The various forms of local models and methods outlined in this book are connected in different ways, and they can be grouped using particular
Local Modelling 31 characteristics or properties (for example, as in Section 2.2). Unwin (368) divides local statistics into three groups comprising those statistics based on (i) non-overlapping sets of calculations, (ii) calculations with respect to an observation and its neighbours, and (iii) local values standardised by global values derived from the entire dataset. A recent important contribution to the characterisation and development of local statistical methods is the paper by Boots and Okabe (54). The authors present local spatial statistical analysis (LoSSA) as an integrative structure and a framework that facilitates the development of new local and global statistics. In their paper, Boots and Okabe (54) aim to formalise what is involved in the implementation of a local statistic and to explicitly consider the main components and limitations of the procedures employed. In particular, the paper focuses on: (i) the nature of spatial subsets, (ii) their relationship to the complete dataset, and (iii) the relationship between a global statistic and the corresponding local statistic, and each of these is summarised below.
2.7.1 The nature of spatial subsets
Different forms of local spaces exist and Boots and Okabe (54) define focused local spaces and unfocused local spaces:
• Focused local space: determined with respect to a given location, an example being a circle drawn around the location.
• Unfocused local space: determined with respect to attributes, rather than locations. An example is a subset of zones with a population density greater than some particular threshold.
A local space may be connected (e.g., sharing boundaries with another local space) or disjointed.
The n data values at sites sican be given by zsi, i = 1, ..., n. Given a local space SLj with data values of sites in the local space zLj1, ..., zLjnLj, Boots and Okabe (54) term the global and local statistic similar in the case where the former is given by F (n; zs1, ..., zsn) and the latter by F (nLj; zLj1, ..., zLjnLj).
In such cases, the application of a common statistical test may be possible.
An important distinction made by Boots and Okabe (54), and other authors, is between global statistics that are decomposable into local statistics (see the local indicators of spatial association defined in Section 4.4 as an example) and those that are not. Furthermore, in the case of a global statistic which is a linear combination of the local statistics, the authors use the term linearly decomposable. Boots and Okabe (54) demonstrate their framework by developing a local variant of the cross K function (see Sections 8.8 and 8.12 for relevant material).
32 Local Models for Spatial Analysis 2.7.2 The relationship of spatial subsets to the complete
dataset
In terms of relationships between global space and local spaces, Boots and Okabe (54) define four conditions:
• Local spaces exhaustively covering global space.
• Local spaces not exhaustively covering global space.
• Overlap between local spaces.
• No overlap between local spaces.
These conditions can be structured and defined as follows:
1. Exhaustive:
• Overlapped — covering
• Non-overlapped — tessellation (zones filling space with, as indi-cated, no gaps and no overlaps)
2. Non-exhaustive
• Overlapped — clumps
• Non-overlapped — incomplete tessellation (e.g., only zones with more than a particular population amount)
• Non-overlapped — islands
Figure 2.6 shows local spaces defined with respect to a set of point locations (variable 1). These include local spaces exhaustively covering global space (local space 1) and local spaces which do not exhaustively cover global space (local space 2). The latter local spaces overlap (in some cases), while the former do not. In the example, variable 1 could represent the location of a school and variable 2 the location of pupils.
As Boots and Okabe (54) note, the relationship between global and local spaces is an important one and has implications for significance testing.
Where there are overlaps between local spaces it will be necessary to adjust significance levels, and this topic is discussed in various contexts in the book.
2.7.3 The relationship between a global statistic and the corresponding local statistic
Boots and Okabe (54) consider different ways of evaluating the local statistic relative to the global statistic. It is possible to evaluate the global and local statistics or just the local statistics. The possibilities can be outlined as shown in Figure 2.7.
Local Modelling 33
Legend
Variable 2 Variable 1
Focused local spaces 2 Focused local spaces 1
FIGURE 2.6: Local spaces.
A model-based case of computation of both the global and local statistics using the same parameters is a second-order point pattern analysis whereby the global estimate of the intensity is used as a null model of complete spatial randomness (CSR; see Section 8.4) (54). Boots and Okabe (54) note that, in the case of black/white cell join counts in a binary map assessed against a CSR model, separate global and local estimates of the probability, pb, of observing a black cell may be made (lower line, second box from left). Section 4.3 summarises the joins count approach, and Section 4.5 provides relevant discussion in a local context. Cases where only the local statistics are calculated include scan statistics (outlined in Section 8.13) and the geographical analysis machine (GAM; again, see Section 8.13).
Geographically weighted regression (Section 5.8) provides another example of a case where local statistics only are computed.
Boots and Okabe (54) give the empirically-based example of computing the global mean and variance of nearest-neighbour distances and using these to evaluate the probability of occurrence of a local nearest-neighbour distance (lower line, third box from left), or the significant local nearest-neighbour
34 Local Models for Spatial Analysis Global and
local statistics calculated
Model based Empirically based
Model based Empirically based
FIGURE 2.7: Evaluation of global and local statistics. After Boots and Okabe (54, p. 372).
distances could be designated as the outliers in a Tukey box-and-whiskers plot (365) of distances (lower line, first box from right).
The contribution of Boots and Okabe (54) is important in providing a means to formally assess the components and implementation of local statistics.
LoSSA also facilitates the development of new procedures.