Statistical Methods for Spatial Data Analysis
Oliver Schabenberger, Carol A. GotwayPreface
The study of statistical methods for spatial data analysis presents challenges
that are fairly unique within the statistical sciences. Like few other areas,
spatial statistics draws on and brings together philosophies, methodologies,
and techniques that are typically taught separately in statistical curricula.
Understanding spatial statistics requires tools from applied statistics, mathematical
statistics, linear model theory, regression, time series, and stochastic
processes. It also requires a different mindset, one focused on the unique characteristics
of spatial data, and additional analytical tools designed explicitly
for spatial data analysis.
When preparing graduate level courses in spatial statistics for the first time,
we each struggled to pull together all the ingredients necessary to present the
material in a cogent manner at an accessible, practical level that did not tread
too lightly on theoretical foundations. This book ultimately began with our
efforts to resolve this struggle. It has its foundations in our own experience,
almost 30 years combined, with the analysis of spatial data in a variety of
disciplines, and in our efforts to keep pace with the new tools and techniques
in this diverse and rapidly evolving field.
The methods and techniques discussed in this text do by no means provide
a complete accounting of statistical approaches in the analysis of spatial data.
Weighty monographs are available on any one of the main chapters. Instead,
our goal is a comprehensive and illustrative treatment of the basic statistical
theory and methods for spatial data analysis. Our approach is mostly modelbased
and frequentist in nature, with an emphasis on models in the spatial,
and not the spectral, domain. Geostatistical methods that developed largely
outside of the statistical mainstream, e.g., kriging methods, can be cast easily
in terms of prediction theory based on statistical regression models. Focusing
on a model formulation allows us to discuss prediction and estimation in the
same general framework. But many derivations and results in spatial statistics
either arise from representations in the spectral domain or are best tackled
in this domain, so spectral representations appear throughout. We added a
section on spectral domain estimation (§4.7) that can be incorporated in a
course together with the background material in §2.5. While we concentrate
on frequentist methods for spatial data analysis, we also recognize the utility
of Bayesian hierarchical models. However, since these models are complex
and intricate, we leave their discussion until Chapter 6, after much of the
foundation of spatial statistics necessary to understand and interpret them
has been developed.
The tools and approaches we consider essential comprise Chapters 1–7.
Chapter 1, while introductory, also provides a first description of the basic
measures of spatial autocorrelation and their role in the analysis of spatial
data. Chapter 2 provides the background and theoretical framework of random
fields necessary for subsequent chapters, particularly Chapters 4 and
5. We begin the heart of statistical methods for spatial data analysis with
mapped point patterns in Chapter 3. Since a good understanding of spatial
autocorrelation is necessary for spatial analysis, estimation and modeling of
the covariance function and semivariogram are treated in detail in Chapter
4. This leads easily into spatial prediction and kriging in Chapter 5. One of
the most important chapters is Chapter 6 on spatial regression. It is unique
in its comprehensiveness, beginning with linear models with uncorrelated errors
and ending with a succinct discussion of Bayesian hierarchical models
for spatial data. It also provides a discussion of model diagnostics for linear
and generalized linear spatial models. Chapter 7 is devoted to the simulation
of spatial data since we believe simulation is an essential component of statistical
methods for spatial data analysis and one that is often overlooked in
most other textbooks in spatial statistics. The chapters on non-stationary covariance
(Chapter 8) and spatio-temporal models (Chapter 9) are primarily a
review of a rapidly evolving and emerging field. These chapters are supplemental
to the core of the text, but will be useful to Ph.D. students in statistics
and others who desire a brief, concise, but relatively thorough overview of
these topics.
The book is intended as a text for a one-semester graduate level course in
spatial statistics. It is assumed that the reader/student is familiar with linear
model theory, the requisite linear algebra, and a good, working knowledge of
matrix algebra. A strong foundation in fundamental statistical theory typically
taught in a first course in probability and inference (e.g., probability
distributions, conditional expectation, maximum likelihood estimation) is essential.
Stochastic process theory is not assumed; the needed ingredients are
discussed in Chapter 2. The text does not require prior exposure to spectral
methods. There are problems at the end of the main chapters that can be
worked into course material.
The material in the book will be supplemented with additional material
provided through the CRC Press Web site (www.crcpress.com). The site will
provide many of the data sets used as examples in the text, software code
that can be used to implement many of the principal methods described and
illustrated in the text, as well as updates and corrections to the text itself.
We welcome additions, corrections, and discussions for this Web page so that
it can make statistical methods for spatial data analysis useful to scientists
across many disciplines.