Spatial Statistics

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Spatial Statistics

• Modified from Dr. YU-FEN LI

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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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Kolmogorov-Smirnov Test (?1)

• A. Situations in which the control and treatment
  groups do not differ in mean, but only in some
  other way. For example consider the datasets
• controlA0.22, -0.87, -2.39, -1.79, 0.37,
  -1.54, 1.28, -0.31, -0.74, 1.72, 0.38, -0.17,
  -0.62, -1.10, 0.30, 0.15, 2.30, 0.19, -0.50,
  -0.09
• treatmentA-5.13, -2.19, -2.43, -3.83, 0.50,
  -3.25, 4.32, 1.63, 5.18, -0.43, 7.11, 4.87,
  -3.10, -5.81, 3.76, 6.31,2.58, 0.07, 5.76, 3.50

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Kolmogorov-Smirnov Test (?1)

• There are then a few situations in which it is a
  mistake to trust the results of a t-test
• Notice that both datasets are approximately
  balanced around zero evidently the mean in both
  cases is "near zero. However there is
  substantially more variation in the treatment
  group which ranges approximately from -6 to 6
  whereas the control group ranges approximately
  from -2½ to 2½. The datasets are different, but
  the t-test cannot see the difference.

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Kolmogorov-Smirnov Test (?1)
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Kolmogorov-Smirnov Test (?1)

• the percentile plot of this data (in red) along
  with the behavior expected for the above
  lognormal distribution (in blue)

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Kolmogorov-Smirnov Test (?2)

• Situations in which the treatment and control
  groups are smallish datasets (say 20 items each)
  that differ in mean, but substantial non-normal
  distribution masks the difference. For example,
  consider the datasets
• controlB1.26, 0.34, 0.70, 1.75, 50.57, 1.55,
  0.08, 0.42, 0.50, 3.20, 0.15, 0.49, 0.95, 0.24,
  1.37, 0.17, 6.98, 0.10, 0.94, 0.38
• treatmentB 2.37, 2.16, 14.82, 1.73, 41.04,
  0.23, 1.32, 2.91, 39.41, 0.11, 27.44, 4.51, 0.51,
  4.50, 0.18, 14.68, 4.66, 1.30, 2.06, 1.19
• These datasets were drawn from lognormal
  distributions that differ substantially in mean.
  The KS test detects this difference, the t-test
  does not. Of course, if the user knew that the
  data were non-normally distributed, s/he would
  know not to apply the t-test in the first place.

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Kolmogorov-Smirnov Test (?2)

• Sorted controlB0.08, 0.10, 0.15, 0.17, 0.24,
  0.34, 0.38, 0.42, 0.49, 0.50, 0.70, 0.94, 0.95,
  1.26, 1.37, 1.55, 1.75, 3.20, 6.98, 50.57

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Kolmogorov-Smirnov Test (?2)
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Kolmogorov-Smirnov Test (?2)
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Kolmogorov-Smirnov Test (?2)
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Kolmogorov-Smirnov Test (?2)
the percentile plot of this data (in red) along
with the behavior expected for the above
lognormal distribution (in blue).
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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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Space as a matrix

• W where wij is some measure of interaction
• adjacency
• decreasing function of distance
• invariant under rotation, displacement
• readily obtained from a GIS

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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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Applications of the W matrix

• Spatial regression
• add spatially lagged terms weighted by W
• Anselins SPACESTAT
• Moran and Geary indices of spatial dependence

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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

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More Discussions on GIS and Spatial Statistics
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Spatial dependence

• The First Law of Geography (Tobler)
• all things are related but nearby things are more
  related than distant things
• Acceptance of the null hypothesis of no spatial
  dependence is always a Type II error
• Hell is a place with no spatial dependence

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It's chilly today in Seattle
Spoken word
Text
Picture
x, y, T
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Spatial heterogeneity

• Uncontrolled variance over the Earths surface
• There is no average place
• Results depend explicitly on bounds
• Places as samples
• Consider the model
• y a bx

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