Spatial Functional Data Research Articles

Gaussian modeling with B-splines for spatial functional data on irregular domains

Functional Data Analysis (FDA) has emerged as a powerful framework for datasets that exhibit continuous variation over specified intervals. Unlike traditional methods, FDA treats data as functions, providing a more comprehensive understanding of their behavior and variability. This approach is particularly beneficial for exploring observations collected over time or space. The Spatial Functional Data Analysis (SFDA) incorporates spatial locations as an additional dimension. This integration enhances the ability to model complex associations between curves in the functional domain. Despite progress in analyzing multivariate spatial data, limited attention has been given to SFDA. This study addresses this gap by proposing a Bayesian Gaussian functional approach based on B-splines. The proposal also addresses challenges posed by irregularly spaced observations in the functional domain, a feature rarely considered in the literature. A simulation study is developed to evaluate performance. Finally, results are explored using a relatively novel real dataset from monitoring stations of temperature in Mexico City.

Spatial Functional Outlier Detection in Multivariate Spatial Functional Data

Multivariate spatial functional data consists of multiple functions of time-dependent attributes observed at each spatial point. This study focuses on detecting spatial outliers in spatial functional data. Firstly, we develop a new method called Mahalanobis Distance Spatial Outlier (MDSO) to detect functional outliers in the data. The method introduces the multivariate functional Mahalanobis semi-distance and multivariate pairwise functional Mahalanobis semi-distance metrics based on the multivariate functional principal components analysis to calculate the dissimilarity between functions at each spatial point. Via simulation, we show that MDSO performs better than the other competing methods. Secondly, MDSO has been extended to detect spatial functional outliers as well. The functional outliers can now be categorized as global or/and local functional outliers. The appropriate number of neighbors and the cut-off point for the degree of isolation are determined via simulation. Finally, we demonstrate the application of the MDSO on a water quality data set obtained from Sungai Klang basin in Malaysia. The results can be used to support the authority in making better decisions on the management of the river basin or other spatial data with time-independent attributes.

Fast Bayesian Functional Regression for Non-Gaussian Spatial Data

Functional generalized linear models (FGLM) have been widely used to study the relations between non-Gaussian response and functional covariates. However, most existing works for FGLM assume independence among observations and therefore they are of limited applicability for correlated data. A particularly important example is spatial functional data, where we observe functions over spatial domains, such as the age population curve or temperature curve at each areal unit. In this paper, we extend FGLM by incorporating spatial random effects. Especially, we study the relationship between the non-Gaussian response variable and functional covariates that are spatially observed. However, such model has computational and inferential challenges. The high-dimensional spatial random effects cause the slow mixing of Markov chain Monte Carlo (MCMC) algorithms. Furthermore, spatial confounding can lead to bias in parameter estimates and inflate their variances. To address these issues, we propose an efficient Bayesian method using a sparse reparameterization of high-dimensional random effects. We also study the average coverage probabilities of the credible intervals of functional parameters. We apply our methods to simulated and real data examples, including malaria incidence data and US COVID-19 data. The proposed method is fast while providing accurate functional estimates.

Climate model selection via conformal clustering of spatial functional data

Climate model selection stands as a critical process in climate science and research. It involves choosing the most appropriate climate models to address specific research questions, simulating climate behaviour, or making projections about future climate conditions. This paper proposes a new approach, using spatial functional data analysis, to asses which of the 18 EURO CORDEX simulation models work better for predicting average temperatures in the Campania region (Italy). The method involves two key steps: first, using functional data analysis to process climate variables and select optimal models by a hierarchical clustering procedure; second, validating the chosen models by proposing a new conformal prediction approach to the anomalies associated to each cluster.

Spatial Functional Data analysis: Irregular spacing and Bernstein polynomials

Spatial Functional Data (SFD) analysis is an emerging statistical framework that combines Functional Data Analysis (FDA) and spatial dependency modeling. Unlike traditional statistical methods, which treat data as scalar values or vectors, SFD considers data as continuous functions, allowing for a more comprehensive understanding of their behavior and variability. This approach is well-suited for analyzing data collected over time, space, or any other continuous domain. SFD has found applications in various fields, including economics, finance, medicine, environmental science, and engineering. This study proposes new functional Gaussian models incorporating spatial dependence structures, focusing on irregularly spaced data and reflecting spatially correlated curves. The model is based on Bernstein polynomial (BP) basis functions and utilizes a Bayesian approach for estimating unknown quantities and parameters. The paper explores the advantages and limitations of the BP model in capturing complex shapes and patterns while ensuring numerical stability. The main contributions of this work include the development of an innovative model designed for SFD using BP, the presence of a random effect to address associations between irregularly spaced observations, and a comprehensive simulation study to evaluate models’ performance under various scenarios. The work also presents one real application of Temperature in Mexico City, showcasing practical illustrations of the proposed model.

Spatially aware self-representation learning for tissue structure characterization and spatial functional genes identification.

Spatially resolved transcriptomics (SRT) enable the comprehensive characterization of transcriptomic profiles in the context of tissue microenvironments. Unveiling spatial transcriptional heterogeneity needs to effectively incorporate spatial information accounting for the substantial spatial correlation of expression measurements. Here, we develop a computational method, SpaSRL (spatially aware self-representation learning), which flexibly enhances and decodes spatial transcriptional signals to simultaneously achieve spatial domain detection and spatial functional genes identification. This novel tunable spatially aware strategy of SpaSRL not only balances spatial and transcriptional coherence for the two tasks, but also can transfer spatial correlation constraint between them based on a unified model. In addition, this joint analysis by SpaSRL deciphers accurate and fine-grained tissue structures and ensures the effective extraction of biologically informative genes underlying spatial architecture. We verified the superiority of SpaSRL on spatial domain detection, spatial functional genes identification and data denoising using multiple SRT datasets obtained by different platforms and tissue sections. Our results illustrate SpaSRL's utility in flexible integration of spatial information and novel discovery of biological insights from spatial transcriptomic datasets.

Bayesian clustering of spatial functional data with application to a human mobility study during COVID-19

The coronavirus (COVID-19) global pandemic has made a significant impact on people’s social activities. Cell phone mobility data provide unique and rich information on studying this impact. The motivating dataset of this study is the daily leaving-home index data at Harris County in Texas provided by SafeGraph. To study changes in daily leaving-home index and how they relate to public policy and sociodemographic variables, we propose a new Bayesian wavelet model for modeling and clustering spatial functional data, where domain partitioning is achieved by operating on the spanning trees. The resulting clusters can have arbitrary shapes and are spatially contiguous in the input domain. An efficient tailored reversible jump Markov chain Monte Carlo algorithm is proposed to implement the model. The method is applied to the spatial functional data of the daily percentages of people who left home. We focus on the time period covering both lockdown and phased reopening in Texas during the COVID-19 pandemic and study the changing behaviors of those functional curves. By linking the clustering results with the sociodemographic information, we identify several covariates of census blocks that have a noticeable impact on the clustering patterns of people’s mobility behaviors.

Spatial functional data modeling of plant reflectances

Plant reflectance spectra, the profile of light reflected by leaves across different wavelengths, supply the spectral signature for a species at a spatial location to enable estimation of functional and taxonomic diversity for plants. We consider leaf spectra as “responses” to be explained spatially. These reflectance spectra are also functions over wavelength that respond to the environment. Our motivating data are gathered for several plant families from the Greater Cape Floristic Region (GCFR) in South Africa and lead us to develop rich novel spatial models that can explain spectra for genera within families. Wavelength responses for an individual leaf are viewed as a function of wavelength, leading to functional data modeling. Local environmental features become covariates. We introduce a wavelength, covariate interaction, since the response to environmental regressors may vary with wavelength, as may variance. Formal spatial modeling enables prediction of reflectances for genera at unobserved locations with known environmental features. We incorporate spatial dependence, wavelength dependence, and space–wavelength interaction (in the spirit of space–time interaction). We implement out-of-sample validation for model selection, finding that the model features above are informative for the functional data analysis. We supply ecological interpretation of the results under the selected model.

Variograms for kriging and clustering of spatial functional data with phase variation

Spatial, amplitude and phase variations in spatial functional data are confounded. Conclusions from the popular functional trace-variogram, which quantifies spatial variation, can be misleading when analyzing misaligned functional data with phase variation. To remedy this, we describe a framework that extends amplitude-phase separation methods in functional data to the spatial setting, with a view towards performing clustering and spatial prediction. We propose a decomposition of the trace-variogram into amplitude and phase components, and quantify how spatial correlations between functional observations manifest in their respective amplitude and phase. This enables us to generate separate amplitude and phase clustering methods for spatial functional data, and develop a novel spatial functional interpolant at unobserved locations based on combining separate amplitude and phase predictions. Through simulations and real data analyses, we demonstrate advantages of our approach when compared to standard ones that ignore phase variation, through more accurate predictions and more interpretable clustering results.

Spatio-temporal characterization of long-term solar resource using spatial functional data analysis: Understanding the variability and complementarity of global horizontal irradiance in Ecuador

Understanding the spatio-temporal variability of the solar resource is crucial to effectively support solar power utilization. Unfortunately, long-term and high-resolved measurements of solar irradiance are generally scarce, challenging the characterization for larger areas. In this paper, we propose a methodology to characterize the spatio-temporal variability of global horizontal irradiance (GHI) at a regional scale using long-term satellite-derived data. Spatial functional data analysis (sFDA) is used to identify areas with similar intra-annual variability patterns. The methodology is applied to a 21-year period data on Ecuador retrieved from the National Solar Radiation Database. Being the first time that sFDA is used for this purpose, the results indicate that it provides an appropriate basis for the interannual variability and complementarity analyses. In Ecuador's mainland, twenty-two subregions with four seasonal patterns are identified. The highest GHI potential (5.4 kWhm−2d−1) with the lowest variability (3.4%) is found in the Inter-Andean valleys. Further, seasonal complementarities between the coast and western Andes are identified. In Galapagos, high values are found over all islands (≥4.8 kWhm−2d−1), characterized by three subregions with one seasonal pattern. Our findings provide the first comprehensive spatio-temporal characterization of GHI in Ecuador, which aims at supporting a sustainable energy transition in the country.

Recent developments in complex and spatially correlated functional data

As high-dimensional and high-frequency data are being collected on a large scale, the development of new statistical models is being pushed forward. Functional data analysis provides the required statistical methods to deal with large-scale and complex data by assuming that data are continuous functions, for example, realizations of a continuous process (curves) or continuous random field (surfaces), and that each curve or surface is considered as a single observation. Here, we provide an overview of functional data analysis when data are complex and spatially correlated. We provide definitions and estimators of the first and second moments of the corresponding functional random variable. We present two main approaches: The first assumes that data are realizations of a functional random field, that is, each observation is a curve with a spatial component. We call them spatial functional data. The second approach assumes that data are continuous deterministic fields observed over time. In this case, one observation is a surface or manifold, and we call them surface time series. For these two approaches, we describe software available for the statistical analysis. We also present a data illustration, using a high-resolution wind speed simulated dataset, as an example of the two approaches. The functional data approach offers a new paradigm of data analysis, where the continuous processes or random fields are considered as a single entity. We consider this approach to be very valuable in the context of big data.

Functional unsupervised classification of spatial biodiversity

Abstract The recognition of spatial heterogeneity as well as of areas of low and high biodiversity through spatial techniques is essential to guide decision-making regarding the conservation and management of natural areas. In this context, reliable maps of biodiversity across sampling sites can be useful tools. Many ecological studies, which have dealt with a spatial approach for biodiversity, have focused only on one specific biodiversity aspect at a time, such as species richness or species evenness, yielding a partial overview of this complex concept. To solve this issue, we propose a spatial functional data analysis approach to diversity profiles for assessing spatial biodiversity and identifying groups of sampling sites which are similar in spatial patterns. Specifically, the functional distance-based LISA algorithm has been extended to the case of diversity profiles in lattice, after smoothing the discretized curves and specifying a suitable distance measure. The proposed spatial clustering algorithm has been applied to a real data set involving tree species diversity in a fully censured plot in the Harvard Forest, New England region. Our approach provides a useful method for identifying areas of low and high biodiversity, with the potential to address the monitoring of environmental policies. Indeed, we think that a classification of diversity profiles, which takes into account the spatial dependence, would permit a more homogeneous partition of sampling stations with a substantial noise reduction in supporting conservation planning.

Filling missing data and smoothing altered data in satellite imagery with a spatial functional procedure

Outliers and missing data are commonly found in satellite imagery. These are usually caused by atmospheric or electronic failures, hampering the correct monitoring of remote-sensing data. To avoid distorted data, we propose a procedure called “spatial functional prediction” (SFP). The SFP procedure consists of the following: (1) aggregating remote-sensing data for reducing the number of missing data and/or outliers; (2) additively decomposing the time series of images into a trend, a seasonal, and an error component; (3) defining the spatial functional data and predicting the trend component using an ordinary kriging; and (4) adding back the seasonal and error components to the predicted trend. The benefits of the SFP procedure are illustrated in the following scenarios: introducing random outliers, random missing data, mixtures of both, and artificial clouds in an extensive simulation study of composite images, and using daily images with real clouds. The following two derived variables are considered: land surface temperature (LST day) and normalized vegetation index (NDVI), which are obtained as remote-sensing data in a region in northern Spain during 2003–2016. The performance of SFP was checked using the root mean squared error (RMSE). A comparison with a procedure based on predicting with thin-plate splines (TpsP) is also made. We conclude that SFP is simpler and faster than TpsP, and provides smaller values of RMSE.

Spatial functional data analysis for regionalizing precipitation seasonality and intensity in a sparsely monitored region: Unveiling the spatio‐temporal dependencies of precipitation in Ecuador

The identification of area‐wise homogeneous precipitation regions helps to unveil similar precipitation patterns and amounts, where similar atmospheric processes at diverse temporal scales are likely to occur. However, although scientifically and socially relevant, the regionalization of precipitation is challenging, specially in areas of complex orography and with sparse monitoring. This limits our understanding of complex spatio‐temporal dependencies and hinders any information‐based resource management decision‐making. Gridded satellite precipitation products are useful in this context, even though they contain bias errors. Spatial functional data analysis (sFDA) is a novel technique that considers time as well as space dependencies by means of spatial autocorrelation and complete time functions, one for each spatial point. Therefore, the aim of this study is to evaluate sFDA as a tool to regionalize seasonality and intensity precipitation patterns, having Ecuador as a case study. The Tropical Rainfall Measuring Mission (TRMM 3B43) satellite precipitation is used to create an exhaustive spatial delineation. To the best of our knowledge, this is the first time that a sFDA regionalization approach is performed on gridded satellite precipitation. The complex orography and heat‐driven atmospheric processes in Ecuador's latitude make it a highly non‐trivial case to test the aforementioned technique. As a result, five relevant regions of precipitation seasonality were spatially delineated and temporally characterized. Three of them were zonally oriented, and two meridional‐wise in the coast. In addition, 20 relevant intensity regions across Ecuador were identified specially in regions with sparse monitoring. The regions were related to regional climate processes. However, limitations were found in regions with important orographic precipitation and locally variability patterns, probably due to the shortcomings of TRMM precipitation quantification. After the successful application of hierarchical regionalization using sFDA in a tropical region with sparse monitoring, it is reasonable to conclude that sFDA is a robust method to detect compact and meaningful homogeneous areas.

A Functional Spatial Analysis Platform for Discovery of Immunological Interactions Predictive of Low-Grade to High-Grade Transition of Pancreatic Intraductal Papillary Mucinous Neoplasms.

Intraductal papillary mucinous neoplasms (IPMNs), critical precursors of the devastating tumor pancreatic ductal adenocarcinoma (PDAC), are poorly understood in the pancreatic cancer community. Researchers have shown that IPMN patients with high-grade dysplasia have a greater risk of subsequent development of PDAC in the remnant pancreas than do patients with low-grade dysplasia. In this study, we built a computational prediction model that encapsulates the spatial cellular interactions in IPMNs that play key roles in the transformation of low-grade IPMN cysts to high-grade cysts en route to PDAC. Using multiplex immunofluorescent images of IPMN cysts, we adopted algorithms from spatial statistics and functional data analysis to create metrics that summarize the spatial interactions in IPMNs. We showed that an ensemble of models learned using these spatial metrics can robustly predict, with high accuracy, (1) the dysplasia grade (low vs high grade) and (2) the risk of a low-grade cyst progressing to a high-grade cyst. We obtained high classification accuracies on both tasks, with areas under the curve of 0.81 (95% confidence interval: 0.71-0.9) for task 1 and 0.81 (95% confidence interval: 0.7-0.94) for task 2. To the best of our knowledge, this is the first application of an ensemble machine learning approach for discovering critical cellular spatial interactions in IPMNs using imaging data. We envision that our work can be used as a risk assessment tool for patients diagnosed with IPMNs and facilitate greater understanding and investigation of the cellular interactions that cause transition of IPMNs to PDAC.

Mantel test for spatial functional data

Statistics for spatial functional data is an emerging field in statistics which combines methods of spatial statistics and functional data analysis to model spatially correlated functional data. Checking for spatial autocorrelation is an important step in the statistical analysis of spatial data. Several statistics to achieve this goal have been proposed. The test based on the Mantel statistic is widely known and used in this context. This paper proposes an application of this test to the case of spatial functional data. Although we focus particularly on geostatistical functional data, that is functional data observed in a region with spatial continuity, the test proposed can also be applied with functional data which can be measured on a discrete set of areas of a region (areal functional data) by defining properly the distance between the areas. Based on two simulation studies, we show that the proposed test has a good performance. We illustrate the methodology by applying it to an agronomic data set.

A penalized regression model for spatial functional data with application to the analysis of the production of waste in Venice province

We propose a method for the analysis of functional data with complex dependencies, such as spatially dependent curves or time dependent surfaces, over highly textured domains. The models are based on the idea of regression with partial differential regularizations. In particular, we consider here two roughness penalties that account separately for the regularity of the field in space and in time. Among the various modelling features, the proposed method is able to deal with spatial domains featuring peninsulas, islands and other complex geometries. Space-time varying covariate information is included in the model via a semi-parametric framework. The proposed method is compared via simulation studies to other spatio-temporal techniques and it is applied to the analysis of the annual production of waste in the towns of Venice province.

On the performance of two clustering methods for spatial functional data

The performance of two clustering strategies for spatially correlated functional data based on the same measure of spatial dependence is examined and compared. In particular, the role of the spatial dependence computed by the trace-variogram function is analyzed. The main features of both procedures is shown through a simulation study based on a variety of practical scenarios easily encountered in the analysis of spatial functional data. An application on real data based on salinity curves is also presented.

Hybrid PET/MR imaging: physics and technical considerations.

In just over a decade, hybrid imaging with FDG PET/CT has become a standard bearer in the management of cancer patients. An exquisitely sensitive whole-body imaging modality, it combines the ability to detect subtle biologic changes with FDG PET and the anatomic information offered by CT scans. With advances in MR technology and advent of novel targeted PET radiotracers, hybrid PET/MRI is an evolutionary technique that is poised to revolutionize hybrid imaging. It offers unparalleled spatial resolution and functional multi-parametric data combined with biologic information in the non-invasive detection and characterization of diseases, without the deleterious effects of ionizing radiation. This article reviews the basic principles of FDG PET and MR imaging, discusses the salient technical developments of hybrid PET/MR systems, and provides an introduction to FDG PET/MR image acquisition.

Hilbertian spatial periodically correlated first order autoregressive models

In this article, we consider Hilbertian spatial periodically correlated autoregressive models. Such a spatial model assumes periodicity in its autocorrelation function. Plausibly, it explains spatial functional data resulted from phenomena with periodic structures, as geological, atmospheric, meteorological and oceanographic data. Our studies on these models include model building, existence, time domain moving average representation, least square parameter estimation and prediction based on the autoregressive structured past data. We also fit a model of this type to a real data of invisible infrared satellite images.