Statistical Analysis of Geographical Information(2)

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Statistical Analysis of Geographical
Information(2)

• Dr. Marina Gavrilova

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Topics

• Autocorrelation
• Line Pattern Analyzers
• Polygon Pattern Analyzers
• Network Pattern Analyzes

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Spatial Autocorrelation of Points

• 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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Measures for Spatial Autocorrelation

• Two popular indices for measuring spatial
  autocorrelation applicable to a point
  distribution Gearys Ratio and Morans 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, wii0 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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SAC (1)

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

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SAC (2)

• 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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Gearys Ratio

• In Gearys Ratio, the similarity attribute
  values between two points is defined
• The computation of Gearys Ratio

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Morans I Index

• In Morans I Index, the similarity attribute
  values between two points is defined
• The computation of Morans I Index

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Gearys Ratio vs. Morans I Index
Numerical scales of Gearys Ratio and Morans I Numerical scales of Gearys Ratio and Morans I Numerical scales of Gearys Ratio and Morans I
Spatial Patterns Gearys C Morans I
Clustered pattern in which adjacent or nearby points show similar characteristics 0ltClt1 I gt 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 1ltClt2 I lt E(I)
E(I) (-1)/(n-1), which n denoting the number of points in distribution E(I) (-1)/(n-1), which n denoting the number of points in distribution E(I) (-1)/(n-1), which n denoting the number of points in distribution
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Scales of Gearys Ratio and Morans I Index

• The indexs scale for Gearys Ratio does not
  correspond to our conventional impression of the
  correlation coefficient of the (-1, 1) scale,
  while the scale of Morans 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 Morans I Index in some empirical
  studies are not bounded by (-1,1), especially the
  upper bound of 1.

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Introduction of Linear Features

• 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 (124,000), a
  polygon object is more appropriate to represent
  the detail of a mountain range.

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Linear Features

• 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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Spatial Attributes of Linear Features

• Linear features can have attributes just like
  other types of features.
• Length
• Orientation and Direction

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Directional Mean

• 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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Spatial Attribute of Network Features (1)

• 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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Spatial Attribute of Network Features (2)

• 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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Railroads Centering at Washington, D.C.
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Length Attribute Analysis of Linear Features

• 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

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

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Summary statistics of route report

• 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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Application Example for Network Analysis

• 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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Highway networks
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Mapping the networks
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Introduction of Polygon Pattern Analyzers

• 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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Spatial Relationships

• 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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Types of Patterns
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Spatial Dependency

• 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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Strength of spatial autocorrelation

• 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

• 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 Morans I
  index, Gearys 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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Neighbor Definition

• 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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Binary Connectivity Matrix

• The cell will either be 0 or 1 in a binary
  matrix.
• cij1 when the i th polygon is adjacent to the j
  th polygon.
• cij0 when the i th polygon is not adjacent to
  the j th polygon.

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Centroid Distance

• 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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Nearest Distances

• 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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Conclusions

• 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