Spatial Correspondence of Areal Distributions

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Spatial Correspondence of Areal Distributions

• Quadrat and nearest-neighbor analysis deal with a
  single distribution of points
• Often, we want to measure the distribution of two
  or more variables
• The coefficient of Areal correspondence and
  chi-square statistics perform these tasks

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Coefficient of Areal Correspondence

• Simple measure of the extent to which two
  distributions correspond to one another
• Compare wheat farming to areas of minimal
  rainfall
• Based on the approach of overlay analysis

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Overlay Analysis

• Two distributions of interest are mapped at the
  same scale and the outline of one is overlaid
  with the other

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Coefficient of Areal Correspondence

• CAC is the ratio between the area of the region
  where the two distributions overlap and the total
  area of the regions covered by the individual
  distributions of the entire region

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Result of CAC

• Where there is no correspondence, CAC is equal to
  0
• Where there is total correspondence, CAC is equal
  to 1
• CAC provides a simple measure of the extent of
  spatial association between two distributions,
  but it cannot provide any information about the
  statistical significance of the relationship

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Resemblance Matrix

• Proposed by Court (1970)
• Advantages over CAC
• Limits are 1 to 1 with a perfect negative
  correspondence given a value of 1
• Sampling distribution is roughly normal, so you
  can test for statistical significance

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Chi-Square Statistic

• Measures the strength of association between two
  distributions
• Class Example
• Relationship between wheat yield and
  precipitation
• Two maps showing high and low yields and high and
  low precipitation

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HIGH PRECIP
HIGH YIELD
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High Precip.
High Yield
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Chi-Square

• By combining distribution on one map we can
  better understand the relationship between the
  two distributions
• In this example we are using a grid
• The finer the grid, the more precise the
  measurement
• Four possibilities exist
• Low rainfall, low yield
• Low rainfall, high yield
• High rainfall, low yield
• High rainfall, high yield

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Chi-Square

• Record the total number of occurrences into a
  table of observed frequencies

WHEAT
High Low
High Low
PRECIP.
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Chi-Square

• Create a table of expected frequencies using
  probability statistics ( High rain of high
  yield cells)
• Row total column total / table total

WHEAT
WHEAT
High Low
High Low
High Low
High Low
PRECIP.
PRECIP.
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Compute Chi-Square

• Therefore, in our example we have

High Low
High Low
High Low
High Low
Observed
Expected
High/High
High/Low
Low/Low
Low/Low
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Interpreting Chi Square

• Zero indicates no relationship
• Large numbers indicate stronger relationship
• Or, a table of significance can be consulted to
  determine if the specific value is statistically
  significant
• The fact that we have shown that there is a
  correlation between variables does NOT mean that
  we have found out anything about WHY this is so. 
  In our analysis we might state our assumptions as
  to why this is so, but we would need to perform
  other analyses to show causation.

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If you dont have Chi-Square values

• Yules Q
• Value of Yules Q always lies between 1 and 1
• Value of 0 indicates no relationship
• Value of 1 indicates a positive relationship
• Value of 1 indicates a negative relationship

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Analysis of Election 2000

• Polygon to Polygon
• Point to Polygon

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Assessing Our Cultural Divide Results from the
2000 Presidential Election Arthur J. Lembo, Jr.
Ph.D. Cornell University Paul Overberg USAToday
ANALYSIS OF SPATIAL AUTOCORRELATION JOIN COUNT
ANALYSIS
ANALYSIS OF SPATIAL CORRESPONDENCE OVERLAY
ANALYSIS
A second analysis was used to determine the
likelihood of a county with urban areas voting
for either candidate. For this study, four
categories were evaluated counties with small
cities (under 50,000), medium sized cities
(50,000 75,000), large sized cities (greater
than 75,000), and no cities. Based on the
percentage of counties won by each candidate
(Gore 22 Bush 78) we computed the random
probability that a city would fall within a Bush
county or a Gore county. This probability
allowed us to determine the expected number of
cities that would be located within Gore counties
or Bush counties. The actual number of cities
located in a Gore county or Bush county was
determined using overlay analysis with ArcView.
Similar to the previous example, z-scores were
computed for each of the categories as
follows where O is the observed number of
cities falling within a county, E is the expected
number of cities falling within a county, p is
the probability of a city falling in a Bush
county, q is the probability of a city falling in
a Gore County, and n are the total number of
cities. Table 2. Cities Falling Inside a
County Won by Either Bush or Gore
Expected Expected Observed Observed
Z Z Gore
Bush Gore Bush Gore
Bush Large ( 75K) 66 238
184 119 267 272 Medium
(50-75K) 54 196
147 98 470 55 Small
(2030 1236 4,998 3 No
City 427 1588 347
1690 18 29
As previously stated, a purely random sample
drawn from a population whose true mean is 0 at
the 95 confidence level would fall within a
z-score range of /- 1.96 in magnitude. Table 2
indicates that each of the z-score values exceed
1.96. Implied from this is that significant
correlation among votes for Al Gore and counties
with cities, and votes for George W. Bush and
counties without cities (rural areas) exists.
Join Count Analysis is a method of spatial
autocorrelation that evaluates the statistical
significance of clustering among neighboring
polygons. Based upon the total number of
counties won by each candidate (Gore 588 Bush
2214), the expected number of adjacent counties
that voted for the same candidate (i.e. two
adjacent counties voting for Bush) was computed .
In addition, the actual number of adjacent
counties that voted for the same candidate was
also computed using spatial analysis techniques
in ArcView GIS. The results were as
follows Table 1. Expected vs. Actual Joins of
Adjacent Counties Voting for the Same Candidate
Expected Actual Expected
Expected Actual Gore/Gore Joins
Gore/Gore Joins Bush/Bush Joins
Bush/Bush Joins 438
879 5516 6253 Assuming
an independent random process, we computed the
z-score, or number of standard deviations away
from the mean for each candidates specified
number of joins (ZGore/Gore 15.47 ZBush/Bush
8.75). A purely random sample drawn from a
population whose true mean is 0 at the 95
confidence level would fall within a z-score
range of /- 1.96 in magnitude. Both numbers
were significantly higher than 1.96, indicating
significant positive spatial autocorrelation.
Therefore, the join count analysis showed that
clustering exists within the county voting
patterns. Inferred from this analysis is the
observation that regionalized voting patterns
existed in the 2000 Presidential Election.
ABSTRACT Although the 2000 Presidential election
was one of the closest in recent history, many
commentators noted that the voting patterns
appeared to exhibit a cultural divide, with
urban areas voting for Al Gore, and rural areas
voting for George W. Bush. Because most of the
comments are based on a subjective view of the
county voting patterns, this project attempts to
provide a quantifiable measure of the voting
patterns exhibited during the 2000 election.
Specifically, we were interested in determining
if a statistically significant clustering pattern
existed based on county-wide results, and if each
candidate won their assumed cultural association
(Gore Urban Bush rural). To test these
hypotheses, two separate spatial analysis methods
were performed on county-wide voting patterns
within the United States. The first method
utilized a principle of spatial autocorrelation
called join count analysis to determine if voting
patterns exhibited evidence of spatial
clustering. The second method used map overlay
to determine the likelihood of cities falling
within either Bush or Gore counties.
Conclusion This analysis provided quantifiable
evidence that positive spatial autocorrelation
(clustering) of voting patterns existed during
the 2000 Presidential Election. Also, the
analysis showed a high statistical correlation
between urbanized areas and county votes for Al
Gore. Further analysis is necessary to better
understand causation (i.e. ethnicity, income,
age), however both analyses indicate that
geographic regions (i.e. urban areas) may have
played a large role in the vote determination for
Election 2000. Data Provided Courtesy of
Election Data Services, and USAToday
Figure 1. Examples of Cities in Relation to the
Distribution of Counties. These examples from
New York and Minnesota show that although Bush
(in red) won a majority of the counties, the
cities appear clustered primarily within the few
counties in which Gore won (in blue). For
example, in Minnesota, a majority of the cities
exist within Hennepin County, while in New York,
virtually every county Gore won has a city within
its border.
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Election 2000 Results

• Join Count Analysis
• Table 1. Expected vs. Actual Joins of Adjacent
  Counties Voting for the Same Candidate
• Expected Actual Expected
  Expected Actual
• Gore/Gore Joins Gore/Gore Joins
  Bush/Bush Joins Bush/Bush Joins
• 438 879
  5516 6253
• ZGore/Gore 15.47 ZBush/Bush 8.75
• Overlay Analysis
• Table 2. Cities Falling Inside a County Won by
  Either Bush or Gore
• Expected Expected Observed
  Observed Z Z
• Gore Bush
  Gore Bush Gore
  Bush
• Large ( 75K) 66 238
  184 119 267
  272
• Medium (50-75K) 54 196
  147 98 470 55
• Small ( 2030 1236
  4,998 3
• No City 427 1588
  347 1690 18
  29

Not mutually exclusive from large cities. We must
account for this
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Election 2000 Results

• There was obvious spatial autocorrelation in the
  way way people voted. That is, Bush counties and
  Gore counties were highly clustered
• Also, there are a very high correlation between
  urbanized counties voting for Gore, and
  non-urbanized counties voting for Bush

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Analysis of Environmental Justice

• Point in Polygon Analysis
• By
• Greg Thorhaugcss620 project Spring 2001

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Erie Chi-Squared
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Summary

• Spatial Data Analysis is possible, through basic
  statistical methods
• More in-depth analysis is possible using spatial
  statistics
• GIS software may be used to prepare data for
  statistical analysis
• Spatial data analysis techniques provide a
  powerful tool for analyzing GIS data, and enable
  users to solve creative problems

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Cross Tabulation

• Assume we have a 9 cell land cover map, one from
  1980 and one from 2000 with three categories A,
  B, and C.
• You can see that the resulting cross tabulation
  provides a pixel, by pixel comparison of the
  interpreted land cover types with the two dates.
  So, for the upper left hand cell, the 1980 land
  use was A, and the 2000 land cover also indicated
  the value of A. Therefore, this is a match
  between the 1980 data and 2000 data. However, in
  the lower right cell you can see that the 1980
  data indicated a value of C, while the 2000 value
  was B. This is not a match, and would indicate
  an error between the two sources.
• We can now quantify the results into a matrix as
  shown below. This matrix, is oftentimes called a
  confusion matrix

Ground Reference Data
Interpreted Land Cover Data
Cross Tabulated Grid
A
B
A
B
B
A
BA
BB
AA
Cross Tabulate
B
C
C
B
B
C
BB
BC
CC
A
A
B
B
A
C
BA
AA
CB
A B C
A B C
2
0
2
0
2
1
0
1
1
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Confusion Matrix
Ground Reference

• The matrix on the right shows the comparison of
  the two hypothetical data sets. The 1980 data
  set and the 2000 data set .
• As an example, geographic features that were
  classified as A on the map in 1980, and actually
  were still be A in 2000, represent the upper left
  hand matrix with the value 2 (there were two
  pixels that met this criteria). This means that
  2 units in the overall map that were A, actually
  is A. Similarly, the same exists for the
  classifications of B and C.
• But, there may have been times where the 1980
  value was A and the 2000 value was B. In this
  case, the 2 represented in the top row of the
  matrix says that there are 2 units of something
  that was A in 1980, but is now B in 2000.
• We can begin to add these number up, by adding an
  additional row and column. But what do these
  numbers tell us?

A B C
A B C
2
0
2
Map Classification
0
2
1
0
1
1
Ground Reference
A B C
A B C
2
0
2
4
Map Classification
0
2
1
3
0
1
1
2
2
5
2

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Comparing the maps

• The bottom row tells us that there were two cells
  that were A, five cells that were B, and two
  cells that were C. The rightmost column tells us
  that we mapped four cells as A, three cells as B,
  and 2 cells as C. Adding up the Diagonal cells
  says there were 5 cells where we actually got it
  right.
• So, the overall map comparison is really a
  function of
• Total cells on the diagonal / total number of
  cells.
• (2 2 1) / (2 2 0 0 2 1 0 11)
  5/9 .55 agreement

Ground Reference
A B C
A B C
2
0
2
4
Map Classification
0
2
1
3
0
1
1
2
2
5
2

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Other Accuracy Assessment

• The total correspondence of our example is 55.
  But, that only tells us part of the story. What
  if we were really interested in classification B?
  Where there changes in classification B? Even
  here, there are two different ways of
  interpreting that question
• If I were interested in mapping all the areas of
  B, how well did I get them all? This is called
  the map Producers Accuracy. That is, how well
  did we produce a map of classification B.
• If I were to use the map to find B, how
  successful would I be? This is called the Map
  Users Accuracy. That is, much confidence should
  a user of the map have for a given
  classification.
• To compute the map users accuracy, we would
  divide the total number correct within a row with
  the total number in the whole row. Staying with
  our example of classification B
• We said that we had two cells where B was
  correct. However, we actually said that there
  were three cells that contained B (in other
  words, we incorrectly called a cell B, when it
  should have been C). Therefore, we have
• 2 correct B values / 3 total values .66 users
  accuracy.
• This means that if we were to use this map and
  look for the classification of B, we would be
  correct 66 of the time.
• To compute the map producers accuracy, we would
  divide the total number of correct within a
  column with the total number in the whole column.
  Staying with our example of classification B
• We said that we had two cells where B was
  correct. However, we actually said that there
  were five cells that should have been B.
  Therefore, we have
• 2 correct B values / 5 total values that should
  be B .4 producers accuracy
• This means that the map produced only 40 of all
  the Bs that were out there.

Ground Reference
A B C
A B C
2
0
2
4
Map Classification
0
2
1
3
0
1
1
2
2
5
2

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User and Producer Accuracy
Users Accuracy
Ground Reference

• To test your understanding of all this, compute
  the users and producers accuracy for
  classifications A and C.
• This also gives us some indication of the nature
  of the errors. For instance, it appears that we
  confused classification A with classification B
  (we said on two occasions that B was A). By
  understanding the nature of the errors, perhaps
  we can go back, look over our process and correct
  for that mistake.

A B C
A B C
2
0
2
4
Map Classification
0
2
1
3
.66
0
1
1
2

2
5
2
Producers Accuracy

.4