f(\norm{\bm{s}})$ in the log-likelihood and
$\frac{\partial f(\norm{\bm{s}})}{\partial \log\sigma}$
in the score function (note that $\sigma$ is estimated on the log-scale)
can be performed in parallel via %the ``multicore'' functions
\code{mclapply} \textit{et al.}\ from the base package \pkg{parallel},
here with \code{cores = 2} processes.

Table~\ref{tab:imdfit_Gaussian} shows the output of \code{twinstim}'s
\code{xtable} method \citep{R:xtable} applied to the above model fit,
providing a table of estimated rate ratios for the endemic and epidemic effects.
The alternative \code{toLatex} method simply translates the
\code{summary} table of coefficients to \LaTeX\ without \code{exp}-transformation.
On the subject-matter level, we can conclude from
Table~\ref{tab:imdfit_Gaussian} that the meningococcal finetype of
serogroup~C is less than half as infectious as the B-type, and that patients in
the age group 3 to 18 years are estimated to cause twice as many secondary infections as infants
aged 0 to 2 years.

<<tab_imdfit_Gaussian, echo=FALSE, results="asis">>=
library("xtable")
print(xtable(imdfit_Gaussian,
             caption="Estimated rate ratios (RR) and associated Wald confidence intervals (CI) for endemic (\\code{h.}) and epidemic (\\code{e.}) terms. This table was generated by \\code{xtable(imdfit\\_Gaussian)}.",
             label="tab:imdfit_Gaussian"),
      sanitize.text.function=NULL, sanitize.colnames.function=NULL,
      sanitize.rownames.function=function(x) paste0("\\code{", x, "}"))
@


\subsection{Model-based effective reproduction numbers}

The event-specific reproduction numbers~\eqref{eqn:R0:twinstim} can be extracted
from fitted \class{twinstim} objects via the \code{R0} method.
For the above IMD model, we obtain the following mean numbers of
secondary infections by finetype:
<<>>=
R0_events <- R0(imdfit_Gaussian)
tapply(R0_events, marks(imdepi_untied)[names(R0_events), "type"], mean)
@
Confidence intervals %for the estimated reproduction numbers $\hat\mu_j$
can be obtained via Monte Carlo simulation,
where Equation~\ref{eqn:R0:twinstim} is repeatedly evaluated
with parameters sampled from the asymptotic multivariate normal distribution of the
maximum likelihood estimate. For this purpose, the \code{R0}-method
takes an argument \code{newcoef}, which is exemplified in \code{help("R0")}.
%% Note that except for (piecewise) constant $f$, computing confidence intervals for
%% $\hat\mu_j$ takes a considerable amount of time since the integrals over the
%% polygons $\bm{R}_j$ have to be solved numerically for each new set of parameters.


\subsection{Interaction functions}

<<imdfit_exponential, results="hide", eval=COMPUTE, include=FALSE>>=
imdfit_exponential <- update(imdfit_Gaussian, siaf = siaf.exponential())
@

<<imdfit_powerlaw, results="hide", eval=COMPUTE, include=FALSE>>=
imdfit_powerlaw <- update(imdfit_Gaussian, siaf = siaf.powerlaw(),
  data = imdepi_untied_infeps,
  start = c("e.(Intercept)" = -6.2, "e.siaf.1" = 1.5, "e.siaf.2" = 0.9))
@

<<imdfit_step4, results="hide", eval=COMPUTE, include=FALSE>>=
imdfit_step4 <- update(imdfit_Gaussian,
  siaf = siaf.step(exp(1:4 * log(100) / 5), maxRange = 100))
@

<<save-fits, include = FALSE, eval = COMPUTE, purl=FALSE>>=
save(imdfit_Gaussian, imdfit_exponential, imdfit_powerlaw, imdfit_step4,
     file = "twinstim-cache.RData", compress = "xz")
@

Figure~\ref{fig:imdfit_siafs} shows several estimated spatial interaction
functions, which can be plotted by, e.g.,
\code{plot(imdfit_Gaussian, "siaf")}.

<<imdfit_siafs, fig.cap="Various estimates of spatial interaction (scaled by the epidemic intercept $\\gamma_0$).", fig.pos="!ht", echo=FALSE>>=
par(mar = c(5,5,1,1))
set.seed(2)  # Monte-Carlo confidence intervals
plot(imdfit_Gaussian, "siaf", xlim=c(0,42), ylim=c(0,5e-5), lty=c(1,3),
     xlab = expression("Distance " * x * " from host [km]"))
plot(imdfit_exponential, "siaf", add=TRUE, col.estimate=5, lty = c(5,3))
plot(imdfit_powerlaw, "siaf", add=TRUE, col.estimate=4, lty=c(2,3))
plot(imdfit_step4, "siaf", add=TRUE, col.estimate=3, lty=c(4,3))
legend("topright", legend=c("Power law", "Exponential", "Gaussian", "Step (df=4)"),
       col=c(4,5,2,3), lty=c(2,5,1,4), lwd=3, bty="n")
@

The estimated standard deviation $\hat\sigma$ of the Gaussian kernel is:
<<>>=
exp(cbind("Estimate" = coef(imdfit_Gaussian)["e.siaf.1"],
          confint(imdfit_Gaussian, parm = "e.siaf.1")))
@

\citet{meyer.held2014} found that a power-law decay of spatial interaction
more appropriately describes the spread of human infectious diseases.
A power-law kernel concentrates on short-range interaction, but also exhibits a
heavier tail reflecting occasional transmission over large distances.
%This result is supported by the power-law distribution of short-time human
%travel \citep{brockmann.etal2006}, which is an important driver of epidemic spread.
To estimate the power law $f(x) = (x+\sigma)^{-d}$, we use the
prepared \code{eps.s = Inf} version of the \class{epidataCS} object,
and update the model as follows:
<<imdfit_powerlaw_echo, echo=TRUE, eval=FALSE, purl=FALSE>>=
<<imdfit_powerlaw>>
@
To reduce the runtime of this example, we specified convenient \code{start} values
for some parameters.
The estimated parameters $(\hat\sigma, \hat d)$ are:
<<>>=
exp(cbind("Estimate" = coef(imdfit_powerlaw)[c("e.siaf.1", "e.siaf.2")],
          confint(imdfit_powerlaw, parm = c("e.siaf.1", "e.siaf.2"))))
@
Sometimes $\sigma$ is difficult to estimate, and also in this example, its
confidence interval is relatively large.
The one-parameter version \code{siaf.powerlaw1} can be used to estimate a
power-law decay with fixed $\sigma = 1$.
A more common option is the exponential kernel $f(x) = \exp(-x/\sigma)$:
<<imdfit_exponential_echo, echo=TRUE, eval=FALSE, purl=FALSE>>=
<<imdfit_exponential>>
@

Table~\ref{tab:iafs} also lists the step function kernel as an alternative, which is
particularly useful for two reasons. First, it is a more flexible approach since it
estimates interaction between the given knots without assuming an overall functional form.
Second, the spatial integrals in the log-likelihood can be computed
analytically for the step function kernel, which therefore offers a quick estimate
of spatial interaction. We update the Gaussian model to use four
steps at log-equidistant knots up to an interaction range of 100 km:
<<imdfit_step4_echo, echo=TRUE, eval=FALSE, purl=FALSE>>=
<<imdfit_step4>>
@
Figure~\ref{fig:imdfit_siafs} suggests that the estimated step function is in
line with the power law.
Note that suitable knots for the step function could also be derived from
quantiles of the observed distances between events and their
potential source events, e.g.:
<<>>=
quantile(getSourceDists(imdepi_untied_infeps, "space"), c(1,2,4,8)/100)
@

For the temporal interaction function $g(t)$, model updates and plots are
similarly possible, e.g., using
\code{update(imdfit_Gaussian, tiaf = tiaf.exponential())}. However, the events
in the IMD data are too rare to infer the time-course of infectivity with
confidence.

<<echo=FALSE, results="asis", eval=FALSE, purl=FALSE>>=
local({
    nSources <- sapply(levels(imdepi$events$type), function (.type) {
        mean(summary(subset(imdepi_untied_infeps, type==.type))$nSources)
    })
    structure(
        paste("Specifically, there are only",
              paste0(round(nSources,1), " (", names(nSources), ")",
                     collapse=" and "),
              "cases on average within the preceding 30 days",
              "(potential sources of infection)."),
        class="Latex")
})
@


\subsection{Model selection}

<<imdfits_AIC>>=
AIC(imdfit_endemic, imdfit_Gaussian, imdfit_exponential, imdfit_powerlaw, imdfit_step4)
@

Akaike's Information Criterion (AIC)
suggests superiority of the power-law vs.\ the exponential, Gaussian, and
endemic-only models. The more flexible step function yields the best AIC value,
but its shape strongly depends on the chosen knots and is not guaranteed to be
monotonically decreasing.
The function \code{stepComponent} -- a wrapper around the \code{step} function
from \pkg{stats} -- can be used to perform AIC-based stepwise
selection within a given model component.
<<imdfit_endemic_sel, results="hide", include=FALSE>>=
## Example of AIC-based stepwise selection of the endemic model
imdfit_endemic_sel <- stepComponent(imdfit_endemic, component = "endemic")
## -> none of the endemic predictors is removed from the model
@


\subsection{Model diagnostics}

The element \code{"fittedComponents"} of a \class{twinstim} object contains
the endemic and epidemic values of the estimated intensity at each event
occurrence. However, plots of the conditional intensity (and
its components) as a function of location or time provide more insight into the
fitted process.
Evaluation of \code{intensity.twinstim} requires the model environment to be
stored with the fit. By default, \code{model = FALSE} in \code{twinstim},
but if the data are still available, the model environment can also be added
afterwards using the convenient \code{update} method:
<<imdfit_powerlaw_model>>=
imdfit_powerlaw <- update(imdfit_powerlaw, model = TRUE)
@

Figure~\ref{fig:imdfit_powerlaw_intensityplot_time} shows
an \code{intensityplot} of
the fitted ``ground'' intensity
$\sum_{k=1}^2 \int_{\bm{W}} \hat\lambda(\bm{s},t,k) \dif \bm{s}$:
%aggregated over both event types:
<<eval=FALSE, purl=FALSE>>=
intensityplot(imdfit_powerlaw, which = "total", types = 1:2)
@
<<imdfit_powerlaw_intensityplot_time, fig.cap="Fitted ``ground'' intensity process aggregated over space and both types.", fig.pos="ht", echo=FALSE>>=
par(mar = c(5,5,1,1), las = 1)
intensityplot(imdfit_powerlaw, "total intensity", tgrid=501, lwd=2,
              xlab="Time [days]", ylab="Intensity")
intensityplot(imdfit_powerlaw, "endemic intensity", tgrid=501, lwd=2,
              add=TRUE, col=2)
text(2500, 0.36, labels="total", col=1, pos=2, font=2)
text(2500, 0.08, labels="endemic", col=2, pos=2, font=2)
@

%% Note that this represents a realization of a stochastic process, since it
%% depends on the occurred events.
The estimated endemic intensity component has also been added to the plot.
It exhibits strong seasonality and a slow negative trend. The proportion
of the endemic intensity is rather constant along time since
no major outbreaks occurred.
This proportion can be visualized separately by specifying
\code{which = "endemic proportion"} in the above call.

<<echo=FALSE, eval=FALSE>>=
meanepiprop <- integrate(intensityplot(imdfit_powerlaw, which="epidemic proportion"),
                         50, 2450, subdivisions=2000, rel.tol=1e-3)$value / 2400
@

Spatial \code{intensityplot}s
as in Figure~\ref{fig:imdfit_powerlaw_intensityplot_space}
can be produced via \code{aggregate = "space"}
and require a geographic representation of \code{stgrid}.
The epidemic proportion is naturally high around clusters of cases and
even more so if the population density is low.
%% The function \code{epitest} offers a model-based global test for epidemicity,
%% while \code{knox} and \code{stKtest} implement related classical approaches
%% \citep{meyer.etal2016}.

<<imdfit_powerlaw_intensityplot_space, fig.cap="Epidemic proportion of the fitted intensity process accumulated over time by type.", fig.subcap=c("Type B.", "Type C."), fig.width=5, fig.height=5, out.width="0.47\\linewidth", fig.pos="p", echo=FALSE>>=
for (.type in 1:2) {
    print(intensityplot(imdfit_powerlaw, aggregate="space", which="epidemic proportion",
                        types=.type, tiles=districtsD, sgrid=1000,
                        # scales=list(draw=TRUE), # default (sp>=2 uses 'sf', with a fallback)
                        xlab="x [km]", ylab="y [km]", at=seq(0,1,by=0.1),
                        col.regions=rev(hcl.colors(10,"Reds"))))
}
@


Another diagnostic tool is the function \code{checkResidualProcess} (Figure~\ref{fig:imdfit_checkResidualProcess}), which transforms the temporal ``residual
process'' in such a way that it exhibits a uniform distribution and lacks serial
correlation if the fitted model describes the true CIF well
\citep[see][Section~3.3]{ogata1988}. % more recent work: \citet{clements.etal2011}

<<imdfit_checkResidualProcess, fig.cap="\\code{checkResidualProcess(imdfit\\_powerlaw)}. The left-hand plot shows the \\code{ecdf} of the transformed residuals with a 95\\% confidence band obtained by inverting the corresponding Kolmogorov-Smirnov test (no evidence for deviation from uniformity). The right-hand plot suggests absence of serial correlation.", results="hide", fig.pos="p", echo=FALSE>>=
par(mar = c(5, 5, 1, 1))
checkResidualProcess(imdfit_powerlaw)
@



\section{Simulation} \label{sec:twinstim:simulation}


%% Simulations from the fitted model are also useful to investigate the
%% goodness of fit.
To identify regions with unexpected IMD dynamics,
\citet{meyer.etal2012} compared the observed numbers of cases by district to the
respective 2.5\% and 97.5\% quantiles of 100 simulations from the selected model.
Furthermore, simulations allow us to investigate the stochastic volatility of the
endemic-epidemic process, to obtain probabilistic forecasts, and
to perform parametric bootstrap of the spatio-temporal point pattern.

The simulation algorithm we apply is described in \citet[Section 4]{meyer.etal2012}.
It requires a geographic representation of the \code{stgrid},
as well as functionality for sampling locations from the spatial kernel
$f_2(\bm{s}) := f(\norm{\bm{s}})$. This is implemented for all predefined
spatial interaction functions listed in Table~\ref{tab:iafs}.
%For instance for the
%power-law kernel, we pass via polar coordinates (with density then proportional
%to $rf(r)$) %, a function also involved in the efficient cubature of
%           %$f_2(\bm{s})$ via Green's theorem)
%and the inverse transformation method with numerical root finding for the
%quantiles.
Event marks are by default sampled from their respective empirical
distribution in the original data.
%but a customized generator can be supplied as argument \code{rmarks}.
The following code runs \emph{a single} simulation over the last year
based on the estimated power-law model:
<<imdsim, results="hide">>=
imdsim <- simulate(imdfit_powerlaw, nsim = 1, seed = 1, t0 = 2191, T = 2555,
  data = imdepi_untied_infeps, tiles = districtsD)
@
This yields an object of the class \class{simEpidataCS}, which extends
\class{epidataCS}. It carries additional components from the
generating model to enable an \code{R0}-method and
\code{intensityplot}s for simulated data.
%All methods for \class{epidataCS} are applicable.
%% The result is simplified in that only the \code{events} instead of a full
%% \class{epidataCS} object are retained from every run to save memory and
%% computation time. All other components, which do not vary between simulations,
%% e.g., the \code{stgrid}, are only stored from the first run.
%% There is a \code{[[}-method for such \class{simEpidataCSlist}s in order to
%% extract single simulations as full \class{simEpidataCS} objects from the
%% simplified structure.
%Extracting a single simulation (e.g., \code{imdsims[[1]]})
Figure~\ref{fig:imdsim_plot} shows the cumulative number of cases from the
simulation appended to the first six years of data.

<<imdsim_plot, fig.cap = "Simulation-based forecast of the cumulative number of cases by finetype in the last two years. The black lines correspond to the observed numbers.", fig.pos="bht", echo=FALSE>>=
.t0 <- imdsim$timeRange[1]
.cumoffset <- c(table(subset(imdepi, time < .t0)$events$type))
par(mar = c(5,5,1,1), las = 1)
plot(imdepi, ylim = c(0, 20), col = c("indianred", "darkblue"),
     subset = time < .t0, cumulative = list(maxat = 336),
     xlab = "Time [days]")
plot(imdsim, add = TRUE, legend.types = FALSE,
     col = adjustcolor(c("indianred", "darkblue"), alpha.f = 0.5),
     subset = !is.na(source),  # exclude events of the prehistory
     cumulative = list(offset = .cumoffset, maxat = 336, axis = FALSE),
     border = NA, density = 0) # no histogram for simulations
plot(imdepi, add = TRUE, legend.types = FALSE,
     col = 1, subset = time >= .t0,
     cumulative = list(offset = .cumoffset, maxat = 336, axis = FALSE),
     border = NA, density = 0) # no histogram for the last year's data
abline(v = .t0, lty = 2, lwd = 2)
@

%% Because we have started simulation at time \code{t0 = 0},
%% no events from \code{data} have been used as the prehistory, i.e.,
%% the first simulated event is necessarily driven by the endemic model component.
A special feature of such simulated epidemics is that the source of each event
is known:
<<strip.white.output=TRUE>>=
table(imdsim$events$source > 0, exclude = NULL)
@
The stored \code{source} value is 0 for endemic events, \code{NA} for events of
the prehistory but still infective at \code{t0},
and otherwise corresponds to the row index of the infective source.
%% Averaged over all 30 simulations, the proportion of events triggered by
%% previous events is
%% Sexpr{mean(sapply(imdsims$eventsList, function(x) mean(x$source > 0, na.rm = TRUE)))}.



%--------------
% BIBLIOGRAPHY
%--------------

<<Rbib, include = FALSE, purl = FALSE>>=
## create automatic references for R packages
knitr::write_bib( # produces UTF-8
    c("memoise", "sp", "polyclip",
      "xtable"),
    file = "twinstim-R.bib", tweak = FALSE, prefix = "R:")
@

% also using the installed bibliography (shared with the Rd system)
\bibliography{references,twinstim-R,\Sexpr{find.package("surveillance")}/REFERENCES}



\end{document}
