\section{Introduction}
\label{sec:Intro}

A criminal activity which has attracted little modeling attention in the statistics literature is that of automobile thefts.
Such data will consist of a set of theft locations, perhaps with associated covariate information for the theft site, e.g., demographic information and criminal activity information. There will also be an associated set of recovery locations for which covariate information is available.  However, recoveries are typically for only a small fraction of thefts so that the set of recovery locations is only a partial set of all of the potential recovery locations.

We are motivated by two real data settings. One consists of a collection of automobile thefts, with a fraction (roughly $10\%$) of recoveries, over the state of Neza in Mexico. The data is a total of 4,016 car theft locations (after deleting
some missing locations) during 2015, over both northern and southern parts of Neza. This dataset is small but is endowed with areal covariate information regarding population features and crime type that can be used for explanation in our modeling strategy. See Figures 1 and 2, and Section 2 for further description. A second dataset consists of car thefts which occurred in Belo Horizonte (Brazil).  It is a much larger dataset, but lacks covariates. This city is 331 km$^2$ in area and has approximately 2.4 million inhabitants. In the period from August, 1, 2000, to July, 31, 2001, the dataset consists of 5,250 \emph{pairs} of theft and recovery locations. See Figure 3 and Section 2 for a more complete description.

It is important to note some limitations of the available data. The low recovery rate for the Neza dataset is disappointing.  Issues such as what happened to the $90\%$ unrecovered vehicles, how different are they from the recovered vehicles (how representative of the total thefts are the $10\%$ we have observed), and what the local law enforcement might do to improve the recovery rate are evidently important and would enable us to enrich the story; unfortunately this information is not available.  
However, the data, as provided, reflects the reality of this type of crime data that police is reporting, and this is what we usually have to cope with. This expected poor nature of the data is a motivation for ad-hoc statistical modeling. Furthermore, the nature of the vehicles stolen - mark, condition, etc., would enable comparison of the subset of those recovered to the subset of those not recovered and might also make a promising story.  However, the only available covariates are the aggregated ones we consider in Section 2 below.  We have no individual vehicle data.
As for the Belo Horizonte data, in fact there were $6339$ thefts during the study window with $5257$ eventually found within the city limits.  So, there is a much higher recovery rate for this data than for the Neza data.  However, we only received the theft locations for the cars that were recovered.  It may be argued that there is potential bias in this subsample of thefts.  We cannot assess this but with nearly $85\%$ of the total thefts included, we can hope that the bias is small.  Another issue is that of a false report, e.g., the owner forgot where the vehicle was parked or the vehicle was borrowed by a friend or relative without informing the owner.  Again, intriguing inference might emerge but this information is another ``individual'' feature that is not supplied.

Acknowledging the foregoing limitations, the contribution here is to take the perspective of crime data analysts/investigators trying to better understand the behavior of car thefts for a specified region.   So, a first issue they might focus on would be to attempt to understand the \emph{point pattern} of car thefts.  They might seek a ``risk'' surface for theft.
In Section 3 below, we offer modeling to provide an intensity surface for the point pattern of thefts to clarify where risk is high, where it is low.
A second issue becomes one of attempting to predict recovery location given theft location.  Evidently, an effective predictive model would help local law enforcement in the process of vehicle recovery.  In Section 4 below, we offer modeling to provide such prediction.

A third issue connects us to Section 5.  We find ourselves in what has been referred to as spatial interaction/origin-destination modeling.  Such modeling is customarily proposed at areal scale.  That is, the study region is partitioned into municipal units, e.g., postcodes, census units, business districts, labor markets.  The observations consist of a pair of areal units, an origin unit and a destination unit. In addition, we would have potential regressors associated with each areal unit and a suitable distance between the units. The origin-destination modeling obtains $\{p_{ij}\}$, the matrix of origin-destination probabilities, e.g., the probability of living in unit $i$ and working in unit $j$ \citep{Chakrabortyetal(13)}, or the probability of a mail originating from unit $i$ and sent to unit $j$ \citep{Banerjeeetal(00)}.  Interest lies in \emph{flows}, the number of people who live in unit $i$ and work in unit $j$, e.g., $n_{i}p_{ij}$ where $n_{i}$ is the number of people living in unit $i$.


Our car theft setting differs in two ways.  First, the data is available at point level and can be viewed as a pair of point patterns.  Second, the recovery point pattern is typically only partially observed.  When a complete pair is observed, we have a geo-coded origin location and a geo-coded destination location; when recovery is missing, we have only a geo-coded origin.  Regardless, we can phrase analogous questions but with no need to aggregate to areal units in order to consider them.  Rather, we build a \emph{joint}  intensity of the form $\lambda(\bm{s}_{o}, \bm{s}_{d})$ over pairs of locations $(\bm{s}_{o}, \bm{s}_{d}) \in D_{o} \times D_{d}$ where $\bm{s}_{o}$ is a theft location over a region $D_{o}$ and $\bm{s}_{d}$ is a recovery location over a region $D_{d}$.  Useful insight can be gleaned from the marginal intensity surface for both theft locations and recovery locations.  To attempt to understand the flow of vehicles from theft location to recovery location, suppose a neighborhood is created, say $B_{o} \subset D_{o}$, as an theft neighborhood and, say $B_{d} \subset D_{d}$, a recovery neighborhood.  Then, we can ask for the predictive distribution of the number of thefts in $B_{o}$ with recovery in $B_{d}$.
Alternatively, we could convert to proportions, e.g., obtaining the proportion of recoveries in $B_{d}$ associated with a theft in $B_{o}$.

Examination of modeling of spatial interaction, also referred to as gravity modeling, has a long history in the literature. \cite{Wilson(75)} provides an early review.
\cite{Fotheringham(83)} presents a more formal discussion. More recent reviews can be found in \cite{RoyThill(03)} and in \cite{LeSagePace(08)}.
Spatial interaction data have become increasingly available due to the wide adoption of location-aware technologies \citep{Guoetal(12)}.
Examination of mobility data also has some history, e.g., \cite{BrownHolmes(71)}, \cite{Simpson(92)} and more recently, \cite{deVriesetal(09)}.
Origin-destination problems involving mobility can be found in, e.g., \cite{Woodetal(10)}; \cite{AdrienkoAdrienko(11)}; \cite{Guoetal(12)}.

For us, mobility refers to the movement of a vehicle from a theft location to a recovery location.
Pertinent to our setting is work of \cite{AssuncaoLopes(07)} and \cite{LopesAssuncao(12)}.  Particularly, the former builds a bivariate linked point process with a joint pairwise interaction function.  Our view is that the theft locations should be viewed as conditionally independent given the intensity function so that either a nonhomogeneous Poisson process or a log-Gaussian Cox process model applies.  The flexibility of the log-Gaussian Cox process  along with the availability of covariates, as with the Neza data, will make it impossible for say a pairwise interaction model to outperform it. A point pattern model incorporating interactions between the points would seem to need some mechanistic motivation.


The connection between origin-destination problems and spatial point processes has been little treated in the literature. \cite{Benesetal(05)} consider statistical analysis of linked point processes, where, in their study, for each case of a disease they have the coordinates of the individual's home and of the reported infection location. However, they used only the distance between the two linked locations.
Again, \cite{AssuncaoLopes(07)} and \cite{LopesAssuncao(12)} consider bivariate linked point processes as point processes with events marked with another spatial event representing origin-destination data types. Their methods are illustrated with the Belo Horizonte data on car theft locations and the eventual car retrieval locations; this data is also analyzed here.

As noted above, three types of issues are considered with regard to automobile theft data and we devote a section below to each. First, the set of theft locations is modeled by using both a nonhomogeneous Poisson process as well as a log-Gaussian Cox process. We demonstrate the benefit of the latter specification. Second, a conditional specification is proposed to provide the distribution of recovery location given theft location.
Third, we consider a joint model, viewing the data as an origin-destination pair, and treating the point pattern as consisting of random pairs of locations.  Because both origin and destination are points in $\mathbb{R}^{2}$, we propose to specify the model as a point pattern over a bounded set $D_{o} \times D_{d}\subset \mathbb{R}^{2}\times \mathbb{R}^{2}$. This approach needs an intensity over $D_{o} \times D_{d}$ linking pairs of locations. In this regard, the modeling can be cast in the context of a marked point pattern.  That is, the recovery location can be viewed as a mark associated with the theft location where the mark lies in a subset of $\mathbb{R}^{2}$. Furthermore, these marks might depend on theft locations themselves, i.e., location dependent marks. A statistical challenge for this specification is to introduce spatial dependence into modeling of marks (recovery locations) defined on $D_{d}$.
We propose a reasonable modeling in Section \ref{sec:Joint}.

The plan of the paper is the following. Section 2 provides a description of the two datasets that motivate this paper. Section 3 presents the statistical approach that models the set of theft locations using both a nonhomogeneous Poisson process as well as a log-Gaussian Cox process. Then Section 4 considers the conditional specification approach that provides the distribution of recovery locations given theft locations. Section 5
supplies the joint modeling approach, viewing the data as an origin-destination pair, and treating the point pattern as consisting of random pairs of locations.  The paper ends with a summary and future work.



\section{Data Description}
\label{sec:Data}

We analyze two datasets consisting of a collection of automobile thefts and recoveries, one for the state of Neza in Mexico, the other in Belo Horizonte (Brazil).
The supplied longitude and latitude information is transformed to eastings and northings on meter scale in Figures 1-3 below.
In the analysis, we transformed from meter scale to kilometer scale.
There may be measurement error in the locations but this is beyond the scope of the data we have to work with.

\subsection{The Neza data}
\label{sec:Neza}

The Ciudad Neza (referred to as Neza in what follows) is a city and municipality adjacent to the northeast corner of Mexico's Federal district.
It is part of the Mexico City metropolitan area.
The region is composed of the North and South parts separated by a single road.
On the east side of this road there is a large park, and on the west side an airport.
In the analysis below, these two regions are separated.
Our dataset contains car theft locations in 2015.
The number of car theft locations is 4,016 after deleting some missing locations.

We also have several areal unit covariates split into two categories.
The first category consists of population types: (1) \texttt{Pop15} - number of individuals 15 years and older, (2) \texttt{Apart} - number of apartments, (3) \texttt{Eco} - number of economically active individuals, (4) \texttt{Employ} - number of employed individuals, hence \texttt{unEmploy} - number of unemployed individuals, (5) \texttt{inBorn} - number of individuals born in the area, hence \texttt{outBorn} - number of individuals born outside the area, (6) \texttt{Health} - number of individuals with health insurance access, hence \texttt{noHealth} - number of individuals without health insurance access and (7) \texttt{Scholar} average of scholarly grade (integer values from $6- 10$).

The second category consists of crime types: (1) \texttt{Extor} - number of extortion crimes, (2) \texttt{Murder} - number of murders, (3) \texttt{Burg} - number of burglaries, (4) \texttt{Shop} - number of shop robberies, (5) \texttt{Public} - number of public transport robberies, (6) \texttt{Street} - number of street robberies, (7) \texttt{Kidnap} - number of kidnappings and (8) \texttt{Total} - total number of infractions (some additional crimes beyond (1) through (7) are included here).
These covariates are provided for 90 disjoint blocks in Neza.
Figure \ref{fig:Map} shows the theft locations for the North and South regions.
22 blocks are located in the North region with the remaining 68 blocks in the South region.  They are indicated in white in the figure.  Of the thefts,
3,327 points (689 points) are observed in the South (North) region.  These locations seem to be spread smoothly over the each region rather than suggesting concentration in ``hot spots.''

\begin{figure}[htbp]
 \begin{minipage}{0.48\hsize}
  \begin{center}
   \includegraphics[width=7.5cm]{PlotNorth.pdf}
  \end{center}
 \end{minipage}
 %
 \hfill
 %
 \begin{minipage}{0.48\hsize}
  \begin{center}
   \includegraphics[width=7.5cm]{PlotSouth.pdf}
  \end{center}
 \end{minipage}
  \caption{Car theft locations in the North region (left) and the South region (right) in Neza ($x$-axis (easting) and $y$-axis (northing) are at meter scale).}
  \label{fig:Map}
\end{figure}


Unfortunately, the recovery locations are observed for only 382 theft locations. 56 of these locations are outside the Neza region.
This is a commonly encountered situation in the context of car theft and recoveries. The set of recovery locations is often quite relative small compared to the set of theft locations, and police have to cope with this. In addition, we are additionally facing a problem of incompleteness of the data. This is typically the reality in car theft data, and what we show in this paper is modeling strategies within these limitations. Note also that the Mexican police are structured so that they also have authority over their city but information from other police departments is sometimes not accessible. This means that the Neza police have control only over the recoveries within the Neza region. This is the reason why we only have data within Neza.  
Figure \ref{fig:Pairs} shows the plot of the recovery locations for observed theft and recovery pairs as well as a histogram of the distance between theft and recovery locations. Fortunately, by looking at Figure \ref{fig:Pairs}  the recoveries tend to be close to the origin or theft location. Thus there should not be a large number of recoveries (or missing recoveries) far from the city of Neza. And this closeness between recovery location and theft location is what motivates our ensuing modeling strategies. In addition, we have no individual vehicle data. However, the police published an internal report in which they described the cars most often stolen in Neza during the period 2013-2016. The list basically included pick-ups. The police confirmed that the black market was not the main aim behind a car theft; perhaps the theft was just needed to move (stolen) goods from one place to another. This could explain why the recoveries within the city tend to be close to the theft locations.

\begin{figure}[htbp]
 \begin{minipage}{0.48\hsize}
  \begin{center}
   \includegraphics[width=7.5cm]{RecoveredNew.pdf}
  \end{center}
 \end{minipage}
 %
 \hfill
 %
 \begin{minipage}{0.48\hsize}
  \begin{center}
   \includegraphics[width=7.5cm]{HistPairsNew.pdf}
  \end{center}
 \end{minipage}
  \caption{Recovery locations (left, $x$-axis (easting) and $y$-axis (northing) are at meter scale) and histogram of the distance between theft and recovery locations (right, $x$-axis (distance) also at meter scale).}
  \label{fig:Pairs}
\end{figure}

\subsection{The Belo Horizonte data}
\label{sec:Belo}

We also examine car theft and recovery point patterns in Belo Horizonte in Brazil (\cite{AssuncaoLopes(07)}).
The dataset contains $6339$ thefts during the study window with $5257$ eventually found within the city limits. So, there is a much higher recovery rate for this data than for the Neza data. However, we only received the theft locations for the cars that were recovered. This dataset does not have any covariate information. It may be argued that there is potential bias in this subsample of thefts. This cannot be assessed but with nearly $85\%$ of the total thefts included, we hope that the bias is small.
The left panel of Figure \ref{fig:MapBH} shows the point patterns of theft and recovery locations.
The point patterns are similar, though recovery points seemed to be a bit more concentrated. The right panel provides the histogram of the distance between theft and recovery locations.  Again, recovery location tends to be near theft location; in fact, 770 pairs (roughly $15\%$) are observed to be within 200m of each other.

\begin{figure}[htbp]
 \begin{minipage}{0.53\hsize}
  \begin{center}
   \includegraphics[width=8cm]{PlotBH.pdf}
  \end{center}
 \end{minipage}
 %
 \hfill
 %
 \begin{minipage}{0.43\hsize}
  \begin{center}
   \includegraphics[width=7cm]{HistBH.pdf}
  \end{center}
 \end{minipage}
  \caption{Car theft and recovery locations (left, $x$-axis (easting) and $y$-axis (northing) are at meter scale) and histogram of the distance between theft and recovery locations (right, $x$-axis (distance) also at meter scale) in Belo Horizonte}
  \label{fig:MapBH}
\end{figure}

\section{Modeling of car thefts}
\label{sec:Model}
\subsection{LGCP and NHPP models for vehicle theft}
\label{sec:LGCPNHPP}

Here we turn to the first issue raised in the Introduction.  Viewing the collection of car thefts as a random point pattern, can a satisfying explanatory model be developed?  We seek to provide an investigator with understanding of the nature of the intensity surface that is driving the point pattern of thefts.  This surface can be viewed as a risk surface for theft enabling clarification of where risk is high, where it is low.

We consider the vehicle theft events in Neza with available covariate information.
Let $\mathcal{S}=\{\bm{s}_{1},\ldots,\bm{s}_{n}\}$ denote the observed point pattern over the study region $D\subset \mathbb{R}^{2}$.  In our case, $\mathcal{S}$ is the set of car theft locations and $D$ is the North or the South region.
We view the theft events as conditionally independent given the intensity and therefore consider a non-homogeneous Poisson process (NHPP) and a log-Gaussian Cox processes (LGCP, \cite{Molleretal(98)}) for modeling theft events.

The LGCP is defined so that the log of the intensity is a Gaussian process (GP), i.e.,
\begin{align}
\log \lambda(\bm{s})=\bm{X}(\bm{s})\bm{\beta}+z(\bm{s}), \quad  \bm{z}(\mathcal{S})\sim \mathcal{N}(\bm{0}, \mathbf{C}_{\bm{z}}), \quad \bm{s}\in D.
\end{align}
where $\bm{X}(\bm{s})$ is a covariate vector at $\bm{s}$ and $z(\bm{s})$ is a Gaussian process.
In particular, the point pattern $\mathcal{S}$ has associated vector $\bm{z}(\mathcal{S})=(z(\bm{s}_{1}), \ldots, z(\bm{s}_{n}))$ which follows an $n$-variate zero mean Gaussian distribution, with covariance matrix $\mathbf{C}_{\bm{z}}=[C(\bm{s}_{i}, \bm{s}_{j})]_{i,j=1,\ldots, n}$.  The component spatial random effects for the intensity surface provide pushing up and pulling down the surface, as appropriate. We assume an exponential covariance function, i.e., $C(\bm{u}, \bm{u}^{'})=\sigma^2 \exp(-\phi \|\bm{u}-\bm{u}^{'}\|)$\footnote{Since we never see observations of the intensity, it is hard to justify or identify a richer covariance function for the spatial random effects.}.

If $\bm{z}(\bm{s})$ is removed from the log intensity, the corresponding NHPP is obtained. NHPP's have a long history in the literature (see, e.g., \cite{Illianetal(08)}).
Furthermore, given $\lambda(\bm{s})$ with $z(\bm{s})$ included, $\mathcal{S}$ again, follows an NHPP with intensity $\lambda(\bm{s})$.
The likelihood takes the form
\begin{align}
\mathcal{L}(\mathcal{S})&\propto \exp\biggl(-\int_{D}\lambda(\bm{u})d\bm{u} \biggl)\prod_{i=1}^{n}\lambda(\bm{s}_{i})
\end{align}

For inference with a LGCP using (2), the stochastic integral inside the exponential need to be approximated. We create $K$ grid cells roughly uniformly over the study region $D$; convergence to the exact posterior distribution when $K \to \infty$ (with grid cell area decreasing to $0$) is guaranteed following \cite{Waagepetersen(04)}.    Then, the approximate likelihood for the LGCP becomes
\begin{align}
\mathcal{L}(\mathcal{S})\propto \exp\biggl(-\sum_{k=1}^{K}\lambda(\bm{u}_{k})\Delta_{k} \biggl)\prod_{k=1}^{K}\lambda(\bm{u}_{k})^{n_{k}}
\end{align}
where $n_{k}$ is the number of points in $k$-th grid, i.e., $\sum_{k}^{K}n_{k}=n$, $\Delta_{k}$ is the area of $k$-th grid (in practice, we standardize $\Delta_{k}$ so that $\sum_{k}^{K}\Delta_{k}=|D|=1$) and $u_{k}$ is the ``representative point" for $k$-th grid (e.g., \cite{MollerWaagepetersen(04)} and \cite{BanerjeeCarlinGelfand(14)}).  In fact, since covariate values for 90 different areal units are available, this discretization is adopted.  In order to implement full inference we work within a Bayesian framework, fitting the model using Markov chain Monte Carlo (see, e.g., \cite{RobertCasella(04)}).   Other model fitting approaches are available, helpful for large point patterns (high dimensional grids).  They include integrated nested Laplace approximation (INLA) \citep{Simpsonetal(16b)}, approximate Gaussian process models, e.g, nearest neighbor Gaussian processes \cite{Dattaetal(16a)} and multi-resolution Gaussian processes \cite{Katzfuss(17)}.

\subsection{Covariate selection}
\label{sec:Covariate}

The spatstat R-package (\cite{BaddeleyTurner(05)}; \cite{Baddeleyetal(13)}) supports the model fitting of spatial point processes, in particular Poisson processes, and related inference and diagnostic tools.
The function \texttt{ppm} fits a spatial point process to an observed point pattern and allows the inclusion of covariates.
Two estimation methods, pseudo-maximum likelihood (\cite{BaddeleyTurner(00)}) and approximate maximum likelihood (\cite{HuangOgata(99)}), are implemented.
Working with the 90 (22 in North and 68 in South) blocks, covariates from those listed in Section 2.1 are chosen by forward and backward selection (\texttt{step} function in R) based on the models fitted by the \texttt{ppm} function\footnote{Using just forward or backward selection produced the same selection}.

We implement model fitting with the NHPP and the LGCP for theft locations in the two separate regions in Neza. Again,  $K_{N}=22$ and $K_{S}=68$ are the number of grid cells for the North and South regions, respectively.
We rescale the northings and eastings dividing by 1,000 and present estimation results at this scale.

Working with the NHPP model, the forward and backward algorithm is implemented (\texttt{step} function) with BIC penalty ($\log(n)$) for each region.  What emerged is $k=1,\ldots, K_{S}$ $(K_{N})$:
\begin{align}
\bm{X}_{S,k}\bm{\beta}&=\beta_{0}+\beta_{1}\texttt{Extor}_{k}+\beta_{2}\texttt{Shop}_{k}  +\beta_{3}\texttt{Street}_{k}+\beta_{4}\texttt{Total}_{k} \nonumber \\
&+\beta_{5}\texttt{Eco}_{k}+\beta_{6}\texttt{Scholar}_{k} \\
\bm{X}_{N,k}\bm{\beta}&=\beta_{0}+\beta_{2}\texttt{Shop}_{k}+\beta_{4}\texttt{Total}_{k}+\beta_{7}\texttt{Apart}_{k}
\end{align}
All covariates are centered and scaled. The same covariate vectors are used for the LGCP model.

Turning to model fitting, Markov chain Monte Carlo is implemented.  For sampling of $\bm{\beta}$ in the LGCP and NHPP, an adaptive random walk MH algorithm (\cite{AndrieuThoms(08)}) is implemented.
Elliptical slice sampling is implemented for the GP in the LGCP (\cite{MurrayAdams(10)}, \cite{MurrayAdamsMacKay(10)}, \cite{Leininger(14)}).
20,000 samples are discarded as burn-in period and a subsequent 20,000 samples are preserved as posterior samples for the LGCP and the NHPP, respectively.
Since, for spatial Gaussian processes, $\phi$ and $\sigma^2$ are not identifiable but the product, $\phi \sigma^2$ is  \citep{Zhang(04)}, an informative prior distribution needs to be adopted for one of them. Here, we assume informative support for $\phi$ and adopt an inverse Gamma distribution for $\sigma^2$ with relatively large variance.  As specific prior settings, we assume $\sigma^2\sim \mathcal{IG}(2,0.1)$ (inverse gamma), $\bm{\beta}\sim \mathcal{N}(\bm{0}, 100\mathbf{I})$ (normal) and $\phi \sim \mathcal{U}[0, 10]$ (uniform), where, after rescaling, the easting and northing, distances are in kilometers.

When the models were fitted, for the South region all coefficients were significant, i.e., a $95\%$ credible interval (CI) doesn't include 0, for the NHPP while $\beta_{5}$ was insignificant under the LGCP. For the North region, again all coefficients were significant for the NHPP while $\beta_{7}$ was insignificant under the LGCP.
The details are omitted but we note that the total number of infractions has a large positive (increasing) effect on theft events for both the North and South regions.
For the South region, the posterior log likelihood for the NHPP can be summarized (posterior mean, $95\%$ credible interval) as $-294.3 \hspace{.2cm} (-298.6, -291.7)$ while that for the LGCP can be summarized as $-228.2 \hspace{.2cm} (-242.4, -216.7)$.   For the North region, the posterior log likelihood for the NHPP can be summarized as $-81.98 \hspace{.2cm} (-.85.44, -80.28)$ while that for the LGCP can be summarized as $-68.47 \hspace{.2cm} (-75.46, -62.68)$.  Since larger likelihood is desired, the LGCP emerges as preferred for both regions.  We provide further support for the LGCP through cross-validation in the next subsection.

\subsection{$p$-thinning cross validation}
\label{sec:CV}
Cross validation is a standard approach for assessing model adequacy and is available for point pattern models with conditionally independent locations given the intensity, hence, for both the NHPP and LGCP (see, \cite{LeiningerGelfand(17)}).

Cross validation is implemented by obtaining a training (fitting) dataset and a testing (validation) dataset using $p$-thinning as proposed by \cite{LeiningerGelfand(17)}.
Let $p$ denote the retention probability, i.e., we delete $\bm{s}_{i}\in \mathcal{S}$ with probability $1-p$.
This produces a training point pattern $\mathcal{S}^{train}$ and a test point pattern $\mathcal{S}^{test}$, which are independent, conditional on $\lambda(\bm{s})$.
In particular, $\mathcal{S}^{train}$ has intensity $\lambda(\bm{s})^{train}=p \lambda(\bm{s})$. We set $p=0.5$ and estimate $\lambda(\bm{s})^{train}$ $\bm{s}\in D$.  Then, the posterior draws of $\lambda^{train}(\bm{s})$ are converted into predictive draws of $\lambda^{test}(\bm{s})$ using $\lambda^{test}(\bm{s})=\frac{1-p}{p}\lambda^{train}(\bm{s})= \lambda^{train}(\bm{s})$.

Let $\{B_{r}\}$ be a collection of subsets of $D$ as a evaluation grid.
For the choice of  $\{B_{r}\}$, \cite{LeiningerGelfand(17)} suggest to draw random subsets of the same size uniformly over $D$.  Specifically, for $q \in (0,1)$, if the area of  each $B_{r}$ is $q|D|$, then $q$ is the \emph{relative} size of each $B_{r}$.   They argue that making the subsets disjoint is time consuming and unnecessary.
Based on the $p$-thinning cross validation, two model performance criteria are considered: (i) predictive interval coverage (PIC) and  (ii) rank probability score (RPS). PIC offers assessment of model adequacy, RPS enables model comparison.
\\\\
\noindent{\bf Predictive Interval Coverage}\\

After the model is fitted to $\mathcal{S}^{train}$, the posterior predictive intensity function can supply posterior predictive point patterns and therefore samples from the posterior predictive distribution of $N(B_{r})$ for each $r$. For the $\ell$-th posterior sample, $\ell = 1,...., L$, the associated predictive residual is defined as
\begin{align}
R_{\ell}^{pred}(B_{r})=N^{test}(B_{r})-N^{(\ell)}(B_{r})
\end{align}
where $N^{test}(B_{r})$ is the number of points of the test data in $B_{r}$.
If the model is adequate, the empirical predictive interval coverage rate, i.e., the proportion of intervals which contain $0$, is expected to be roughly the nominal level of coverage; below, $90\%$ nominal coverage is chosen. Empirical coverage much less than the nominal suggests model inadequacy; predictive intervals are too optimistic.  Empirical coverage much above, for example $100\%$, is also undesirable.  It suggests that the model is introducing more uncertainty than needed.
\\\\

\noindent{\bf Rank Probability Score}\\

\cite{GneitingRaftery(07)} propose the continuous rank probability score (CRPS).
This score is derived as a proper scoring rule and enables a criterion  for assessing the precision of a predictive distribution for continuous variables.
In our context, we seek to compare a predictive distribution to an observed count.
\cite{Czadoetal(09)} and references therein discuss rank probability scores (RPS) for count data.
Intuitively, a good model will provide a predictive distribution that is very concentrated around the observed count.
While the RPS has a challenging formal computational form, it is directly amenable to Monte Carlo integration.  In particular, for a given $B_{r}$, the RPS is calculated as
\begin{align}
\text{RPS}(F, N^{test}(B_{r}))&=\frac{1}{L}\sum_{\ell=1}^{L}|N^{(\ell)}(B_{r})-N^{test}(B_{r})| \nonumber \\
&-\frac{1}{2L^2}\sum_{\ell=1}^{L}\sum_{\ell^{'}=1}^{L}|N^{(\ell)}(B_{r})-N^{(\ell^{'})}(B_{r})|
\end{align}
Summing over the collection of $B_{r}$ gives a model comparison criterion.  Smaller values of the sum are preferred.










The results of model validation are presented for the Neza data using predictive interval coverage and ranked probability score.
We set $p=0.5$ for dividing into training and test datasets.
Figure \ref{fig:MVTheft} shows the PIC with 90$\%$ nominal level and the RPS for both regions.
Here, $w$ denotes the number of randomly selected blocks for model comparison.
As for the choice of ${B_{r}}$, since the total number of grid cells for this dataset is small, here we choose $w=1,\ldots, 10$ grids from the 22 grids in the North and 68 grids in the South, rather than choosing ${B_{r}}$ with respect to a rate $q$.
Again, the LGCP outperforms the NHPP, more so for the South, the larger dataset.

\begin{figure}[htbp]
 \begin{minipage}{0.48\hsize}
  \begin{center}
   \includegraphics[width=7cm]{PICnorth.pdf}
  \end{center}
 \end{minipage}
 %
 \hfill
 %
 \begin{minipage}{0.48\hsize}
  \begin{center}
   \includegraphics[width=7cm]{PICsouth.pdf}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.48\hsize}
  \begin{center}
   \includegraphics[width=7cm]{RPSnorth.pdf}
  \end{center}
 \end{minipage}
 %
 \hfill
 %
 \begin{minipage}{0.48\hsize}
  \begin{center}
   \includegraphics[width=7cm]{RPSsouth.pdf}
  \end{center}
 \end{minipage}
  \caption{PIC (top) with $90\%$ nominal level (dashed line) and RPS (bottom) for the North (left) and South (right) regions: NHPP ($\bullet$) and LGCP ($\blacktriangle$). $w$ is the number of randomly selected blocks for model comparison.}
  \label{fig:MVTheft}
\end{figure}



Finally, Figures \ref{fig:LGCPsouth} and \ref{fig:LGCPnorth} display the results of nominally $50\%$ held out counts, comparing the observed with the posterior predictive intensity surface estimated by using the retained counts for the South and the North regions, respectively.
Altogether, the posterior predictive intensity surfaces well explain the distribution of held out counts for both regions.

\begin{figure}[htbp]
 \begin{minipage}{0.48\hsize}
  \begin{center}
   \includegraphics[width=8cm]{Neza_South_Count.pdf}
  \end{center}
 \end{minipage}
 %
 \hfill
 %
 \begin{minipage}{0.48\hsize}
  \begin{center}
   \includegraphics[width=8cm]{Neza_South_Int.pdf}
  \end{center}
 \end{minipage}
  \caption{Held out counts (left) and posterior predictive intensity surface (right) in the South region.}
  \label{fig:LGCPsouth}
\end{figure}

\begin{figure}[htbp]
 \begin{minipage}{0.48\hsize}
  \begin{center}
   \includegraphics[width=8cm]{Neza_North_Count.pdf}
  \end{center}
 \end{minipage}
 %
 \hfill
 %
 \begin{minipage}{0.48\hsize}
  \begin{center}
   \includegraphics[width=8cm]{Neza_North_Int.pdf}
  \end{center}
 \end{minipage}
  \caption{Held out counts (left) and posterior predictive intensity surface (right) in the North region.}
  \label{fig:LGCPnorth}
\end{figure}

\section{Conditioning recovery location on theft location}
\label{sec:Conditioning}

We turn to a second issue with regard to vehicle theft. How can we predict recovery location given theft location?  Evidently, an effective predictive model would help local law enforcement in the process of vehicle recovery.  For the analysis of recovery locations, a conditional density specification given theft location is considered.
We do not have to specify a set in which our recovery locations are considered; we can include some recovery points located outside the Neza region.
Also, we do not have to split the Neza region for this analysis.
Furthermore, we can allow the theft location to determine not only the mean for the recovery location but also the uncertainty in the recovery location.


We denote by $\bm{s}_{R}$ a recovery location and by $\bm{s}_{T}$ a theft location with $\mathcal{S}_{T}=\{\bm{s}_{T,1},\ldots, \bm{s}_{T,n}\}$ and $\mathcal{S}_{R}=\{\bm{s}_{R,1},\ldots, \bm{s}_{R,m}\}$ where $m<n$.
We denote the conditional density specification for recovery location $\bm{s}_{R}$ given a theft location $\bm{s}_{T}$ as $f_{R}(\bm{s}_{R}|\bm{s}_{T})$.

Combined with the marginal point pattern model for theft locations in the previous section, a joint model is created for theft location and recovery location. In this way, we can employ all of the theft data and all of the available recovery data.  This model would be a \emph{partially} marked point pattern in the sense that when we have an associated recovery location, it becomes the mark for that location while all of the theft locations without a recovery location have a missing mark. We would then be modeling marks given locations rather than locations given marks.
However, it is not a \emph{joint} specification in the sense of viewing the data as a point pattern of pairs of locations over a bounded set in $\mathbb{R}^2 \times \mathbb{R}^2$.  This model is deferred to the next section.

Let $\mathcal{S}_{T}^{*}=\{\bm{s}_{T,1}^{*},\ldots, \bm{s}_{T,m}^{*}\}$ be the theft locations corresponding to recovery points, i.e., $\bm{s}_{T,j}^{*}$ is the corresponding theft location for the recovery point $\bm{s}_{R,j}$ for $j=1,\ldots, m$. For $j=1,\ldots, m$,
\begin{align}
f_{R}(\bm{s}_{R,j}|\bm{s}_{T,j}^{*})\propto |\Sigma(\bm{s}_{T,j}^{*})|^{-1/2}\exp\biggl(-(\bm{s}_{R,j}-\bm{s}_{T,j}^{*})'\Sigma(\bm{s}_{T,j}^{*})^{-1}(\bm{s}_{R,j}-\bm{s}_{T,j}^{*})\biggl), \label{eq:CD}
\end{align}
$\Sigma(\bm{s}_{T,j}^{*})$ is $2\times 2$ covariance kernel dependent on theft location $\bm{s}_{T,j}^{*}$.
A benchmark specification would assume a constant covariance kernel across theft locations, i.e.,
\begin{align}
\Sigma(\bm{s}_{T,j}^{*})=\Sigma=\begin{pmatrix}
           \sigma_{1}^2 & \rho \sigma_{1}\sigma_{2} \\
           \rho \sigma_{1}\sigma_{2} & \sigma_{2}^2
           \end{pmatrix}
\end{align}
A locally adaptive covariance kernel can be also considered, for example, employing the spatially varying covariance kernel in \cite{Higdonetal(99)},
\begin{align}
\Sigma(\bm{s}_{T})^{\frac{1}{2}}=&\sigma \begin{pmatrix}
                                      \biggl(\frac{\sqrt{4A^{2}+\|\psi(\bm{s}_{T}) \|^{4} \pi^{2}}}{2\pi}+\frac{\|\psi(\bm{s}_{T}) \|^2}{2} \biggl)^{\frac{1}{2}} & 0 \\
                                      0  & \biggl(\frac{\sqrt{4A^{2}+\|\psi(\bm{s}_{T}) \|^{4} \pi^{2}}}{2\pi}-\frac{\|\psi(\bm{s}_{T}) \|^2}{2} \biggl)^{\frac{1}{2}}
                                      \end{pmatrix} \nonumber \\
                                      &\times \begin{pmatrix}
                                      \cos(\alpha(\bm{s}_{T})) & \sin(\alpha(\bm{s}_{T})) \\
                                      -\sin(\alpha(\bm{s}_{T})) & \cos(\alpha(\bm{s}_{T}))
                                      \end{pmatrix}
\end{align}
where $\|\psi(\bm{s}_{T}) \|^2=\psi_{x}(\bm{s}_{T})^2+\psi_{y}(\bm{s}_{T})^2$, $\alpha(\bm{s}_{T})=\tan^{-1}\psi_{y}(\bm{s}_{T})/\psi_{x}(\bm{s}_{T})$ and $A=3.5$ as fixed in \cite{Higdonetal(99)}.
$\psi_{x}(\bm{s})$ and $\psi_{y}(\bm{s})$ are independent Gaussian processes with mean 0 and common Gaussian covariance function $C(\bm{s}_{T,i}, \bm{s}_{T,j})=\exp(-\phi^{*} \|\bm{s}_{T,i}-\bm{s}_{T,j}\|^2)$.
They introduce spatial dependence in $\Sigma(\bm{s}_{T})$.
$\phi^{*}$ is a tuning parameter which determines the spatial decay of the Gaussian processes. We fix this parameter at several different values in the ensuing analysis.

As a last remark here, there is no evident way to introduce spatial covariates such as those noted in the previous section into the conditional model.  The mean for the recovery location should be the theft location; a regression specification here is not sensible.  Furthermore, with the flexibility of a location dependent covariance matrix to accommodate direction and dispersion, more flexibility could not be gained by attempting to insert covariate information associated with $\bm{s}_{T}$ into $\Sigma(\bm{s}_{T})$.

\subsection{Results}
\label{sec:ResultCondtional}

For recovery locations, we implement conditional density specification with constant and spatially varying covariance kernels for both datasets.
For the constant covariance kernel parameters, we assume $\sigma_{1}^2, \sigma_{2}^2\sim \mathcal{IG}(2,0.1)$ and $\rho \in \mathcal{U}[-1,1]$.
The first 10,000 samples are discarded and the subsequent 10,000 samples are retained as posterior samples.
For the spatially varying covariance kernel parameter, we assume $\sigma^2\sim \mathcal{IG}(2,0.1)$.
For this model, three fixed $\phi^{*}$ values are considered: (i) $\phi^{*}=30$, (ii) $\phi^{*}=10$ and (iii) $\phi^{*}=1$.
The first 20,000 samples are discarded and the subsequent 20,000 samples are retained as posterior samples.

Computation for the Neza dataset is manageable.  However, the number of theft locations in Belo Horizonte is a bit large (in terms of matrix inversion and determinant calculation) to sample the Gaussian processes at all $\bm{s}_{T}$.
So, the study region is approximated by using 305 disjoint regular grid cells. $\Sigma(\bm{s}_{T})$ is evaluated at the nearest grid centroid.
For sampling the Gaussian processes $\psi_{x}(\bm{s})$ and $\psi_{y}(\bm{s})$, elliptical slice sampling is implemented.
The estimation results are given in Table \ref{tab:RealCond}.
The spatially varying covariance model fits better than the constant covariance kernel model with respect to the loglikelihood, preferring the larger values of $\phi^{*}$ (weaker spatial dependence in the $\Sigma(\cdot)$'s) for the Neza, less so for Belo Horizonte.


\begin{table}[htbp]
\caption{Estimation results for the conditional model specifications}
\small
\begin{tabular}{lcccccccccc}
\hline
\hline
  &  \multicolumn{4}{c}{Neza} & & \multicolumn{4}{c}{Belo Horizonte}   \\
\hline
   & Mean & Stdev &  $95\%$ Int & IF & & Mean & Stdev &  $95\%$ Int & IF  \\
\hline
 \multicolumn{2}{c}{Constant} & & & &&&&&\\
 $\sigma_{1}$ & 2.360 & 0.084 & [2.199, 2.542] & 13 &  & 3.012 & 0.028 & [2.957, 3.071] & 13 \\
 $\sigma_{2}$ & 2.142 & 0.076 & [2.001, 2.303] & 9 &  & 3.953 & 0.038 & [3.875, 4.028] & 11\\
 $\rho$ & -0.421 & 0.041 & [-0.502, -0.334] & 8 &  &0.039 & 0.013 & [0.013, 0.067] & 17\\
 like & -961.9 & 1.214 & [-965.0, -960.5] & 14 &  &-18269 & 1.231 & [-18272, -18268] & 12\\
\hline
 \multicolumn{2}{c}{Spatial} & & & &&&&&\\
 \multicolumn{2}{c}{(i) $\phi^{*}=30$}  & & & &&&&&\\
 $\sigma$ & 1.527 & 0.050 & [1.430, 1.628] & 226 && 2.796 & 0.019 & [2.757, 2.835] & 45 \\
 like & -744.2 & 15.28 & [-770.3, -711.4] & 649 &&-16632 & 16.15 & [-16666, -16603] & 730 \\
 \multicolumn{2}{c}{(ii) $\phi^{*}=10$}  & & & &&&&&\\
 $\sigma$ & 1.532 & 0.049 & [1.436, 1.631] & 242 && 2.796 & 0.019 & [2.756, 2.837] & 32 \\
 like & -746.6 & 14.56 & [-775.0, -717.0] & 681 &&-16631 & 16.13 & [-16660, -16598] & 457\\
 \multicolumn{2}{c}{(iii) $\phi^{*}=1$}  & & & &&&&&\\
 $\sigma$ & 1.693 & 0.047 & [1.604, 1.788] & 47 && 2.790 & 0.020 & [2.752, 2.830] & 36 \\
 like & -822.9 & 8.519 & [-838.8, -805.7] & 259 &&-16613 & 16.93 & [-16656, -16586] & 672 \\
\hline
\hline
\end{tabular}
\label{tab:RealCond}
\end{table}


Lastly here, model performance is compared by calculating \emph{bivariate} CRPS.  Following Section \ref{sec:CV}, let $\{\bm{s}_{T,h}^{test},\bm{s}_{R,h}^{test}\}_{h=1}^{H}$ be the randomly selected test samples for evaluating predictive performance and $\{\bm{s}_{T,j}^{train},\bm{s}_{R,j}^{train}\}_{j=1}^{m-H}$ be the remaining training samples for parameter estimation.
The bivariate continuous rank probability score (CRPS)  \citep{Gneitingetal(08)} values a bivariate distribution $F(\cdot|\bm{s}_{T}^{test})$ more if it is more concentrated around $\bm{s}_{R}^{test}$.  The criterion is calculated through Monte Carlo integrations, using draws from $F(\cdot|\bm{s}_{T}^{test})$, as
\begin{align}
CRPS(F(\cdot|\bm{s}_{T,h}^{test}), \bm{s}_{R,h}^{test})=\frac{1}{L}\sum_{\ell=1}^{L}\| \bm{s}_{R,h}^{(\ell)}-\bm{s}_{R,h}^{test}\|-\frac{1}{2L^{2}}\sum_{\ell=1}^{L} \sum_{\ell^{'}=1}^{L}\|\bm{s}_{R,h}^{(\ell)}- \bm{s}_{R,h}^{(\ell^{'})}. \|
\label{eq:biCRPS}
\end{align}
Here, $\bm{s}_{R,h}^{(\ell)}$ are samples from $F(\cdot|\bm{s}_{T}^{test})$.  Fitting within a Bayesian framework enables posterior samples from $F(\cdot|\bm{s}_{T}^{test})$ using posterior samples of the model parameters.  In particular, for construction of the bivariate predictive distribution with spatially varying kernel, given the $\ell$-th posterior sample of $\psi_{x}(\bm{s})$, $\psi_{y}(\bm{s})$ at training samples $\{\bm{s}_{T,j}^{train}\}_{j=1}^{m-H}$ and $\sigma$, we generate $\psi_{x}(\bm{s})$, $\psi_{y}(\bm{s})$ at test samples $\{\bm{s}_{T,h}^{test}\}_{h=1}^{H}$ from the conditional Gaussian distribution and calculate $\Sigma(\bm{s}_{T,h}^{test})$ for $h=1,\ldots, H$. as defined in (\ref{eq:biCRPS}).

For Neza (Belo Horizonte), $H=80$ (2625) samples are randomly held out for testing, and parameter values are estimated for the remaining 302 (2625) training dataset.
Then, given test theft locations $\{\bm{s}_{T,h}^{test}\}_{h=1}^{H}$, the predictive conditional densities are calculated for the corresponding recovery locations.  That is, for the spatially varying kernel, we predict $\psi_{x}(\bm{s}_{T,h}^{test})$ and $\psi_{y}(\bm{s}_{T,h}^{test})$ given posterior samples $\sigma$ and $\psi_{x}(\bm{s})$ and $\psi_{y}(\bm{s})$.
For Neza, the estimated bivariate CRPS's for the 80 test pairs are 2.947 ($\Sigma$ constant), 2.624 ($\phi^{*}=30$), 2.636 ($\phi^{*}=10$) and 2.962 ( $\phi^{*}=1$). The spatially varying kernel models with $\phi^{*}=30$ and $\phi^{*}=10$ show similar performance, being preferred to the spatially varying kernel model with $\phi^{*}=1$ and constant kernel model.
For Belo Horizonte, $H=2,625$ (50$\%$ of the total number of points) are randomly held out.
The estimated bivariate CRPS for 2,625 test pairs are 156.43 ($\Sigma$ constant), 152.59 ($\phi^{*}=30$), 152.29 ($\phi^{*}=10$) and 152.64 ($\phi^{*}=1$).  The spatially varying kernel models are indistinguishable and slightly outperform the constant kernel model.

Since the question of interest here is prediction of recovery location given theft location, some illustrative posterior predictive densities are demonstrated for such recovery with held out recovery locations.
Figure \ref{fig:CDNZ} shows the conditional density $f_{R}(\cdot|\cdot)$ defined in (\ref{eq:CD}) for some pairs in Neza
(using $\phi^{*}=30$, which, above, gave the best predictive performance).
For ID 2, 32, 49 and 50, theft locations are in the north region, and their conditional densities are close to be uncorrelated densities.
On the other hand, conditional densities for some pairs in the south regions, e.g., ID 196, 301, 332 and 346, show different shapes for the kernels.
Figure \ref{fig:CDBH} shows the conditional density $f_{R}(\cdot|\cdot)$ for some pairs in Belo Horizonte (under $\phi^{*}=10$, which gave the best predictive performance).
Conditional densities for some pairs, e.g., ID 33, 302 and 429, show different shapes for the kernels.
These results suggest that the shapes of conditional densities are location dependent, particularly showing sensitivity when the theft locations are near the boundary of the region.

\begin{figure}[htbp]
  \begin{center}
   \includegraphics[width=15cm]{OD_Conditional.pdf}
  \end{center}
  \caption{Predictive conditional density $f_{R}(\cdot|\cdot)$ for selected pairs in Neza with $\phi^{*}=30$.}
  \label{fig:CDNZ}
\end{figure}

\begin{figure}[htbp]
  \begin{center}
   \includegraphics[width=15cm]{OD_Conditional_BH.pdf}
  \end{center}
  \caption{Predictive conditional density $f_{R}(\cdot|\cdot)$ for selected pairs in Belo Horizonte with $\phi^{*}=10$.}
  \label{fig:CDBH}
\end{figure}


\section{Joint Point Pattern Modeling}
\label{sec:Joint}
Here, linking the theft location point pattern and the recovery location point pattern is considered.  As noted in the Introduction, we find ourselves in an origin-destination setting but at point-referenced scale rather than areal unit scale.  We build a \emph{joint}  intensity of the form $\lambda(\bm{s}_{o}, \bm{s}_{d})$ over pairs of locations $\lambda(\bm{s}_{o}, \bm{s}_{d}) \in D_{o} \times D_{d}$ where $\bm{s}_{o}$ is a theft location and $\bm{s}_{d}$ is a recovery location.
A marginal intensity surface for both theft locations and for recovery locations emerges.
In addition, to attempt to understand the flow of vehicles from theft location to recovery location, we consider a theft neighborhood, say $B_{o} \in D_{o}$ and a recovery neighborhood say $B_{d} \in D_{d}$.  Then, we can ask for the predictive distribution of the number of thefts in $B_{o}$ with recovery in $B_{d}$.  In fact, $D_{d}$ can be partitioned into several neighborhoods to see the flow from $B_{o}$ into each.

Now, a LGCP is introduced for \emph{pairs} of locations as a joint point process model over $D_{o} \times D_{d} \subset \mathbb{R}^{2}\times \mathbb{R}^{2}$.  In fact, $D_{o}=D_{d}=D$ is taken in the sequel.
We denote observed pairs as $\mathcal{S}_{P}=\{\bm{s}_{P,1}, \ldots, \bm{s}_{P,m} \}=\{(\bm{s}_{R,1}, \bm{s}_{T,1}^{*}), \ldots, (\bm{s}_{R,m}, \bm{s}_{T,m}^{*}) \}$; $R$ denotes recovery, $T$ denotes theft.
The intensity function for observed pairs is defined as
\begin{align}
\log \lambda(\bm{s}_{R}, \bm{s}_{T}^{*})&=\bm{X}_{R}(\bm{s}_{R})\bm{\beta}_{R}+\bm{X}_{T}(\bm{s}_{T}^{*})\bm{\beta}_{T} \nonumber \\
&+\eta (\bm{s}_{R}-\bm{s}_{T}^{*})'\Sigma(\bm{s}_{T}^{*})^{-1}(\bm{s}_{R}-\bm{s}_{T}^{*})+z_{R}(\bm{s}_{R})+z_{T}(\bm{s}_{T}^{*}),  \label{pairsintensity}
 \\
\bm{z}_{R}&  \sim \mathcal{N}(\bm{0}, \mathbf{C}_{\bm{z}_{R}}), \quad \bm{z}_{T} \sim \mathcal{N}(\bm{0}, \mathbf{C}_{\bm{z}_{T}}).
\end{align}
Here, $z_{R}(\bm{s})$ and $z_{T}(\bm{s})$ are mean $0$ GP's with covariance functions $C_{R}$ and $C_{T}$, respectively. $\bm{z}_{R}=(z(\bm{s}_{R,1}), \ldots, z(\bm{s}_{R,m}))$, $\bm{z}_{T}=(z(\bm{s}_{T,1}^{*}), \ldots, z(\bm{s}_{T,m}^{*}))$, and $\mathbf{C}_{\bm{z}_{R}}=[C_{R}(\bm{s}_{R,i},\bm{s}_{R,j})]_{i,j=1, \ldots, m}$ ($\mathbf{C}_{\bm{z}_{T}}=[C_{T}(\bm{s}_{T,i}^{*},\bm{s}_{T,j}^{*})]_{i,j=1, \ldots, m}$).
Exponential covariance functions are assumed for $C_{R}$ and $C_{T}$, i.e., $C_{R}(\bm{u}, \bm{u}^{'})=\sigma_{R}^2\exp(-\phi_{R}\|\bm{u}-\bm{u}^{'}\|)$ and $C_{T}(\bm{u}, \bm{u}^{'})=\sigma_{T}^2\exp(-\phi_{T}\|\bm{u}-\bm{u}^{'}\|)$.

The first two terms of the log intensity introduce recovery location and theft location covariates, the third term provides a local (spatially varying) distance measure between the recovery location and the theft location, $\bm{s}_{R}$ and $\bm{s}_{T}^{*}$ through $\Sigma(\bm{s}_{T}^{*})$, the spatially varying kernel presented in the previous section. $\eta$ is the \emph{critical} parameter; it captures the dependence between the point patterns.  If it is not significant, then the joint intensity factors into an intensity for the theft point pattern times an intensity for the recovery point pattern.  In fact, $\eta$ is expected to be negative, i.e., the recovery locations are more observed near the corresponding theft locations. In addition, the local $\Sigma(\bm{s}_{T}^{*})$ enables directional preference for $\bm{s}_{R}$ near the boundaries of $D$.  The fourth and fifth terms provide recovery location and theft location random effects using Gaussian processes.  Without them, the analogue of an NHPP is available; with them, we have the analogue of a LGCP.   This joint specification only employs the complete pairs in the data and will need a large number of pairs in order to learn about the local random effects adjustments.

As above, the likelihood is approximated by gridding $D$ into $K$ blocks.   Now, employing $K \times K$ blocks for $D \times D$, we obtain
\begin{align}
\mathcal{L}(\mathcal{S}_{P})&\propto \exp\biggl(-\int_{D}\int_{D}\lambda(\bm{u}_{R}, \bm{u}_{T})d\bm{u}_{T}d\bm{u}_{R} \biggl)\prod_{j=1}^{m}\lambda(\bm{s}_{R, j}, \bm{s}_{T,j}^{*}) \\
&\approx \exp\biggl(-\sum_{k=1}^{K}\sum_{k^{'}=1}^{K} \lambda(\bm{u}_{R,k}, \bm{u}_{T,k^{'}})\Delta_{T, k}\Delta_{T, k^{'}} \biggl)\prod_{k=1}^{K}\prod_{k^{'}=1}^{K}\lambda(\bm{u}_{R, k}, \bm{u}_{T, k^{'}})^{n_{kk^{'}}}
\end{align}
where $\sum_{k=1}^{K}\sum_{k^{'}=1}^{K}n_{kk^{'}}=m$. Note that $\bm{u}_{R,k}=\bm{u}_{T,k}$ for $k=1,\ldots, K$.

\subsection{Results}
\label{sec:ResultJoint}

We demonstrate results only for the Belo Horizonte data because of the large number of pairs of points (again, $5250$ points).  The small number of pairs for the Neza region (only $68$ in the South) precludes informative model fitting for \eqref{pairsintensity}.

Without covariates, the intensity model for the Belo Horizonte data becomes
\begin{align}
\log \lambda(\bm{s}_{R}, \bm{s}_{T}^{*})&=\beta_{0}+\eta (\bm{s}_{R}-\bm{s}_{T}^{*})'\Sigma(\bm{s}_{T}^{*})^{-1}(\bm{s}_{R}-\bm{s}_{T}^{*})+z_{R}(\bm{s}_{R})+z_{T}(\bm{s}_{T}^{*}), \\
\bm{z}_{R}(\mathcal{S}_{R})&  \sim \mathcal{N}(\bm{0}, \mathbf{C}_{R}(\mathcal{S}_{R},\mathcal{S}_{R})), \quad \bm{z}_{T}(\mathcal{S}_{T}^{*}) \sim \mathcal{N}(\bm{0}, \mathbf{C}_{T}(\mathcal{S}_{T}^{*},\mathcal{S}_{T}^{*})).
\end{align}
$K=305$ grids are taken, the as same as in conditional density specification.
Without covariates, two models are fitted: (i) a LGCP without the spatially varying distance measure (LGCP-Ind, i.e., $\eta=0$) and (ii) a LGCP with this measure (LGCP-Dep).  For priors we assume $\sigma_{R}^2, \sigma_{T}^2\sim \mathcal{IG}(2, 0.1)$, $\beta_{0}\sim \mathcal{N}(0, 100)$ and $\phi_{R}, \phi_{T}\sim \mathcal{U}[0, 10]$.
For sampling $\beta_{0}$, an adaptive random walk MH algorithm is implemented.
Elliptical slice sampling is adopted for $\psi_{x}(\bm{s})$, $\psi_{y}(\bm{s})$, $z_{R}(\bm{s})$ and $z_{T}(\bm{s})$. We fixed the tuning parameter $\phi^{*}=1$.
For high dimensional grids, approximate Gaussian process models can provide efficient process sampling, e.g., nearest neighbor Gaussian processes \citep{Dattaetal(16a)} and multi-resolution Gaussian processes \citep{Katzfuss(17)}.
\cite{ShirotaBanerjee(18)} propose scalable inference for Gaussian Cox process models.
Integrated nested Laplace approximation (INLA) for the LGCP \citep{Simpsonetal(16b)} is another option.
However, the elliptical slice sampling approach is user friendly with no tuning required, enabling easy implementation for moderate size cases.
20,000 samples are discarded as the burn-in period and the subsequent 20,000 samples are retained as posterior samples.
The likelihood value for the LGCP-Dep (-10789 [-10829, -10744]) is much larger than that of the LGCP-ind (-14348 [-14381, -14319]), demonstrating the superiority of LGCP-Dep.
Furthermore, the estimated value of $\eta$ is significantly negative (-0.044 [-0.046, -0.043]) as expected.




 %



We demonstrate the predictive flow of recovery locations from theft locations.
Four subregions are created, each of  which is composed of $G=25$ grid cells, around four locations: $L_{1}=(612500, 7797500)$, $L_{2}=(612500, 780500)$, $L_{3}=(605000, 7797500)$ and $L_{4}=(605000, 7805000)$.
The proportion of predictive intensities and counts are compared for the same theft subregion, i.e., $p_{int}(B_{d}|B_{o})=\frac{\lambda(B_{d}, B_{o})}{\lambda(D_{d}, B_{o})}$  and $p_{count}(B_{d}|B_{o})=\frac{N(B_{d}, B_{o})}{N(D_{d}, B_{o})}$ where $B_{d}\subset D_{d}$ and $B_{o}\subset D_{o}$ and the $\lambda$s are integrated intensities.
Figure \ref{fig:JointPreBH} looks at two origin regions. The left two panels are associated with the southeast origin region, a high intensity region.  The resulting predictive distribution shows that the flow is highly concentrated in that region, in agreement with the actual held out recoveries. The right two panels are associated with the northeast region, a lower intensity region.  The resulting predictive distribution shows more flow from that region to the other three regions, again in agreement with the held out recoveries.
Hence, the nature of the concentration of recovery locations is dependent on theft locations; our model is able to capture this dependence.

\begin{figure}[htbp]
  \begin{center}
   \includegraphics[width=15cm]{JointPre_BH_Count.pdf}
  \end{center}
  \caption{Proportion of held out counts and predictive intensities on four subregions for the theft regions around $L_{1}$ (left two panels: southeast region) and $L_{2}$ (right two panels: northeast region).}
  \label{fig:JointPreBH}
\end{figure}

\section{Summary and Future Work}
\label{sec:Summary}
We have considered a little-studied problem for point patterns namely the setting where we have a point pattern of origins over $D \subset \mathbb{R}^2$ (in our case, locations of car thefts) and an associated partial point pattern of destinations, again over $D \subset \mathbb{R}^2$ (in our case locations of car recoveries). A marginal approach is proposed for modeling the theft locations using NHPP's and LGCPs along with a conditional regression specification for predicting recovery locations given theft locations.  Also, a joint modeling approach is proposed where we view the point pattern as a version of an origin-destination pair and specify a model over a subset of $\mathbb{R}^2 \times \mathbb{R}^2$.

A potential follow-on analysis here would return to Section 4 and the partially marked point pattern specification.
When an associated recovery location is available under this model, it becomes the mark for that location. On the other hand, all of the theft locations without a recovery location have a missing mark.  We would then be modeling marks given locations rather than locations given marks.  This conditioning direction opens up the possibility of preferential sampling \citep[][]{Diggleetal(10)}.  The question of whether the theft location influences the probability of recovery can be examined.

It is worth emphasizing that our approaches here can be applied to other origin-destination problems where the origins and destinations are provided at point level, as geo-coded locations.  We have noted that working at the highest spatial resolution provides a more clear picture of the origin surface, the destination surface, and the dependence between the surfaces than working at areal unit scales.  In this regard, with larger datasets, the dependence might be also included between the $z_{R}(\bm{s}_{R})$ process and the $z_{T}(\bm{s}_{T})$ process through say coregionalization \citep[][]{BanerjeeCarlinGelfand(14)}. This would further illuminate the dependence structure between the two surfaces.

One path for future work will investigate a much different application.  We will examine economic labor force data where, for an individual, we have the location where she/he resides as well as the location where she/he works.  Working with metropolitan areas will provide much larger point patterns with much more demanding model fitting.  Future work with theft-recovery data would introduce consideration of time, i.e., we will have not only the location of the theft but also the time of the theft.  Similarly, we have not only a location for the recovery but as well the time of the recovery, with an implicit order in time for the latter relative to the former.  Unfortunately, at present, neither of the datasets provide time information needed to enable such investigation.
\section*{Acknowledgements}
\label{sec:Ack}
This work partially funded by Grant MTM2016-78917-R from the Spanish Ministry
of Science and Education, and Grant P1-1B2015-40 from University Jaume I. The
work of Shinichiro Shirota was supported in part by the Nakajima Foundation. The
authors thank Renato Assun{\c{c}}{\~{a}}o for providing the datasets and shape file of Belo
Horizonte. The authors also thank the Nezahualcoyotl Town Hall, Arturo Arango, and Direccion General de Seguridad Ciudadana (in Mexico) for providing the Neza dataset. The computational results are obtained by using Ox version 7.1 (Doornik (2007)).

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