Spatial Statistics

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Spatial Statistics

• YU-FEN LI
• 6/6/2006 6/13/2006

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Point Pattern Descriptors

• Central tendency
• Mean Center (Spatial Mean)
• Weighted Mean Center
• Median Center (Spatial Median) not used widely
  for its ambiguity
• Consider n points

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Central tendency Mean Center (Spatial Mean)

• The two means of the coordinates define the
  location of the mean center as

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Central tendency Weighted Mean Center

• The two means of the coordinates define the
  location of the mean center as
• where is the weight at point i

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Point Pattern Descriptors

• Dispersion and Orientation
• Standard distance
• Weighted standard distance
• Standard deviational ellipse

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Dispersion and Orientation Standard Distance

• How points deviate from the mean center
• Recall population standard deviation
• is the mean center,

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Dispersion and Orientation Weighted Standard
Distance

• Points may have different attribute values that
  reflect the relative importance
• is the weighted mean center,

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Dispersion and Orientation Standard
Deviational Ellipse

• Standard distance is a good single measure of the
  dispersion of the incidents around the mean
  center, but it does not capture any directional
  bias
• The standard deviational ellipse gives dispersion
  in two dimensions and is defined by 3 parameters
• Angle of rotation
• Dispersion along major axis
• Dispersion along minor axis

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Dispersion and Orientation Standard
Deviational Ellipse

• Basic concept is to
• Find the axis going through maximum dispersion
  (thus derive angle of rotation)
• Calculate standard deviation of the points along
  this axis (thus derive the length of major axis)

• Calculate standard deviation of points along the
  axis perpendicular to major axis (thus derive the
  length of minor axis)

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Statistical Methods in GIS

• Point pattern analyzers
• Location information only
• Line pattern analyzers
• Location Attribute information
• Polygon pattern analyzers
• Location Attribute information

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POINT PATTERN ANALYZERS

• Two primary approaches
• Quadrat Analysis
• based on observing the frequency distribution or
  density of points within a set of grids
• Nearest Neighbor Analysis
• based on distances of points

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Quadrat Analysis (QA)

• Point Density approach
• The density measured by QA is compared with it of
  a random pattern

RANDOM
CLUSTERED
UNIFORM/ DISPERSED
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Quadrat Analysis (QA)
Exhaustive census
Random sampling
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Quadrat Analysis (QA)

• Apply uniform or random grid over area (A) with
  size of quadrats given by
• where r of points
• width of square quadrat is
• radius of circular quadrat is

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Quadrat Analysis (QA) --Frequency distribution
comparison

• Treat each cell as an observation and count the
  number of points within it
• Compare observed frequencies in the quadrats with
  expected frequencies that would be generated by
• a random process (modeled by the Poisson
  distribution)
• a clustered process (e.g. one cell with r
  points, n-1 cells with 0 points) (n number of
  quadrats)
• a uniform process (e.g. each cell has r/n
  points)
• The standard Kolmogorov-Smirnov (K-S) test for
  comparing two frequency distributions can then be
  applied

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Quadrat Analysis (QA) -- Kolmogorov-Smirnov (K-S)
Test

• The test statistic D is simply given by
• where Oi and Ei are the observed and expected
  cumulative proportions of the ith category in the
  two distributions.
• i.e. the largest difference (irrespective of
  sign) between observed cumulative frequency and
  expected cumulative frequency

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Quadrat Analysis (QA) -- Kolmogorov-Smirnov (K-S)
Test

• The critical value at the 5 level is given by
• where n is the number of quadrats
• in a two-sample case -- where n1 and n2 are the
  numbers of quadrats in the two sets of
  distributions

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Quadrat Analysis Variance-Mean Ratio (VMR)

• Test if the observed pattern is different from a
  random pattern (generated from a Poisson
  distribution which mean variance)
• Treat each cell as an observation and count the
  number of points within it, to create the
  variable X
• Calculate variance and mean of X, and create the
  variance to mean ratio variance / mean

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Quadrat Analysis Variance-Mean Ratio (VMR)

• For an uniform distribution, the variance is
  zero.
• we expect a variance-mean ratio close to 0
• For a random distribution, the variance and mean
  are the same.
• we expect a variance-mean ratio around 1
• For a clustered distribution, the variance is
  relatively large
• we expect a variance-mean ratio above 1

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Significance Test for VMR

• the mean of the observed distribution
• , where xi is the number
  of points in a quadrat, ni is the number of
  quadrats with xi points, and n is the total
  number of quadrats

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Weakness of Quadrat Analysis

• Results may depend on quadrat size and
  orientation
• Is a measure of dispersion, and not really
  pattern, because it is based primarily on the
  density of points, and not their arrangement in
  relation to one another
• Results in a single measure for the entire
  distribution, so variations within the region are
  not recognized (could have clustering locally in
  some areas, but not overall)

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Weakness of Quadrat Analysis

• For example, quadrat analysis cannot distinguish
  between these two, obviously different, patterns

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Nearest-Neighbor Index (NNI)

• Uses distances between points as its basis.
• Compares the observed average distance between
  each point and its nearest neighbors with the
  expected average distance that would occur if the
  distribution were random
• NNI r obs / r exp
• For random pattern, NNI 1
• For clustered pattern, NNI lt 1
• For dispersed pattern, NNI gt 1

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Nearest-Neighbor Index (NNI) Significance test
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Nearest-Neighbor Index (NNI)

• Advantages
• NNI takes into account distance
• No quadrat size problem to be concerned with
• However, NNI not as good as might appear --
• Index highly dependent on the boundary for the
  area
• its size and its shape (perimeter)
• Fundamentally based on only the mean distance
• Doesnt incorporate local variations (could have
  clustering locally in some areas, but not
  overall)
• Based on point location only and doesnt
  incorporate magnitude of phenomena at that point

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Nearest-Neighbor Index (NNI)

• An adjustment for edge effects available but
  does not solve all the problems

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Nearest-Neighbor Index (NNI)

• Some alternatives to the NNI are
• the G and F functions, based on the entire
  frequency distribution of nearest neighbor
  distances, and
• the K function based on all interpoint distances.

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Spatial Autocorrelation

• Most statistical analyses are based on the
  assumption that the values of observations in
  each sample are independent of one another
• Positive spatial autocorrelation violates this,
  because samples taken from nearby areas are
  related to each other and are not independent

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Spatial Autocorrelation

• In ordinary least squares regression (OLS), for
  example, the correlation coefficients will be
  biased and their precision exaggerated
• Bias implies correlation coefficients may be
  higher than they really are
• They are biased because the areas with higher
  concentrations of events will have a greater
  impact on the model estimate
• Exaggerated precision (lower standard error)
  implies they are more likely to be found
  statistically significant
• they will overestimate precision because, since
  events tend to be concentrated, there are
  actually a fewer number of independent
  observations than is being assumed.

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Spatial Autocorrelation

• Several measures available
• Join Count Statistic
• Morans I
• Gearys Ratio C
• General (Getis-Ord) G
• Anselins Local Index of Spatial Autocorrelation
  (LISA)

Discuss them later
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LINE PATTERN ANALYZERS

• Two general types of linear features
• Vectors (lines with arrows)
• Networks
• Spatial attributes of linear features
• Length
• Orientation and Direction
• Spatial attribute of network features
• Connectivity or Topology

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Spatial Attributes of Linear Features -- Length

• Linear distance

(x1,y1 )
c
a
(x1,y2 )
(x2,y2 )
b
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Spatial Attributes of Linear Features -- Length

• Great circle distance D of locations A and B
• where
• a and b are the latitude readings of locations A
  and B
• ?? is the absolute difference in longitude
  between A and B

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Spatial Attributes of Linear Features
Orientation and Direction

• Orientation
• Directional
• e.g. West-East orientation
• Non-directional (from to )
• e.g. To describe a fault line --
• from location y to location x
• from location x to
  location y
• Direction
• Dependent on the beginning and ending locations
• from location y to location x
• ? from location x to
  location y

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Directional Statistics Directional Mean
Directional Mean Average direction of a set of
vectors
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Directional Statistics Directional Mean
Y


?
X
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Directional Statistics Circular Variance

• Shows the angular variability of the set of
  vectors

Y
X
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Directional Statistics Circular Variance

• For a set of n vectors,
• , all vectors have the same direction
  or no circular variability
• , all vectors are in opposite
  directions

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Network Analysis

• Connectivity how different links are connected
• Vertices junctions or nodes
• Links/edges the lines joining the vertices

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Connectivity Matrix (C)

• Cij 1 if direct connect between i and j
• Cij 0, otherwise

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Connectivity Matrix (C)

• C1 direct
• C2 number of 2 step paths from i to j
• Example from i to k to j is a 2 step path with
  one intermediate vertex k
• C3 number of 3 step paths from i to j
• Example from i to k to m to j is a 3 step path
  with two intermediate vertices

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Network as a matrix
C2 C1 C1 C3 C2 C1 C4 C3 C1 C5 C4
C1 .
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Minimally connected network

• Each vertex is connected to the network, and
  there are no superfluous linkages
• The minimum number of edges needed to create a
  network is V-1, one less than the number of
  vertices in the network i.e, eminV-15

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Maximally connected network

• Nonplanar
• the maximum number of edges is

• Directional

emax V(V-1)

• Non-directional

emax V(V-1)/2
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Maximally connected network

• Planar --
• the maximum number of edges is emax 3(V-2)

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Gamma Index

• Gamma index provides useful basic ratio for
  evaluating the relative connectivity of an entire
  network
• Ratio between the number of edges actually in a
  given network and the maximum number possible in
  that network
• ? actual edges/maximum edges
• minimally connected network is
• ? (V-1) / 3(V-2)

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Alpha Index

• compares the number of actual (fundamental)
  "circuits" with the maximum number of all
  possible fundamental circuits
• ? (E - V 1) / (2V - 5), where 2V - 5 the
  maximum number of fundamental circuits

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Diameter

• the number of linkages or steps needed to connect
  the two most remote nodes in the network
• the better connected the network, the lower the
  diameter

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POLYGON PATTERN ANALYZERS

• We will discuss the use of spatial statistics to
  describe and measure spatial patterns formed by
  geographic objects that are associated with areas
  or polygons.

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Spatial Autocorrelation (SA) Spatial Weights
Matrices

• SA measures the degree of sameness of attribute
  values among areal units (or polygons) within
  their neighborhood
• Different ways of specifying spatial relationships

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Neighborhood Definitions Adjacency Criterion

• Immediate (first-order) neighbors of X
• Rooks case

• Queens case

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Neighborhood Definitions Binary Connectivity
Matrix

• C connectivity matrix with elements cij ,
• cij 1 if the ith polygon is adjacent to the jth
  polygon
• cij 0 if the ith polygon is NOT adjacent to the
  jth polygon
• Symmetrical cij cji
• Not efficiency

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Neighborhood Definitions Stochastic Matrix

• Row-standardized matrix (stochastic matrix)
• Assume each neighbor exerts the same amount of
  influence
• W spatial weights matrix with elements wij ,

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Neighborhood Definitions Distance between
polygon centroids

• For example,
• Within a radius of 1 mile
• Adjacency measure is just a binary representation
  of the distance measure
• 1 zero distance between two neighboring units

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Spatial Weights Matrices Centroid Distances

• dij represents the distance between areal units i
  and j
• Weight
• Inversely proportional to the distance
• Weight
• Distance-decay spatial relationships diminish
  more than just proportionally to the distance

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Spatial Autocorrelation (SA)

• Univariate handle one variable and evaluate how
  that variable is correlated over space
• Several measures available
• Global measures SA stable across the study
  region
• Join Count Statistic measure the magnitude of
  SA among polygons with binary nominal data
• Morans I Index
• Gearys Ratio C
• G statistic

For interval or ratio data
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Spatial Autocorrelation (SA)

• Several measures available
• Local measures may not stable over the study
  region
• Local version of the G statistic
• Local Index of Spatial Autocorrelation (LISA)
  local version of Morans I and Gearys Ratio C

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Spatial Autocorrelation (SA)Joint Count
Statistics

• Binary attribute data
• WW
• BW
• BB
• Compare the observed numbers of joints of various
  types (BB,WW, BW) with those expected from a
  random pattern

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Global spatial autocorrelation statistic --
Morans I

• xi is the value of interval or ratio variable in
  areal unit i,
• W is the sum of all elements of the spatial
  weights matrix (i.e. W??wij), and
• n is the number of areal units

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Global spatial autocorrelation statistic --
Morans I

• I ranges from 1 to 1
• If no spatial autocorrelation exists,
• lt 0
• inversely related to n
• Z-test

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Global spatial autocorrelation statistic
Gearys Ratio

• xi is the value of interval or ratio variable in
  areal unit i,
• W is the sum of all elements of the spatial
  weights matrix (i.e. W??wij), and
• n is the number of areal units

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Global spatial autocorrelation statistic --
Gearys Ratio

• C ranges from 0 to 2
• C0 indicates a perfect positive spatial
  autocorrelation when all neighboring values are
  the same
• C2 indicates an extremely negative spatial
  autocorrelation
• E(C)1, not affected by n
• Z-test

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Global spatial autocorrelation statistic
General G Statistic

• Morans I Gearys C cannot tell HH vs LL as
  they are concerned with only whether neighboring
  values are similar or not
• The general G-statistic
• where wij(d)1 if areal unit j is within d from
  areal unit i o.w. wij(d)0.
• Z-test

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Local spatial autocorrelation statistic LISA

• Local Index of Spatial Autocorrelation (LISA)
  local version of Morans I and Gearys Ratio C
• Local Moran statistic for areal unit i
• High clustering of similar values (all high or
  all low)
• Low clustering of dissimilar values

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Local spatial autocorrelation statistic LISA

• Local Gearys Ratio C for areal unit i
• Low clustering of similar values (all high or
  all low)
• High clustering of dissimilar values

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Local spatial autocorrelation statistic local
G-statistic

• Local G-statistic for areal unit i
• Standard Scores

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Local spatial autocorrelation statistic local
G-statistic

• Interpretation of standard scores for