# e-book Local Models for Spatial Analysis, Second Edition

Three distance based approaches are covered next: The average nearest neighbor ANN , the K and L functions, and the pair correlation function. An average nearest neighbor ANN analysis measures the average distance from each point in the study area to its nearest point. In the following example, the average nearest neighbor for all points is 1.

For example, the point closest to point 1 is point 9 which is 2.

An extension of this idea is to plot the ANN values for different order neighbors, that is for the first closest point, then the second closest point, and so forth. For example, the ANN for the first closest neighbor is 1. The shape of the ANN curve as a function of neighbor order can provide insight into the spatial arrangement of points relative to one another. In the following example, three different point patterns of 20 points are presented. The black ANN line is for the first point pattern single cluster ; the blue line is for the second point pattern double cluster and the red line is for the third point pattern.

The bottom line black dotted line indicates that the cluster left plot is tight and that the distances between a point and all other points is very short. This is in stark contrast with the top line red dotted line which indicates that the distances between points is much greater. Note that the way we describe these patterns is heavily influenced by the size and shape of the study region. If the region was defined as the smallest rectangle encompassing the cluster of points, the cluster of points would no longer look clustered.

How differently would you describe the point pattern in both cases?

### 2nd Edition

An important assumption that underlies our interpretation of the ANN results is that of stationarity of the underlying point process i. If the point is not stationary, then it will be difficult to assess if the results from the ANN analysis are due to interactions between the points or due to changes in some underlying factor that changes as a function of location.

The average nearest neighbor ANN statistic is one of many distance based point pattern analysis statistics. Another statistic is the K-function which summarizes the distance between points for all distances. The calculation of K is fairly simple: it consists of dividing the mean of the sum of the number of points at different distance lags for each point by the area event density. We then count the number of points events inside each circle. In our example, the stores appear to be more clustered than expected at distances greater than 12 km.

The transformation is calculated as follows:. This graph makes it easier to compare K with K expected at lower distance values.

## ISBN 13: 9781439829196

Here, we observe distances between stores greater than expected under CSR up to about 5 km. The concept of 1 st order effects and 2 nd order effects is an important one. It underlies the basic principles of spatial analysis. Tree distribution can also be influenced by 2 nd order effects such as seed dispersal processes where the process is independent of location and, instead, dependent on the presence of other trees.

Density based measurements such as kernel density estimations look at the 1 st order property of the underlying process.

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Distance based measurements such as ANN and K-functions focus on the 2 nd order property of the underlying process. Preface 1 Introduction to GIS 1. I Working with spatial data 2 Feature Representation 2.

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Raster 2. Field 2. Chapter 11 Point Pattern Analysis It provides an overview of a range of different approaches that have been developed and employed within Geographical Information Science GIScience. Starting with first principles, the author introduces users of GISystems to the principles and application of some widely used local models for the analysis of spatial data, including methods being developed and employed in geography and cognate disciplines.

He discusses the relevant software packages that can aid their implementation and provides a summary list in Appendix A. Presenting examples from a variety of disciplines, the book demonstrates the importance of local models for all who make use of spatial data. Taking a problem driven approach, it provides extensive guidance on the selection and application of local models. Accedi Cerca ordine. This can easily lead to poor analysis, for example, when considering disease transmission which can happen at work or at school and therefore far from the home.

## Local Models for Spatial Analysis

The spatial characterization may implicitly limit the subject of study. For example, the spatial analysis of crime data has recently become popular but these studies can only describe the particular kinds of crime which can be described spatially. This leads to many maps of assault but not to any maps of embezzlement with political consequences in the conceptualization of crime and the design of policies to address the issue.

This describes errors due to treating elements as separate 'atoms' outside of their spatial context.

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The fallacy is about transferring individual conclusions to spatial units. The ecological fallacy describes errors due to performing analyses on aggregate data when trying to reach conclusions on the individual units. For example, a pixel represents the average surface temperatures within an area. Ecological fallacy would be to assume that all points within the area have the same temperature.

## Spatial analysis - Wikipedia

This topic is closely related to the modifiable areal unit problem. A mathematical space exists whenever we have a set of observations and quantitative measures of their attributes. For example, we can represent individuals' incomes or years of education within a coordinate system where the location of each individual can be specified with respect to both dimensions. The distance between individuals within this space is a quantitative measure of their differences with respect to income and education.

However, in spatial analysis, we are concerned with specific types of mathematical spaces, namely, geographic space. In geographic space, the observations correspond to locations in a spatial measurement framework that capture their proximity in the real world. The locations in a spatial measurement framework often represent locations on the surface of the Earth, but this is not strictly necessary.

A spatial measurement framework can also capture proximity with respect to, say, interstellar space or within a biological entity such as a liver. The fundamental tenet is Tobler's First Law of Geography : if the interrelation between entities increases with proximity in the real world, then representation in geographic space and assessment using spatial analysis techniques are appropriate. The Euclidean distance between locations often represents their proximity, although this is only one possibility.

There are an infinite number of distances in addition to Euclidean that can support quantitative analysis. For example, "Manhattan" or " Taxicab " distances where movement is restricted to paths parallel to the axes can be more meaningful than Euclidean distances in urban settings. In addition to distances, other geographic relationships such as connectivity e. It is also possible to compute minimal cost paths across a cost surface; for example, this can represent proximity among locations when travel must occur across rugged terrain.

Spatial data comes in many varieties and it is not easy to arrive at a system of classification that is simultaneously exclusive, exhaustive, imaginative, and satisfying. Fingelton . Urban and Regional Studies deal with large tables of spatial data obtained from censuses and surveys. It is necessary to simplify the huge amount of detailed information in order to extract the main trends.

Multivariable analysis or Factor analysis , FA allows a change of variables, transforming the many variables of the census, usually correlated between themselves, into fewer independent "Factors" or "Principal Components" which are, actually, the eigenvectors of the data correlation matrix weighted by the inverse of their eigenvalues.

This change of variables has two main advantages:. Using multivariate methods in spatial analysis began really in the s although some examples go back to the beginning of the century and culminated in the s, with the increasing power and accessibility of computers. In , in a groundbreaking study, British geographers used FA to classify British towns. Since the vectors extracted are determined by the data matrix, it is not possible to compare factors obtained from different censuses. A solution consists in fusing together several census matrices in a unique table which, then, may be analyzed.

This, however, assumes that the definition of the variables has not changed over time and produces very large tables, difficult to manage. Spatial autocorrelation statistics measure and analyze the degree of dependency among observations in a geographic space. These statistics require measuring a spatial weights matrix that reflects the intensity of the geographic relationship between observations in a neighborhood, e.

Classic spatial autocorrelation statistics compare the spatial weights to the covariance relationship at pairs of locations. Spatial autocorrelation that is more positive than expected from random indicate the clustering of similar values across geographic space, while significant negative spatial autocorrelation indicates that neighboring values are more dissimilar than expected by chance, suggesting a spatial pattern similar to a chess board.

The possibility of spatial heterogeneity suggests that the estimated degree of autocorrelation may vary significantly across geographic space. Local spatial autocorrelation statistics provide estimates disaggregated to the level of the spatial analysis units, allowing assessment of the dependency relationships across space. Spatial stratified heterogeneity, referring to the within-strata variance less than the between strata-variance, is ubiquitous in ecological phenomena, such as ecological zones and many ecological variables.

Spatial stratified heterogeneity of an attribute can be measured by geographical detector q -statistic: . The value of q is within [0, 1], 0 indicates no spatial stratified heterogeneity, 1 indicates perfect spatial stratified heterogeneity. The value of q indicates the percent of the variance of an attribute explained by the stratification. The q follows a noncentral F probability density function. Spatial interpolation methods estimate the variables at unobserved locations in geographic space based on the values at observed locations.

Basic methods include inverse distance weighting : this attenuates the variable with decreasing proximity from the observed location. Kriging is a more sophisticated method that interpolates across space according to a spatial lag relationship that has both systematic and random components.