A spatio-temporal hybrid Strauss hardcore point process for forest fire occurrences

We propose a new point process model that combines, in the spatio-temporal setting, both multi-scaling by hybridization and hardcore distances. Our so-called hybrid Strauss hardcore point process model allows different types of interaction, at different spatial and/or temporal scales, that might be of interest in environmental and biological applications. The inference and simulation of the model are implemented using the logistic likelihood approach and the birth-death Metropolis-Hastings algorithm. Our model is illustrated and compared to others on two datasets of forest fire occurrences respectively in France and Spain.


Introduction
In point process modeling, most of the existing models yield point patterns with mainly single-structure, but only a few with multi-structure. Interactions with single-structure are often classified into three classes: randomness, clustering and inhibition. Among the inhibition processes is the hardcore process. It has some hardcore distance h in which distinct points are not allowed to come closer than a distance h apart. This type of interaction can be modeled by Gibbs point processes as the hardcore or Strauss hardcore point processes and also by Cox point processes as Matérn's hardcore (Matérn, 1960;1986) or Matérn thinned Cox point processes (Andersen and Hahn, 2016). Here, we focus on the former, i.e. Gibbs models implemented by a hardcore component as in the Strauss hardcore model. The form of Strauss hardcore density indicates that the hardcore parameter only rules at least the distance between points, and has no effect on the interaction terms of the density (Dereudre and Lavancier, 2017, sect. 2.3).
In several domains, there exist point patterns with hardcore distances that have to be modeled. Spatial point patterns with hardcore property can be found in capillaries studies (Mattfeldt et al., 2006;2007;2009), in texture synthesis (Hurtut et al., 2009), in forest fires (Turner, 2009), in cellular networks (Taylor et al., 2012 andYing et al., 2014), in landslides (Das and Stein, 2016), in modern and contemporary architecture and art (Stoyan, 2016) and in location clustering econometrics (Sweeney and Gomez-Antonio, 2016).
There also exist point patterns with either clustering and inhibition like hardcore interactions at different scales simultaneously (Badreldin et al., 2015;Andersen andHahn, 2016 andWang et al., 2020). Wang et al. (2020, sect. 2.4) investigated the effect of the hardcore distance on spatial patterns of trees by comparing the pair correlation function curves for different values of hardcore distances in the fitted hybrid Geyer hardcore model. Raeisi et al. (2019) review spatial and spatio-temporal point processes that model both inhibition and clustering at different scales. Such multi-structure interactions can be modeled by the spatial hybrid Gibbs point process (Baddeley et al, 2013). In this paper, we aim to extend the spatial Strauss hardcore point process (Ripley, 1988) to the spatio-temporal framework and introduce a multi-scale version of it using a hybridization approach. We use this model to describe one of the most complex phenomena from the spatio-temporal modeling point of view: forest fire occurrences.
The complexity of forest fire occurrences is due in particular to the existence of multi-scale structures and hardcore distances in space and time. For instance, spatio-temporal variations of fire occurrences depend on the spatial distribution of current land use and weather conditions. Changes in vegetation due to forest fires burnt areas further affect the probability of fire occurrences during the regeneration period leading to the existence of hardcore distances in space-time. The multi-scale structure of clustering and inhibition in the spatiotemporal pattern of forest fire occurrences is discussed in Gabriel  This paper is organized as follows. In Section 2 we introduce in the spatiotemporal framework the notations and definitions of Gibbs point processes in order to introduce our multi-scale version of the Strauss hardcore model. Section 3 is devoted to the inference of our model. It describes techniques to determine the irregular parameters (hardcore and interaction distances) and the logisticlikelihood approach generalized to the spatio-temporal setting to estimate the regular parameters (strength of interactions). Section 4 illustrates the goodnessof-fit of the logistic likelihood approach on simulated patterns of our model obtained by an extended Metropolis-Hastings algorithm. Finally, in Section 5, we illustrate our model on two monthly and yearly records of forest fires in France and Spain and compare our results to those obtained by two other spatial point process models.

Towards multi-scale Strauss hardcore point processes
Gibbs models are flexible point processes that allow the specification of point interactions via a probability density defined with respect to the unit rate Poisson point process. These models allow us to characterize a form of local or Markovian dependence amongst events. Gibbs point processes contain a large class of flexible and natural models that can be applied for: • Postulating the interaction mechanisms between pairs of points, • Taking into account clustering, randomness, or inhibition structures, • Combining several structures at different scales with the hybridization approach.
where r, q > 0 are spatial and temporal radius and ||·|| denotes the Euclidean distance in R 2 and |·| denotes the usual distance in R. Note that C q r (u, v) is a cylinder with centre (u, v), radius r, and height 2q.
A finite Gibbs point process is a finite simple point process defined with a density f (x) on W that satisfies the hereditary condition, i.e. f (x) > 0 ⇒ f (y) > 0 for all y ⊂ x. Throughout the paper, we suppose that W is a bounded set in R 2 × R, and all Gibbs models are defined with respect to a unit rate Poisson point process on W .
A closely related concept to density functions is Papangelou conditional intensity function (Papangelou, 1974) which is a characterization of Gibbs point processes useful for inferring the model parameters and simulating related point patterns. The Papangelou conditional intensity of a spatio-temporal point process on W with density f is defined, for (u, v) ∈ W , by with a/0 := 0 for all a ≥ 0 (Cronie and van Lieshout, 2015). The Papangelou conditional intensity is very useful to describe local interactions in a point pattern and leads to the notion of a Markov point process which is the basis for the implementation of MCMC algorithms used for simulating of Gibbs models. We say that the point process has "interactions of range R at (ξ, t)" if points further than R away from (ξ, t) do not contribute to the conditional intensity at (ξ, t). A spatio-temporal Gibbs point process X has a finite interaction range R if the Papangelou conditional intensity satisfies for all configurations x of X and all (u, v) ∈ W, where C R R (u, v) denotes the cylinder of radius R > 0 and height 2R > 0 centered at (u, v) (see Iftimi et al. (2018) for a well-detailed discussion). Spatio-temporal Gibbs models usually have finite interaction range property (spatio-temporal Markov property) and are called in this case Markov point processes (van Lieshout 2000). The finite range property of a spatio-temporal Gibbs model implies that the probability to insert a point (u, v) into x depends only on some cylindrical neighborhood of (u, v). Finally, a spatio-temporal Gibbs point process is said to be locally stable if is stochastically dominated by a Poisson point processs, that is if there exists λ constant < ∞ such that for any (u, v) ∈ W and for any configuration x of X, λ((u, v)|x) ≤ λ constant . We further refer to Dereudre (2019) for a more formal introduction of Gibbs point processes.
Here, we first review spatio-temporal Gibbs models and then extend the spatial Strauss hardcore model to the spatio-temporal and multi-scale context. A Gibbs point process model explicitly postulates that interactions traduce dependencies between the points of the pattern. The hardcore interaction is one of the simplest type of interaction, which forbids points being too close to each other. The homogeneous spatio-temporal hardcore point process is defined by the density

Single-scale Gibbs point process models
where c > 0 is a normalizing constant, λ > 0 is an activity parameter, h s , h t > 0 are, respectively, the spatial and the temporal hardcore distances and n(x) is the number of points in x. The Papangelou conditional intensity of a homogeneous spatio-temporal hardcore point process for The homogeneous spatio-temporal Strauss point process is defined by density where 0 < γ ≤ 1, and the Papangelou conditional intensity of the model is and where n[C q r (y, z); x] = (ξ,t)∈x 1{||y − ξ||≤ r, |z − t|≤ q} is the number of points in x which are in a cylinder C q r (y, z). Although the Strauss point process was originally intended as a model of clustering, it can only be used to model inhibition, because the parameter γ cannot be greater than 1. If we take γ > 1, the density function of Strauss model is not integrable, so it does not define a valid probability density.
As mentioned, the Strauss point process model only achieves the inhibition structure. In the spatial framework, two ways are introduced to overcome this problem that we extend to the spatio-temporal framework hence defining two new spatio-temporal Gibbs point process models.
A first way is to consider an upper bound for the number of neighboring points that interact. In this case, Raeisi et al. (2021) defined a homogeneous spatio-temporal Geyer saturation point process by density where s is a saturation parameter and A second way is to introduce a hardcore condition to the Strauss density (6). Hence, we can define a Strauss hardcore model in the spatio-temporal context. Definition 1. We define the spatio-temporal Strauss hardcore point process as the point process with density where 0 < h s < r, 0 < h t < q and γ > 0.
Note that, contrary to the case of a spatio-temporal Strauss process (6), for Strauss hardcore processes the interaction parameter γ can assume any nonnegative value, in particular a value larger than 1. If the interaction parameter γ is bigger than one, we have an attraction effect, whereas for γ smaller than one there is a repulsion tendency. If γ = 1 a spatio-temporal hardcore process (4) is obtained.
The Papangelou conditional intensity of a homogeneous spatio-temporal Strauss hardcore point process for We can define inhomogeneous versions of all above models by replacing the constant λ by a function λ(ξ, t), inside the product operator over (ξ, t) ∈ x, that expresses a spatio-temporal trend, which can be a function of the coordinates of the points and depends on covariate information.

Multi-scale Gibbs point process models
Iftimi et al. (2018) extended the hybrid approach for an area-interaction model to the spatio-temporal framework where the density is given by where (r j , q j ) are pairs of irregular parameters of the model and γ j are interaction parameters, j = 1, . . . , m.
In the same way, Raeisi et al. (2021) defined a spatio-temporal multi-scale Geyer saturation point process with density where c > 0 is a normalizing constant, λ ≥ 0 is a measurable and bounded function, γ j > 0 are the interaction parameters.
Similarly, a hybrid version of spatio-temporal Strauss model can be defined by hybridization.
Note that we called the model (14) hybrid rather than multi-scale. The model (14) can cover inhibition structure because 0 < γ j < 1, ∀j ∈ {1, . . . , m}. However, it can take into account clustering if one of densities in hybrid is the one of a hardcore process.

Hybrid Strauss hardcore point process
The hybrid Gibbs point process models do not necessarily include m same Gibbs point process models (see Baddeley et al., 2015 sect. 13.8). Badreldin et al. (2015) applied a spatial hybrid model including a hardcore density to model strong inhibition at very short distances, Geyer density for cluster structure in short to medium distances and a Strauss density for a randomness structure in larger distances to the spatial pattern of the halophytic species distribution in an arid coastal environment. Wang et al. (2020) fitted a spatial hybrid Geyer hardcore point process on the tree spatial distribution patterns. In this section, we extend this type of hybrids to the spatio-temporal context. Definition 3. We define the spatio-temporal hybrid Strauss hardcore point process with density where 0 < h s < r 1 < · · · < r m , 0 < h t < q 1 < · · · < q m and γ j > 0 for all j = 1, ..., m.
The model could be used to model clustering patterns with a softer attraction between the points like a pattern with a combination of interaction terms that show repulsion between the points at a small scale and attraction between the points at a larger scale.
The Papangelou conditional intensity of an inhomogeneous spatio-temporal hybrid Strauss hardcore process is then, for Because, the conditional intensity of Gibbs models including a hardcore interaction term takes the value zero at some locations, we can rewrite it as where m((u, v)|x) takes only the values 0 and 1, and λ + ((u, v)|x) > 0 everywhere.
The spatio-temporal hybrid Strauss hardcore point process (15) is a Markov point process in Ripley-Kelly's (1977) sense at interaction range max{r m , q m } and is also locally stable. These can be shown as in Iftimi et al. (2018) and Raeisi et al. (2021). The Markov property allows us to design MCMC simulation algorithms and local stability property ensures the convergence of simulation algorithms. The spatio-temporal hybrid Strauss hardcore point process (15) is well-defined due to the log-linearity of its conditional intensity (see Section 3) and satisfying in Markov and local stability properties (Vasseur et al., 2020) and moreover, it exists also in R d × R (Dereudre 2019).

Inference
Gibbs point process models involve two types of parameters: regular and irregular parameters. A parameter is called regular if the log likelihood of density is a linear function of that parameter otherwise it is called irregular. Typically, regular parameters determine the 'strength' of the interaction, while irregular parameters determine the 'range' of the interaction. As an example, in the Strauss hardcore point process (10), the trend parameter λ and the interaction γ are regular parameters and the interaction distances r and q and the hardcore distances h s and h t are irregular parameters.
To determine the interaction distances r and q, there are several practical techniques, but no general statistical theory available. A useful technique is the maximum profile pseudo-likelihood approach (Baddeley and Turner, 2000). In the spatio-temporal framework, Iftimi et al. (2018) and Raeisi et al. (2021) selected feasible range of irregular parameters by analyzing the behavior of some summary statistics and the goodness-of-fit of several models with different combinations of parameters.
The hardcore interaction term m(·|x) in the conditional intensity (17) does not depend on the other parameters of the densities of Gibbs point processes. This implies that it can first be estimated and kept fixed for the sequel (Baddeley et al., 2019, p. 26). In the spatial framework, the maximum likelihood estimate of the hardcore distance in m(·|x) corresponds to the minimum interpoint distance (Baddeley et al., 2013, Lemma 7). The generalization to the spatio-temporal context with a cylindrical hardcore structure implies to consider a multi-objective minimization problem over the spatial and temporal hardcore distances h s and h t . The choice of our hardcore parameters needs to analyze the Pareto front of feasible solutions on the graph of spatial and temporal interpoint distances. We refer the reader to Ehrgott (2005) for a description of multi-criteria optimization and the definition of Pareto optimality. To estimate the hardcore distance h s and h t , we consider a feasible solution on the Pareto front as large as possible and with a ratio consistent with our knowledge of interaction mechanisms in practice. Indeed, in the spatial case, there is a unique solution for the minimum interpoint distance. In the spatio-temporal case, we search h s and h t minimizing respectively the interpoint distances in space and time for a same couple of points. Because the couple (h s ; h t ) does not exist in general. we thus have to consider the feasible solutions on the Pareto front (see Figure 1). Regular parameters can be estimated using the pseudo-likelihood method (Baddeley and Turner, 2000) or logistic likelihood method (Baddeley et al., We assume that θ = (log γ 1 , log γ 2 , . . . , log γ m ) is the logarithm of interaction parameters in spatio-temporal hybrid Strauss hardcore point process (15). To estimate θ, due to (17) , v), x)). Hence, the logarithm of the Papangelou conditional intensity of the spatio-temporal hybrid Strauss hardcore point process for (u, v) ∈ W which satisfies in hardcore condition, i.e. m((u, v)|x) = 1 in (16), is corresponding to a linear model in θ with offset log By considering a set of dummy points d from an independent Poisson process with intensity function ρ, we obtain by defining the Bernoulli variables Y ((ξ, t)) = 1 {(ξ,t)∈x} for (ξ, t) ∈ x ∪ d that the logit of P (Y ((ξ, t))) = 1 is equal to log λ θ ((ξ,t)|x\(ξ,t)) ρ(ξ,t) . Under regularity conditions, conditional on x ∪ d, the log-logistic likelihood corresponds to the logistic regression model with responses Y ((ξ, t)) = 1 {(ξ,t)∈x} and offset term − log ρ(ξ, t) and admits a unique maximum which already implemented by using standard software for GLMs. The term λ(ξ, t) may also depend on a parameter, say β. We shall consider λ(ξ, t) = β in the simulation study and λ(ξ, t) = exp(β ⊺ Z(ξ, t)) where Z(ξ, t) is a matrix of covariates. In summary we have the following algorithm for data and dummy points such that m(·|x) = 1.

Algorithm
• Generate a set of dummy points according to a Poisson process with intensity function ρ and merge them with all the data points in x to construct the set of quadrature points (ξ k , t k ) ∈ W = S × T , • Obtain the response variables y k (1 for data points, 0 for dummy points), • Compute the values S((ξ k , t k ), x) of the vector of sufficient statistics at each quadrature point, • Fit a logistic regression model with explanatory variables S((ξ k , t k ), x) and Z(ξ k , t k ), and offset log(1/ρ(ξ k , t k )) on the responses y k to obtain estimatesθ for the S-vector andβ for covariate-vector, • Return the parameter estimatorγ = exp(θ) andβ.

Simulation study
Due to the markovian property of the spatio-temporal hybrid Strauss hardcore point process (15), its Papangelou conditional intensity at a point thus depends only on that point and its neighbors in x. Hence, we can design simulation approach by Markov chain Monte Carlo algorithms. Gibbs point process models can be simulated a birth-death Metropolis-Hastings algorithm that typically requires only computation of the Papangelou conditional intensity (Møller and Waagepetersen, 2004). Raeisi et al. (2021) extended the birth-death Metropolis-Hastings algorithm to the spatio-temporal context that we adapt here for simulating the spatio-temporal hybrid Strauss hardcore point process.
We implement the estimation and simulation algorithms in R (R Core Team, 2016) and generate simulations of three stationary spatio-temporal hybrid Strauss    Table 1. The spatial and temporal radii r and q, spatial and temporal hardcores h s and h t , are treated as known parameters. We generate 100 simulations of each specified model. Boxplots of parameter estimates λ, γ 1 , and γ 2 obtained from the logistic likelihood estimation method for each model are shown in Figure 2. The red horizontal lines represent the true parameter values. Point and interval parameter estimates λ, γ 1 , and γ 2 are reported in Table 2. Most of the estimated parameter values are close to the true values for three models. Due to visual and computational comparisons, we conclude that the logistic likelihood approach performs well for spatio-temporal hybrid Strauss hardcore point processes.

Application
In this section, we aim to model the interactions of forest fire occurrences across a range of spatio-temporal scales. We compare the relevance of our model on two datasets in the center of Spain and south of France, w.r.t. the ones of two widely used models: the inhomogeneous Poisson process and the log-Gaussian Cox process (LGCP).  Figure 3 (right) shows a seasonal effect with notably large numbers of fires in summer that could be caused by high temperatures and low precipitations in this period and also by human activities. As the spatio-temporal inhomogeneity is notably driven by covariates, we include in our analysis four spatial covariates: elevation, orientation, slope and land use (available in spatstat) and two spatio-temporal covariates: monthly maximum temperatures ( • C) and total precipitations (mm), available on WorldClim database 1 . The covariates are known on a spatial grid with pixels of 4 km×4 km, resulting in a total of 10, 000 pixels. Figure 4 illustrates the environmental covariates, which are considered fixed during our temporal period, and the climate covariates in January 2007.  The Prométhée database 2 was created to gather several informations relative to the wildfire in French Mediterranean regions. It uses a system of coordinates specially designed for fire management: the DFCI coordinates (DFCI is the French acronym for "Défense de la Foret Contre les Incendies", i.e. it refers to all processes concerning forest defense facing to wildfires). The fire ignition locations are therefore available with a spatial resolution of 4 km 2 . The Prométhée has already been used in various academic works devoted to the point process theory, see for example Gabriel

Inference
We consider an inhomogeneous hybrid Strauss hardcore model with Papangelou conditionl intensity where the spatio-temporal trend is log-linearly related to spatial (Z S k ) and spatio-temporal (Z ST l ) covariates as in For Promethee we have also a term αt. The inference of the hybrid Strauss hardcore model involves different approaches according to the kind of parameters: empirical and ad-hoc for the irregular parameters, likelihood-based for the regular parameters.
Irregular parameter estimation There are two types of irregular parameters: the hardcore distances and the nuisance parameters. The hardcore distances can be chosen among all feasible solutions on the Pareto front of spatial and temporal interpoint distances. For clmfires dataset, according to Figure 6, we choose on the Pareto front the feasible solution in our case that gives non-zero values for the two hardcore distances i.e. h s = 0.35 km and h t = 1 month. For Prométhée dataset, due to the nature of the dataset, we consider h s = 2000 meter and h t = 1 year. There is no common method for estimating the nuisance parameters m, r j and q j , j = 1, . . . , m. A preliminary spatio-temporal exploratory analysis of the interaction ranges based on the inhomogeneous pair correlation function, the maximum nearest neighbor distance and the temporal auto-correlation function, allowed us to set 25 configurations of reasonable range for the nuisance parameters. We select the optimal irregular parameters according to the Akaike's Information Criterion (AIC) of the fitted model after the regular parameter estimation step (Raeisi et al., 2021).

Regular parameter estimation
We consider the logistic likelihood method investigated in Section 3 to estimate the regular parameters. The dummy points are simulated from a Poisson process with intensity ρ(ξ, t) = Cλ P (ξ, t)/v, with C = 4 by a classical rule of thumb in the logistic likelihood approach and v is volume of spatio-temporal region. In order to satisfy the hardcore condition in (17), we remove dummy points at spatial and temporal distances respectively less than h s and h t .

Results
For the clmfires (resp. Promethee) data we select m = 6 (resp. m = 4) spatial and temporal interaction distances. Table 3 (resp. Table 4) provides them and the related regular parameters.

Model validation
The goodness-of-fit of the hybrid Strauss hardcore models is accomplished by simulations from the fitted model. The first diagnostic can be formulated by summary statistics of point processes. As the second-order characteristics carry most of the information on the spatio-temporal structure (Stoyan, 1992 ; Gonzalez et al., 2016), we only consider the pair correlation function (g-function). We generate n sim = 99 simulations from the fitted hybrid Strauss hardcore models and compute the corresponding second-order summary statistics g i (u, v), i = 1, . . . , n sim , for fixed spatio-temporal distances (u, v). We then build upper and lower envelopes:  and compare the summary statistics obtained from the data, g obs (u, v), to the pointwise envelopes. If it lies outside the envelopes at some spatio-temporal distances (u, v), then we reject at these distances the hypothesis that our data come from our fitted model. Figure 7 shows the spatio-temporal inhomogeneous g-function computed on our dataset from clmfires (blue) and the envelopes obtained from the fitted model (light grey); g obs (u, v) lies inside the envelopes for all (u, v), meaning that the hybrid Strauss hardcore model is suitable for the data. In addition, we compute global envelopes and p-value of the spatio-temporal g-functions based on the Extreme Rank Length (ERL) measure defined in Myllymäki et al. (2017) and implemented in the R package GET (Myllymäki and Mrkvička, 2020). For each point pattern, we consider the long vector T i , i = 1, . . . , n sim (resp. T obs ) merging the g i (·, v) (resp. g obs (·, v)) estimates for all considered values h t . The ERL measure of vector T i (resp. T obs ) of length n st is defined as where R i is the vector of pointwise ordered ranks and ≺ is an ordering operator (Myllymäki et al., 2017;Myllymäki and Mrkvička, 2020). The final p-value is obtained by For clmfires and Prométhée, due to the global p-values p erl = 0.15, 0.08 respectively and the absence of significant regions, that corresponds here to pairs of spatial and temporal distances where the statistics is significantly above or below the envelopes (see Figure 8, Figure 9 and GET package), we conclude that our hybrid Strauss hardcore model can not be rejected.

Model comparison
We are interested to evaluate the performance of the hybrid Strauss hardcore model over other models. Hence, we fit an inhomogeneous Poisson point process and a LGCP to two wildfire datasets.

Poisson process
To evaluate the performance of the hybrid Strauss hardcore model, we generated 99 simulations from two inhomogeneous spatio-temporal Poisson point processes with estimated intensity λ/(4×4×1) for clmfires dataset and λ/(2000×2000× 1) for Prométhée dataset by rpp function in stpp package (Gabriel et al., 2013). Figure 10 is the plots of upper and lower of the global p-value for 99 simulated Poisson point process patterns with value 0.01 related to clmfires (up) and Prométhée datasets (down) that confirms the advantage the hybrid Strauss hardcore point process over the inhomogeneous Poisson point process for our two forest fire datasets.

LGCP
For more comparison, we fit a LGCP on clmfires and Prométhée datasets by INLA approach. We consider a LGCP with stochastic intensity consists a space-month resolution for incorporating covariate information. The model is considered for a Gaussian space-time effect W (ξ, t), whose spatial component is always based on the flexible yet computationally convenient Matérn-like spatial Gauss-Markov random fields arising as approximate solutions to certain stochastic partial differential equations. By adding the Gaussian space-time effect W (ξ, t), we define a LGCP with a stochastic intensity log Λ(ξ, t) rather than log λ(ξ, t).
For Promethee we have a term αt. The inference is based on the Integrated Nested Laplace Approximation (Rue et al., 2009) and its implementation in R-INLA. We refer the reader to Opitz et al. (2020) for a thorough description of the implementation of very similar models and data. Figure 11 is the plots of upper and lower of the global p-value for 99 simulated point patterns from LGCP with values 0.08, 0.01 related to clmfires (up) and Prométhée datasets (down) that confirms the advantage the hybrid Strauss hardcore point process over the LGCP for our two forest fire datasets.
In summary, the global envelope tests may be used for model comparison by comparing p-values and concluding that the model with the highest p-value provides the best fit. Use consider this test to compare the Inhomogeneous Poisson, Log-Gaussian Cox, and hybrid Strauss hardcore processes on the two datasets. The p-values are given in Table 5 that is shown a good candidate can be Gibbs point processes based on hybridization of Strauss and hardcore densities to model the forest fire occurrences. Figure 10: Estimated pair correlation functionĝ obs , lower E L and upper E U bounds, differences E obs −E L and E U −E obs of the 99% global rank envelope (ERL) from two inhomogeneous Poisson point processes related to clmfires (up) and Prométhée datasets (down).

Conclusion
In this paper, we introduced the spatio-temporal Strauss hardcore point process. The Strauss hardcore model is a Gibbs model for which points are pushed to be at a hardcore distance apart and repel up to a interaction distance which is larger than the hardcore distance. As in Raeisi et al. (2021), inference and simulation of the model were performed with logistic likelihood and birthdeath Metropolis-Hasting algorithm, respectively. A multi-scale version of the model was introduced and applied to wildfires to take into account structural complexity of forest fire occurrences in space and time. We based our model validation on both pointwise and global envelopes and p-value based on the Extreme Rank Length (ERL) measure of the spatio-temporal inhomogeneous Figure 11: Estimated pair correlation functionĝ obs , lower E L and upper E U bounds, differences E obs − E L and E U − E obs of the 99% global rank envelope (ERL) from two LGCPs related to clmfires (up) and Prométhée datasets (down).
pair correlation function. Note that, here, our criterion for choosing the best fitted model was to compare the extracted AIC value of the GLM of conditional intensity implemented in R. The composite AIC criterion for spatial Gibbs models introduced by Daniel  However, the composite AIC requires to estimate the variance-covariance matrix of the logistic score and the sensitivity matrix which should be extended to the spatio-temporal framework that is a full-blown work and also involves efficient implementation.
Our model could be suitable in other environmental and ecological frameworks, when we want to deal with the complexity of mechanisms governing attraction and repulsion of entities (particles, cells, plants. . . ).
In spatio-temporal Gibbs point process models, the heterogeneity can be captured by estimating a non-constant trend. This spatio-temporal trend is typically considered as a function of covariates by estimating fixed effects in a generalized linear model as we carried out it in this paper and also done in Iftimi et al. (2018) and Raeisi et al. (2021). A different approach consists in considering Gibbs models with both random and fixed effects (e.g. see Illian and Hendrichsen, 2010) to take into account complex patterns of spatio-temporal heterogeneity. Vihrs et al. (2021) proposed a new modeling approach for this case and embedded spatially structured Gaussian random effects in trend function of a pairwise interaction process. They introduced the spatial log-Gaussian Cox Strauss point process to capture both structures; aggregation in small-scale and repulsion in large-scale. Rather than spatial pairwise interaction processes in single-scale, we can focus on models derived from the multi-scale classes of combinations of Gibbs and log-Gaussian Cox point processes in space and time. Indeed, in a work in progress, we aim to propose to embed spatio-temporally structured Gaussian random effects in the Gibbs trend function. Due to the hierarchical structure of such models, we can formulate and estimate them within a Bayesian hierarchical framework, using the INLA approach.