Dr. Marina Gavrilova

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Autocorrelation

Line Pattern Analyzers

Polygon Pattern Analyzers

Network Pattern Analyzes

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Spatial autocorrelation coefficients measure and test how clustered/dispersed the point locations are with respect to their attribute values.

Spatial autocorrelation of a set of points refers to the degree of similarity between points or events occurring at these points and points or evens in nearby locations.

With the spatial autocorrelation coefficient, we can measure:◦ The proximity of location◦ The similarity of the characteristics of these locations.

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Two popular indices for measuring spatial autocorrelation applicable to a point distribution: Geary’s Ratio and Moran’s I Index.

sij representing the similarity of point i ’s and point j ’s attributes.

wij representing the proximity of point i ’s and point j ’s locations, wii=0 for all points.

xi representing the value of the attribute of interest for point i .

n representing the total number of points.

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The spatial autocorrelation coefficient (SAC) is proportional to the weighted similarity of the point attribute values.

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SAC

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The spatial weights in the computations of the spatial autocorrelation coefficient may take on a form other than a distance-based format. For example:

wij can take a binary form of 1 or 0, depending on whether point i and point j are spatially adjacent.

If tow regions share a common boundary, the two centroids of these regions can be defined as spatially adjacent wij = 1; otherwise wij = 0.

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In Geary’s Ratio, the similarity attribute values between two points is defined

The computation of Geary’s Ratio

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In Moran’s I Index, the similarity attribute values between two points is defined

The computation of Moran’s I Index

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Numerical scales of Geary’s Ratio and Moran’s I

Spatial Patterns Geary’s C Moran’s I

Clustered pattern in which adjacent or nearby points show similar characteristics

0<C<1 I > E(I)

Random pattern in which points do not show particular patterns of similarity

C ~ = 1 I ~ = E(I)

Dispersed pattern in which adjacent or nearby points show different characteristics

1<C<2 I < E(I)

E(I) = (-1)/(n-1), which n denoting the number of points in distribution

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The index’s scale for Geary’s Ratio does not correspond to our conventional impression of the correlation coefficient of the (-1, 1) scale, while the scale of Moran’s I resembles more closely the scale conventional correlation measure:

The value for no spatial autocorrelation is not zero but -1/n-1;

The values of Moran’s I Index in some empirical studies are not bounded by (-1,1), especially the upper bound of 1.

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In a vector GIS database, linear features are best described as line objects. The representation of geographic features by geographic objects is scale dependent.

For instance, on a small-scale map (1: 1,000,000), a mountain range may be represented by a line showing its approximate location. When a large geographic scale is adopted (1:24,000), a polygon object is more appropriate to represent the detail of a mountain range.

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Some linear features do not have to be connected to each other to form a network. Each of these linear segments can be interpreted alone. Examples include extensive features such as mountain ranges and touchdown paths of tornados.

Besides linear geographic features, line objects in a GIS environment can represent phenomena or events that have beginning locations and ending locations. For example, we often use lines with arrow to show wind direction and magnitudes.

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Linear features can have attributes just like other types of features.

Length

Orientation and Direction

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Direction mean is similar to the concept of an average in classical statistics. It shows the general direction of a set of vectors. It can be simplified to 1 unit in length (unit vectors).

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In a network database, linear features are linked together topologically.

The length of a network can be defined as the aggregated length of individual segments of links.

Orientation or direction is also essential. For example, the flow direction of tributaries of river network should relatively consistent if the watershed is not very large or is elongated in shape.

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Connectivity of a network is how many different links or edges are connected to each other.

Connectivity matrix store and represent how different links are joined together. The labels of the columns and rows in the connectivity matrix are the IDs or the links in the network. If two links are directly joined to each other, the cell have a value of 1. Otherwise, the value will be 0.

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The spatial dataset for the application example is the Breeding Bird Survey Routes of North America from the National Atlas.

The database includes routes for the annual bird survey. Routes for the survey are represented as polyline segments.

The data describing the Continental Divide – the Rocky Mountains from the National Atlas is also used.

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Breeding bird survey routes at 100 miles and between 100 and 200 miles from the Continental Divide

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From these descriptive statistics, it is quite obvious that the routes closer to the Continental Divide have a slightly higher degree of geometric complexity than those farther away.

Still, we would like to confirm if the difference in the mean is due to sampling error or to some systematic processes by performing the difference-of-means test.

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We use a dataset modified from the shape file of major U.S. interstate highways included in the dissemination of ArcView GIS by ESRI.

The data theme is Roads_rt.shp with 147 line segments, which represent major interstate highways and some state highways.

The data must conform to the properties of a planar graph. When two lines cross each other, a vertex will be created. However, the highway data do not need meet it.

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The spatial patterns of geographic objects and phenomena are often the result of physical of cultural-human processes taking place on the surface of the earth.

Spatial pattern is a static concept since a pattern only show how geographic objects distribute at one given time.

Spatial process is a dynamic concept because it depicts and explains how the distribution of geographic objects comes to exist and may change over time.

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A spatial pattern can generally categorized as clustered, dispersed, or random.

In clustered case, darker shades representing a certain characteristic appear to cluster on the western side.

In dispersed case, countries with darker shades appear to be spaced evenly.

In random case, there may be no particular systematic structure or mechanism controlling the way these polygons are distributed.

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In classifying spatial patterns of polygons as either clustered, dispersed, or random, we can focus on how various polygons are arranged spatially.

We can measure the similarity or dissimilarity of any pair of neighboring polygons, or polygons within a given neighborhood.

When these similarities and dissimilarities are summarized for the entire spatial pattern, we essentially measure the magnitude of spatial autocorrelation, or spatial dependency.

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In addition to its type or nature, spatial autocorrelation can be measured by its strength.

Strong spatial autocorrelation means that the attribute values of adjacent geographic objects are strongly related.

If attribute values of adjacent geographic objects do not appear have a clear order or a relationship, the distribution is said to have a weak spatial autocorrelation, or a random pattern.

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Joint Count Statistics can be used to measure the magnitude of spatial autocorrelation among polygons with binary nominal data.

For interval or ratio data, we may use Moran’s I index, Geary’s Ratio C, and G-statistic.

These global measures assume that the magnitude of the spatial autocorrelation is reasonably stable across the study region.

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Elements in spatial weight matrices are often used as weights in the calculation of spatial autocorrelation statistics or in the spatial regression models.

The neighboring polygons of X: First order neighbors, high order neighbors.

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The cell will either be 0 or 1 in a binary matrix. ◦ cij=1 when the i th polygon is adjacent to the j th

polygon. ◦ cij=0 when the i th polygon is not adjacent to the j

th polygon.

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There are several ways to measure the distance between any two polygons. A very popular practice is to use the centroid of the polygon to represent the polygon.

There are different ways to determine the centroid of a polygon.

In general, the shape of the polygon affects the location of its centroid. Polygons with unusual shapes may generate centroids that are located in undesirable locations.

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One method to determine the distance between any two features is based on the distance of their nearest parts.

An interesting situation involving the distance of nearest parts occurs when the two features are adjacent to each other. When this is the case, the distance between two features is 0.

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Autocorrelation is needed to understand the relationship between locations and observed variables

Line Pattern Analyzers and Polygon Pattern Analyzers are used to udenrstand complex spatial processes

Network Pattern analysis can be performed using advanced mathematical modeling tools

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