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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Statistics in Practice
- Sprache: Englisch
A state-of-the-art presentation of optimum spatio-temporal sampling design - bridging classic ideas with modern statistical modeling concepts and the latest computational methods.
Spatio-temporal Design presents a comprehensive state-of-the-art presentation combining both classical and modern treatments of network design and planning for spatial and spatio-temporal data acquisition. A common problem set is interwoven throughout the chapters, providing various perspectives to illustrate a complete insight to the problem at hand.
Motivated by the high demand for statistical analysis of data that takes spatial and spatio-temporal information into account, this book incorporates ideas from the areas of time series, spatial statistics and stochastic processes, and combines them to discuss optimum spatio-temporal sampling design.
Spatio-temporal Design: Advances in Efficient Data Acquisition:
- Provides an up-to-date account of how to collect space-time data for monitoring, with a focus on statistical aspects and the latest computational methods
- Discusses basic methods and distinguishes between design and model-based approaches to collecting space-time data.
- Features model-based frequentist design for univariate and multivariate geostatistics, and second-phase spatial sampling.
- Integrates common data examples and case studies throughout the book in order to demonstrate the different approaches and their integration.
- Includes real data sets, data generating mechanisms and simulation scenarios.
- Accompanied by a supporting website featuring R code.
Spatio-temporal Design presents an excellent book for graduate level students as well as a valuable reference for researchers and practitioners in the fields of applied mathematics, engineering, and the environmental and health sciences.
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Seitenzahl: 573
Veröffentlichungsjahr: 2012
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Table of Contents
Series Page
Title Page
Copyright
Dedication
Contributors
Foreword
Chapter 1: Collecting Spatio-Temporal Data
1.1 Introduction
1.2 Paradigms in Spatio-Temporal Design
1.3 Paradigms in Spatio-Temporal Modeling
1.4 Geostatistics and Spatio-Temporal Random Functions
1.5 Types of Design Criteria and Numerical Optimization
1.6 The Problem Set: Upper Austria
1.7 The Chapters
Acknowledgments
References
Chapter 2: Model-Based Frequentist Design for Univariate and Multivariate Geostatistics
2.1 Introduction
2.2 Design for Univariate Geostatistics
2.3 Design for Multivariate Geostatistics
2.4 Application: Austrian Precipitation Data Network
2.5 Conclusions
References
Chapter 3: Model-Based Criteria Heuristics for Second-Phase Spatial Sampling
3.1 Introduction
3.2 Geometric and Geostatistical Designs
3.3 Augmented Designs: Second-Phase Sampling
3.4 A Simulated Annealing Approach
3.5 Illustration
3.6 Discussion
References
Chapter 4: Spatial Sampling Design by Means of Spectral Approximations to the Error Process
4.1 Introduction
4.2 A Brief Review on Spatial Sampling Design
4.3 The Spatial Mixed Linear Model
4.4 Classical Bayesian Experimental Design Problem
4.5 The Smith and Zhu Design Criterion
4.6 Spatial Sampling Design for Trans-Gaussian Kriging
4.7 The Spatdesign Toolbox
4.8 An Example Session
4.9 Conclusions
References
Chapter 5: Entropy-Based Network Design Using Hierarchical Bayesian Kriging
5.1 Introduction
5.2 Entropy-Based Network Design Using Hierarchical Bayesian Kriging
5.3 The Data
5.4 Spatio-Temporal Modeling
5.5 Obtaining a Staircase Data Structure
5.6 Estimating the Hyperparameters and the Spatial Correlations between Gauge Stations
5.7 Spatial Predictive Distribution Over the 445 Areas Located in the 18 Districts of Upper Austria
5.8 Adding Gauge Stations Over the 445 Areas Located in the 18 Districts of Upper Austria
5.9 Closing Down an Existing Gauge Station
5.10 Model Evaluation
Appendix 5.1: Hierarchical Bayesian Spatio-Temporal Modeling (or Kriging)
Appendix 5.2: Some Estimated Parameters
Acknowledgments
References
Chapter 6: Accounting for Design in the Analysis of Spatial Data
6.1 Introduction
6.2 Modeling Approaches
6.3 Analysis of the Austrian Precipitation Data
6.4 Discussion
References
Chapter 7: Spatial Design for Knot Selection in Knot-Based Dimension Reduction Models
7.1 Introduction
7.2 Handling Large Spatial Datasets
7.3 Dimension Reduction Approaches
7.4 Some Basic Knot Design Ideas
7.5 Illustrations
7.6 Discussion and Future Work
References
Chapter 8: Exploratory Designs for Assessing Spatial Dependence
8.1 Introduction
8.2 Spatial Links
8.3 Measures of Spatial Dependence
8.4 Models for Areal Data
8.5 Design Considerations
8.6 Discussion
8.7 Appendix 8.1: Code
Acknowledgments
References
Chapter 9: Sampling Design Optimization for Space-Time Kriging
9.1 Introduction
9.2 Methodology
9.3 Upper Austria Case Study
9.4 Discussion and Conclusions
Appendix 9.1: Code
Acknowledgment
References
Chapter 10: Space-Time Adaptive Sampling and Data Transformations
10.1 Introduction
10.2 Adaptive Sampling Network Design
10.3 Predictive Information Based on Data Transformations
10.4 Application to Upper Austria Temperature Data
10.5 Summary
Acknowledgments
References
Chapter 11: Adaptive Sampling Design for Spatio-Temporal Prediction
11.1 Introduction
11.2 Review of Spatial and Spatio-Temporal Adaptive Designs
11.3 The Stationary Gaussian Model
11.4 The Dynamic Process Convolution Model
11.5 Upper Austria Rainfall Data Example
11.6 Discussion
11.7 Appendix 11.1
References
Chapter 12: Semiparametric Dynamic Design of Monitoring Networks for Non-Gaussian Spatio-Temporal Data
12.1 Introduction
12.2 Semiparametric Non-Gaussian Space-Time Dynamic Design
12.3 Application: Upper Austria Precipitation
12.4 Discussion
Acknowledgments
References
Chapter 13: Active Learning for Monitoring Network Optimization
13.1 Introduction
13.2 Statistical Learning From Data
13.3 Support Vector Machines and Kernel Methods
13.4 Active Learning
13.5 Active Learning with SVMs
13.6 Case Studies
13.7 Conclusions
Acknowledgments
References
Chapter 14: Stationary Sampling Designs Based on Plume Simulations
14.1 Introduction
14.2 Plumes: From Random Fields to Simulations
14.3 Cost Functions
14.4 Optimisation
14.5 Results
14.6 Discussion
Acknowledgments
References
Index
Statistics in Practice
Human and Biological Sciences
Earth and Environmental Sciences
Industry, Commerce and Finance
Statistics in Practice
Series Advisory Editors
Marian Scott
University of Glasgow, UK
Stephen Senn
CRP-Santé, Luxembourg
Wolfgang Jank
University of Maryland, USA
Founding Editor
Vic Barnett
Nottingham Trent University, UK
Statistics in Practice is an important international series of texts which provide detailed coverage of statistical concepts, methods and worked case studies in specific fields of investigation and study.
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This edition first published 2013
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Library of Congress Cataloging-in-Publication Data
Spatio-temporal design : advances in efficient data acquisition / edited by Jorge Mateu, Department of Mathematics of the University Jaume I of Castellon, Spain, Werner G. Müller, Department of Applied Statistics, Johannes Kepler University Linz, Austria.
pages cm.—(Statistics in practice)
ISBN 978-0-470-97429-2 (hardback)
1. Sampling (Statistics) 2. Spatial analysis (Statistics) I. Mateu, Jorge, editor of compilation.
II. Müller, W. G. (Werner G.), editor of compilation.
QA276.6.S63 2013
001.4′33–dc23
ISBN: 978-0-470-97429-2
To Eva and Evelyn
Contributors
Foreword
Imagine driving a car that has not been built yet. Design the car: look at principles of combustion, stability, and comfort; consider the manufacturing tools available, including the individuals who will build the car; and keep it within budget. Spatio-temporal sampling design has some of the same features: the principles of stratification, replication, and randomization are important; measuring instruments have to be bought or built and possibly moved around the spatio-temporal domain by field teams; and there is still a bottom line to adhere to.
In this edited volume of chapters on spatio-temporal sampling design, by and large the long-term design focus has been on ‘driving the car,’ that is, data analysis and inference are very much on the minds of the designers. There, it is the spatio-temporal variability that is the coin of the realm. Controlling this variability allows for more precise inferences and a greater likelihood of detecting ‘signals’ in the data. Importantly, relating this benefit to a cost, allows an efficient allocation of a study's resources.
Readers of the book's chapters will find a myriad of techniques for linking the design with the analysis, and by far the majority of the authors concentrate on model-based designs. The models are statistical and require some knowledge of the underlying spatio-temporal variability, presumably from a pilot study. (Build a prototype and test drive it!) Statistical analyses require assumptions, and an important one in spatio-temporal design is that the spatio-temporal sampling inadequacies not be confounded with sources of variability due to the process being studied. A small amount of randomization in the design can be very prudent, like putting a spare tire in the back of the car. So too can a sampling protocol that includes samples very close together in space and time to disentangle measurement error from microscale variation.
One of the strengths of the book is that the editors asked the authors to use a common dataset, namely rainfall, temperature, and grassland-usage measurements in Upper Austria. Readers can see how different chapters' design criteria relate to this dataset. While it is not large in size, there are many scientific applications where sampling is limited (e.g., computer experiments, and wellington-boots-on-the-ground field studies).
To see which design approaches scale up to massive datasets, it is usually better to simulate first from a known process and determine what can be recovered from noisy, incompletely sampled, but massive data. In the geosciences, these are sometimes called Observing System Simulation Experiments (OSSEs), and they are used to mimic the data explosion coming from satellite remote-sensing instruments. This spatio-temporal data explosion is also coming from the mobile devices we carry around as we move through our space-time continuum; crowd-sourcing of this sort offers new statistical sampling challenges to build an accurate information base, and then a knowledge base for making important societal decisions.
The authors of the chapters in this book are eminent in their field, and the editors have meticulously framed a state-of-the-art snapshot for 2012. This is a fertile area for future research, but we should not forget the bottom line, that sampling is costly.
Noel CressieUniversity of Wollongong, Australiaand The Ohio State University, USA
Chapter 1
Collecting Spatio-Temporal Data
Jorge Mateu1 and Werner G. Müller2
1Department of Mathematics, University of Jaume I of Castellon, Spain
2Department of Applied Statistics, Johannes Kepler University Linz, Austria
1.1 Introduction
In this volume we intend to provide a comprehensive state-of-the-art presentation combining both classical and modern treatments of network design and planning for spatial and spatio-temporal data acquisition. A common problem set is interwoven throughout the chapters, providing various perspectives to illustrate a complete insight to the problem at hand. Motivated by the high demand for statistical analysis of data that takes spatial and spatio-temporal information into account, this book incorporates ideas from the areas of time series, spatial statistics and stochastic processes, and combines them to discuss optimum spatio-temporal sampling design.
The past has seen, perhaps initiated by Gribik et al. (1976), a great number of statistical papers devoted to the purely spatial aspect of sampling design mainly in the context of monitoring networks. Other early papers include those by Caselton and Zidek (1984), Olea (1984) and Fedorov and Müller (1988); book-length treatments are given by Müller (1998, 2007) and de Gruijter et al. (2006). An excellent recent overview over this literature is provided by Zidek and Zimmerman (2010) and we will take the liberty in this introduction to draw heavily from their structure and exposition, albeit complementing it with aspects that enter due to the additional temporal component. Another excellent review paper is that by Dobbie et al. (2008), who provide some material on spatio-temporal aspects. As those two texts are comprehensive we can restrict ourselves to a brief exposition with the goal and emphasis to lead the readers into the more substantial subsequent chapters.
1.2 Paradigms in Spatio-Temporal Design
An important clarification that needs to be made before thinking about spati(o-tempor)al design is whether we assume the randomness of the observations to stem from stochastic disturbances or from the sampling process itself. This leads to the distinction of so-called model-based and design-based (rather than this common but confusing expression we prefer to call them probability-based) inferences and their respective design procedures.
The probability-based paradigm is rooted in classical sampling theory and assumes the ability of defining a population explicitly and a respective randomness in the design. These methods aim at restoring unobserved observations and more importantly general attributes of the spatial population, such as total means (cf. de Gruijter and ter Braak 1990) and variances (cf. Fewster 2011). Probability-based inferences of these attributes are bias-free and allow uncertainty assessments under mild assumptions. The corresponding design techniques reach from the benchmark random sampling to stratified, two-stage, cluster or sequential random sampling with a multitude of variants (Stehman 1999) that all lend themselves to straightforward extensions into incorporating a temporal dimension (Brus and de Gruijter 2011). An excellent account of the latter can be found in Part IV of de Gruijter et al. (2006), which can in general be considered the most definitive text for the probability-based design paradigm.
The model-based paradigm on the other hand requires a statistical model to describe the data-generating spatio-temporal process. Here we typically assume that observations stem from a random field generally given by
1.1
where s denotes a spatial location, t a time point, x some potentially space and time dependent regressors, and η a parametrized trend model (a nearly encyclopedic reference for these type of processes is Cressie and Wikle 2011). Note that the random element here is the error ϵ rather than the design mechanism. This allows to assign meaning to purely geometric designs, such as regular grids or space-filling lattices, that are common in applications. Another advantage here is that by borrowing inference strength from the model we can make meaningful inferences from rather small samples and for very specific aspects derived from the model parameters, such as threshold exceedances, times of trend-reversals, local outliers, etc.
We believe that unless a reasonable modeling is out of reach the model-based approach offers more flexibility and statistical power, which is why most of the contributions in this volume will fall into this category. However, this issue has been the subject of considerable debate in the literature and further details are provided in Papritz and Webster (1995), Brus and de Gruijter (1997), Stevens (2006), and an overview is given in Table 1.1 of Dobbie et al. (2008). A general discussion that goes beyond the spatio-temporal realm can be found in Thompson (2002). Recently, there also have been attempts to fuse the two paradigms, Brus and de Gruijter (2012) for instance employ probability-based sampling for the spatial coordinates, whereas they build a time series model and use a respective design for the temporal trend. How to include probability-based design in a hierarchical statistical modeling framework is surveyed in Cressie et al. (2009).
1.3 Paradigms in Spatio-Temporal Modeling
Another dichotomy clearly shows when one examines the spati(o-tempor)al modeling literature. Historically, two schools have somewhat independently developed, one based on discrete time series model analogies and the other one derived from generalizations of stochastic process methodologies. The former is much used by geographers and economists particularly in the advent of what was termed ‘new economic geography’ and was consequently referred to as ‘spatial econometrics’ (for perhaps the earliest full exposition see Anselin 1988; a recent account of the history of the field thereafter by the same author can be found in Anselin 2010). The latter school stems from the theory of regionalized variables developed among mining scientists and geologists and has consequently been named ‘geostatistics’ (a book-length treatment is provided by Chilès and Delfiner 1999). Comparative discussions on these two paradigms can be found in the encyclopedic Cressie (1993) and more recently in Griffith and Paelinck (2007), Hae-Ryoung et al. (2008) and Haining et al. (2010).
Both of these modeling views are encompassed by the random field (1.1) and can be solely distinguished by the nature of the indexing variables s and t. In spatial econometrics spatial econometrics we usually assume the s's to form a discrete geographic lattice and their relationships are usually described in the form of a so-called spatial weight or link matrix W. Various types of dependence structures can be modeled by assigning particular forms of η and covariances of ϵ employing W, such as the common simultaneous and conditionally autoregressive regression models (SAR and CAR), the latter being spatial manifestations of Gaussian Markov random fields (GMRF; see Rue and Held 2005 for a definitive text).
Despite this divide there have lately been successful attempts to merge those two spatio-temporal modeling paradigms. While previously only results for regular sampling schemes were available (cf. Griffith and Csillag 1993), Lindgren et al. (2011) provide an explicit link for arbitrary lattices, thus opening the issue for the question of sampling design. In a discussion to this article Müller and Waldl (2011) indeed uncover relationships between the respective designs that will allow to exploit properties from both paradigms.
A great number of spatio-temporal extensions of these models exist particularly for GRF; see Cressie and Wikle (2011) for an extensive review and Baxevani et al. (2011) for a particular representation using velocity fields. GMRF are usually extended by modeling them in discrete time, so-called spatial panel models (see e.g., Elhorst 2012 for a recent survey); a continuous time extension of spatial panels is given in Oud et al. (2012).
1.4 Geostatistics and Spatio-Temporal Random Functions
Geostatistical research has typically analyzed random fields, in which every spatio-temporal location can be seen as a point on . While from a mathematical point of view , from a physical perspective it would make no sense to consider spatial and temporal aspects in the same way, due to the significant differences between the two axes of coordinates. Therefore, while the time axis is ordered intrinsically (as it exists in the past, present and future), the same does not occur with the spatial coordinates.
1.4.1 Relevant Spatio-Temporal Concepts
The spatio-temporal r.f. Z(s, t) is said to have a spatially stationary covariance function if, for any two pairs (si, ti) and (sj, tj) on , the covariance C((si, ti), (sj, tj)) only depends on the distance between the locations si and sj and the times ti and tj. And the spatio-temporal r.f. Z(s, t) is said to have a temporarily stationary covariance function if, for any two pairs (si, ti) and (sj, tj) on , the covariance C((si, ti), (sj, tj)) only depends on the distance between the times ti and tj and the spatial locations si and sj. If the spatio-temporal r.f. Z(s, t) has a stationary covariance function in both spatial and temporal terms, then it is said to have a stationary covariance function. In this case, the covariance function can be expressed as
1.2
A spatio-temporal r.f. Z(s, t) has a separable covariance function if there is a purely spatial covariance function Cs(si, sj) and a purely temporal covariance function Ct(ti, tj) such that
1.3
for any pair of spatio-temporal locations (si, ti) and (sj, tj) .
A spatio-temporal r.f. Z(s, t) has a fully symmetrical covariance function if
1.4
for any pair of spatio-temporal locations (si, ti) and (sj, tj) .
Separability is a particular case of complete symmetry and, as such, any test to verify complete symmetry can be used to reject separability. In the case of stationary spatio-temporal covariance functions, the condition of full symmetry reduces to
1.5
A spatio-temporal r.f. has a compactly supported covariance function if, for any pair of spatio-temporal locations (si, ti) and , the covariance function C((si, ti), (sj, tj)) tends towards zero when the spatial or temporal distance is sufficiently large.
If C(si − sj, ti − tj) depends only on the distance between positions, that is, , the r.f., apart from being stationary, is also isotropic in space and time. Note that if the covariance function of a stationary r.f. is isotropic in space and time, then it is fully symmetrical.
The spatio-temporal variogram is defined as the function
1.6
where V is the variance, and half this quantity is called a semivariogram.
In the case of a r.f. with a zero mean,
1.7
Whenever it is possible to define the covariance function and the variogram, they will be related by means of the following expression
1.8
If the spatio-temporal r.f. Z(s, t) has an intrinsically stationary variogram in both space and time, then it is said to have an intrinsically stationary variogram. In this case, the variogram can be expressed as
1.9
The marginals and 2γ(h, · ) are called purely spatial and purely temporal variograms, respectively.
A r.f. Z(s, t) is strictly stationary if its probability distribution is translation invariant. Second-order stationarity is a less demanding condition than strict stationarity. A spatio-temporal r.f. Z(s, t) is second-order stationary in the broad sense or weakly stationary if it has a constant mean and the covariance function depends on h and u.
1.10
If ρ(h, u) is a correlation function on , then its marginal functions ρ(0, u) and ρ(h, 0) will respectively be the spatial correlation function on and the temporal correlation function on .
1.4.2 Properties of the Spatio-Temporal Covariance and Variogram Functions
1.11
1.12
with .
In case of stationarity, the above results reduce to functions depending on spatial and temporal lags. Another seminal result that characterizes covariance functions is that given in Bochner (1933). A function C(h, u) defined on is a stationary covariance function if, and only if, it has the following form
1.13
where the function F is a non-negative distribution function with a finite mean defined on , which is known as a spectral distribution function. Therefore, the class of stationary spatio-temporal covariance functions on is identical to the class of Fourier transforms of non-negative distribution functions with finite means on that domain. If the function C can also be integrated, then the spectral distribution function F is absolutely continuous and the representation (1.13) simplifies to
1.14
where f is a non-negative, continuous and integrable function that is known as a spectral density function. The covariance function C and the spectral density function f then form a pair of Fourier transforms , and
1.15
The decomposition (1.13) can be specialized for fully symmetrical covariance functions. Let C( · , · ) be a continuous function defined on , then C( · , · ) is a fully symmetrical stationary covariance function if, and only if, the following decomposition is possible
1.16
where F is the non-negative and symmetrical spectral distribution function defined on .
Cressie and Huang (1999) provide a theorem for characterizing the class of stationary spatio-temporal covariance functions under the additional hypothesis of integrability. Let C( · , · ) be a continuous, bounded, symmetrical and integrable function defined on , then C( · , · ) is a stationary covariance function if, and only if, in view of ,
1.17
Both the covariance function and the spectral density function are important tools for characterizing random stationary spatio-temporal fields. Mathematically speaking, both functions are closely related as a pair of Fourier transforms. Furthermore, the spectral density function is particularly useful in situations where there is no explicit expression of the covariance function. Stein (2005) shows the benefit of using smooth covariance functions far from the origin, which can be tested by verifying whether their spectral densities have derivatives of certain orders.
1.4.3 Spatio-Temporal Kriging
Kriging is aimed at predicting an unknown point value Z(s0, t0) at a point (s0, t0) that does not belong to the sample. To do so, all the information available about the regionalized variable is used, either at the points in the entire domain or in a subset of the domain called the neighborhood.
Assume that the value of the r.f. has been observed on a set of n spatio-temporal locations {Z(s1, t1), ..., Z(sn, tn)}. We now want to predict the value of the r.f. on a new spatio-temporal location (s0, t0), for which we use the linear predictor
1.18
constructed from the random variables Z(si, ti). As in the spatial case, spatio-temporal kriging equations will depend on the degree of stationarity attributed to the r.f. that supposedly generates the observed realization. The most widely used kriging techniques in the spatio-temporal case are simple spatio-temporal kriging, ordinary spatio-temporal kriging, and universal spatio-temporal kriging. In the case of simple spatio-temporal kriging we assume that Z(s, t) is a second-order stationary spatio-temporal r.f., with a constant and known mean μ(s, t), constant and known variance C(0, 0), and a known covariance function C(h, u). The kriging equations (n equations with n unknown elements) are of the form
1.19
from which we obtain the values λi that minimize the prediction error variance, which is given by
1.20
In the case of ordinary spatio-temporal kriging, the constant mean μ(s, t) is not known, and the covariance function C(h, u) is known, under second-order stationarity. In the case of an intrinsic r.f. the variance is unbounded. In these two cases, simple kriging cannot be performed as the mean cannot be subtracted. We must therefore impose a condition of unbiasedness. In these situations, ordinary spatio-temporal kriging equations can be expressed, in the first case, in terms of the covariance function, and in the second case, in terms of the semivariogram, as there is no covariance at the origin.
In the universal kriging approach, assume Z(s, t) is a r.f. with drift, and so the mean of the r.f. is not constant, but depends on the pairs (s, t). In this situation the so-called condition of unbiasedness is substantially affected. In this case, the r.f. can be disaggregated into two components: one deterministic μ(s, t) and the other stochastic e(s, t) which can be treated as an intrinsically stationary r.f. with zero expectation,
1.21
We can assume that the mean, even unknown, can be expressed locally by
1.22
where are p known functions, ah constant coefficients, and p the number of terms used in the approximation. It must be taken into account that this expression is only valid locally. In this case, the equations that yield the prediction of the weights are obtained from the prediction error conditions of zero expectation and minimum variance.
1.4.4 Spatio-Temporal Covariance Models
One key stage in the spatio-temporal prediction procedure is choosing the covariance function (covariogram or semivariogram) that models the structure of the spatio-temporal dependence of the data. However, while the semivariogram is normally chosen for this purpose in the spatial case, in a spatio-temporal framework the covariance function is the most commonly chosen tool. By referring to a valid covariographic spatio-temporal model, we are implicitly stating that the covariance function must be positive-definite. The purely spatial and temporal covariance models have been widely studied and there is a long list of those which can be used to model spatial or spatio-temporal dependence that guarantee the (spatial or temporal) covariance function is positive-definite. However, this is not the case in the spatio-temporal scenario, in which constructing valid spatio-temporal covariance models is one of the main research activities. In addition, while it is difficult to demonstrate that a spatial or temporal function is positive-definite, it is even more so when seeking to determine valid spatio-temporal covariance models. For this reason, many authors began to study how to combine valid spatial and temporal models to obtain (valid) spatio-temporal covariance models.
By way of introduction, the first approximations to modeling spatio-temporal dependence using covariance functions were nothing more than generalizations of the stationary models used in the spatial scenario. In this sense, early studies often modeled the spatio-temporal covariance using metric models by defining a metric in space and time that allowed researchers to directly use isotropic models that are valid in the spatial case. Such metric models were characterized by being nonseparable, isotropic and stationary. The next step in this initial stage consisted of configuring spatio-temporal covariance functions by means of the sum or product of a spatial covariance and a temporal covariance, both of which were stationary, giving rise to separable, isotropic and stationary models. Later, realizing the limitations of the two procedures detailed above in terms of capturing the spatio-temporal dependence that really exists in the large majority of the phenomena studied, interest shifted towards including the interaction of space and time, in covariance models, giving rise to the so-called nonseparable models (while remaining isotropic and stationary). It is worth highlighting the nonseparable models developed by Jones and Zhang (1997), Cressie and Huang (1999), Brown et al. (2000), De Cesare et al. (2001a, b), De Iaco et al. (2001, 2002a, b, 2003), Gneiting (2002), Ma (2002, 2003a, c, 2005a, b, c), Fernández-Casal et al. (2003), Kolovos et al. (2004), and Stein (2005) among others.
Development continued with the search for nonseparable spatio-temporal, spatially anisotropic and/or temporally asymmetrical models such as those described in Fernández Casal et al. (2003), Porcu et al. (2006), and Mateu et al. (2007). Finally, we can cite some recent approaches to the problem of modeling nonstationary covariance functions, such as those made by Ma (2002, 2003b), Fuentes et al. (2005), Stein (2005), Chen et al. (2006), Porcu et al. (2006, 2007a, b, 2009), Mateu et al. (2007), Porcu and Mateu (2007) and Gregori et al. (2008).
1.4.5 Parametric Estimation of Spatio-Temporal Covariograms
The empirical determination of the covariance function or the variogram of a spatio-temporal process can be generalized naturally using the procedures for merely spatial processes. Let Z( · , · ) be an intrinsically stationary process observed on a set of n spatio-temporal pairs {(s1, t1), ..., (sn, tn)}. Two direct and popular alternatives to obtain an estimation of the variogram 2γ( · , · ) [and its covariance function C( · , · ), if the process is also second-order stationary] are the classical estimator based on the method-of-moments (MoM), and the robust estimator proposed by Cressie and Hawkins (1980).
This MoM estimator for the variogram is given in its most general form by
1.23
Although the classical estimation method has the advantage of being easy to calculate, it also has some practical drawbacks, such as not being robust in the case of extreme values. In order to avoid this problem, and following and extending Cressie and Hawkins (1980) we have the following variogram estimator for the spatio-temporal case
As in the spatial case, these estimators of the covariance function or variogram of the process do not generally fulfill the condition of being positive-definite or conditionally negative definite, respectively. For this reason, in practice we select a parametric model of covariance or variogram that we already know is valid, and estimate the parameters of the covariance function or variogram that best fits the values of the empirical estimator.
Any of the procedures for least squares (LS) estimation used in the spatial case can be easily generalized to the spatio-temporal case. Let us assume that the process being analyzed has been decomposed such that either the original process or the process of residuals after modeling its mean, is intrinsically stationary. In this case, we can estimate the parameters that define the semivariogram model chosen using ordinary (OLS), generalized (GLS) or weighted (WLS) least squares. But the OLS procedure does not take into account the behavior of the semivariogram (or covariance function) near the origin and, above all, does not consider the possibility of the values of the semivariogram being correlated, which could have an adverse effect on the estimations of the vector of parameters as the amount of data increases. This last problem is normally solved by resorting to GLS, but it is also true that in order to determine the covariance matrix, which is normally quite large, we should work in a Gaussian context and proceed using iterative methods (cf. Müller 1999). A compromise between efficiency and computation is provided by the WLS method, which is the most popular LS estimation method. The WLS method consists of (iteratively) minimizing the expression
1.24
where is a diagonal matrix, the elements of which, ωi, are approximately the variances of .
It is a well-known fact that the maximum likelihood (ML) method requires knowing the distribution of the r.f. underlying the phenomenon under study. However, only the case of Gaussian random functions has been developed in the literature (Mardia and Marshall 1984). With this in mind, if the spatio-temporal process is Gaussian, then it is possible to use the ML method, which is also more efficient than the least squares procedures when estimating the parameters of the valid covariance function (or semivariogram) chosen. ML estimators have desirable properties for spatial analysis, such as consistency and asymptotic normality in the field of extendible domains (Mardia and Marshall 1984). However, they also have one significant limitation: even in the merely spatial case they entail high computation costs as the number of parameters the model includes is high. The most critical part of the estimation process is calculating the determinant and the inverse of the covariance matrix. An alternative to ML is given by the restricted maximum likelihood (RML), in which the estimators are obtained by applying the ML method to the so-called ‘error contrasts’.
As the problem with the ML procedure is computational, it is natural that research has focused on searching for approaches to the likelihood function that require less than stages and which have desirable statistical properties. However, another line of research has resorted to exploiting the special structure of the covariance matrix in order to enhance computation. In this sense, Zimmerman (1989) indicates that in certain sampling procedures it is possible to compute ML more efficiently. Another way of simplifying the number of operations in the context of lattice data is to represent the process spectrally (cf. Whittle 1954). Kaufman et al. (2008) focus directly on the covariance matrix and propose calculating likelihood with a reduced version of the covariance matrix instead of the original matrix. Furrer et al. (2006) use this technique to interpolate observations rather than estimate the parameters.
The computational problem was once again addressed recently using likelihood approaches, such as composite likelihood. According to Varin and Vidoni (2005), there are two types of composite likelihood. The first includes the so-called ‘subsetting methods’ and is based on marginal functions or ‘subset pieces’ of full likelihood. The second is based on the so-called ‘mission methods’, obtaining composite likelihood by omitting components from the full likelihood. The latter intuitively yield computational benefits insofar as they do not consider certain ‘pieces of likelihood’ when computing composite likelihood. Three types of composite likelihood have been proposed in the literature to approach full likelihood when working with a large set of spatial data (see Vecchia 1988, Curriero and Lele 1999, Caragea and Smith 2006, and the modified version of Stein et al. 2004).
Vecchia (1988) proposes factorizing the full likelihood into a product of conditional densities to later reduce the size of the conditioning sets. Stein et al. (2004) suggest that it might be more efficient to carry out the above procedure in blocks. Stein (2005) extended this approach to the spatio-temporal field when the data are measured in monitoring stations at regular intervals of time.
The Weighted Composite Likelihood (WCL) technique (cf. Lindsay 1988) defines a general estimation method when working with large sets of observations and has been applied in a large number of fields of science in recent years (particularly in agronomy). It is easy to appreciate that the approximations in Vecchia (1988), Stein et al. (2004), Stein (2005), and Caragea and Smith (2006) are special cases of composite likelihood. This approach is the starting point of the procedure conceived by Bevilacqua et al. (2012). Using WCL to estimate the parameters of a valid covariographic or semivariographic model aims to obtain accurate estimates comparable with those obtained by ML, but with lower computation costs (cf. Bevilacqua et al. 2012). This issue is of vital importance in the spatio-temporal field, as there is usually a sufficient number of observations that make the ML prohibitive in terms of computation.
1.5 Types of Design Criteria and Numerical Optimization
When it comes to actually determining the spatio-temporal design we have a variety of choices and face a multitude of constraints. Perhaps it should be first stressed that many of the decisions we are going to make will have a considerable influence on the designs we yield but it is the decision to design in the first place, that will impact the quality of our inference the most. As Ver Hoef (2012) puts it: ‘The main point [...] is that any search for a good design is better than relying on a randomly chosen one, [...]’. We (and he) argue for designs that are robust with respect to choice that need to be made like prior guesses of parameters and spatial correlations and insensitive to the constraints imposed, ideally even those stemming from cost considerations.
1.25
with ϕ being a suitable scalar function. The result ξ* is referred to as a (ϕ)-optimal design. For spatio-temporal designs the time dimension is sometimes treated separately (cf. de Gruijter et al. 2006) since there are often specific restrictions as to whether and how temporal changes in the spatial configuration of measurement sites are allowed (see Wikle and Royle 1999 for an early description of the problem). Sometimes one even has to take into account measurement devices that move in space according to given trajectories (Ucinski and Chen 2005). In other cases one might want to exploit information gained from early stages of the data collecting process as time progresses leading to so-called sequential or adaptive designs (Berliner et al 1999).
The choice of a particular design criterion ϕ clearly reflects the purpose of a study. Is it (cf. Overton and Stehman 1996):
exploratory in nature to gain a first understanding of the phenomenon under study?
confirmatory aimed at estimating global or local characteristics or model parameters?
solely used for most accurate prediction purposes?
For the first in this list one usually constructs designs that are filling the space (and sometimes period) available in some efficient way. One could now look for a design that is trying to get good coverage of every point in the space D. Let us for this purpose define a metric for the (inverse) distance between a given point s and a design ξ, i.e
which can be considered as a measure of (lack of) coverage of a design ξ for the point s. Consequently we can use the Lq average over the space D
which forms a design criterion to be minimized over ξ (cf. Royle and Nychka 1998; a probability-based concept with a similar purpose was devised in Stevens and Olsen 2004).
As a particular choice, it is natural to attempt to make the maximum distance from all the points in D to their closest points in ξ as small as possible. This is achieved by letting p → ∞ and q → ∞ giving
1.26
and we call the resulting design
a minimax distance design. If we instead seek for a design that wants to achieve a high spread solely amongst its support points, we can similarly to above define a criterion that Lq averages the (inverse) interdistances for maximization, i.e.
Then a design that is seeking for a high spread amongst its support points within the design region must attempt to make the smallest distance between neighboring points in ξ as large as possible. That is ensured by the maximin design criterion which corresponds to letting q → ∞,
1.27
and we call the resulting design
a maximin distance design, first introduced by Johnson et al. (1990). For more details on space-fillingness refer to Pronzato and Müller (2012).
For the second point in the above list, that is improving the quality of the estimation of the model parameters or derived characteristics it is customary to base the design criterion on the corresponding Fisher information matrix M(ξ, β) for the parameters β. A large theory of optimal experimental design is built around those criteria (Atkinson et al. 2007), however most of it covers the case of independent errors ϵ, which is usually violated in spatial studies. Some attempts have been made to extend the theory into the correlated error case (cf. Fedorov 1996 and Müller and Pázman 2003), but the respective criteria are still popular for spatial design even without proper adaptations and despite lacking theoretical justification they seem to fare well. The most common choice for a criterion here is
and resulting designs are termed D-optimal, although occasionally other scalar functions of M(ξ, β) such as the trace (Ver Hoef 2012) are preferred for simplicity (see Nowak 2010 for a survey). Sometimes rather than estimating trend parameters the focus could be on fitting parametrized variograms γ(s, s′, θ) and estimation of their parameters, which consequently leads to designs based upon information matrices M′(ξ, θ) (Müller and Zimmerman 1999). A compound version taking into account both purposes
1.28
with a weighting factor 0 ≤ α ≤ 1 was proposed by Müller and Stehlík (2010). A Bayesian approach for the same problem was proposed in Diggle and Lophaven (2006) and extended in Nowak et al. (2010).
When it comes to providing most accurate prediction for unobserved locations and times it is naturally to base a design criterion on the so-called kriging variance , the mean squared prediction error of the best linear unbiased predictor of the random field (see McBratney and Webster 1981 for some early suggestions and Gao et al. 1996 for an efficient formulation). Most commonly the concrete design criterion employed for prediction is
which amounts to minimizing the maximum prediction variance over the design space, which in design theory is often called G-optimal. Some authors prefer to use the average variance (cf. Brus and Heuvelink 2007) but all variants basically yield space-filling designs. It has, however, to be noted that the kriging variance undervalues the uncertainty of the prediction, when—as is common—the variogram parameters θ are estimated together with β from the same data (see also Marchant and Lark 2007 for a discussion of this issue). Therefore Zhu and Stein (2006) and Zimmerman (2006) instead calculate their optimal prediction designs based on the modified kriging variance
1.29
resulting in much less space-filling and more patchy optimal designs. Note that designs for criteria of type (1.29) are computationally much more difficult to obtain than those of (1.28) since they require evaluations at all points of the design space. However, for uncorrelated regression there exist a certain duality between D- and G-optimality via the celebrated general equivalence theorem by Kiefer and Wolfowitz (1960). This formed the basis of the promising investigations relating criteria ϕDα and given in Müller et al. (2012).
Finally, particularly from a Bayesian viewpoint, it may be desirable to reduce the uncertainty associated to an (a posteriori) distribution. This naturally leads to the so-called entropy criterion, which is given a comprehensive exposition in Le and Zidek (2006) and offers a quite general unified approach at least in the Gaussian case. Note that for most of the above methods the application in the spatio-temporal setting will require adjustments for specific purposes, such as allowing for adapting sampling (Marchant and Lark 2006), ensuring robustness from model assumptions (Wiens 2005) or knowledge of variograms (Spöeck and Pilz 2010).
Finding exact optimum designs is obviously a challenging computational problem and most of the mentioned criteria are nonconvex functions of the design. Thus the algorithms known from the classical design theory (cf. Wynn 1970 and Fedorov 1971) either require some straightforward heuristic adaptions (cf. Brimkulov et al. 1980) or taylor-cut numerical routines. Recently, stochastic search algorithms such as spatial simulated annealing (van Groeningen and Stein 1998) and genetic algorithms (Ruiz-Cárdenas et al. 2010) or hybrids (Guedes et al. 2011) have been considered; a comparison of some of these methods with classic procedures can be found in Baume et al. (2010). An approach based on Markov chain Monte Carlo (MCMC) calculations first proposed by Müller (1999) seems to be particularly suitable for finding dynamic spatio-temporal designs (cf. Wikle and Royle 2005).
Available software implementation of the above discussed algorithms and methods is scarce, and we can only find something in R. Some indications can be found in Bivand et al. (2008); concrete useful packages are ‘fields’ by Furrer et al. (2011) (model-based) and ‘spsurvey’ by Kincaid and Olsen (2012) (design-based).
1.6 The Problem Set: Upper Austria
It was a cornerstone of this project to make the different approaches to design assembled in this volume comparable on an identical problem set. A natural choice was the region of Oberösterreich (Upper Austria), which has the city of Linz, where the university of one of the authors is located, as its capital and was also featured in Müller (1998, 2007). Since we wanted to encourage the use of methods for both discretely and continuously indexed random fields two complementary datasets were provided to the authors.
1.6.1 Climatic Data
This dataset contains climatic data measured at 37 stations irregularly placed over the region provided from http://www.zamg.ac.at/fix/klima/oe71-00/klima2000/klimadaten_oesterreich_1971_frame1.htm. Here, we have (incomplete) monthly data from 1994 to 2009 on average temperature and total rainfall. Due to some missing observations, however, not all of the stations could be effectively used. A map of the region with the respective locations of the measurement stations is displayed in Figure 1.1.
Figure 1.1 The sampling locations of the climatic dataset within the region of Upper Austria.
Both of these climatic indicators have been frequently analyzed by geostatistical methods in the past, a recent showcase analysis for rainfall is found in Grimes and Pardo-Igúzquiza (2010). A kriging variance based early study for the optimal design of a rain gauge network was undertaken by Papamichail and Metaxa (1996), whereas Zimmermann et al. (2010) provide a mixed probability and model-based approach for the related issue of throughfall monitoring. For the current dataset (July 1994) the results of a straightforward kriging analysis employing an exponential semivariogram can be found in Figures 1.2 (temperature) and 1.3 (rainfall), respectively.
Figure 1.2 Average temperature map for July 1994 in Upper Austria.
Figure 1.3 Average rainfall map for July 1994 in Upper Austria.
To get an impression of the temporal character of this dataset the rainfall measurements for Linz/Stadt from 1994 to 2009 are displayed in Figure 1.4. The seasonality is less pronounced here than in the temperature series, which require particular modeling.
Figure 1.4 Monthly rainfall in Linz/Stadt.
1.6.2 Grassland Usage
The second part of the dataset contains information on the grassland usage in the 444 municipal districts of Upper Austria over the years 1995, 1999, 2003, 2005, 2007 and 2008. Those data were processed from a set that was provided by the ‘Abteilung Statistik’ of the ‘Amt der oberösterreichischen Landesregierung’ (www.land-oberoesterreich.gv.at/statistik), a regionalized and extended version of what is provided by Statistics Austria (http://www.statistik.at/web_de/statistiken/land_und_forstwirtschaft/agrarstruktur_flaechen_ertraege/bodennutzung/index.html). The districts were represented by a lattice of local coordinates (easting and northing) of the respective centroids and the boundary information is contained in shape-file (http://doris.ooe.gv.at/index.asp?MenuID=4; see Figure 1.5).
Figure 1.5 The 444 municipal districts of Upper Austria.
Specifically we have provided the authors with the following variables: longitude, latitude (in local coordinates), LBBGG (identification number of municipality), BEZNR (identification number for district), FLKM2 (area of the region in square kilometers), ALTITUDE (elevation of the district's capital), and R95,R99,R03,R05,R07,R08 [log (area of arable land +1) − log (area of grassland +1)], i.e. practically the log of the ratio of the two areas for the periods 1995 (Figure 1.6) until 2008, making it scalefree. The latter variables should give some indication of the characteristic of interest to environmental scientists and policy makers, as to whether grassland is increasingly substituted by arable land leading to numerous economic and ecological impacts (see Vellinga et al. 2004 and Carlier et al. 2005 for a statistical analysis). The indicator is thus well suited for a spatio-temporal analysis and a positive temporal trend as well as some short range spatial autocorrelations can be expected (Yang et al. 2006). Furthermore some scientists postulated some interactions between climate change and land use (Thornley and Cannell 1997) thus allowing speculation about integrating some of our climatic variables into a potential spatio-temporal grassland usage model.
Figure 1.6 Grassland usage: log (area of arable land +1) − log (area of grassland +1) in Upper Austria 1995.
The altitudes were not directly available, but were read off the respective Wikipedia pages (http://de.wikipedia.org/wiki/Liste_der_Gemeinden_in_Oberoesterreich). Some authors have used a full digital elevation model such as provided by http://gdem.ersdac.jspacesystems.or.jp/; a corresponding plot can be found in Figure 1.7. As the local coordinate system (MGI Austria GK Central projection system based on Gauss-Krüger) was not sufficient for all purposes a transformation to the WGS84 geographical coordinates was performed (see http://spatial-analyst.net/book/uppera, where some more useful information on the case study can be found). A Google-Earth depiction of the area is presented in Figure 1.8.
Figure 1.7 A perspective contour plot of Upper Austria.
Figure 1.8 A Google-Earth view of observation regions and sites.
Particularly in the environmental application areas we can naturally find most of the examples of spatio-temporal monitoring network design, see Vašát et al. (2009), Corwin et al. (2010), Brus and de Gruijter (2011), and Creelman and Risk (2011) for soil assessment, Reed and Minsker (2004) and Masoumi and Kerachian (2010) for groundwater and Murtojärvi et al. (2011) for sea water quality, Romary et al. (2011) and Wu and Bocquiet (2011) for air pollution, Abida et al. (2008) and Melles et al. (2011) for radioactive releases, Martínez et al. (2008) for solar radiation, Hooten et al. (2009) and Fewster (2011) for ecological surveys, Stehman (2009) for land cover evaluation and Barabesi et al. (2012) for forestry among the most recent. Note, however, that the problem of where and when to collect spatio-temporal data is also of relevance in many other fields, for example geophysics (Maurer et al. 2010), engineering (Ucinski and Patan 2010), industrial production (Borgoni et al. 2010), agriculture (Heuvelink and Egmond 2010), social sciences (Kumar 2007) and signal transmission (Rogerson et al. 2004). Furthermore the proposed techniques could even be fruitful for solving problems that are just loosely related to data collection (Guhaniyogi et al. (2011).
1.7 The Chapters
Let us now give a quick overview of the contents of the book in terms of the individual chapters.
In the geostatistical context, the quality of inferences are affected substantially by the spatial configuration of the network of sites where measurements are taken. An extensive literature exists on spatial network design encompassing a variety of design objectives and statistical perspectives. Chapter 2 (Model-based frequentist design for univariate and multivariate geostatistics), written by Dale L. Zimmerman and Jie Li, considers spatial network design for four design objectives from a frequentist, parametric model-based perspective. The design objectives considered pertain to optimal estimation of model parameters and optimal spatial prediction, both for the univariate case (one spatially varying quantity of interest) and multivariate case (two or more such quantities).
Chapter 3 (Model-based criteria heuristics for second-phase spatial sampling), written by Eric M. Delmelle, discusses several objectives related to second-phase spatial sampling which is the process of collecting additional measurements of a spatial variable of interest. Different criteria exist to guide the location of these new measurements, and they generally are nonlinear. The author focuses on simulated annealing, a heuristic method which facilitates the process of finding a suitable sampling set among candidate locations.
As with any statistical prediction method, a general goal is to obtain as good predictions as possible for the complete area of investigation. One possibility to formalize this goal is to try to select the sampling locations in such a way that the sum of all kriging mean square errors of prediction becomes a minimum over the area of investigation. Notably, this is a very complicated optimization problem and becomes still more complicated by the fact that the covariance matrix enters the kriging mean square error of prediction in its inverse form. This design problem has been tackled in a number of recent research papers. Gunter Spöck and Jürgen Pilz in Chapter 4 (Spatial sampling design by means of spectral approximations to the error process) first consider the integrated kriging variance averaged over the area of investigation and then, as a combined design criterion, the averaged expected lengths of predictive intervals are considered. Finally it is shown how these design criteria can be generalized to spatial variables having skewed distributions. In such context a criterion for spatial sampling design with Box–Cox transformed spatial variables is proposed. Mathematically speaking, they approximate the investigated spatial random field by means of a large regression model consisting of cosine-sine Bessel surface harmonics with random amplitudes. This approximating regression model is a direct result of the polar spectral representation theorem for isotropic random fields. The authors show that kriging prediction in the original random field model is equivalent to trend prediction in this approximating Bayesian linear regression model.
Assessing the changes both spatially and temporally of data are highly important for monitoring purposes. Many spatio-temporal data are collected from monitoring stations. However, the selection of these stations is often influenced by administrative, political, and other pragmatic considerations. Thus it is important to evaluate if an existing station is statistically unnecessary. It is also crucial to find a location such that a new monitoring station upon it can greatly improve the data modeling. These are considered as environmental network design problems. Most of the approaches that tackle this type of problems can be classified into the following three categories: (1) geometry-based; (2) probability-based; (3) model-based. In Chapter 5 (Entropy-based network design using hierarchical Bayesian kriging), written by Baisuo Jin, Yuehua Wu, and Baiqi Miao, the attention is paid to the entropy-based design within Category (3). Although the authors cannot escape from the ‘curse of dimensionality’, they take advantage of recent increases in computational speed and numerical advances (e.g. MCMC) that allow the implementation of Bayesian space-time dynamical models in a hierarchical framework. Such specifications provide simple strategies for incorporating complicated space-time interactions at different stages of the model's hierarchy, and the models are feasible to implement in high dimensions.
In spatial statistics, the data locations and the measurement process are commonly stochastically dependent. In Chapter 6 (Accounting for design in the analysis of spatial data) Brian J. Reich and Montserrat Fuentes identify and distinguish between two such situations: informative sampling and informative missingness. Informative or preferential sampling occurs when measurement locations are selected in a way that depends on the underlying process. For example, air pollution monitors may be placed in locations with high air pollution levels. On the other hand, informative missingness occurs when the measurement locations are fixed, but observations are censored in a way that is informative about the underlying process. An approach that can be used in both situations is the shared variable model which introduces random effects that are shared between the data location and the measurement processes. Conditioned on the random effects, the data location and measurement processes are assumed to be independent. These methods can be naturally implemented in a Bayesian framework using MCMC methods.
Chapter 7 (Spatial design for knot selection in knot-based dimension reduction models), written by Alan E. Gelfand, Sudipto Banerjee and Andrew O. Finley, investigates the challenge of fitting hierarchical models for large spatial datasets. The authors propose the use of approximation through dimension reduction models, and work in the setting of spatial random effects specified through Gaussian processes. Such specification requires development of knot locations, whence the spatial design problem emerges. More precisely, they have a post-data collection, predate analysis design problem, and need to specify knots in order to fit desired spatial models. They propose an implementation of knot design based upon an average predictive variance criterion, and show how to implement the design and model fitting in a two-step process.
