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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...