arXiv subjects

Francisco Cuevas-Pacheco

Publications and source records attributed to Francisco Cuevas-Pacheco.

14 recordsLinked to original sources

arXiv2026

$K$-functions for point processes on complex surfaces

The $K$-function is a fundamental summary statistic for assessing clustering or regularity of point processes in two or three dimensional Euclidean space. In practice, however, many planar point patterns arise from projecting locations of objects on a surface in three dimensional space to two dimensional space. For example, when events or objects occur in a landscape their elevation is often ignored. This can lead to erroneous conclusions regarding properties of the point process generating the point pattern. There is not a unique way to extend the classical $K$-function to point patterns on a complex surface. In this paper we propose, explore, and discuss several approaches in terms of their theoretical and computational properties. The best performing approach, coined the surface area $K$-function, can be viewed is an analogue of the classical $K$-function replacing counts of points in Euclidean balls with counts of points in surface geodesic balls. However, an important distinction is that the argument of our surface area $K$-function is area instead of radius of geodesic balls. The performances of the various surface $K$-functions are compared in applications to simulated and real data.

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arXiv2026

Fast Stochastic Nearest Neighbor Pairwise Composite Likelihood for Massive Spatial Datasets

Weighted pairwise composite likelihoods based on nearest-neighbor (NN) pairs provide a scalable alternative to full likelihood inference for spatial random fields, but can remain expensive when moderately large NN neighborhoods are needed. We propose a stochastic acceleration that constructs the deterministic NN candidate graph and evaluates only a randomized subset of its pairwise contributions. We consider two thinning designs: Bernoulli thinning, which controls the retained-pair budget in expectation, and fixed-budget thinning, which enforces an exact budget through target-wise sampling without replacement. Simulation studies for Matérn covariance models suggest that, in the settings considered, retaining two pairs per observation provides a stable statistical--computational compromise. The stochastic NN pairwise estimators provide a faster alternative to a Vecchia-type Gaussian approximation when substantial reductions in covariance-fitting time are desired and a modest loss of efficiency is acceptable. This trade-off is especially favorable for the mean, scale, and sill parameters, while the main efficiency loss is concentrated on smoothness estimation. In an application to July average temperature over the western--central United States, based on 2.5 million WorldClim observations, the proposed estimators achieve predictive accuracy essentially indistinguishable from the Vecchia benchmark, with substantially shorter covariance-fitting time.

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arXiv2026

Regularized estimation for highly multivariate spatial Gaussian random fields

Estimating covariance parameters for multivariate spatial Gaussian random fields is computationally challenging, as the number of parameters grows rapidly with the number of variables, and likelihood evaluation requires operations of order $\mathcal{O}((np)^3)$. In many applications, however, not all cross-dependencies between variables are relevant, suggesting that sparse covariance structures may be both statistically advantageous and practically necessary. We propose a LASSO-penalized estimation framework that induces sparsity in the Cholesky factor of the multivariate Matérn correlation matrix, enabling automatic identification of uncorrelated variable pairs while preserving positive semidefiniteness. Estimation is carried out via a projected block coordinate descent algorithm that decomposes the optimization into tractable subproblems, with constraints enforced at each iteration through appropriate projections. Regularization parameter selection is discussed for both the likelihood and composite likelihood approaches. We conduct a simulation study demonstrating the ability of the method to recover sparse correlation structures and reduce estimation error relative to unpenalized approaches. We illustrate our procedure through an application to a geochemical dataset with $p = 36$ variables and $n = 3998$ spatial locations, showing the practical impact of the method and making spatial prediction feasible in a setting where standard approaches fail entirely.

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arXiv2026

Fast simulation of Gaussian random fields with flexible correlation models in Euclidean spaces

The efficient simulation of Gaussian random fields with flexible correlation structures is fundamental in spatial statistics, machine learning, and uncertainty quantification. In this work, we revisit the \emph{spectral turning-bands} (STB) method as a versatile and scalable framework for simulating isotropic Gaussian random fields with a broad range of covariance models. Beyond the classical Matérn family, we show that the STB approach can be extended to two recent and flexible correlation classes that generalize the Matérn model: the Bummer-Tricomi model, which allows for polynomially decaying correlations and long-range dependence, and the Gauss-Hypergeometric model, which admits compactly supported correlations, including the Generalized Wendland family as a special case. We derive exact stochastic representations for both families: a Beta-prime mixture formulation for the Kummer-Tricomi model and complementary Beta- and Gasper-mixture representations for the Gauss-Hypergeometric model. These formulations enable exact, numerically stable, and computationally efficient simulation with linear complexity in the number of spectral components. Numerical experiments confirm the accuracy and computational stability of the proposed algorithms across a wide range of parameter configurations, demonstrating their practical viability for large-scale spatial modeling. As an application, we use the proposed STB simulators to perform parametric bootstrap for standard error estimation and model selection under weighted pairwise composite likelihood in the analysis of a large climate dataset.

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arXiv2026

Kriging for large datasets via penalized neighbor selection

Kriging is a fundamental tool for spatial prediction, but its computational complexity of $O(N^3)$ becomes prohibitive for large datasets. While local kriging using $K$-nearest neighbors addresses this issue, the selection of $K$ typically relies on ad-hoc criteria that fail to account for spatial correlation structure. We propose a penalized kriging framework that incorporates LASSO-type penalties directly into the kriging equations to achieve automatic, data-driven neighbor selection. We further extend this to adaptive LASSO, using data-driven penalty weights that account for the spatial correlation structure. Our method determines which observations contribute non-zero weights through $\ell_1$ regularization, with the penalty parameter selected via a novel criterion based on effective sample size that balances prediction accuracy against information redundancy. Numerical experiments demonstrate that penalized kriging automatically adapts neighborhood structure to the underlying spatial correlation, selecting fewer neighbors for smoother processes and more for highly variable fields, while maintaining prediction accuracy comparable to global kriging at substantially reduced computational cost.

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arXiv2024

Composite likelihood inference for space-time point processes

The dynamics of a rain forest is extremely complex involving births, deaths and growth of trees with complex interactions between trees, animals, climate, and environment. We consider the patterns of recruits (new trees) and dead trees between rain forest censuses. For a current census we specify regression models for the conditional intensity of recruits and the conditional probabilities of death given the current trees and spatial covariates. We estimate regression parameters using conditional composite likelihood functions that only involve the conditional first order properties of the data. When constructing assumption lean estimators of covariance matrices of parameter estimates we only need mild assumptions of decaying conditional correlations in space while assumptions regarding correlations over time are avoided by exploiting conditional centering of composite likelihood score functions. Time series of point patterns from rain forest censuses are quite short while each point pattern covers a fairly big spatial region. To obtain asymptotic results we therefore use a central limit theorem for the fixed timespan - increasing spatial domain asymptotic setting. This also allows us to handle the challenge of using stochastic covariates constructed from past point patterns. Conveniently, it suffices to impose weak dependence assumptions on the innovations of the space-time process. We investigate the proposed methodology by simulation studies and applications to rain forest data.

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arXiv2023

Pairwise interaction function estimation of Gibbs point processes using basis expansion

The class of Gibbs point processes (GPP) is a large class of spatial point processes able to model both clustered and repulsive point patterns. They are specified by their conditional intensity, which for a point pattern $\mathbf{x}$ and a location $u$, is roughly speaking the probability that an event occurs in an infinitesimal ball around $u$ given the rest of the configuration is $\mathbf{x}$. The most simple and natural class of models is the class of pairwise interaction point processes where the conditional intensity depends on the number of points and pairwise distances between them. This paper is concerned with the problem of estimating the pairwise interaction function non parametrically. We propose to estimate it using an orthogonal series expansion of its logarithm. Such an approach has numerous advantages compared to existing ones. The estimation procedure is simple, fast and completely data-driven. We provide asymptotic properties such as consistency and asymptotic normality and show the efficiency of the procedure through simulation experiments and illustrate it with several datasets.

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arXiv2021

A convolution type model for the intensity of spatial point processes applied to eye-movement data

Estimating the first-order intensity function in point pattern analysis is an important problem, and it has been approached so far from different perspectives: parametrically, semiparametrically or nonparametrically. Our approach is close to a semiparametric one. Motivated by eye-movement data, we introduce a convolution type model where the log-intensity is modelled as the convolution of a function $β(\cdot)$, to be estimated, and a single spatial covariate (the image an individual is looking at for eye-movement data). Based on a Fourier series expansion, we show that the proposed model \rev{can be viewed as a} log-linear model with an infinite number of coefficients, which correspond to the spectral decomposition of $β(\cdot)$. After truncation, we estimate these coefficients through a penalized Poisson likelihood. We illustrate the efficiency of the proposed methodology on simulated data and on eye-movement data.

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arXiv2021

The $\mathcal{F}$-family of covariance functions: A Matérn analogue for modeling random fields on spheres

The Mat{é}rn family of isotropic covariance functions has been central to the theoretical development and application of statistical models for geospatial data. For global data defined over the whole sphere representing planet Earth, the natural distance between any two locations is the great circle distance. In this setting, the Mat{é}rn family of covariance functions has a restriction on the smoothness parameter, making it an unappealing choice to model smooth data. Finding a suitable analogue for modelling data on the sphere is still an open problem. This paper proposes a new family of isotropic covariance functions for random fields defined over the sphere. The proposed family has a parameter that indexes the mean square differentiability of the corresponding Gaussian field, and allows for any admissible range of fractal dimension. Our simulation study mimics the fixed domain asymptotic setting, which is the most natural regime for sampling on a closed and bounded set. As expected, our results support the analogous results (under the same asymptotic scheme) for planar processes that not all parameters can be estimated consistently. We apply the proposed model to a dataset of precipitable water content over a large portion of the Earth, and show that the model gives more precise predictions of the underlying process at unsampled locations than does the Mat{é}rn model using chordal distances.

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arXiv2020

Karhunen-Loève Expansions for Axially Symmetric Gaussian Processes: Modeling Strategies and $L^2$ Approximations

Axially symmetric processes on spheres, for which the second-order dependency structure may substantially vary with shifts in latitude, are a prominent alternative to model the spatial uncertainty of natural variables located over large portions of the Earth. In this paper, we focus on Karhunen-Loève expansions of axially symmetric Gaussian processes. First, we investigate a parametric family of Karhunen-Loève coefficients that allows for versatile spatial covariance functions. The isotropy as well as the longitudinal independence can be obtained as limit cases of our proposal. Second, we introduce a strategy to render any longitudinally reversible process irreversible, which means that its covariance function could admit certain types of asymmetries along longitudes. Then, finitely truncated Karhunen-Loève expansions are used to approximate axially symmetric processes. For such approximations, bounds for the $L^2$-error are provided. Numerical experiments are conducted to illustrate our findings.

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arXiv2019

Structured space-sphere point processes and $K$-functions

This paper concerns space-sphere point processes, that is, point processes on the product space of $\mathbb R^d$ (the $d$-dimensional Euclidean space) and $\mathbb S^k$ (the $k$-dimen\-sional sphere). We consider specific classes of models for space-sphere point processes, which are adaptations of existing models for either spherical or spatial point processes. For model checking or fitting, we present the space-sphere $K$-function which is a natural extension of the inhomogeneous $K$-function for point processes on $\mathbb R^d$ to the case of space-sphere point processes. Under the assumption that the intensity and pair correlation function both have a certain separable structure, the space-sphere $K$-function is shown to be proportional to the product of the inhomogeneous spatial and spherical $K$-functions. For the presented space-sphere point process models, we discuss cases where such a separable structure can be obtained. The usefulness of the space-sphere $K$-function is illustrated for real and simulated datasets with varying dimensions $d$ and $k$.

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arXiv2019

Regularized estimation for highly multivariate log Gaussian Cox processes

Statistical inference for highly multivariate point pattern data is challenging due to complex models with large numbers of parameters. In this paper, we develop numerically stable and efficient parameter estimation and model selection algorithms for a class of multivariate log Gaussian Cox processes. The methodology is applied to a highly multivariate point pattern data set from tropical rain forest ecology.

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arXiv2019

Second-order variational equations for spatial point processes with a view to pair correlation function estimation

Second-order variational type equations for spatial point processes are established. In case of log linear parametric models for pair correlation functions, it is demonstrated that the variational equations can be applied to construct estimating equations with closed form solutions for the parameter estimates. This result is used to fit orthogonal series expansions of log pair correlation functions of general form.

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arXiv2018

Log Gaussian Cox processes on the sphere

A log Gaussian Cox process (LGCP) is a doubly stochastic construction consisting of a Poisson point process with a random log-intensity given by a Gaussian random field. Statistical methodology have mainly been developed for LGCPs defined in the $d$-dimensional Euclidean space. This paper concerns the case of LGCPs on the $d$-dimensional sphere, with $d=2$ of primary interest. We discuss the existence problem of such LGCPs, provide sufficient existence conditions, and establish further useful theoretical properties. The results are applied for the description of sky positions of galaxies, in comparison with previous analysis based on a Thomas process, using simple estimation procedures and making a careful model checking. We account for inhomogeneity in our models, and as the model checking is based on a thinning procedure which produces homogeneous/isotropic LGCPs, we discuss its sensitivity.

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