Spatiotemporal Covariance Structure Research Articles

Covariate-dependent spatio-temporal covariance models

Geostatistical regression models are widely used in environmental and geophysical sciences to characterize the mean and dependence structures for spatio-temporal data. Traditionally, these models account for covariates solely in the mean structure, neglecting their potential impact on the spatio-temporal covariance structure. This paper addresses a significant gap in the literature by proposing a novel covariate-dependent covariance model within the spatio-temporal random-effects model framework. Our approach integrates covariates into the covariance function through a Cholesky-type decomposition, ensuring compliance with the positive-definite condition. We employ maximum likelihood for parameter estimation, complemented by an efficient expectation conditional maximization algorithm. Simulation studies demonstrate the superior performance of our method compared to conventional techniques that ignore covariates in spatial covariances. We further apply our model to a PM2.5 dataset from Taiwan, highlighting wind speed’s pivotal role in influencing the spatio-temporal covariance structure. Additionally, we incorporate wind speed and sunshine duration into the covariance function for analyzing Taiwan ozone data, revealing a more intricate relationship between covariance and these meteorological variables.

Representing natural climate variability in an event attribution context: Indo-Pakistani heatwave of 2022

Attribution of extreme climate events to global climate change as a result of anthropogenic greenhouse gas emissions has become increasingly important. Extreme climate events arise at the intersection of natural climate variability and a forced response of the Earth system to anthropogenic greenhouse gas emissions, which may alter the frequency and severity of such events. Accounting for the effects of both natural climate variability and the forced response to anthropogenic climate change is thus central for the attribution. Here, we investigate the reproducibility of probabilistic extreme event attribution results under more explicit representations of natural climate variability. We employ well-established methodologies deployed in statistical Earth System Model emulators to represent natural climate variability as informed from its spatio-temporal covariance structures. Two approaches towards representing natural climate variability are investigated: (1) where natural climate variability is treated as a single component; and (2) where natural climate variability is disentangled into its annual and seasonal components. We showcase our approaches by attributing the 2022 Indo-Pakistani heatwave to human-induced climate change. We find that explicit representation of annual and seasonal natural climate variability increases the overall uncertainty in attribution results considerably compared to established approaches such as the World Weather Attribution Initiative. The increase in likelihood of such an event occurring as a result of global warming differs slightly between the approaches, mainly due to different assessments of the pre-industrial return periods. Our approach that explicitly resolves annual and seasonal natural climate variability indicates a median increase in likelihood by a factor of 41 (95% range: 6-603). We find a robust signal of increased likelihood and intensification of the event with increasing global warming levels across all approaches. Compared to its present likelihood, under 1.5 °C (2 °C) of global near-surface air temperature increase relative to pre-industrial temperatures, the likelihood of the event would be between 2.2 to 2.5 times (8 to 9 times) higher. We note that regardless of the different statistical approaches to represent natural variability, the outcomes on the conducted event attribution are similar, with minor differences mainly in the uncertainty ranges. Possible reasons for differences are evaluated, including limitations of the proposed approach for this type of application, as well as the specific aspects in which it can provide complementary information to established approaches.

Various ways of using empirical orthogonal functions for climate model evaluation

Abstract. We present a framework for evaluating multi-model ensembles based on common empirical orthogonal functions (common EOFs) that emphasize salient features connected to spatio-temporal covariance structures embedded in large climate data volumes. This framework enables the extraction of the most pronounced spatial patterns of coherent variability within the joint dataset and provides a set of weights for each model in terms of the principal components which refer to exactly the same set of spatial patterns of covariance. In other words, common EOFs provide a means for extracting information from large volumes of data. Moreover, they can provide an objective basis for evaluation that can be used to accentuate ensembles more than traditional methods for evaluation, which tend to focus on individual models. Our demonstration of the capability of common EOFs reveals a statistically significant improvement of the sixth generation of the World Climate Research Programme (WCRP) Climate Model Intercomparison Project (CMIP6) simulations in comparison to the previous generation (CMIP5) in terms of their ability to reproduce the mean seasonal cycle in air surface temperature, precipitation, and mean sea level pressure over the Nordic countries. The leading common EOF principal component for annually or seasonally aggregated temperature, precipitation, and pressure statistics suggests that their simulated interannual variability is generally consistent with that seen in the ERA5 reanalysis. We also demonstrate how common EOFs can be used to analyse whether CMIP ensembles reproduce the observed historical trends over the historical period 1959–2021, and the results suggest that the trend statistics provided by both CMIP5 RCP4.5 and CMIP6 SSP245 are consistent with observed trends. An interesting finding is also that the leading common EOF principal component for annually or seasonally aggregated statistics seems to be approximately normally distributed, which is useful information about the multi-model ensemble data.

Detecting relevant changes in the spatiotemporal mean function

For a spatiotemporal process , where denotes the set of spatial locations and the time domain, we consider the problem of testing for a change in the sequence of mean functions . In contrast to most of the literature, we are not interested in arbitrarily small changes but only in changes with a norm exceeding a given threshold. Asymptotically distribution free tests are proposed, which do not require the estimation of the long‐run spatiotemporal covariance structure. In particular, we consider a fully functional approach and a test based on the cumulative sum paradigm, investigate the large sample properties of the corresponding test statistics and study their finite sample properties by means of simulation study.

Principal component analysis for river network data: Use of spatiotemporal correlation and heterogeneous covariance structure

AbstractSpatiotemporal measurements observed through river networks have two distinct characteristics: a spatiotemporal correlation under the flow‐connected structure and the existence of heterogeneous covariances, which require a careful approach to implement principal component analysis (PCA). This paper focuses on developing a PCA method to reflect the unique characteristics of river networks. We propose a novel method combining flow‐directed PCA and geographically weighted PCA for the domain of river networks. The strengths of our approach are that it can (i) reduce dimensionality for streamflow data while effectively removing correlation among them and (ii) identify the group structure of data. It is possible to find essential patterns and sources of variation that may not be disclosed due to the attributes of flow‐connected networks. We apply the proposed method to the daily monitoring records of total organic carbon in the Geum River catchment area in South Korea. The results show that the proposed method successfully adjusts for the topological structure of the network and temporal correlation among observations while considering the spatial heterogeneity, enabling a more concrete understanding of monitoring networks.

Hybrid Modeling of Regional COVID-19 Transmission Dynamics in the U.S.

The fast transmission rate of COVID-19 worldwide has made this virus the most important challenge of year 2020. Many mitigation policies have been imposed by the governments at different regional levels (country, state, county, and city) to stop the spread of this virus. Quantifying the effect of such mitigation strategies on the transmission and recovery rates, and predicting the rate of new daily cases are two crucial tasks. In this paper, we propose a hybrid modeling framework which not only accounts for such policies but also utilizes the spatial and temporal information to characterize the pattern of COVID-19 progression. Specifically, a piecewise susceptible-infected-recovered (SIR) model is developed while the dates at which the transmission/recover rates change significantly are defined as "break points" in this model. A novel and data-driven algorithm is designed to locate the break points using ideas from fused lasso and thresholding. In order to enhance the forecasting power and to describe additional temporal dependence among the daily number of cases, this model is further coupled with spatial smoothing covariates and vector auto-regressive (VAR) model. The proposed model is applied to several U.S. states and counties, and the results confirm the effect of "stay-at-home orders" and some states' early "re-openings" by detecting break points close to such events. Further, the model provided satisfactory short-term forecasts of the number of new daily cases at regional levels by utilizing the estimated spatio-temporal covariance structures. They were also better or on par with other proposed models in the literature, including flexible deep learning ones. Finally, selected theoretical results and empirical performance of the proposed methodology on synthetic data are reported which justify the good performance of the proposed method.

Supervised linear classification of Gaussian spatio-temporal data

In this article we focus on the problem of supervised classifying of the spatio-temporal Gaussian random field observation into one of two classes, specified by different mean parameters. The main distinctive feature of the proposed approach is allowing the class label to depend on spatial location as well as on time moment. It is assumed that the spatio-temporal covariance structure factors into a purely spatial component and a purely temporal component following AR(p) model. In numerical illustrations with simulated data, the influence of the values of spatial and temporal covariance parameters to the derived error rates for several prior probabilities models are studied.

Frequency-Dependent Estimation of Effective Spatial Degrees of Freedom

AbstractClimate variability occurs over wide ranges of spatial and temporal scales. It exhibits a complex spatial covariance structure, which depends on geographic location (e.g., tropics vs extratropics) and also consists of a superposition of (i) components with gradually decaying positive correlation functions and (ii) teleconnections that often involve anticorrelations. In addition, there are indications that the spatial covariance structure depends on frequency. Thus, a comprehensive assessment of the spatiotemporal covariance structure of climate variability would require an extensive set of statistical diagnostics. Therefore, it is often desirable to characterize the covariance structure by a simple summarizing metric that is easy to compute from datasets. Such summarizing metrics are useful, for example, in the context of comparisons between climate models or between models and observations. Here we introduce a frequency-dependent version of a simple measure of the effective spatial degrees of freedom. The measure is based on the temporal variance of the global average of some climate variable, and its novel aspect consists in its frequency dependence. We also provide a clear geometric interpretation of the measure. Its easy applicability is demonstrated using near-surface temperature and precipitation fields obtained from a paleoclimate model simulation. This application reveals a distinct scaling behavior of the spatial degrees of freedom as a function of frequency, ranging from monthly to millennial scales.

Spatio-temporal Cokriging method for assimilating and downscaling multi-scale remote sensing data

No single satellite remote sensing system is able to provide the observations on the Earth's surface at both high spatial and high temporal resolution due to the general trade-off between orbit revisit frequency and satellite sensor's spatial resolution. This paper presents a spatio-temporal Cokriging (ST-Cokriging) method for assimilating remote sensing data sets acquired by multiple remote sensing systems with different temporal sampling frequencies and different spatial resolutions. By extending the traditional Cokriging technique from a sole spatial domain to a spatio-temporal domain, we derived and implemented ST-Cokriging algorithm that explicitly takes the spatial covariance, temporal covariance and spatio-temporal covariance structures within and between different data sets into account. Compared with previous downscaling methods, such as, STARFM and FSDAF, our ST-Cokriging method produces more accurate and reliable assimilation results for the heterogeneous region, with associated uncertainty estimates. This method has been implemented into a software package using Python language within ArcGIS environment. The advantages and effectiveness of our ST-Cokriging method have been demonstrated through an application example, in which MODIS images (daily, 250 m and 500 m spatial resolution) and Landsat TM/ETM+ images (16 days revisit cycle, 30 m) are assimilated to generate daily spectral bands and NDVI images at 30 m spatial resolution. Our validation and accuracy assessments indicate that our ST-Cokriging method can effectively fill in data gaps due to clouds and generate reliable assimilation results and uncertainty estimates at both high spatial resolution and high temporal frequency

Spatio-temporal modeling of an environmental trivariate vector combining air and soil measurements from Ireland

In environmental sciences, it is very common to observe spatio-temporal multiple data concerning several correlated variables which are measured in time over a monitored spatial domain. In multivariate Geostatistics, the evaluation of their behavior is often based on the knowledge of the spatio-temporal multivariate covariance structure. Since this last is often unknown it has to be estimated and modeled. In this paper, a spatio-temporal multivariate analysis of three relevant environmental indicators, which include 10-centimeter soil temperature, minimum and maximum air temperature, is proposed. This study is of particular interest for its reflection in ecology and the lack of information due to the presence of monitoring networks for soil and air variables characterized by different levels of spatial and temporal detail. A space–time linear coregionalization model (ST-LCM) with suitable models for the latent components of the variables under study is selected by using a simple procedure.

Propagation of BOLD Activity Reveals Task-dependent Directed Interactions Across Human Visual Cortex.

It has recently been shown that large-scale propagation of blood-oxygen-level-dependent (BOLD) activity is constrained by anatomical connections and reflects transitions between behavioral states. It remains to be seen, however, if the propagation of BOLD activity can also relate to the brain's anatomical structure at a more local scale. Here, we hypothesized that BOLD propagation reflects structured neuronal activity across early visual field maps. To explore this hypothesis, we characterize the propagation of BOLD activity across V1, V2, and V3 using a modeling approach that aims to disentangle the contributions of local activity and directed interactions in shaping BOLD propagation. It does so by estimating the effective connectivity (EC) and the excitability of a noise-diffusion network to reproduce the spatiotemporal covariance structure of the data. We apply our approach to 7T fMRI recordings acquired during resting state (RS) and visual field mapping (VFM). Our results reveal different EC interactions and changes in cortical excitability in RS and VFM, and point to a reconfiguration of feedforward and feedback interactions across the visual system. We conclude that the propagation of BOLD activity has functional relevance, as it reveals directed interactions and changes in cortical excitability in a task-dependent manner.

Numerical methods for the deterministic second moment equation of parabolic stochastic PDEs

Numerical methods for stochastic partial differential equations typically estimate moments of the solution from sampled paths. Instead, we shall directly target the deterministic equations satisfied by the mean and the spatio-temporal covariance structure of the solution process. In the first part, we focus on stochastic ordinary differential equations. For the canonical examples with additive noise (Ornstein–Uhlenbeck process) or multiplicative noise (geometric Brownian motion) we derive these deterministic equations in variational form and discuss their well-posedness in detail. Notably, the second moment equation in the multiplicative case is naturally posed on projective–injective tensor product spaces as trial–test spaces. We then propose numerical approximations based on Petrov–Galerkin discretizations with tensor product piecewise polynomials and analyze their stability and convergence in the natural tensor norms. In the second part, we proceed with parabolic stochastic partial differential equations with affine multiplicative noise. We prove well-posedness of the deterministic variational problem for the second moment, improving an earlier result. We then propose conforming space-time Petrov–Galerkin discretizations, which we show to be stable and quasi-optimal. In both parts, the outcomes are validated by numerical examples.

Geostatistical Analysis and Mitigation of the Atmospheric Phase Screens in Ku-Band Terrestrial Radar Interferometric Observations of an Alpine Glacier

Terrestrial radar interferometry (TRI) can measure displacements at high temporal resolution, potentially with high accuracy. An application of this method is the observation of the surface flow velocity of steep, fast-flowing aglaciers. For these observations, the main factor limiting the accuracy of TRI observations is the spatial and temporal variabilities in the distribution of atmospheric water vapor content, causing a phase delay [atmospheric phase screen (APS)] whose magnitude is similar to the displacement phase. This contribution presents a geostatistical analysis of the spatial and temporal behaviors of the APS in Ku-Band TRI. The analysis is based on the assumption of a separable spatiotemporal covariance structure, which is tested empirically using variogram analysis. From this analysis, spatial and temporal APS statistics are estimated and used in a two-step procedure combining regression-Kriging with generalized least squares (GLS) inversion to estimate a velocity time-series. The performance of this method is evaluated by cross-validation using phase observations on stable scatterers. This analysis shows a considerable reduction in residual phase variance compared with the standard approach of combining the linear models of APS stratification and interferogram stacking.

Assessing the performance of the BARCAST climate field reconstruction technique for a climate with long-range memory

Abstract. The skill of the state-of-the-art climate field reconstruction technique BARCAST (Bayesian Algorithm for Reconstructing Climate Anomalies in Space and Time) to reconstruct temperature with pronounced long-range memory (LRM) characteristics is tested. A novel technique for generating fields of target data has been developed and is used to provide ensembles of LRM stochastic processes with a prescribed spatial covariance structure. Based on different parameter setups, hypothesis testing in the spectral domain is used to investigate if the field and spatial mean reconstructions are consistent with either the fractional Gaussian noise (fGn) process null hypothesis used for generating the target data, or the autoregressive model of order 1 (AR(1)) process null hypothesis which is the assumed temporal evolution model for the reconstruction technique. The study reveals that the resulting field and spatial mean reconstructions are consistent with the fGn process hypothesis for some of the tested parameter configurations, while others are in better agreement with the AR(1) model. There are local differences in reconstruction skill and reconstructed scaling characteristics between individual grid cells, and the agreement with the fGn model is generally better for the spatial mean reconstruction than at individual locations. Our results demonstrate that the use of target data with a different spatiotemporal covariance structure than the BARCAST model assumption can lead to a potentially biased climate field reconstruction (CFR) and associated confidence intervals.

A comprehensive and systematic evaluation framework for a parsimonious daily rainfall field model

The spatial distribution of rainfall has a significant influence on catchment dynamics and the generation of streamflow time series. However, there are few stochastic models that can simulate long sequences of stochastic rainfall fields continuously in time and space. To address this issue, the first goal of this study was to present a new parsimonious stochastic model that produces daily rainfall fields across the catchment. To achieve parsimony, the model used the latent-variable approach (because this parsimoniously simulates rainfall occurrences as well as amounts) and several other assumptions (including contemporaneous and separable spatiotemporal covariance structures). The second goal was to develop a comprehensive and systematic evaluation (CASE) framework to identify model strengths and weaknesses. This included quantitative performance categorisation that provided a systematic, succinct and transparent method to assess and summarise model performance over a range of statistics, sites, scales and seasons. The model is demonstrated using a case study from the Onkaparinga catchment in South Australia. The model showed many strengths in reproducing the observed rainfall characteristics with the majority of statistics classified as either statistically indistinguishable from the observed or within 5% of the observed across the majority of sites and seasons. These included rainfall occurrences/amounts, wet/dry spell distributions, annual volumes/extremes and spatial patterns, which are important from a hydrological perspective. One of the few weaknesses of the model was that the total annual rainfall in dry years (lower 5%) was overestimated by 15% on average over all sites. An advantage of the CASE framework was that it was able to identify the source of this overestimation was poor representation of the annual variability of rainfall occurrences. Given the strengths of this continuous daily rainfall field model it has a range of potential hydrological applications, including drought and flood risk.

Spatio-temporal Modelling of Forest Growth Spanning 50 Years – The Effects of Different Thinning Strategies

Models for tree growth in the literature include spatial models and dynamic models. Spatial models have been mostly restricted to Gaussian variograms and comparisons at single time points while dynamic models ignore tree competition due to spatial proximity. Here we model tree growth in forest plots with a spatio-temporal covariance structure and compare plots with different thinnings. Tree growth is measured by diameter at breast height. A model was fitted using a sum-metric spatio-temporal variogram. Fixed effects of age, number of neighbours and polygon area were also considered. Predicted values for models were computed using regression-kriging and mean squared error of prediction was used to compare models and thinning strategies. Over time the correlation of a single tree followed a wave variogram while the spatio-temporal anisotropy parameter captured the changing spatial wave intensity. Thinned plots clearly outperformed unthinned plots in terms of total tree growth.

Analysis of space–time patterns of rainfall events during 1996–2008 in Yilan County (Taiwan)

Understanding local precipitation patterns is essential to water resource management and flood mitigation. Precipitation patterns can vary in space and time depending on factors from different spatial scales such as local topographical changes and macroscopic atmospheric circulation. This study applied the two-stage classification method to distinguish the space–time patterns of local precipitations in the two identified distinct synoptic conditions, i.e. summer and autumn, from 24 gauges during 1996–2008 in Yilan County, Taiwan. The proposed method classifies the synoptic and local conditions for the space–time rainfall patterns by using K-means coupled with empirical orthogonal function analysis, and hierarchical ascending clustering method respectively. The proposed two-stage classification method considers not only the magnitude and the space–time distribution of rainfall events, but also the associated synoptic conditions. The results identified three primary patterns of extreme and two patterns of normal events in both seasons. Regarding the extreme events from typhoons, wind directions and the frontal accompanied effect are major contributors to the magnitude and spatial distribution of rainfall events in the summer and autumn, respectively. Spatiotemporal covariance structures are used to characterize the variability of normal events, showing the increasing frequency of wide spatial and temporal ranges from the summer to autumn. In summary, the proposed classification analysis provides patterns associated with distinct underlying physical mechanisms and space–time characteristics. The general characteristics of rainfall patterns can provide insights for the hydrological modeling of local catchments under different climatic scenarios.

Joint Composite Estimating Functions in Spatiotemporal Models

SummaryModelling of spatiotemporal processes has received considerable attention in recent statistical research. However, owing to the high dimensionality of the data, the joint modelling of spatial and temporal processes presents a great computational challenge, in both likelihood-based and Bayesian approaches. We propose a joint composite estimating function approach to estimating spatiotemporal covariance structures. This substantially reduces the computational complexity and is more efficient than existing composite likelihood methods. The novelty of the proposed joint composite estimating function is rooted in the construction of three sets of estimating functions from spatial, temporal and spatiotemporal cross-pairs, which results in overidentified estimating functions. Thus, we form a joint inference function in a spirit that is similar to Hansen’s generalized method of moments. We show that under practical scenarios the estimator proposed is consistent and asymptotically normal. Simulation studies prove that our method performs well in finite samples. Finally, we illustrate the joint composite estimating function method by estimating the spatiotemporal dependence structure of airborne particulates (PM10) in the north-eastern USA over a 32-month period.

Probabilistic and Ensemble Representations of the Uncertainty in an IR/Microwave Satellite Precipitation Product

AbstractWhile current satellite techniques are theoretically capable of producing precipitation estimates to image pixel resolutions, significant uncertainty is present in such high-resolution products. This uncertainty is frequently difficult to characterize using scalar measures of additive error. This paper describes the development of a methodology to more fully represent the uncertainty in satellite precipitation retrievals. The methodology derives conditional probability distribution functions of rainfall on a pixel-by-pixel basis. This array of distribution functions is then combined with a simple model of the spatiotemporal covariance structure of the uncertainty in the precipitation field to stochastically generate an ensemble precipitation product. Each element of the ensemble represents an equiprobable realization of the precipitation field that is consistent with the original satellite data while containing a random element commensurate with the uncertainty in that field. The technique has been tested using data from the Tropical Rainfall Measuring Mission (TRMM) Texas and Florida Underflight Experiment (TEFLUN-B) field campaign.

Modeling transport effects on ground‐level ozone using a non‐stationary space–time model

AbstractThis article presents a novel autoregressive space–time model for ground‐level ozone data, which models not only spatio–temporal dynamics of hourly ozone concentrations, but also relationships between ozone concentrations and meteorological variables. The proposed model has a non‐separable spatio–temporal covariance function that depends on wind speed and wind direction, and hence is non‐stationary in both time and space. Ozone concentration for a given location and time is assumed to be directly influenced by ozone concentrations at neighboring locations at the previous time, via a weight function of space–time dynamics caused by wind speed and wind direction.To our knowledge, the proposed method is the first one to incorporate the transport effect of ozone into the spatio–temporal covariance structure. Moreover, it uses a computationally efficient space–time Kalman filter and can compute optimal spatio–temporal prediction at any location and at anytime very fast for given meteorological conditions. Ozone data from Taipei are used for illustration, in which the model parameters are estimated by maximum likelihood. Copyright © 2004 John Wiley & Sons, Ltd.