arXiv subjects

Christopher J. Geoga

Publications and source records attributed to Christopher J. Geoga.

13 recordsLinked to original sources

arXiv2026

Bigraphical Mat\'ern-Whittle (BMW) Processes for Fast Inference of Big Multivariate Spatial Data on General Domains

Large spatial data sets now record many correlated variables at many thousands of locations, often on domains where Euclidean distance misrepresents proximity. The central difficulty is modelling the cross-variable dependence jointly while retaining variable-level interpretation. We introduce the bigraphical Mat'ern-Whittle process, a multivariate Gaussian process that resolves this with two graphs. A spatial graph generates the Mat'ern structure of each variable through a fractional power of a graph Laplacian, so the process is valid on any topology, with per-variable range, smoothness and amplitude. A directed acyclic variable graph encodes the scientific structure: we prove that each absent edge yields an exact conditional independence between the corresponding fields. We further prove that the operator determinant does not involve the cross-dependence coefficients, which keeps matrix-free likelihood evaluation and Bayesian learning of the variable graph tractable at scale. Estimation requires only sparse matrix-vector products and scales to tens of millions of space-variable pairs. In simulations the method recovered parameters and graphs accurately, remained robust under misspecification, and halved held-out prediction error on a non-convex domain. In a spatial transcriptomics section with 19,809 cells and 1,122 genes, fitted in 75 minutes on a laptop, borrowing across the learned gene graph reduced held-out prediction error by 50 to 91 percent. Theoretical challenges, such as the achievable efficiency of estimating the variance of the nugget, are also explored.

stat.ME↗

arXiv2026

Linear-cost Polyharmonic Spline Interpolation of Arbitrary Degree

We introduce a simple and performant approach for rapidly and accurately performing polyharmonic spline (PHS) interpolation using a combination of the fast multipole method (FMM) from computational electrostatics and the Vecchia approximation from the Gaussian process and sparse approximate inverse literatures. Using basic properties about Hadamard products and low-rank matrices, we demonstrate that an FMM with two kernels, the logarithmic and distance kernels, results in fast PHS interpolation for all orders. Furthermore, we demonstrate the exceptional performance of sparse inverse approximation methods with the Mat\'ern covariance model for preconditioning. Combined with careful management of disallowed subspaces, we describe a procedure for obtaining prediction weights using preconditioned conjugate gradient that converges in less than $15$ iterations, even for problem sizes with over one million points. As a result, thin-plate spline interpolation---a particularly popular method that does not require parameter tuning---that matches the fully dense $\mathcal{O}(n^3)$ computation in accuracy can be done at the cost of approximately $50-60$ FMMs. A high-performance software library for odd-order PHS interpolation in two dimensions is made available as a companion to this work.

math.NA↗

arXiv2026

A Gaussian Process Model of 3D Udder Point Clouds for Teat Length Phenotyping in Dairy Cows

Phenotyping conformation traits is important for dairy cattle breeding and management. Although large-scale phenotyping is possible with 3D imaging technologies, manual annotation of anatomical landmarks and long run times prevent full pipeline automation. In particular, the morphological heterogeneity of cow udders makes automated detection of teat landmarks challenging. To address this limitation, we propose and evaluate a method for teat length estimation from udder point clouds with a Gaussian process. We model the vertical coordinates as the sum of a Gaussian process representing the udder floor and an unknown function representing the teat. Since the udder floor process is smooth and has a significantly wider dependence lengthscale than the teat function, this model allows separating the two terms needed for teat landmark definition. To ensure computational feasibility, we implement a low-rank approximation of the covariance matrix, reducing the computational complexity of the method from $\mathcal{O}(n^3)$ to $\mathcal{O}(n)$. This approach is both faster and more accurate than existing methods, reducing RMSE by a factor of two. It is also more robust to uncommon udder morphologies, making it better suited for automated phenotyping of large numbers of individuals.

stat.AP↗

arXiv2026

Fast and accurate conditioning for large-scale Gaussian process prediction problems

Gaussian Process (GP) models provide a flexible framework for prediction and uncertainty quantification. For most covariance functions, however, exact GP prediction with $n$ points scales as $\mathcal{O}(n^3)$, making it prohibitively expensive for large datasets or large numbers of prediction points. While nearest neighbor-based prediction can work well in certain settings, non-pathological circumstances (like measurement noise, for example) can severely restrict its efficiency. This work presents a complementary approach where one conditions on carefully designed linear combinations of data, which is particularly effective in the setting of jointly predicting many values in large connected regions of the data domain. For kernel functions that are smooth away from the origin and simple prediction domains, this method can be exponentially convergent in the number of linear combinations $r$ used for conditioning. The procedure costs $\mathcal{O}(T r^2)$ work, where $T$ is the cost of solving a linear system with the data covariance matrix, and so in many cases can be computed in linear or near-linear cost by exploiting rank structure in well-behaved covariance matrices. At the cost of $\mathcal{O}(nr^2)$ additional precomputation work, this approach can also provide predictions at arbitrary points of a designated region in $\mathcal{O}(1)$ online work, making it particularly attractive for problems where prediction points are not known in advance. After establishing favorable theoretical properties, we provide several example applications to problems in prediction and matrix approximation.

stat.CO↗

arXiv2025

High-Resolution Retrieval of Atmospheric Boundary Layers with Nonstationary Gaussian Processes

The atmospheric boundary layer (ABL) plays a critical role in governing turbulent exchanges of momentum, heat moisture, and trace gases between the Earth's surface and the free atmosphere, thereby influencing meteorological phenomena, air quality, and climate processes. Accurate and temporally continuous characterization of the ABL structure and height evolution is crucial for both scientific understanding and practical applications. High-resolution retrievals of the ABL height from vertical velocity measurements is challenging because it is often estimated using empirical thresholds applied to profiles of vertical velocity variance or related turbulence diagnostics at each measurement altitude, which can suffer from limited sampling and sensitivity to noise. To address these limitations, this work employs nonstationary Gaussian process (GP) modeling to more effectively capture the spatio-temporal dependence structure in the data, enabling high-quality -- and, if desired, high-resolution -- estimates of the ABL height without reliance on ad-hoc parameter tuning. By leveraging Vecchia approximations, the proposed method can be applied to large-scale datasets, and example applications using full-day vertical velocity profiles comprising approximately $5$M measurements are presented.

physics.ao-ph↗

arXiv2025

Fast nonparametric spectral density estimation from irregularly sampled data

We introduce a nonparametric spectral density estimator for continuous-time and continuous-space processes measured at fully irregular locations. Our estimator is constructed using a weighted nonuniform Fourier sum whose weights yield a high-accuracy quadrature rule with respect to a user-specified window function. The resulting estimator significantly reduces the aliasing seen in periodogram approaches and least squares spectral analysis, sidesteps the dangers of ill-conditioning of the nonuniform Fourier inverse problem, and can be adapted to a wide variety of irregular sampling settings. We describe methods for rapidly computing the necessary weights in various settings, making the estimator scalable to large datasets. We then provide a theoretical analysis of sources of bias, and close with demonstrations of the method's efficacy, including for processes that exhibit very slow spectral decay and are observed at up to a million locations in multiple dimensions.

stat.ME↗

arXiv2024

Fast Adaptive Fourier Integration for Spectral Densities of Gaussian Processes

The specification of a covariance function is of paramount importance when employing Gaussian process models, but the requirement of positive definiteness severely limits those used in practice. Designing flexible stationary covariance functions is, however, straightforward in the spectral domain, where one needs only to supply a positive and symmetric spectral density. In this work, we introduce an adaptive integration framework for efficiently and accurately evaluating covariance functions and their derivatives at irregular locations directly from \textit{any} continuous, integrable spectral density. In order to make this approach computationally tractable, we employ high-order panel quadrature, the nonuniform fast Fourier transform, and a Nyquist-informed panel selection heuristic, and derive novel algebraic truncation error bounds which are used to monitor convergence. As a result, we demonstrate several orders of magnitude speedup compared to naive uniform quadrature approaches, allowing us to evaluate covariance functions from slowly decaying, singular spectral densities at millions of locations to a user-specified tolerance in seconds on a laptop. We then apply our methodology to perform gradient-based maximum likelihood estimation using a previously numerically infeasible long-memory spectral model for wind velocities below the atmospheric boundary layer.

stat.CO↗

arXiv2024

Fast Machine-Precision Spectral Likelihoods for Stationary Time Series

We provide in this work an algorithm for approximating a very broad class of symmetric Toeplitz matrices to machine precision in $\mathcal{O}(n \log n)$ time with applications to fitting time series models. In particular, for a symmetric Toeplitz matrix $\mathbf{\Sigma}$ with values $\mathbf{\Sigma}_{j,k} = h_{|j-k|} = \int_{-1/2}^{1/2} e^{2 \pi i |j-k| \omega} S(\omega) \mathrm{d} \omega$ where $S(\omega)$ is piecewise smooth, we give an approximation $\mathbf{\mathcal{F}} \mathbf{\Sigma} \mathbf{\mathcal{F}}^H \approx \mathbf{D} + \mathbf{U} \mathbf{V}^H$, where $\mathbf{\mathcal{F}}$ is the DFT matrix, $\mathbf{D}$ is diagonal, and the matrices $\mathbf{U}$ and $\mathbf{V}$ are in $\mathbb{C}^{n \times r}$ with $r \ll n$. Studying these matrices in the context of time series, we offer a theoretical explanation of this structure and connect it to existing spectral-domain approximation frameworks. We then give a complete discussion of the numerical method for assembling the approximation and demonstrate its efficiency for improving Whittle-type likelihood approximations, including dramatic examples where a correction of rank $r = 2$ to the standard Whittle approximation increases the accuracy of the log-likelihood approximation from $3$ to $14$ digits for a matrix $\mathbf{\Sigma} \in \mathbb{R}^{10^5 \times 10^5}$. The method and analysis of this work applies well beyond time series analysis, providing an algorithm for extremely accurate solutions to linear systems with a wide variety of symmetric Toeplitz matrices whose entries are generated by a piecewise smooth $S(\omega)$. The analysis employed here largely depends on asymptotic expansions of oscillatory integrals, and also provides a new perspective on when existing spectral-domain approximation methods for Gaussian log-likelihoods can be particularly problematic.

math.NA↗

arXiv2023

A Scalable Method to Exploit Screening in Gaussian Process Models with Noise

A common approach to approximating Gaussian log-likelihoods at scale exploits the fact that precision matrices can be well-approximated by sparse matrices in some circumstances. This strategy is motivated by the \emph{screening effect}, which refers to the phenomenon in which the linear prediction of a process $Z$ at a point $\mathbf{x}_0$ depends primarily on measurements nearest to $\mathbf{x}_0$. But simple perturbations, such as i.i.d. measurement noise, can significantly reduce the degree to which this exploitable phenomenon occurs. While strategies to cope with this issue already exist and are certainly improvements over ignoring the problem, in this work we present a new one based on the EM algorithm that offers several advantages. While in this work we focus on the application to Vecchia's approximation (1988), a particularly popular and powerful framework in which we can demonstrate true second-order optimization of M steps, the method can also be applied using entirely matrix-vector products, making it applicable to a very wide class of precision matrix-based approximation methods.

stat.ME↗

arXiv2022

Scalable Computations for Nonstationary Gaussian Processes

Nonstationary Gaussian process models can capture complex spatially varying dependence structures in spatial datasets. However, the large number of observations in modern datasets makes fitting such models computationally intractable with conventional dense linear algebra. In addition, derivative-free or even first-order optimization methods can be slow to converge when estimating many spatially varying parameters. We present here a computational framework that couples an algebraic block-diagonal plus low-rank covariance matrix approximation with stochastic trace estimation to facilitate the efficient use of second-order solvers for maximum likelihood estimation of Gaussian process models with many parameters. We demonstrate the effectiveness of these methods by simultaneously fitting 192 parameters in the popular nonstationary model of Paciorek and Schervish using 107,600 sea surface temperature anomaly measurements.

stat.CO↗

arXiv2022

Fitting Matérn Smoothness Parameters Using Automatic Differentiation

The Matérn covariance function is ubiquitous in the application of Gaussian processes to spatial statistics and beyond. Perhaps the most important reason for this is that the smoothness parameter $ν$ gives complete control over the mean-square differentiability of the process, which has significant implications for the behavior of estimated quantities such as interpolants and forecasts. Unfortunately, derivatives of the Matérn covariance function with respect to $ν$ require derivatives of the modified second-kind Bessel function $\mathcal{K}_ν$ with respect to $ν$. While closed form expressions of these derivatives do exist, they are prohibitively difficult and expensive to compute. For this reason, many software packages require fixing $ν$ as opposed to estimating it, and all existing software packages that attempt to offer the functionality of estimating $ν$ use finite difference estimates for $\partial_ν\mathcal{K}_ν$. In this work, we introduce a new implementation of $\mathcal{K}_ν$ that has been designed to provide derivatives via automatic differentiation (AD), and whose resulting derivatives are significantly faster and more accurate than those computed using finite differences. We provide comprehensive testing for both speed and accuracy and show that our AD solution can be used to build accurate Hessian matrices for second-order maximum likelihood estimation in settings where Hessians built with finite difference approximations completely fail.

stat.CO↗

arXiv2020

Flexible nonstationary spatio-temporal modeling of high-frequency monitoring data

Many physical datasets are generated by collections of instruments that make measurements at regular time intervals. For such regular monitoring data, we extend the framework of half-spectral covariance functions to the case of nonstationarity in space and time and demonstrate that this method provides a natural and tractable way to incorporate complex behaviors into a covariance model. Further, we use this method with fully time-domain computations to obtain bona fide maximum likelihood estimators---as opposed to using Whittle-type likelihood approximations, for example---that can still be computed efficiently. We apply this method to very high-frequency Doppler LIDAR vertical wind velocity measurements, demonstrating that the model can expressively capture the extreme nonstationarity of dynamics above and below the atmospheric boundary layer and, more importantly, the interaction of the process dynamics across it.

stat.ME↗

arXiv2018

Scalable Gaussian Process Computations Using Hierarchical Matrices

We present a kernel-independent method that applies hierarchical matrices to the problem of maximum likelihood estimation for Gaussian processes. The proposed approximation provides natural and scalable stochastic estimators for its gradient and Hessian, as well as the expected Fisher information matrix, that are computable in quasilinear $O(n \log^2 n)$ complexity for a large range of models. To accomplish this, we (i) choose a specific hierarchical approximation for covariance matrices that enables the computation of their exact derivatives and (ii) use a stabilized form of the Hutchinson stochastic trace estimator. Since both the observed and expected information matrices can be computed in quasilinear complexity, covariance matrices for MLEs can also be estimated efficiently. After discussing the associated mathematics, we demonstrate the scalability of the method, discuss details of its implementation, and validate that the resulting MLEs and confidence intervals based on the inverse Fisher information matrix faithfully approach those obtained by the exact likelihood.

stat.CO↗