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Dr. Marina Gavrilova
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
Introduction

Distribution Descriptors: One Variable

Relationship Descriptors: Two Variables

Point Pattern Descriptors

Point Pattern Analyzers

Autocorrelation
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Statistics classification
 Classified by function:
◦ Description statistics
◦ Inferential statistics

Classified by areas of application:
◦ Classical statistics: sociology, political science,
medicine and engineering.
◦ Spatial statistics: based on classical and extended
to the spatially referenced data.
◦ Geostatistics: one kind of Spatial statistics and
originated in geo-science.
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
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A certain phenomenon occurs: Random
process or Systematic Process?
Soil Example:
◦ Hypothesis – soil fertility of a farm is low
◦ To test the hypothesis, gather more data about
the soil.
◦ Collect a sample of soil for further examination
instead of the entire population.
◦ Observation: each examined location; Sample
size: number of observations selected.
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A region can be partitioned in many ways
based on the given criteria. USA: States
boundaries, census geography.
Modifiable Area Unit Problem (MAUP)
include:
◦ Scale effect: Analyze data at multiple levels of
spatial resolution results in inconsistency.
◦ Zoning effect: Analyze data derived from different
zonal systems with similar number of areal units
results in inconsistency.
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Spatial autocorrelation represents the
nature of geography and, consequently, will
almost always be present in spatial data.
Tober “First Law of Geography”:
“All things related to each other, but closer
things are related more”.
Butterfly Effect: Butterfly flapping in China
may cause a hurricane landfall in the US due
to spatial propagation of air disturbances.
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Mode: The value that occurs most frequently
in a set of data or called the modal value. If
two or more categories have the highest
frequency, then data is bimodal or
multimodal.
Median: The middle value after all values are
sorted in ascending or descending order.
Mean or Average: n observation, each with
an observed value xi then the simple
arithmetic mean is defined as
n
x
x
i
i 1
n
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
Grouped or weighted mean: if data values are
grouped into classes, then all data within each group
are represented by on value as the overall value in
that class. A mean derived from the grouped data is
called a grouped mean or a weighted mean.
If xi is the midpoint of the i th class (k classes
together) with fi as the number of data values in that
class (frequency), the weighted mean:
k
fx
i i
xw 
i 1
k
f
i
i 1
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While mean is a good measure of the central
tendency of a set of data, it captures no
information about how the values are concentrated
or scattered around the mean.
Range, Minimum, Maximum, and Percentiles:
◦ Range = Maximum-Minumum
◦ Percentiles are the corresponding data values that have
certain percentages of the data smaller than these values.
Data Xa and Xb have the same median 7, different 25th (3
for Xa and -5 for Xb )
Xa = 1 3 5 7 9 11 13
Xb = -11 -5 1 7 13 19 25
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
Mean Deviation: unlike the dispersion
measures discussed so far using one or a few
data values in the series, the mean deviation
takes into account all data values. It is
calculated by summing all the differences that
individual data values have from the mean
and then dividing this sum by the number of
n
observation.
x x
D

i
i 1
n
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
Variance and Standard Deviation: Another way
to avoid the offsets caused by adding positive
and negative deviations from the mean
together is to square all deviations from the
mean before summing them.
n
 (x  )
n
2
i
 
2
i 1
n


( xi   ) 2
i 1
n
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
Weighted Variance and Weighted Standard
Deviation.
fi is the frequency for the i th group or class,
xi is the midpoint value in the i th group,
xw is the weighted mean, and
k is the number of groups.
k
 
2
w

f i ( xi  x w ) 2
i 1
n
f
i
i 1
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Relationship Descriptors:
Two Variables
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
The mean and its variations address the issue of
location, where the observations distribute along
the continuous value line. Median and mode
consider this central tendency issue. Variance,
standard deviation, and percentiles address the
issue of dispersion. Skewness deals with direction
clustering. Kurtosis addresses the issue of
concentration. All these measures focus on the
distribution of the values using one variable at a
time.
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Mean, standard variable cannot measure the
relationships between different
distributions quantitatively.
One of statistics is based on the concept
correlation measures statistically the
direction and strength of the relationship
between two sets of data or two variables
for a number of observation. Regression
measures the dependence of one variable
on another.
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
Education is traditionally regarded as an
asset. It enriches a person’s life in many
ways. We usually believe that education and
income are somewhat related and change in
the same direction. If we recognize the
value of education in eventually achieving a
higher income, it would be nice to know
how strong this relationship is, that is, how
these aspects of life are related or
correlated.
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Each relationship has two important aspects: the
direction and strength of the relationship. Between
two related variable, the relationship is typically
measured as correlation– a statistical measure
indicating how values in one variable are related to
values in the other variable.
Positive or direct correlation
Negative or inverse correlation
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Trend analysis is a technique measuring the
trend, while correlation is a statistical
measure of two variables.
Trend analysis addresses the dependence of
one variable on another.
Going beyond the strength and direction of
the relationship, trend analysis allow us to
model the relationship and to estimate
likely value of one variable based on the
value of another variable.
Models that are constructed with this
technique are known as regression models.
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Simple linear regression model or bivariate
regression model: Using a straight line to model
the relationship between tow variables. Here are
an example. A regression between median
household income and median house value for
51 states.
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Some phenomena may be modeled by the
regression reasonable well, and others may not.
Regression model assumes a linear relationship
between the variable. If the relationship is not
linear or if the two variables have weak or no
relationship, then the model will perform poorly.
A multivariate regression model, which can
accommodate multiple independent variables.
Under either circumstance, we may have committed
a model specification error.
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Point Pattern Descriptors
◦ Central Tendency
◦ Dispersion and Orientation

Point Pattern Analyzers
◦
◦
◦
◦
Quadrant Analysis
Nearest-Neighbor Analysis
Spatial Autocorrelation of Points
K-Function
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Point pattern descriptors cover:
The methods for determining the overall
patterns of a given set of points.
Measures used to describe the magnitude
of spatial dispersion of a given set of
points.
How the direction bias of a set of points can
be extracted statistically.
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A set of point descriptors provide certain
descriptive information on the distribution of a set
of points.
Central tendency information, mean centers,
weighted mean centers, and median centers
provide a good summary of how a set of points
distributes in the geographic space.
To describe the spatial dispersion characteristics of
a set of points, the measures of standard distance
and standard ellipse will be discussed. These
measures indicate the spatial variation and
orientation of a point distribution.
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The mean center, or spatial mean, is a central
or average location of a set of points. For n
points xmc and ymc are the coordinates of the
mean center, xi and yi are the coordinates of
point i, and n is the number of points.
n
n
x  y
i
( xmc , y mc )  (
i 1
n
,
i 1
n
i
)
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The weighted mean center of a distribution of
points can be found by multiplying the x- and
y- coordinates of each point by the weight
assigned to each observation or location.
◦ wi is the weight at point i
n
n
w x w y
i i
( x wmc , y wmc )  (
i 1
n
i
,
i 1
n
w w
i
i 1
i
)
i
i 1
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Two sets of points may occupy the same
geographic space and may be interrelated.
For example, one set of points represents the
location of forest fires and the other the
locations of camping cabins in a wildlife region.
They may have the same overall locations, but
forest fire have a more dispersed spatial pattern
than cabins.
In additional to spatial central tendency, it may
be interesting to evaluate the magnitude of
dispersion of locations and the orientation of
the spatial distribution.
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Similar to those in classical statistics, the
population standard deviation,  ,or the
sample standard deviation, S, can be
computed as:
n


( xi   )
i 1
n
n
2
S

( xi  x ) 2
i 1
n 1
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Points in a distribution may have different
attribute values that reflect the relative
importance of different point observation.
◦ Wi is the weight for point i, and
◦ (xwmc, ywmc) is the weighted spatial mean.
n
SD 

n
wi ( xi  x wmc ) 
2
i 1

wi ( yi  y wmc ) 2
i 1
n
w
i
i 1
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
The standard distance circle is a very effective
visualization tool to show the spatial spread of a
set of point location.
A logical extension of the standard distance circle
is the standard deviational ellipse. It can capture
the directional bias in a point distribution. Three
components are needed to describe it:
◦ An angle of rotation
◦ Deviation along the major axis
◦ Deviation along the minor axis
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To fully understand the various states and
dynamics of a particular geographic phenomenon,
an analyst must be able to detect spatial patterns
from the point distributions and to track the
changes in point patterns at different time.
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Quadrant Analysis allows analysts to determine if a
point distribution is similar to a random pattern
using a spatial sampling framework.
Nearest Neighbor Analysis compares the average
distance between nearest neighbors in a set of
points to that of a theoretical pattern.
Spatial autocorrelation coefficients measure how
similar neighboring points are.
K-function analysis can identify and evaluate the
clustering of points at different spatial scales, or
extents.
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Quadrant Analysis evaluates a point
distribution by examining how its density
changes over space.
The density measured by Quadrant Analysis
is then compared with the density of a
theoretically constructed random pattern to
see if the point distribution in question is
more clustered or more dispersed than the
random pattern.
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A regular square grid and a number of points
falling in some squares.
The square are referred to as quadrants, which are
essentially sampling units in spatial statistical
jargon.
Circle is the most geometrically compact shape,
however circles cannot cover the entire geographic
space unless they overlap.
In an extremely clustered point pattern, all or most
of the points fall inside one or a few squares only.
In an extremely dispersed pattern referred to as a
uniform pattern or a triangular lattice, all squares
contain similar number of points.
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Statistically, Quadrant Analysis will achieve
a fair evaluation of the density across the
study area if it applies a large enough
number of randomly generated quadrants.
An optimal size of quadrant can be
calculated by 2A/r . A is the area of study
area, and r is the number of points in the
distribution.
Once the quadrant size for a point
distribution is determined, Quadrant
Analysis can proceed to establish the
frequency distribution of the number of
points for all quadrant.
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Besides using K-S statistics to test if the
observed pattern is different from a random
pattern, one may perform the VarianceMean Ratio Test by taking advantage of a
specific statistical property of the Position
distribution.
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Quadrant Analysis is useful in comparing
an observed point pattern to a random or
theoretically known distribution. However, it
has certain limitations.
The analysis captures information on the
points within each quadrant, but no
information on points between quadrants is
used in the analysis. As a result, Quadrant
Analysis may be insufficient to distinguish
between certain point pattern in the
following figures.
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Visually, the two patterns are different. Using
Quadrat Analysis, however, the two patterns yield
the same result.
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Nearest Neighbor Statistic is derived from the
average distance between points and each of their
nearest neighbors.
The second-ordered neighbor statistic uses the
distance of the second nearest neighbors. Higherordered neighbors can be defined in similar ways.
Ordered Statistics can evaluate the pattern at
different spatial scales.
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While both Quadrant Analysis and Nearest
Neighbor Analysis test point distribution, they
utilize different spatial concepts.
◦ Quadrant Analysis tests a point distribution with
the points per area concept using quadrants as
sampling units.
◦ Nearest Neighbor Analysis uses the concept of
area per point.
Both methods are similar in sense that the
observed pattern is compared with some
know distribution (random pattern).
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How Nearest Neighbor Analysis
works.

In a homogeneous region, the most uniform
pattern formed by a set of points occurs
when this region is partitioned into a set of
identical hexagons with a point at its center.
The distance between points will be
1.075 A/ n
, where A is the area of the region and n is the
number of points.
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
R statistic is the ratio of the observed average
distance between nearest neighbors of a point
distribution and the expected average nearest
neighbor distance. It is also the nearest
neighbor statistic.
robs
R
rexp

robs is the observed average distance between
nearest neighbors and rexp is the expected
average distance between nearest neighbors as
determined by the theoretical pattern.
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d1=d13
d2=d23
d3=d32
d4=d43
(For point 1, the nearest neighbor is 3)
robs
d


i
n
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By selecting the seven
largest cities in Ohio,
we can compute their
nearest neighbor
distance and the
observed average
nearest neighbor
distance robs
=51.82miles.
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Nearest Neighbor Analysis has been extended to
accommodate the second, third, and other higherorder neighbor definitions. When two points are
not immediate nearest neighbors but rather the
second nearest neighbors, the way distances are
computed between them will need to be adjusted
accordingly.
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The second-order nearest neighbor statistic R2 is
robs/rexp .
d
robs 



i
n
di is the distance between i and its second nearest
neighbor.
The expected nearest neighbor distance in the
denominator of the R2 statistic is similar to the
first-order expected distance, the constant change
from 0.5 to 0.75.
rexp  0.75
A
n
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
Standard error estimate for second-order
nearest neighbor distance A
SEr  0.2722

n2
Generally, for k-order neighbor statistic,
 (k ),  (k ) are the constants for expected
distance and standard error, respectively.
1
2
A
rexp (k )   1 (k )
n
SEr (k ) 
 2 (k )
n2
A
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Another statistic that can offer some insights and
is more parsimonious to evaluate if the magnitude
of clustering is uniform over different spatial scales
is K-function analysis. It is an extension of the
ordered neighbor statistics. For a set of point in a
region, the K-function analysis involves following
steps:
Select a distance increment or spatial lab, d, that is
analogous to the unit reflecting the change in the
spatial scale.
Set the iteration number g=1 to begin the process.
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Around each point i in a region, create a circular
buffer with a radius of h, where h=d*g. Therefore,
the buffer will have a size d in the first iteration
and 2d in the second and so on.
For each point, count the number of points falling
within its buffer of size h and denote that count as
n(h).
Increase the radius of the buffer by d.
Repeat steps 3, 4, and 5 by increasing h until g=r
or g=D/d.
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Figure in next slide uses only four points to
illustrate the procedure.
◦ Only three rings or buffers were created instead
of the full range up to D. For a give h, we count
the number of points within the buffers centered
at all points. Point A is rather dispersed from
other points, and therefore the counts are
relatively low for buffers with small h. For point B,
the point is in the middle of the cluster, and
therefore the point count are relatively high with
the small buffers, but the increases in point
counts are substantial with large h’s. For Point C
and D, the points themselves are apart from the
cluster.
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The relationship between point counts and
the spatial lag from empirical observation
can be compared with a known patter, most
likely a random pattern.
In a random pattern, point counts increase
with increasing h but in no particular
pattern.
K-function detect clustering at different
scales by comparing the relationship
between point counts and the size of h to
that in a random distribution.
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The number of points within the
buffer with a lag h, as follows:
n ( h) 
 I (d
h
i
ij
), i  j,
j
◦ i and j are the indices of points.
◦ dij is the distance between the two points i,
j.
◦ Ih is an indicator function such that Ih=1 if
dij<h and Ih=0 otherwise
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Sharing similar problems with other spatial
statistical and analytical techniques, the K-function
is also subject to the boundary problems.
Image that a point is located rather close to the
edge of the study region. When buffers are formed
around the point, a significant proportion of
buffers will be outside of the study area and thus
will distort the probability of finding a point within
the vicinity of h.
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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.
n
n
 s w
ij
SAC 
ij
i 1 j 1
n
n
 w
ij
i 1
j 1
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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
sij  ( xi  y j ) 2
The computation of Geary’s Ratio
C
 w ( x  x )
2 w  ( x  x )
(n  1)
ij
i
2
j
2
ij
i
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In Moran’s I Index, the similarity attribute
values between two points is defined
sij  ( xi  x )( x j  x )
The computation of Moran’s I Index
I
n
 w ( x  x )( x  x )
 w  ( x  x )
ij
i
j
2
ij
i
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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 0<C<1
points show similar characteristics
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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Distribution Descriptors using single variable and
Relationship Descriptors using two (or more)
variables are typical statistical tools.
Point Pattern Descriptors and Point Pattern
Analyzers can be used to study more deep
patterns of the data, in combination with various
representations (spatial, grid, k-mean, ellipse
etc)
Autocorrelation analysis is sued to understand
further data relationship in respect to distance
between spatial locations
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