part-3.qmd

# Models for Spatial Data

The first two parts of this book contain a considerable
amount of concepts that one could classify as "models for spatial
data", including:

* how numbers relate to real-world phenomena (@sec-cs)
* how coordinates are defined in different spaces (@sec-cs, @sec-spherical)
* simple feature geometries (@sec-geometries), how straight lines between points can be used to define linestrings and polygons
* the set of geometry types (@sec-simplefeatures)
* support and the way feature attributes can relate to geometries (@sec-featureattributes)
* how simple tesselations can describe space-time volumes (@sec-datacube)
* how these concepts can be made operational using data science software (@sec-sf)

The third and largest part of this book is dedicated to _statistical_
modelling of spatial data. The scientific discipline _statistics_
is concerned with describing and understanding variability in
observations, and predicting future observations. Observations are
often modelled as

$$\mbox{observed}=\mbox{explained}+\mbox{remainder}$$

where "remainder" refers to variation that could not be explained
by predictors, including measurement error but also lack of fit
or variation caused by model misspecification. For spatial data,
a further term is often helpful, as in

$$\mbox{observed}=\mbox{explained} + \mbox{smooth} + \mbox{remainder}$$

where "smooth" refers to variation that is not explained by external
predictors but that clearly shows "smooth" spatial patterns, as
opposed to the "rough" remainder which does not do this.  Such a
"smooth" term can for instance be modelled by base functions
in coordinates (splines, smoothers) or as a random term that is
spatially correlated.

@sec-modelling introduces statistical modelling of spatial data, as a
preparation to the subsequent chapters but also highlighting a number
of relevant aspects that are not elaborated on in later chapters. It
tries to bridge these chapters with concepts from the first part
of this book, in particular support (@sec-featureattributes).

It is now obvious that a complete and comprehensive treatment of
the topic of statistical models for spatial data that also includes
instructions about the use of computational software in a single book
is an impossible task. The `spatstat` book [@baddeley2015spatial]
has around 850 pages for only spatial point patterns and R.  This part
focuses on the three "classical" spatial statistics topics: analysis
of point patterns (@sec-pointpatterns), geostatistical data
(Chapters [-@sec-interpolation] and [-@sec-stgeostatistics]), and lattice (areal)
data (Chapters [-@sec-area]-[-@sec-spatecon]).  Where possible we
attempt to refer to further literature on methods and software
implementations in R.