arXiv subjects

Mark D. Risser

Publications and source records attributed to Mark D. Risser.

At least 19 recordsLinked to original sources

arXiv2026

A General Kernel Framework for Non-CND Distance Measures Using |D|-Dimensional Sparse Landmark Embeddings

Kernel methods, and Gaussian Processes (GPs) in particular, require a Hilbertian distance measure---one whose square is conditionally negative definite (CND)---to guarantee positive semi-definiteness (PSD) of the kernel matrix; a condition that fails for many natural input spaces, including smooth manifolds and spaces of probability distributions. We propose the Sparse Landmark Embedding (SLE) kernel, which eliminates this requirement entirely. Each input is embedded into a sparse feature vector via compactly supported bump functions centered at all |D| training points; applying any standard PSD kernel in this embedding space yields a kernel that is provably PSD for arbitrary distance measures. The compact support automatically controls embedding sparsity, keeping kernel matrices well-conditioned and computationally tractable despite the high ambient dimension. We provide theoretical guarantees on PSD, sparsity, stability, and universal approximation, and demonstrate, using geodesic and Wasserstein distances, that the SLE kernel matches or substantially exceeds domain-specific baselines in both predictive accuracy and uncertainty quantification.

stat.ML↗

arXiv2026

gp2Scale: A Class of Compactly Supported Non-Stationary Kernels and Distributed Computing for Exact Gaussian Processes on 10 Million Data Points

Despite a large corpus of recent work on scaling up Gaussian processes, a stubborn trade-off between computational speed, prediction and uncertainty quantification accuracy, and customizability persists. This is because the vast majority of existing methodologies exploit various levels of approximations that lower accuracy and limit the flexibility of kernel and noise-model designs -- an unacceptable drawback at a time when expressive non-stationary kernels are on the rise in many fields. Here, we propose a methodology we term \emph{gp2Scale} that scales exact Gaussian processes to more than 10 million data points without relying on inducing points, kernel interpolation, or neighborhood-based approximations, and instead leveraging the existing capabilities of a GP: its kernel design. Highly flexible, compactly supported, and non-stationary kernels lead to the identification of naturally occurring sparse structure in the covariance matrix, which is then exploited for the calculations of the linear system solution and the log-determinant for training. We demonstrate our method's functionality on several real-world datasets and compare it with state-of-the-art approximation algorithms. Although we show superior approximation performance in many cases, the method's real power lies in its agnosticism toward arbitrary GP customizations -- core kernel design, noise, and mean functions -- and the type of input space, making it optimally suited for modern Gaussian process applications.

cs.LG↗

arXiv2026

Wasserstein-type Gaussian Process Regressions for Input Measurement Uncertainty

Gaussian process (GP) regression is widely used for uncertainty quantification, yet the standard formulation assumes noise-free covariates. When inputs are measured with error, this errors-in-variables (EIV) setting can lead to optimistically narrow posterior intervals and biased decisions. We study GP regression under input measurement uncertainty by representing each noisy input as a probability measure and defining covariance through Wasserstein distances between these measures. Building on this perspective, we instantiate a deterministic projected Wasserstein ARD (PWA) kernel whose one-dimensional components admit closed-form expressions and whose product structure yields a scalable, positive-definite kernel on distributions. Unlike latent-input GP models, PWA-based GPs (\PWAGPs) handle input noise without introducing unobserved covariates or Monte Carlo projections, making uncertainty quantification more transparent and robust.

stat.ME↗

arXiv2026

GGMPs: Generalized Gaussian Mixture Processes

Conditional density estimation is complicated by multimodality, heteroscedasticity, and strong non-Gaussianity. Gaussian processes (GPs) provide a principled nonparametric framework with calibrated uncertainty, but standard GP regression is limited by its unimodal Gaussian predictive form. We introduce the Generalized Gaussian Mixture Process (GGMP), a GP-based method for multimodal conditional density estimation in settings where each input may be associated with a complex output distribution rather than a single scalar response. GGMP combines local Gaussian mixture fitting, cross-input component alignment and per-component heteroscedastic GP training to produce a closed-form Gaussian mixture predictive density. The method is tractable, compatible with standard GP solvers and scalable methods, and avoids the exponentially large latent-assignment structure of naive multimodal GP formulations. Empirically, GGMPs improve distributional approximation on synthetic and real-world datasets with pronounced non-Gaussianity and multimodality.

cs.LG↗

arXiv2025

Spatial scale-aware tail dependence modeling for high-dimensional spatial extremes

Extreme events over large spatial domains may exhibit highly heterogeneous tail dependence characteristics, yet most existing spatial extremes models yield only one dependence class over the entire spatial domain. To accurately characterize "data-level dependence'' in analysis of extreme events, we propose a mixture model that achieves flexible dependence properties and allows high-dimensional inference for extremes of spatial processes. We modify the popular random scale construction that multiplies a Gaussian random field by a single radial variable; we allow the radial variable to vary smoothly across space and add non-stationarity to the Gaussian process. As the level of extremeness increases, this single model exhibits both asymptotic independence at long ranges and either asymptotic dependence or independence at short ranges. We make joint inference on the dependence model and a marginal model using a copula approach within a Bayesian hierarchical model. Three different simulation scenarios show close to nominal frequentist coverage rates. Lastly, we apply the model to a dataset of extreme summertime precipitation over the central United States. We find that the joint tail of precipitation exhibits non-stationary dependence structure that cannot be captured by limiting extreme value models or current state-of-the-art sub-asymptotic models.

stat.ME↗

arXiv2025

Compactly-supported nonstationary kernels for computing exact Gaussian processes on big data

The Gaussian process (GP) is a widely used probabilistic machine learning method with implicit uncertainty characterization for stochastic function approximation, stochastic modeling, and analyzing real-world measurements of nonlinear processes. Traditional implementations of GPs involve stationary kernels (also termed covariance functions) that limit their flexibility, and exact methods for inference that prevent application to data sets with more than about ten thousand points. Modern approaches to address stationarity assumptions generally fail to accommodate large data sets, while all attempts to address scalability focus on approximating the Gaussian likelihood, which can involve subjectivity and lead to inaccuracies. In this work, we explicitly derive an alternative kernel that can discover and encode both sparsity and nonstationarity. We embed the kernel within a fully Bayesian GP model and leverage high-performance computing resources to enable the analysis of massive data sets. We demonstrate the favorable performance of our novel kernel relative to existing exact and approximate GP methods across a variety of synthetic data examples. Furthermore, we conduct space-time prediction based on more than one million measurements of daily maximum temperature and verify that our results outperform state-of-the-art methods in the Earth sciences. More broadly, having access to exact GPs that use ultra-scalable, sparsity-discovering, nonstationary kernels allows GP methods to truly compete with a wide variety of machine learning methods.

stat.ML↗

arXiv2025

Granger causal inference for climate change attribution

Climate change detection and attribution (D&A) is concerned with determining the extent to which anthropogenic activities have influenced specific aspects of the global climate system. D&A fits within the broader field of causal inference, the collection of statistical methods that identify cause and effect relationships. There are a wide variety of methods for making attribution statements, each of which require different types of input data and each of which are conditional to varying extents. Some methods are based on Pearl causality (experimental interference) while others leverage Granger (predictive) causality, and the causal framing provides important context for how the resulting attribution conclusion should be interpreted. However, while Granger-causal attribution analyses have become more common, there is no clear statement of their strengths and weaknesses and no clear consensus on where and when Granger-causal perspectives are appropriate. In this prospective paper, we provide a formal definition for Granger-based approaches to trend and event attribution and a clear comparison with more traditional methods for assessing the human influence on extreme weather and climate events. Broadly speaking, Granger-causal attribution statements can be constructed quickly from observations and do not require computationally-intesive dynamical experiments. These analyses also enable rapid attribution, which is useful in the aftermath of a severe weather event, and provide multiple lines of evidence for anthropogenic climate change when paired with Pearl-causal attribution. Confidence in attribution statements is increased when different methodologies arrive at similar conclusions. Moving forward, we encourage the D&A community to embrace hybrid approaches to climate change attribution that leverage the strengths of both Granger and Pearl causality.

stat.AP↗

arXiv2025

Data-driven upper bounds and event attribution for unprecedented heatwaves

The last decade has seen numerous record-shattering heatwaves in all corners of the globe. In the aftermath of these devastating events, there is interest in identifying worst-case thresholds or upper bounds that quantify just how hot temperatures can become. Generalized Extreme Value theory provides a data-driven estimate of extreme thresholds; however, upper bounds may be exceeded by future events, which undermines attribution and planning for heatwave impacts. Here, we show how the occurrence and relative probability of observed events that exceed a priori upper bound estimates, so-called "impossible" temperatures, has changed over time. We find that many unprecedented events are actually within data-driven upper bounds, but only when using modern spatial statistical methods. Furthermore, there are clear connections between anthropogenic forcing and the "impossibility" of the most extreme temperatures. Robust understanding of heatwave thresholds provides critical information about future record-breaking events and how their extremity relates to historical measurements.

physics.data-an↗

arXiv2024

A Unifying Perspective on Non-Stationary Kernels for Deeper Gaussian Processes

The Gaussian process (GP) is a popular statistical technique for stochastic function approximation and uncertainty quantification from data. GPs have been adopted into the realm of machine learning in the last two decades because of their superior prediction abilities, especially in data-sparse scenarios, and their inherent ability to provide robust uncertainty estimates. Even so, their performance highly depends on intricate customizations of the core methodology, which often leads to dissatisfaction among practitioners when standard setups and off-the-shelf software tools are being deployed. Arguably the most important building block of a GP is the kernel function which assumes the role of a covariance operator. Stationary kernels of the Matérn class are used in the vast majority of applied studies; poor prediction performance and unrealistic uncertainty quantification are often the consequences. Non-stationary kernels show improved performance but are rarely used due to their more complicated functional form and the associated effort and expertise needed to define and tune them optimally. In this perspective, we want to help ML practitioners make sense of some of the most common forms of non-stationarity for Gaussian processes. We show a variety of kernels in action using representative datasets, carefully study their properties, and compare their performances. Based on our findings, we propose a new kernel that combines some of the identified advantages of existing kernels.

stat.ML↗

arXiv2024

A flexible class of priors for orthonormal matrices with basis function-specific structure

Statistical modeling of high-dimensional matrix-valued data motivates the use of a low-rank representation that simultaneously summarizes key characteristics of the data and enables dimension reduction. Low-rank representations commonly factor the original data into the product of orthonormal basis functions and weights, where each basis function represents an independent feature of the data. However, the basis functions in these factorizations are typically computed using algorithmic methods that cannot quantify uncertainty or account for basis function correlation structure a priori. While there exist Bayesian methods that allow for a common correlation structure across basis functions, empirical examples motivate the need for basis function-specific dependence structure. We propose a prior distribution for orthonormal matrices that can explicitly model basis function-specific structure. The prior is used within a general probabilistic model for singular value decomposition to conduct posterior inference on the basis functions while accounting for measurement error and fixed effects. We discuss how the prior specification can be used for various scenarios and demonstrate favorable model properties through synthetic data examples. Finally, we apply our method to two-meter air temperature data from the Pacific Northwest, enhancing our understanding of the Earth system's internal variability.

stat.ME↗

arXiv2023

Explaining the unexplainable: leveraging extremal dependence to characterize the 2021 Pacific Northwest heatwave

In late June, 2021, a devastating heatwave affected the US Pacific Northwest and western Canada, breaking numerous all-time temperature records by large margins and directly causing hundreds of fatalities. The observed 2021 daily maximum temperature across much of the U.S. Pacific Northwest exceeded upper bound estimates obtained from single-station temperature records even after accounting for anthropogenic climate change, meaning that the event could not have been predicted under standard univariate extreme value analysis assumptions. In this work, we utilize a flexible spatial extremes model that considers all stations across the Pacific Northwest domain and accounts for the fact that many stations simultaneously experience extreme temperatures. Our analysis incorporates the effects of anthropogenic forcing and natural climate variability in order to better characterize time-varying changes in the distribution of daily temperature extremes. We show that greenhouse gas forcing, drought conditions and large-scale atmospheric modes of variability all have significant impact on summertime maximum temperatures in this region. Our model represents a significant improvement over corresponding single-station analysis, and our posterior medians of the upper bounds are able to anticipate more than 96% of the observed 2021 high station temperatures after properly accounting for extremal dependence.

stat.AP↗

arXiv2022

Exact Gaussian Processes for Massive Datasets via Non-Stationary Sparsity-Discovering Kernels

A Gaussian Process (GP) is a prominent mathematical framework for stochastic function approximation in science and engineering applications. This success is largely attributed to the GP's analytical tractability, robustness, non-parametric structure, and natural inclusion of uncertainty quantification. Unfortunately, the use of exact GPs is prohibitively expensive for large datasets due to their unfavorable numerical complexity of $O(N^3)$ in computation and $O(N^2)$ in storage. All existing methods addressing this issue utilize some form of approximation -- usually considering subsets of the full dataset or finding representative pseudo-points that render the covariance matrix well-structured and sparse. These approximate methods can lead to inaccuracies in function approximations and often limit the user's flexibility in designing expressive kernels. Instead of inducing sparsity via data-point geometry and structure, we propose to take advantage of naturally-occurring sparsity by allowing the kernel to discover -- instead of induce -- sparse structure. The premise of this paper is that GPs, in their most native form, are often naturally sparse, but commonly-used kernels do not allow us to exploit this sparsity. The core concept of exact, and at the same time sparse GPs relies on kernel definitions that provide enough flexibility to learn and encode not only non-zero but also zero covariances. This principle of ultra-flexible, compactly-supported, and non-stationary kernels, combined with HPC and constrained optimization, lets us scale exact GPs well beyond 5 million data points.

stat.ML↗

arXiv2020

Bayesian inference for high-dimensional nonstationary Gaussian processes

In spite of the diverse literature on nonstationary spatial modeling and approximate Gaussian process (GP) methods, there are no general approaches for conducting fully Bayesian inference for moderately sized nonstationary spatial data sets on a personal laptop. For statisticians and data scientists who wish to learn about spatially-referenced data and conduct posterior inference and prediction with appropriate uncertainty quantification, the lack of such approaches and corresponding software is a significant limitation. In this paper, we develop methodology for implementing formal Bayesian inference for a general class of nonstationary GPs. Our novel approach uses pre-existing frameworks for characterizing nonstationarity in a new way that is applicable for small to moderately sized data sets via modern GP likelihood approximations. Posterior sampling is implemented using flexible MCMC methods, with nonstationary posterior prediction conducted as a post-processing step. We demonstrate our novel methods on two data sets, ranging from several hundred to several thousand locations, and compare our methodology with related statistical methods that provide off-the-shelf software. All of our methods are implemented in the freely available BayesNSGP software package for R.

stat.CO↗

arXiv2020

The effect of geographic sampling on evaluation of extreme precipitation in high resolution climate models

Traditional approaches for comparing global climate models and observational data products typically fail to account for the geographic location of the underlying weather station data. For modern high-resolution models, this is an oversight since there are likely grid cells where the physical output of a climate model is compared with a statistically interpolated quantity instead of actual measurements of the climate system. In this paper, we quantify the impact of geographic sampling on the relative performance of high resolution climate models' representation of precipitation extremes in Boreal winter (DJF) over the contiguous United States (CONUS), comparing model output from five early submissions to the HighResMIP subproject of the CMIP6 experiment. We find that properly accounting for the geographic sampling of weather stations can significantly change the assessment of model performance. Across the models considered, failing to account for sampling impacts the different metrics (extreme bias, spatial pattern correlation, and spatial variability) in different ways (both increasing and decreasing). We argue that the geographic sampling of weather stations should be accounted for in order to yield a more straightforward and appropriate comparison between models and observational data sets, particularly for high resolution models. While we focus on the CONUS in this paper, our results have important implications for other global land regions where the sampling problem is more severe.

stat.AP↗

arXiv2019

Detected changes in precipitation extremes at their native scales derived from in situ measurements

The gridding of daily accumulated precipitation -- especially extremes -- from ground-based station observations is problematic due to the fractal nature of precipitation, and therefore estimates of long period return values and their changes based on such gridded daily data sets are generally underestimated. In this paper, we characterize high-resolution changes in observed extreme precipitation from 1950 to 2017 for the contiguous United States (CONUS) based on in situ measurements only. Our analysis utilizes spatial statistical methods that allow us to derive gridded estimates that do not smooth extreme daily measurements and are consistent with statistics from the original station data while increasing the resulting signal to noise ratio. Furthermore, we use a robust statistical technique to identify significant pointwise changes in the climatology of extreme precipitation while carefully controlling the rate of false positives. We present and discuss seasonal changes in the statistics of extreme precipitation: the largest and most spatially-coherent pointwise changes are in fall (SON), with approximately 33% of CONUS exhibiting significant changes (in an absolute sense). Other seasons display very few meaningful pointwise changes (in either a relative or absolute sense), illustrating the difficulty in detecting pointwise changes in extreme precipitation based on in situ measurements. While our main result involves seasonal changes, we also present and discuss annual changes in the statistics of extreme precipitation. In this paper we only seek to detect changes over time and leave attribution of the underlying causes of these changes for future work.

stat.AP↗

arXiv2019

A probabilistic gridded product for daily precipitation extremes over the United States

Gridded data products, for example interpolated daily measurements of precipitation from weather stations, are commonly used as a convenient substitute for direct observations because these products provide a spatially and temporally continuous and complete source of data. However, when the goal is to characterize climatological features of extreme precipitation over a spatial domain (e.g., a map of return values) at the native spatial scales of these phenomena, then gridded products may lead to incorrect conclusions because daily precipitation is a fractal field and hence any smoothing technique will dampen local extremes. To address this issue, we create a new "probabilistic" gridded product specifically designed to characterize the climatological properties of extreme precipitation by applying spatial statistical analyses to daily measurements of precipitation from the GHCN over CONUS. The essence of our method is to first estimate the climatology of extreme precipitation based on station data and then use a data-driven statistical approach to interpolate these estimates to a fine grid. We argue that our method yields an improved characterization of the climatology within a grid cell because the probabilistic behavior of extreme precipitation is much better behaved (i.e., smoother) than daily weather. Furthermore, the spatial smoothing innate to our approach significantly increases the signal-to-noise ratio in the estimated extreme statistics relative to an analysis without smoothing. Finally, by deriving a data-driven approach for translating extreme statistics to a spatially complete grid, the methodology outlined in this paper resolves the issue of how to properly compare station data with output from earth system models. We conclude the paper by comparing our probabilistic gridded product with a standard extreme value analysis of the Livneh gridded daily precipitation product.

stat.AP↗

arXiv2018

Nonstationary Spatial Prediction of Soil Organic Carbon: Implications for Stock Assessment Decision Making

The Rapid Carbon Assessment (RaCA) project was conducted by the US Department of Agriculture's National Resources Conservation Service between 2010-2012 in order to provide contemporaneous measurements of soil organic carbon (SOC) across the US. Despite the broad extent of the RaCA data collection effort, direct observations of SOC are not available at the high spatial resolution needed for studying carbon storage in soil and its implications for important problems in climate science and agriculture. As a result, there is a need for predicting SOC at spatial locations not included as part of the RaCA project. In this paper, we compare spatial prediction of SOC using a subset of the RaCA data for a variety of statistical methods. We investigate the performance of methods with off-the-shelf software available (both stationary and nonstationary) as well as a novel nonstationary approach based on partitioning relevant spatially-varying covariate processes. Our new method addresses open questions regarding (1) how to partition the spatial domain for segmentation-based nonstationary methods, (2) incorporating partially observed covariates into a spatial model, and (3) accounting for uncertainty in the partitioning. In applying the various statistical methods we find that there are minimal differences in out-of-sample criteria for this particular data set, however, there are major differences in maps of uncertainty in SOC predictions. We argue that the spatially-varying measures of prediction uncertainty produced by our new approach are valuable to decision makers, as they can be used to better benchmark mechanistic models, identify target areas for soil restoration projects, and inform carbon sequestration projects.

stat.AP↗

arXiv2017

Spatially-Dependent Multiple Testing Under Model Misspecification, with Application to Detection of Anthropogenic Influence on Extreme Climate Events

The Weather Risk Attribution Forecast (WRAF) is a forecasting tool that uses output from global climate models to make simultaneous attribution statements about whether and how greenhouse gas emissions have contributed to extreme weather across the globe. However, in conducting a large number of simultaneous hypothesis tests, the WRAF is prone to identifying false "discoveries." A common technique for addressing this multiple testing problem is to adjust the procedure in a way that controls the proportion of true null hypotheses that are incorrectly rejected, or the false discovery rate (FDR). Unfortunately, generic FDR procedures suffer from low power when the hypotheses are dependent, and techniques designed to account for dependence are sensitive to misspecification of the underlying statistical model. In this paper, we develop a Bayesian decision theoretic approach for dependent multiple testing and a nonparametric hierarchical statistical model that flexibly controls false discovery and is robust to model misspecification. We illustrate the robustness of our procedure to model error with a simulation study, using a framework that accounts for generic spatial dependence and allows the practitioner to flexibly specify the decision criteria. Finally, we apply our procedure to several seasonal forecasts and discuss implementation for the WRAF workflow.

stat.AP↗