1. Introduction
A spatial point process is a stochastic process that governs the location of events in a geographic space (Diggle [
1]). A point pattern is a collection of locations in a given area and is interpreted as the realization of a point process (Illian et al. [
2]). This type of pattern can be analyzed when georeferenced data are available and the exact locations of the events of interest are known; for example, the geographic coordinates of trees infected by a particular pest or the location of occurrences of a disease. Unlike the applications in geostatistics, in which the locations are fixed and the interest lies on the measurement of some random response, the very location of the occurrences is the random object in point processes. In the early literature on spatial point processes, it was common to assume that patterns were realizations of stationary and isotropic point processes, which implies that the statistical properties of the point process are invariant under spatial translations and rotations. However, these assumptions are often unrealistic, since in many cases there may be local or directional influences in the correlation structure of the spatial random process.
Several authors have explored nonstationarity and anisotropy in point processes. Häbel et al. [
3] introduced anisotropic pair-potential functions based on Ripley’s
K-function. Gabriel [
4] proposed a similar extension in a spatiotemporal context. Hahn and Vedel Jensen [
5] unified nonstationary model classes under a framework of hidden second-order stationary processes. Konstantinou and Särkkä [
6] extended Ripley’s
K-function to three dimensions using a pairwise interaction Markov field model to accommodate anisotropic data. Rajala et al. [
7] devised a two-stage non-parametric method for quantifying geometric anisotropy. Sormani et al. [
8] formulated a comprehensive framework for addressing geometric anisotropy, including directional versions of Ripley’s
K-function, wavelet transforms, and spectral analysis, along with a method based on projecting Fry points onto the unit sphere. Martin et al. [
9] introduced a new family of geometric anisotropic random fields and derived anisotropic point processes from them. Kwon et al. [
10] utilized Hawkes process models, a class of self-exciting point process capturing event-triggering effects, to analyze earthquake occurrences. Dong et al. [
11] devised another non-stationary spatiotemporal self-exciting point process, incorporating a neural network-based kernel to capture spatially varying triggering effects. Biard and Saussereau [
12] applied a fractional Poisson process to situations where variances increase over time according to a defined power law. Roveri et al. [
13] introduced a method to analyze and synthesize point patterns with adaptive density and correlations using a locally stationary point process. Most of these methodologies have the advantage of not requiring specific model assumptions. However, they have the disadvantage of not indicating what the intensity function would be if inherent anisotropy were taken into account, and they do not allow for an evaluation of the influence of covariates.
In geostatistics, various methods address nonstationarity/anisotropy in spatial modeling, such as process convolution models (Higdon et al. [
14], Risser and Calder [
15]), kernel/smoothing methods (Fuentes [
16]), and basis functions (Pintore and Holmes [
17], Stephenson et al. [
18]). Spatial deformation allows the data to control the behavior of the fitted process without specifying mechanisms. Sampson and Guttorp [
19] proposed a non-parametric method for estimating spatial covariance globally using a deformation technique. It maps coordinates from the geographical
G-space to the deformed
D-space (where stationarity holds), locating monitoring stations with multidimensional scaling and interpolating via thin plate splines. The inclusion of a non-linear deformation in the correlation function necessarily implies the loss of stationarity of the process and allows for anisotropic formulation, encompassing a wider range of models for dependence between locations. Challenges include multiple mappings in
D-space and non-bijective predictions. Schmidt and O’Hagan [
20] introduced a bi-dimensional Gaussian process to handle uncertainty in spatial deformation estimation.
This paper aims to apply spatial deformation to spatial point processes in order to handle anisotropy and nonstationarity. Challenges arise when adapting deformation to point processes based on Cox processes (see, for example, Diggle [
1]), such as the problem of dealing with infinite dimensionality and the intractability of likelihood functions for non-homogeneous Poisson processes. Discretizing the
G-space to address this issue is a common approach, involving forming partitions with constant intensities and using the centroids as representative points of each partition fraction (Moller et al. [
21]; Brix and Diggle [
22]; Paez and Diggle [
23]). The discretization procedure also impacts the use of the deformation process (DP), which inherently operates in continuous space but is well-defined in finite dimensions in discretized space.
The Log–Gaussian Cox process is a standard model in point process analysis where the logarithmic function links an intensity function to a Gaussian process (GP). Adams et al. [
24] used a logistic link for isotropic scenarios, and Gonçalves and Gamerman [
25] applied a probit link for efficient sampling. Both bounded link functions require an extra parameter to account for the maximal intensity. An evaluation of the effect of different link functions on the intensity function estimation in anisotropic/nonstationary situations is also carried out in this paper.
This study presents an extended spatial Cox process for point process analysis for estimating intensity function with consideration for anisotropy and nonstationarity through spatial deformation. The novelty lies in using deformations to handle anisotropy in spatial point processes. The research employs the geometry-aware Hamiltonian Monte Carlo (HMC) method for sampling and evaluates different link functions across various scenarios. The methodology allows for data-driven guidance for estimating spatial anisotropy/nonstationarity behavior. It incorporates a multi-dimensional Gaussian process as a prior specification for the DP, adopting a fully Bayesian approach that assigns a probability distribution to spatial deformation as a latent process and quantifying estimation uncertainty. The method resolves the non-identifiability of the deformation, and notably, the proposed model can also accommodate intensity regressors.
The presentation of the paper is organized as follows:
Section 2 describes the proposed Cox process, considering the spatial deformation through the multi-dimensional Gaussian process.
Section 3 presents an analysis of the performance of the proposed model and estimation method, considering both synthetic and real data.
Section 3.1 presents the results of simulation exercises for the comparison of the proposed anisotropic formulation to an isotropic one.
Section 3.2 shows the estimation through the proposed anisotropic model compared to an isotropic formulation for the
Spodoptera fruguperda pest data in an agricultural area of corn production in the city of Cascavel, Brazil.
Section 3.3 provides favourable comparative results of our formulation against an alternative anisotropic specification.
Section 4 discusses the results and indicates current and future extensions of this work.
2. Methodology
As previously mentioned, the well-established deformation technique involves mapping geographic coordinates from the original space (
G-space) to a new latent space (
D-space), ensuring stationarity and isotropy. This work employs an approach similar to previous studies, such as Schmidt and O’Hagan [
20] and Morales et al. [
26], where the random function
transforms original coordinates from
G-space to the new
D-space, following a prior
l-variate Gaussian process
where
h is a prior mean function,
is a
covariance matrix and
is the vector of parameters for any valid correlation function
The covariance structure of the GP is essential for representing the behavior of the function d. It controls the prior warping degree of the mapping. Details on the specification of GP hyperparameters will be discussed later in the text.
The hierarchical structure for the proposed extension for the Cox process is given by
where PP indicates a Poisson process with intensity function
,
is an appropriate link function,
(if the supremum exists (for
functions bounded to
, for example
probit or
logit link functions, it is necessary to estimate
because the intensity function can take values greater than 1. For upper unbounded
functions such as the exponential,
and we can fix
.));
is an intercept,
is a GP with mean 0, the precision is
and the vector of parameters is
for any valid correlation function
where
and
are any points, again, in
G.
In this study, we will analyze the probit () and logarithmic link functions. The bounded range of the cumulative standard normal function to requires estimating , whereas for the unbounded exponential function (logarithmic link), and its estimation is not required.
The prior mean for
d is assumed as the identity function (
) to express prior indifference on how
D differs from
G. The diagonal matrix
indicates deformation variability across dimensions. For example, when
G denotes geographical coordinates (
) and
D has the same dimension (
), higher/lower
values allow more/less deformation along the horizontal axis (
) or vertical axis (
). Additionally, increased correlation range in (
2) implies smoother deformation among nearby regions. While both
and
d are Gaussian processes, from now on we use “GP” for
and “deformation process (DP)” for
d, in order to clarify which process we are referring to.
The likelihood function of the PP in (
3) is given by the Janossy density (see, for example, D. J. Daley [
27], p. 122) and is expressed as
with respect to a PP dominating measure with a constant intensity function, the integral corresponds to a Lebesgue integral. As previously mentioned, the likelihood function in expression (
8) depends on infinite-dimensional quantities like
and
d, which is a problem for making inferences mainly concerning the integral on all
G-space.
When estimating model hyperparameters, fixing some or using informative priors might be necessary due to hierarchical complexity. The phenomenon being examined requires a complex, highly-structured hierarchical model, needing meticulous attention to fully comprehend and accurately elicit all its components. For and , a balanced deformation degree is sought to avoid folding or a loss of anisotropy. It is well known that for any and for any . Therefore, a reasonable strategy when using the probit link function is to set and to favor parameter identification in Equation (4). A Gamma prior is assigned to , possibly centered on some empirically-set intensity or with a non-informative specification. Under the log link, vague improper uniform prior for and non-informative Gamma prior for are used.
For the GP
, for the sake of simplicity, we adopted the exponential correlation function
.
follows a Gamma(
,
) prior, centered on a value that produces low correlation only for higher distances in the geographic region. Defining the density of
as the Radon–Nikodym derivative of
with respect to a suitable dominating measure
we have
with
,
,
,
denoting the prior densities described above,
and
as described in expressions (4) and (6), and
as in expression (
8). The joint posterior distribution is proportional to (
9).
A usual approach to dealing with infinite-dimensional quantities
and
d in this likelihood is to discretize the space of interest. By discretizing the geographical space into a partition
where
for every
, it is possible to define piece-wise constant intensities and assume that for each sub-region
the centroid
is a representative point. The greater the number of occurrences, the more refined the discretization could be in effectively disseminating information, leading to more accurate estimates to more adequately characterize the intensity function. Defining
, the integral in (
8) can be rewritten as
where
is the Lebesgue measure (for
, for example, it represents the area) of the
z-th region; now it is possible to perform inferences for the discretized space.
Defining
for
the total count of events in
,
, the posterior density for the model with space discretization based on expression (
9) is given by
where
,
(where
is the dimension of
D-space),
is the matrix-variate normal density (see, for example, Morales et al. [
26]),
is the multivariate normal density with a zero mean vector and covariance matrix
with elements
for
and
and
is the likelihood function (
8) by applying the discretization method, resulting in
For the developments presented in
Section 3, we focus on two-dimensional (2D)
G-space (latitude and longitude), but the concepts presented here could naturally extend to other dimensions like time (1D) or 3D, when altitude is included. The
D-space dimension is set as 2, as in
Section 3.1 and
Section 3.2, balancing deformation and computational complexity.
The interpolation of the process
d at coordinates
,
,
may be required. This is carried out by applying the standard conditional distribution properties of multivariate normal combined with matrix-variate and multivariate normal relationships (e.g., Gupta and Nagar [
28]). Suppose
where
;
for
and
where
denotes each coordinate to be interpolated;
for
and
, with
denoting each centroid coordinate from the partition; and
for
and
; and finally
. Therefore, the distribution of
is given by
where
and
. The codes provided in
Appendix A can be used to obtain a sample of size
L from the posterior distribution of
. Specifically, for each
l-th sample of
, a sample from the posterior distribution of
can be generated from:
where
and the interpolation is based on the mean distribution.
Function
d is unidentifiable, since all the transformations of the coordinates in
D-space which keep the same distances between locations are observationally equivalent. Scaling all distances by a constant is also equivalent. For the practical application of the deformation model in the Bayesian paradigm, fixing the locations of two points in
D-space is recommended (Damian et al. [
29], Morales et al. [
26], Sampson and Meiring [
30]). We anchor
and
to prevent translation or rotation.
Bayesian inference is implemented via Markov Chain Monte Carlo (MCMC). Sampling of the posterior samples was handled by applying Hamiltonian Monte Carlo (HMC). Non-centered parameterizations (see, for example, Papaspiliopoulos et al. [
31]) were employed for both the Gaussian process
and deformation process
d. This choice also enhances favorable a posteriori geometry and computational efficiency. The analyses were accomplished on R version 4.0.3 (R Core Team (2020)) and some MCMC parts were developed on C++ and Rcpp library to make an interface between R and C++. The Rstan package (Carpenter et al. [
32]) provided comparable results and was used to make an interface. The Stan codes used in all subsequent analyses are available in
Appendix A.
4. Discussion
The proposed point process analysis model, incorporating anisotropy and nonstationarity via DP, outperformed models lacking these considerations in both simulated and real data analyses. Criteria assessing fit quality showed marked improvements. Specifically, in a scenario observing varying oscillation frequencies, models accommodating deformation showed better discrimination between higher and lower frequency zones than those without. Models omitting deformation failed to distinguish the intensity regions accurately. Incorporating deformation notably enhanced all model selection criteria, favoring the model that accounted for it. This approach, addressing anisotropy and nonstationarity via spatial deformation (rather than other specific pattern across the entire space), proved beneficial because the change in the intensity function regime occurred mainly in the relevant subregions, by deforming the geographic space differently across regions. Moreover, specifying a parametrized anisotropy and/or nonstationarity formulation for these intensity function behavior would be insufficient to capture these types of changes in the intensity function. Comparisons between the proposed anisotropic model via deformation and the geometric anisotropic model indicated favorable results for the proposed formulation. This was observed in both artificially generated data and real data concerning the Spodoptera frugiperda pest.
Deformation behavior in
Spodoptera frugiperda pest control data shows increased spatial correlation along the horizontal direction (in rotated coordinates), aligning with findings from Nava et al. [
37], highlighting higher anisotropy at a
angle in non-rotated data. This is consistent with agricultural practices, mirroring the direction of planting machines. Estimated infestation levels enable targeted pesticide application, aiding environmental and consumer health by prioritizing doses in specific regions and prevailing directions.
Different link functions may suit each case, especially when assessing the behavior of the intensity function. In scenarios with low overall intensity but sudden spikes, the logarithm link function can better capture peaks. Alternatively, the probit link function might perform better for frequent sudden intensity function decays. When the intensity function behaves erratically, fitting both and selecting the best performance based on model criteria proves relevant in practical terms.
An extended model includes covariates influencing the point pattern by substituting
with
in expression (
3), where
represents some form of square root precision matrices. Similarly, for some applications offsets
might be needed for a comparison of varying population sizes across space. For instance, defining
in expression (
3) allows the data
Y to be rewritten as
.
The isotropic model can be obtained as the special case of our DP when the main diagonal of tends to zero. Model comparison may then be also obtained by a mixture model with prior weight for isotropy and complementary prior weight for the DP component. Posterior inference on might prove useful for assessing the relevance of the anisotropic component.
The work of this paper considered only discretized space. Recent sampling methods using Poisson thinning offer crucial extensions towards inference in continuous space. Gonçalves and Gamerman [
25] utilized Poisson thinning for non-homogeneous Poisson processes, employing a spatial Cox model driven by a Gaussian process. Their MCMC algorithm allowed for direct simulation, resembling our model with a
probit link function. Extending the methods in this direction merges the benefits of continuous surfaces, improves the intensity estimation for nonstationary/anisotropic processes, and are the subject of our interest for future work.