Geostatistical Functional Data Analysis
Description
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.
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Persons
Jorge Mateu is Full Professor of Statistics at the Department of Mathematics of University Jaume I of Castellon. His research focuses on stochastic processes with a particular interest in spatial and spatio-temporal point processes and geostatistics.
Ramón Giraldo is Full Professor of Statistics at the Department of Statistics at the Universidad Nacional de Colombia. His research focuses on non-parametric statistics, functional data analysis, and spatial and spatio-temporal geostatistics.
Content
List of Contributors xiii
Foreword xvi
1 Introduction to Geostatistical Functional Data Analysis 1
Jorge Mateu and Ramón Giraldo
1.1 Spatial Statistics 1
1.2 Spatial Geostatistics 7
1.2.1 Regionalized Variables 7
1.2.2 Random Functions 7
1.2.3 Stationarity and Intrinsic Hypothesis 9
1.3 Spatiotemporal Geostatistics 12
1.3.1 Relevant Spatiotemporal Concepts 12
1.3.2 Spatiotemporal Kriging 16
1.3.3 Spatiotemporal Covariance Models 17
1.4 Functional Data Analysis in Brief 18
References 22
Part I Mathematical and Statistical Foundations 27
2 Mathematical Foundations of Functional Kriging in Hilbert Spaces and Riemannian Manifolds 29
Alessandra Menafoglio, Davide Pigoli, and Piercesare Secchi
2.1 Introduction 29
2.2 Definitions and Assumptions 30
2.3 Kriging Prediction in Hilbert Space: A Trace Approach 33
2.3.1 Ordinary and Universal Kriging in Hilbert Spaces 33
2.3.2 Estimating the Drift 36
2.3.3 An Example: Trace-Variogram in Sobolev Spaces 37
2.3.4 An Application to Nonstationary Prediction of Temperatures Profiles 39
2.4 An Operatorial Viewpoint to Kriging 42
2.5 Kriging for Manifold-Valued Random Fields 45
2.5.1 Residual Kriging 45
2.5.2 An Application to Positive Definite Matrices 47
2.5.3 Validity of the Local Tangent Space Approximation 49
2.6 Conclusion and Further Research 53
References 53
3 Universal, Residual, and External Drift Functional Kriging 55
Maria Franco-Villoria and Rosaria Ignaccolo
3.1 Introduction 56
3.2 Universal Kriging for Functional Data (UKFD) 56
3.3 Residual Kriging for Functional Data (ResKFD) 58
3.4 Functional Kriging with External Drift (FKED) 60
3.5 Accounting for Spatial Dependence in Drift Estimation 61
3.5.1 Drift Selection 62
3.6 Uncertainty Evaluation 62
3.7 Implementation Details in R 64
3.7.1 Example: Air Pollution Data 64
3.8 Conclusions 69
References 71
4 Extending Functional Kriging When Data Are Multivariate Curves: Some Technical Considerations and Operational Solutions 73
David Nerini, Claude Manté, and Pascal Monestiez
4.1 Introduction 73
4.2 Principal Component Analysis for Curves 74
4.2.1 Karhunen-Loève Decomposition 74
4.2.2 Dealing with a Sample 76
4.3 Functional Kriging in a Nutshell 78
4.3.1 Solution Based on Basis Functions 79
4.3.2 Estimation of Spatial Covariances 81
4.4 An Example with the Precipitation Observations 82
4.4.1 Fitting Variogram Model 83
4.4.2 Making Prediction 83
4.5 Functional Principal Component Kriging 85
4.6 Multivariate Kriging with Functional Data 88
4.6.1 Multivariate FPCA 91
4.6.2 MFPCA Displays 93
4.6.3 Multivariate Functional Principal Component Kriging 94
4.6.4 Mixing Temperature and Precipitation Curves 96
4.7 Discussion 98
4.A Appendices 100
4.A.1 Computation of the Kriging Variance 100
References 102
5 Geostatistical Analysis in Bayes Spaces: Probability Densities and Compositional Data 104
Alessandra Menafoglio, Piercesare Secchi, and Alberto Guadagnini
5.1 Introduction and Motivations 104
5.2 Bayes Hilbert Spaces: Natural Spaces for Functional Compositions 105
5.3 A Motivating Case Study: Particle-Size Data in Heterogeneous Aquifers -Data Description 108
5.4 Kriging Stationary Functional Compositions 110
5.4.1 Model Description 110
5.4.2 Data Preprocessing 112
5.4.3 An Example of Application 113
5.4.4 Uncertainty Assessment 116
5.5 Analyzing Nonstationary Fields of FCs 119
5.6 Conclusions and Perspectives 123
References 124
6 Spatial Functional Data Analysis for Probability Density Functions: Compositional Functional Data vs. Distributional Data Approach 128
Elvira Romano, Antonio Irpino, and Jorge Mateu
6.1 FDA and SDA When Data Are Densities 130
6.1.1 Features of Density Functions as Compositional Functional Data 131
6.1.2 Features of Density Functions as Distributional Data 135
6.2 Measures of Spatial Association for Georeferenced Density Functions 138
6.2.1 Identification of Spatial Clusters by Spatial Association Measures for Density Functions 139
6.3 Real Data Analysis 141
6.3.1 The SDA Distributional Approach 143
6.3.2 The Compositional-Functional Approach 145
6.3.3 Discussion 147
6.4 Conclusion 149
Acknowledgments 151
References 151
Part II Statistical Techniques for Spatially Correlated Functional Data 155
7 Clustering Spatial Functional Data 157
Vincent Vandewalle, Cristian Preda, and Sophie Dabo-Niang
7.1 Introduction 157
7.2 Model-Based Clustering for Spatial Functional Data 158
7.2.1 The Expectation-Maximization (EM) Algorithm 160
7.2.1.1 E Step 161
7.2.1.2 M Step 161
7.2.2 Model Selection 161
7.3 Descendant Hierarchical Classification (HC) Based on Centrality Methods 162
7.3.1 Methodology 164
7.4 Application 165
7.4.1 Model-Based Clustering 167
7.4.2 Hierarchical Classification 169
7.5 Conclusion 171
References 172
8 Nonparametric Statistical Analysis of Spatially Distributed Functional Data 175
Sophie Dabo-Niang, Camille Ternynck, Baba Thiam, and Anne-Françoise Yao
8.1 Introduction 175
8.2 Large Sample Properties 178
8.2.1 Uniform Almost Complete Convergence 180
8.3 Prediction 181
8.4 Numerical Results 184
8.4.1 Bandwidth Selection Procedure 184
8.4.2 Simulation Study 185
8.5 Conclusion 193
8.A Appendix 194
8.A.1 Some Preliminary Results for the Proofs 194
8.A.2 Proofs 196
8.A.2.1 Proof of Theorem 8.1 196
8.A.2.2 Proof of Lemma A.3 196
8.A.2.3 Proof of Lemma A.4 196
8.A.2.4 Proof of Lemma A.5 201
8.A.2.5 Proof of Lemma A.6 201
8.A.2.6 Proof of Theorem 8.2 202
References 207
9 A Nonparametric Algorithm for Spatially Dependent Functional Data: Bagging Voronoi for Clustering, Dimensional Reduction, and Regression 211
Valeria Vitelli, Federica Passamonti, Simone Vantini, and Piercesare Secchi
9.1 Introduction 211
9.2 The Motivating Application 212
9.2.1 Data Preprocessing 214
9.3 The Bagging Voronoi Strategy 216
9.4 Bagging Voronoi Clustering (BVClu) 218
9.4.1 BVClu of the Telecom Data 221
9.4.1.1 Setting the BVClu Parameters 221
9.4.1.2 Results 223
9.5 Bagging Voronoi Dimensional Reduction (BVDim) 223
9.5.1 BVDim of the Telecom Data 225
9.5.1.1 Setting the BVDim Parameters 225
9.5.1.2 Results 227
9.6 Bagging Voronoi Regression (BVReg) 231
9.6.1 Covariate Information: The DUSAF Data 232
9.6.2 BVReg of the Telecom Data 234
9.6.2.1 Setting the BVReg Parameters 234
9.6.2.2 Results 235
9.7 Conclusions and Discussion 236
References 239
10 Nonparametric Inference for Spatiotemporal Data Based on Local Null Hypothesis Testing for Functional Data 242
Alessia Pini and Simone Vantini
10.1 Introduction 242
10.2 Methodology 244
10.2.1 Comparing Means of Two Functional Populations 244
10.2.2 Extensions 248
10.2.2.1 Multiway FANOVA 249
10.3 Data Analysis 250
10.4 Conclusion and FutureWorks 256
References 258
11 Modeling Spatially Dependent Functional Data by Spatial Regression with Differential Regularization 260
Mara S. Bernardi and Laura M. Sangalli
11.1 Introduction 260
11.2 Spatial Regression with Differential Regularization for Geostatistical Functional Data 264
11.2.1 A Separable Spatiotemporal Basis System 265
11.2.2 Discretization of the Penalized Sum-of-Square Error Functional 268
11.2.3 Properties of the Estimators 271
11.2.4 Model Without Covariates 273
11.2.5 An Alternative Formulation of the Model 274
11.3 Simulation Studies 274
11.4 An Illustrative Example: Study of the Waste Production in Venice Province 278
11.4.1 The Venice Waste Dataset 278
11.4.2 Analysis of Venice Waste Data by Spatial Regression with Differential Regularization 279
11.5 Model Extensions 282
References 283
12 Quasi-maximum Likelihood Estimators for Functional Linear Spatial Autoregressive Models 286
Mohamed-Salem Ahmed, Laurence Broze, Sophie Dabo-Niang, and Zied Gharbi
12.1 Introduction 286
12.2 Model 288
12.2.1 Truncated Conditional Likelihood Method 291
12.3 Results and Assumptions 293
12.4 Numerical Experiments 298
12.4.1 Monte Carlo Simulations 298
12.4.2 Real Data Application 305
12.5 Conclusion 312
12.A Appendix 313
Proof of Proposition 12.A.1 313
Proof of Theorem 12.1 314
Proof of Theorem 12.2 317
Proof of Theorem 12.3 319
Proof of Lemma 12.A.2 322
Proof of Lemma 12.A.3 322
Proof of Lemma 12.A.5 323
References 325
13 Spatial Prediction and Optimal Sampling for Multivariate Functional Random Fields 329
Martha Bohorquez, Ramón Giraldo, and Jorge Mateu
13.1 Background 329
13.1.1 Multivariate Spatial Functional Random Fields 329
13.1.2 Functional Principal Components 330
13.1.3 The Spatial Random Field of Scores 331
13.2 Functional Kriging 332
13.2.1 Ordinary Functional Kriging (OFK) 332
13.2.2 Functional Kriging Using Scalar Simple Kriging of the Scores (FKSK) 333
13.2.3 Functional Kriging Using Scalar Simple Cokriging of the Scores (FKCK) 333
13.3 Functional Cokriging 336
13.3.1 Cokriging with Two Functional Random Fields 336
13.3.2 Cokriging with P Functional Random Fields 338
13.4 Optimal Sampling Designs for Spatial Prediction of Functional Data 340
13.4.1 Optimal Spatial Sampling for OFK 341
13.4.2 Optimal Spatial Sampling for FKSK 341
13.4.3 Optimal Spatial Sampling for FKCK 342
13.4.4 Optimal Spatial Sampling for Functional Cokriging 343
13.5 Real Data Analysis 344
13.6 Discussion and Conclusions 348
References 348
Part III Spatio-Temporal Functional Data 351
14 Spatio-temporal Functional Data Analysis 353
Gregory Bopp, John Ensley, Piotr Kokoszka, and Matthew Reimherr
14.1 Introduction 353
14.2 Randomness Test 355
14.3 Change-Point Test 359
14.4 Separability Tests 362
14.5 Trend Tests 365
14.6 Spatio-Temporal Extremes 369
References 373
15 A Comparison of Spatiotemporal and Functional Kriging Approaches 375
Johan Strandberg, Sara Sjöstedt de Luna, and Jorge Mateu
15.1 Introduction 375
15.2 Preliminaries 376
15.3 Kriging 378
15.3.1 Functional Kriging 378
15.3.1.1 Ordinary Kriging for Functional Data 378
15.3.1.2 Pointwise Functional Kriging 380
15.3.1.3 Functional Kriging Total Model 381
15.3.2 Spatiotemporal Kriging 382
15.3.3 Evaluation of Kriging Methods 384
15.4 A Simulation Study 385
15.4.1 Separable 385
15.4.2 Non-separable 390
15.4.3 Nonstationary 391
15.5 Application: Spatial Prediction of Temperature Curves in the Maritime Provinces of Canada 394
15.6 Concluding Remarks 400
References 400
16 From Spatiotemporal Smoothing to Functional Spatial Regression: a Penalized Approach 403
Maria Durban, Dae-Jin Lee, María del Carmen Aguilera Morillo, and Ana M. Aguilera
16.1 Introduction 403
16.2 Smoothing Spatial Data via Penalized Regression 404
16.3 Penalized Smooth Mixed Models 407
16.4 P-spline Smooth ANOVA Models for Spatial and Spatiotemporal data 409
16.4.1 Simulation Study 411
16.5 P-spline Functional Spatial Regression 413
16.6 Application to Air Pollution Data 415
16.6.1 Spatiotemporal Smoothing 416
16.6.2 Spatial Functional Regression 416
Acknowledgments 421
References 421
Index 424
1
Introduction to Geostatistical Functional Data Analysis
Jorge Mateu1 and Ramón Giraldo2
1Department of Mathematics, University Jaume I of Castellon, Spain
2Department of Statistics, National University of Colombia, Bogota, Colombia
1.1 Spatial Statistics
Spatial statistics has developed rapidly during the last 30 years. We have seen an interesting progress both in theoretical developments and in practical studies. Some early applications were in mining, forestry, and hydrology. It seems to be honest to remark that the increasing availability of computer power and skillful computer software has stimulated the ability to solve increasingly complex problems. Clearly, these problems have some common elements: they were all of a spatial nature. Some theory was available, for example the random function theory as developed by Yaglom and others in the 1960s. But that was largely insufficient to find generic solutions for the whole class of problems, and hence, the applications required a new theory. Thereupon some far-reaching theories have been developed: image reconstruction, Markov random fields, point process statistics, geostatistics, and random sets, to mention just a few. As a next stage, these theories were applied successfully to new disciplinary problems leading to modifications and extensions of mathematical and statistical procedures. We therefore notice a general scientific process that has occurred in the field of spatial statistics: well-defined problems with a common character were suddenly on the agenda, and data availability and intensive discussion with practical and disciplinary researchers resulted in new theoretical developments. Often, it is difficult to say which was first, and what followed, but we see different theoretical models developed for different applications.
Spatial statistics has hence emerged as an important new field of science. One of the peculiarities is its power for visualization. A common cold-water fear of many statisticians and mathematicians to analyze images, to communicate their results by maps, and to have to trust information in pictures was overcome. It has led to interesting theories and better and more objective procedures for dealing with spatial variation. Following Wittgenstein, we could state that we needed some geniuses to tackle the obvious. Now, many results of a spatial statistical analysis could be communicated smoothly toward the nonstatistical audience, like a disciplinary scientist, a policy-maker, or an interested student. They, in turn, were able to judge whether a problem was solved, whether a policy measure was relevant or was inspired by the beautiful pictures expressing deep thoughts on relevant issues.
The role in policy-making may be once more stressed. It is known that many policy-makers are inclined to make a decision on the basis of a well developed, well organized, and well understandable figure. They find it (rightly so!) rather boring to use long lists of statistical data. But as political decisions affect us all, it puts another responsibility on the back of statisticians: to make statistically sound maps. It is often hard to say what that should be, but at the very least, we should be able to generate pictures, maps, and graphs that rely on good data and that show important aspects for decision-making.
In this way, spatial statistics has become a refreshing wind in statistics. We do not need to do well much longer on difficult equations, long lists of data, and tables with simulated controlled scenarios. But, to be clear on the back of all these nice pictures a sound science with sometimes difficult and tedious derivations and deep thoughts are still required to make serious progress.
Spatial statistics recognizes and exploits the spatial locations of data when designing for, collecting, managing, analyzing, and displaying such data. Spatial data are typically dependent, for which there are classes of spatial models available that allow process prediction and parameter estimation. Spatially arranged measurements and spatial patterns occur in a surprisingly wide variety of scientific disciplines. The origins of human life link studies of the evolution of galaxies, the structure of biological cells, and settlement patterns in archaeology. Ecologists study the interactions among plants and animals. Foresters and agriculturalists need to investigate plant competition and account for soil variations in their experiments. The estimation of rainfall and of ore and petroleum reserves is of prime economic importance. Rocks, metals, and tissue and blood cells are all studied at a microscopic level. Geology, soil science, image processing, epidemiology, crop science, ecology, forestry, astronomy, atmospheric science, or simply any discipline that works with data collected from different spatial locations, need to develop models that indicate when there is dependence between measurements at different locations. Spatiotemporal variability is a relatively new area within Spatial Statistics, which explains the scarcity of space-time statistical tools 20 years ago. There has been a growing realization in the last decade that knowing where data were observed could help enormously in answering the substantive questions that precipitated their collection. One of the most powerful tools for spatial data analysis is the map. For example, in military applications, the battlespace is mapped for command and control. The sensors are both in situ and remote, and they generate spatially distributed data of many different kinds. Producing a statistically optimal map, together with measures of map uncertainty, which is always up to date, is a complicated task. Once these types of statistical problems are solved, a geographic information system, or GIS, is well suited to forming the decision-making maps.
Spatial statistics can be considered a natural generalization of signal processing to higher dimensions. In traditional signal processing, one has a signal dependent on a scalar variable , which may belong to a discrete set or which may be continuous. Spatial statistics is concerned with cases in which is a multidimensional index of dimension . In most practical examples , though much of the basic theory and methodology is the same whatever the dimension. Although the models and methods of spatial statistics have not developed as rapidly as those for one-dimensional signal processing, there have nevertheless been substantial new developments in recent years. Standard and modern references on spatial statistics include the books of [1-4] among others.
Following Cressie [5], spatial data can be thought of as resulting from observations on the stochastic process , where is possibly a random set in . If we believe that the roots of statistical science are in data, we can classify spatial areas according to the type of observations encountered. Thus, (i) if is a fixed subset of and is a random vector at location , we are dealing with geostatistical data; (ii) if is a fixed (regular or irregular) collection of countably many points of and is a random vector at location , we are dealing with lattice data; (iii) if is a point process in and is a random vector at location , we are dealing with point patterns; (iv) if is a point process in and is itself a random set, we are dealing with spatial objects. Geostatistical-type problems are distinguished most clearly from lattice-and point-pattern-type problems by the ability of the spatial index to vary continuously over a subset of . A space-time process can be denoted by , where each of , , and is possibly random.
Spatial statistics is one of the major methodologies of environmental statistics. Its applications include producing spatially smoothed or interpolated representations of air pollution fields, calculating regional average means or regional average trends based on data at a finite number of monitoring stations, and performing regression analyses with spatially correlated errors to assess the agreement between observed data and the predictions of some numerical model. The notion of proximity in space is implicitly or explicitly present in the environmental sciences. Proximity is a relative notion, relative to the spatial scale of the scientific investigation. When a spatial dimension is present in an environmental study, the statistician's job is to create a statistical framework within which one carries out defensible inferences on processes and parameters of interest. These modeling and inference strategies are not always easy to do, but are never impossible. If statistics is to continue to be the broker of variability, it must address difficult questions such as those found in the environmental sciences, otherwise, it will become marginalized as a discipline. Problems in the environmental sciences are inherently spatial (and temporal), observational in nature, and have experimental units that are highly variable.
In the last decade, spatial statistics has undergone enormous development in the area of statistical modeling. It started slowly, building from models that were purely descriptive of spatial dependence. Then, it became apparent that the process of interest was usually hidden by measurement error and that the principal goal should be inference on the hidden process from the noisy data. It has only been in the last few years that the full potential for hierarchical spatial statistical modeling has been glimpsed. There is an enormous amount of flexibility in hierarchical statistical models, such as the opportunity to account for...
