Geostatistical Functional Data Analysis (Wiley Series in Probability and Statistics)

✍ Scribed by Jorge Mateu (editor), Ramon Giraldo (editor)

• Publisher: Wiley
• Year: 2021
• Tongue: English
• Leaves: 450
• Edition: 1
• Category: Library

✦ Synopsis

Geostatistical Functional Data Analysis

Explore the intersection between geostatistics and functional data analysis with this insightful new reference

Geostatistical Functional Data Analysis presents a unified approach to modelling functional data when spatial and spatio-temporal correlations are present. The Editors link together the wide research areas of geostatistics and functional data analysis to provide the reader with a new area called geostatistical functional data analysis that will bring new insights and new open questions to researchers coming from both scientific fields. This book provides a complete and up-to-date account to deal with functional data that is spatially correlated, but also includes the most innovative developments in different open avenues in this field.

Containing contributions from leading experts in the field, this practical guide provides readers with the necessary tools to employ and adapt classic statistical techniques to handle spatial regression. The book also includes:

• A thorough introduction to the spatial kriging methodology when working with functions
• A detailed exposition of more classical statistical techniques adapted to the functional case and extended to handle spatial correlations
• Practical discussions of ANOVA, regression, and clustering methods to explore spatial correlation in a collection of curves sampled in a region
• In-depth explorations of the similarities and differences between spatio-temporal data analysis and functional data analysis

Aimed at mathematicians, statisticians, postgraduate students, and researchers involved in the analysis of functional and spatial data, Geostatistical Functional Data Analysis will also prove to be a powerful addition to the libraries of geoscientists, environmental scientists, and economists seeking insightful new knowledge and questions at the interface of geostatistics and functional data analysis.

✦ Table of Contents

Cover
Title Page
Copyright
Contents
List of Contributors
Foreword
Chapter 1 Introduction to Geostatistical Functional Data Analysis
1.1 Spatial Statistics
1.2 Spatial Geostatistics
1.2.1 Regionalized Variables
1.2.2 Random Functions
1.2.3 Stationarity and Intrinsic Hypothesis
1.3 Spatiotemporal Geostatistics
1.3.1 Relevant Spatiotemporal Concepts
1.3.2 Spatiotemporal Kriging
1.3.3 Spatiotemporal Covariance Models
1.4 Functional Data Analysis in Brief
References
Part I Mathematical and Statistical Foundations
Chapter 2 Mathematical Foundations of Functional Kriging in Hilbert Spaces and Riemannian Manifolds
2.1 Introduction
2.2 Definitions and Assumptions
2.3 Kriging Prediction in Hilbert Space: A Trace Approach
2.3.1 Ordinary and Universal Kriging in Hilbert Spaces
2.3.2 Estimating the Drift
2.3.3 An Example: Trace‐Variogram in Sobolev Spaces
2.3.4 An Application to Nonstationary Prediction of Temperatures Profiles
2.4 An Operatorial Viewpoint to Kriging
2.5 Kriging for Manifold‐Valued Random Fields
2.5.1 Residual Kriging
2.5.2 An Application to Positive Definite Matrices
2.5.3 Validity of the Local Tangent Space Approximation
2.6 Conclusion and Further Research
References
Chapter 3 Universal, Residual, and External Drift Functional Kriging
3.1 Introduction
3.2 Universal Kriging for Functional Data (UKFD)
3.3 Residual Kriging for Functional Data (ResKFD)
3.4 Functional Kriging with External Drift (FKED)
3.5 Accounting for Spatial Dependence in Drift Estimation
3.5.1 Drift Selection
3.6 Uncertainty Evaluation
3.7 Implementation Details in R
3.7.1 Example: Air Pollution Data
3.8 Conclusions
References
Chapter 4 Extending Functional Kriging When Data Are Multivariate Curves: Some Technical Considerations and Operational Solutions
4.1 Introduction
4.2 Principal Component Analysis for Curves
4.2.1 Karhunen–Loève Decomposition
4.2.2 Dealing with a Sample
4.3 Functional Kriging in a Nutshell
4.3.1 Solution Based on Basis Functions
4.3.2 Estimation of Spatial Covariances
4.4 An Example with the Precipitation Observations
4.4.1 Fitting Variogram Model
4.4.2 Making Prediction
4.5 Functional Principal Component Kriging
4.6 Multivariate Kriging with Functional Data
4.6.1 Multivariate FPCA
4.6.2 MFPCA Displays
4.6.3 Multivariate Functional Principal Component Kriging
4.6.4 Mixing Temperature and Precipitation Curves
4.7 Discussion
4.A.1 Computation of the Kriging Variance
References
Chapter 5 Geostatistical Analysis in Bayes Spaces: Probability Densities and Compositional Data
5.1 Introduction and Motivations
5.2 Bayes Hilbert Spaces: Natural Spaces for Functional Compositions
5.3 A Motivating Case Study: Particle‐Size Data in Heterogeneous Aquifers – Data Description
5.4 Kriging Stationary Functional Compositions
5.4.1 Model Description
5.4.2 Data Preprocessing
5.4.3 An Example of Application
5.4.4 Uncertainty Assessment
5.5 Analyzing Nonstationary Fields of FCs
5.6 Conclusions and Perspectives
References
Chapter 6 Spatial Functional Data Analysis for Probability Density Functions: Compositional Functional Data vs. Distributional Data Approach
6.1 FDA and SDA When Data Are Densities
6.1.1 Features of Density Functions as Compositional Functional Data
6.1.2 Features of Density Functions as Distributional Data
6.2 Measures of Spatial Association for Georeferenced Density Functions
6.2.1 Identification of Spatial Clusters by Spatial Association Measures for Density Functions
6.3 Real Data Analysis
6.3.1 The SDA Distributional Approach
6.3.2 The Compositional–Functional Approach
6.3.3 Discussion
6.4 Conclusion
Acknowledgments
References
Part II Statistical Techniques for Spatially Correlated Functional Data
Chapter 7 Clustering Spatial Functional Data
7.1 Introduction
7.2 Model‐Based Clustering for Spatial Functional Data
7.2.1 The Expectation–Maximization (EM) Algorithm
7.2.1.1 E Step
7.2.1.2 M Step
7.2.2 Model Selection
7.3 Descendant Hierarchical Classification (HC) Based on Centrality Methods
7.3.1 Methodology
7.4 Application
7.4.1 Model‐Based Clustering
7.4.2 Hierarchical Classification
7.5 Conclusion
References
Chapter 8 Nonparametric Statistical Analysis of Spatially Distributed Functional Data
8.1 Introduction
8.2 Large Sample Properties
8.2.1 Uniform Almost Complete Convergence
8.3 Prediction
8.4 Numerical Results
8.4.1 Bandwidth Selection Procedure
8.4.2 Simulation Study
8.5 Conclusion
8.A.1 Some Preliminary Results for the Proofs
8.A.2 Proofs
8.A.2.1 Proof of Theorem 8.1
8.A.2.2 Proof of Lemma A.3
8.A.2.3 Proof of Lemma A.4
8.A.2.4 Proof of Lemma A.5
8.A.2.5 Proof of Lemma A.6
8.A.2.6 Proof of Theorem 8.2
References
Chapter 9 A Nonparametric Algorithm for Spatially Dependent Functional Data: Bagging Voronoi for Clustering, Dimensional Reduction, and Regression
9.1 Introduction
9.2 The Motivating Application
9.2.1 Data Preprocessing
9.3 The Bagging Voronoi Strategy
9.4 Bagging Voronoi Clustering (BVClu)
9.4.1 BVClu of the Telecom Data
9.4.1.1 Setting the BVClu Parameters
9.4.1.2 Results
9.5 Bagging Voronoi Dimensional Reduction (BVDim)
9.5.1 BVDim of the Telecom Data
9.5.1.1 Setting the BVDim Parameters
9.5.1.2 Results
9.6 Bagging Voronoi Regression (BVReg)
9.6.1 Covariate Information: The DUSAF Data
9.6.2 BVReg of the Telecom Data
9.6.2.1 Setting the BVReg Parameters
9.6.2.2 Results
9.7 Conclusions and Discussion
References
Chapter 10 Nonparametric Inference for Spatiotemporal Data Based on Local Null Hypothesis Testing for Functional Data
10.1 Introduction
10.2 Methodology
10.2.1 Comparing Means of Two Functional Populations
10.2.2 Extensions
10.2.2.1 Multiway FANOVA
10.3 Data Analysis
10.4 Conclusion and Future Works
References
Chapter 11 Modeling Spatially Dependent Functional Data by Spatial Regression with Differential Regularization
11.1 Introduction
11.2 Spatial Regression with Differential Regularization for Geostatistical Functional Data
11.2.1 A Separable Spatiotemporal Basis System
11.2.2 Discretization of the Penalized Sum‐of‐Square Error Functional
11.2.3 Properties of the Estimators
11.2.4 Model Without Covariates
11.2.5 An Alternative Formulation of the Model
11.3 Simulation Studies
11.4 An Illustrative Example: Study of the Waste Production in Venice Province
11.4.1 The Venice Waste Dataset
11.4.2 Analysis of Venice Waste Data by Spatial Regression with Differential Regularization
11.5 Model Extensions
References
Chapter 12 Quasi‐maximum Likelihood Estimators for Functional Linear Spatial Autoregressive Models
12.1 Introduction
12.2 Model
12.2.1 Truncated Conditional Likelihood Method
12.3 Results and Assumptions
12.4 Numerical Experiments
12.4.1 Monte Carlo Simulations
12.4.2 Real Data Application
12.5 Conclusion
References
Chapter 13 Spatial Prediction and Optimal Sampling for Multivariate Functional Random Fields
13.1 Background
13.1.1 Multivariate Spatial Functional Random Fields
13.1.2 Functional Principal Components
13.1.3 The Spatial Random Field of Scores
13.2 Functional Kriging
13.2.1 Ordinary Functional Kriging (OFK)
13.2.2 Functional Kriging Using Scalar Simple Kriging of the Scores (FKSK)
13.2.3 Functional Kriging Using Scalar Simple Cokriging of the Scores (FKCK)
13.3 Functional Cokriging
13.3.1 Cokriging with Two Functional Random Fields
13.3.2 Cokriging with P Functional Random Fields
13.4 Optimal Sampling Designs for Spatial Prediction of Functional Data
13.4.1 Optimal Spatial Sampling for OFK
13.4.2 Optimal Spatial Sampling for FKSK
13.4.3 Optimal Spatial Sampling for FKCK
13.4.4 Optimal Spatial Sampling for Functional Cokriging
13.5 Real Data Analysis
13.6 Discussion and Conclusions
References
Part III Spatio–Temporal Functional Data
Chapter 14 Spatio–temporal Functional Data Analysis
14.1 Introduction
14.2 Randomness Test
14.3 Change‐Point Test
14.4 Separability Tests
14.5 Trend Tests
14.6 Spatio–Temporal Extremes
References
Chapter 15 A Comparison of Spatiotemporal and Functional Kriging Approaches
15.1 Introduction
15.2 Preliminaries
15.3 Kriging
15.3.1 Functional Kriging
15.3.1.1 Ordinary Kriging for Functional Data
15.3.1.2 Pointwise Functional Kriging
15.3.1.3 Functional Kriging Total Model
15.3.2 Spatiotemporal Kriging
15.3.3 Evaluation of Kriging Methods
15.4 A Simulation Study
15.4.1 Separable
15.4.2 Non‐separable
15.4.3 Nonstationary
15.5 Application: Spatial Prediction of Temperature Curves in the Maritime Provinces of Canada
15.6 Concluding Remarks
References
Chapter 16 From Spatiotemporal Smoothing to Functional Spatial Regression: a Penalized Approach
16.1 Introduction
16.2 Smoothing Spatial Data via Penalized Regression
16.3 Penalized Smooth Mixed Models
16.4 P‐spline Smooth ANOVA Models for Spatial and Spatiotemporal data
16.4.1 Simulation Study
16.5 P‐spline Functional Spatial Regression
16.6 Application to Air Pollution Data
16.6.1 Spatiotemporal Smoothing
16.6.2 Spatial Functional Regression
Acknowledgments
References
Index