1. Inferential Spatial Statistics: Introduction to Concepts
Population
Infer
Sample
Today:
Review standard statistical inference.
Examine the concept of Spatial Randomness.
Define a random point pattern.
Next Time
Using inferential spatial statistics to analyze point patterns
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2. Spatial Analysis: successive levels of sophistication
Spatial data description: classic GIS capabilities
Spatial queries & measurement,
buffering, map layer overlay
Exploratory Spatial Data Analysis (ESDA):
searching for patterns and possible explanations
GeoVisualization through data graphing and mapping
Descriptive spatial statistics
Spatial statistical analysis and hypothesis testing
Are data “to be expected” or are they “unexpected” relative to some statistical model, usually of a random process
Spatial modeling or prediction
Constructing models (of processes) to predict spatial outcomes (patterns)  
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3. Descriptive & Inferential Statistical Analysis
Last time we discussed descriptive statistics for spatial analysis
Concerned with obtaining summary measures to describe a set of data
For example, the mean and the standard deviation,
the centroid and the standard distance
This time we will discuss inferential statistics
begin by reviewing standard (non-spatial) inferential statistics
then look at inferential spatial statistics
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4. Standard Statistical Inference:
Inferential statistics
Concerned with making inferences:
from a sample(s) about a population(s)
from observed patterns about underlying processes
I hope you are already familiar with standard (non-spatial) inferential statistics.
I will quickly review the main ideas.
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5. Populations and Samples
Sample: a part (subset) of the population for which we have data.
The sample is used to make inferences about the population.
Population: all occurrences of a particular phenomena
You are a sample of the population of all people in the world.
Infer
We draw conclusions about the population from the sample.
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6. From Lecture #2 on Spatial Analysis Process, Pattern and Analysis
Often, we cannot observe the process, so we have to infer the process by observing the pattern
From the sample, we infer the process in the population.
Create
Processes
Patterns
Population
Infer
Sample
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7. The Importance of the Sample
It depends upon the sample!
If we get sample, the conclusions are good.
Sample is representative of the population
If we get sample, the conclusions are not good.
Sample is a not representative of the population.
How “good “ (or “accurate” or “true”) are our inferences or conclusions?
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8. The Requirement of a Random Sample
All statistical inference is based on the assumption (requirement) that you have a random sample
What is a random sample?
A sample chosen such that every member of the population has an equal chance (probability) of being included
Doesn’t guarantee a representative sample
Could be really unlucky and get

9. statistics are estimates of parameters
Some Definitions
Sample
Subset of population for which we have data
Statistics
Numbers calculated from the sample
Population
All occurences
Parameters
Numbers calculated from the population
statistics are estimates of parameters
We can calculate the statistic because we have data for samples. We cannot calculate the parameter because we do not have data for entire population.
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10. Example: Are girls more intelligent than boys?
Sample of boys
IQ* = 115
Sample of girls
IQ* = 130
*IQ = Intelligence Quotient
Ha! Ha! Girls are more intelligent than boys.
Here is the proof!
No! No! It depends on the samples we have.
The sample statistics are different, but the population parameters may be the same!
Who is correct?
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11. Briggs Henan University 2010
How do we decide who is correct? The Null Hypothesis and the Alternative Hypothesis
Assume that in the population the average (mean) IQ of girls is the same as the average IQ of boys This is called the Null Hypothesis: --there is no difference between girls and boys in the population The Alternative Hypothesis: --in the population, girls are smarter than boys
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12. Choosing between Null and Alternative
In our two samples:
The difference between the sample means was 15
Ask the question: if the population means are the same, how probable is it that, from sampling variation alone, I would get a difference of 15 points between sample means?
If this is reasonable probable (or likely), accept the Null Hypothesis
If this is highly improbable (highly unlikely), reject the Null and accept the Alternative Hypothesis
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13. Briggs Henan University 2010
How do I calculate the probability of getting a difference of 15? We use the sampling distribution. What is this?
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14. All girls All boys Random samples Random samples
(the population of girls)
All boys
(the population of boys)
Random
samples
Random
samples
For every pair of samples, calculate the mean of each, and then the difference between these means.
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15. The Sampling Distribution
If we have a thousand sample pairs, we have a thousand values for
We can draw a frequency distribution showing how often or frequently different values occur
The sampling distribution is simply the frequency distribution for some value calculated each time from many, many, many samples.
The calculated value is called the test statistic
-1.96
2.5%
1.96
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16. Using the Sampling Distribution
-1.96
2.5%
1.96
Here, a sample difference of 15 is quite likely:
Conclusion: Accept the Null.
Boys and Girls are the same 15 15
Here, a sample difference of 15 is very unlikely:
Conclusion:
Reject the Null
Accept the Alternative
Girls are smarter than boys
The probability should be less than
5% (.05) to reject the null hypothesis.
This probability is called the statistical significance of the test.
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17. Calculating a Test Statistic
To find the exact probability of getting a difference of 15 between the girls and boys we calculate a test statistic
a test statistic is: a number, calculated from a sample statistic, whose sampling distribution is known
That is, we know the shape of the frequency distribution of the test statistic when multiple samples are taken
In the case of the difference between two sample means the test statistic is:
Note: test statistics always have “degrees of freedom” which are calculated from the sample size (N)
It is a Normal Frequency
Distribution if the sample sizes
are greater than 30.
S2g =variance for girls
S2b =variance for boys

18. Test Statistic for Normal Frequency Distribution
-1.96
2.5%
1.96
To reject the Null Hypothesis, the Z test statistic should have a value greater than (or less than -1.96).
There is less than a 5% chance that, in the population, the means are the same.
Conclusion:
Reject the Null
Accept the Alternative
Girls are smarter than boys
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19. Standard Error: Standard Deviation of the Sampling Distribution
2.5%
-1.96
1.96
Smaller standard error
Test statistic for the difference between two means:
Larger standard error
Standard error for the difference between two means
Standard error very important
Approximately, it tells you how far, on average, the sample statistic is away from the population parameter
Thus, it is a measure of sampling variability or error
The larger the standard error, the more difficult it is to reject the Null Hypothesis
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20. Briggs Henan University 2010
Reporting the Results of a Statistical Significance Test: many ways to say the same thing!
When we use a test statistic and its sampling distribution we say that we are conducting a statistical significance test
We reject the null hypothesis if there are less than 5 chances in 100 that it is true
We say the results are “statistically significant at the 5% level”
Or we say the results are “significant at the 95% confidence level”
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21. The Normal or Gaussian Probability Distribution.
This is the sampling distribution for tests involving differences between means.
Why is it this shape?
-1.96
2.5%
1.96
If the null hypothesis is true,
what would be the average value of the differences between the sample means?
It would be zero (0)
We expect many small difference values and few big differences
Values would be concentrated around mean
We expect as many negative differences as positive differences
Symmetrical—same on each side of the mean
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22. How do we find the Sampling Distribution and Test Statistic?
Two methods:
By mathematical theory:
test statistics and sampling distributions already known through theory
common distributions are Z (Normal), Chi-square, and F distributions
By computer simulation
The computer is used to “simulate” multiple samples, and we use these to draw a frequency distribution
As with our “boys and girls” example
Very common in spatial statistics
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23. Spatial Statistical Inference
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24. Spatial Statistical Inference: Null and Alternative Hypotheses
Null Hypothesis:
The spatial pattern is random
IRP/CSR: independent random process/complete spatial randomness
Alternative Hypothesis:
The spatial pattern is not random
It may be clustered or dispersed
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25. What do we mean by spatially random?
UNIFORM/DISPERSED
CLUSTERED
Random: a point is equally likely to occur at any location, and the position of a point is not affected by the position of any other point.
Uniform: every point is as far from other points as possible: “likely to be distant”
Clustered: every point is close to other points: “likely to be close”

26. Is it Spatially Random? Difficult to know!
Fact: Two times as many people sit “on the corners” rather than opposite at tables in a restaurant
Conclusion: psychological preference for nearness
In actuality: an outcome to be expected from a random process: two ways to sit opposite, but four ways to sit on the corners
From O’Sullivan and Unwin p.69
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27. High Peak district biomass index: ratio of remotely sensed data spectral bands B3 and B4
Spatially clustered
Geographically random

28. Why Processes differ from Random
Processes differ from random in two primary ways
Variation in the study area
Diseases cluster because people cluster (e.g. cancer)
Cancer cases cluster ‘cos chemical plants cluster
First order effect
Interdependence of the points themselves
Diseases cluster ‘cos people catch them from others who have the disease (colds)
Second order effect
In practice, it is very difficult to distinguish these two effects merely by the analysis of spatial data
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29. Briggs Henan University 2010
Bank Robberies—First Order or Second Order effect?
Bank robberies are clustered
First order--because banks are clustered
Bank robbery
Banks
Bank Robberies
In lecture on Spatial Analysis we called this the effect of “non-uniformity of space”
Could there also be a second order effect?
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30. Briggs Henan University 2010
Remember our data on software and telecommunications industries in Dallas?
We can think of this data as a sample.
We can use statistical inference to test if the spatial pattern is clustered, or “random” (no pattern)
We will look at the actual tests later.
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31. Spatial Statistical Hypothesis Testing: Simulation Approach
Because of the complexity of spatial processes, it is often difficult to derive theoretically a test statistic with known probability distribution
Instead, we often use computer simulations
We take multiple samples from a random spatial pattern, the spatial statistic we are using is calculated for each sample, and then a frequency distribution is drawn
This simulated sampling distribution is used to measure the probability
of obtaining our actual
observed spatial statistic
Empirical frequency distribution from 500 random patterns (“samples”)
Our observed value:
--highly unlikely to have occurred if the process was random
--conclude that process is not random

32. Software for Spatial Statistics
ArcGIS 9 The most common GIS Software, but $$$$!
Spatial Statistics Tools for point and polygon analysis
Spatial Analyst tools for density kernel
GeoStatistical Analyst Tools for interpolation of continuous surface data
CrimeStat III download from
Standalone package, free for government and education use
Calculates values for spatial statistics but no GIS graphics
Good documentation and explanation of measures and concepts
OpenGeoDA, Geographic Data Analysis by Luc Anselin now at Arizona State
Download from:
Runs on Vista and Windows 7 (also MAC and UNIX)
Earlier version called GeoDA runs only on XP (0.9.5i_6)
Easy to use and has good graphic capabilities
R Open Source statistical package,
originally on UNIX but now has MS Windows version
Has the most extensive set of spatial statistical analyses
Difficult to use
Need to learn it if you are going to do major work in this area
S-Plus the only commercial statistical package with extensive support for spatial statistics
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33. Briggs Henan University 2010
References
O’Sullivan and Unwin Geographic Information Analysis New York: John Wiley, 1st ed. 2003, 2nd ed. 2010
Jay Lee and David Wong Statistical Analysis with ArcView GIS New York: Wiley, 1st ed (all page references are to this book), 2nd ed. 2005
Unfortunately, these books are based on old software (Avenue scripts used with ArcView 3.x) and no longer work in the current version of ArcGIS 9 or 10.
Ned Levine and Associates CrimeStat III Washington: National Institutes of Justice, 2010
Available as pdf
download from:
Arthur J. Lembo at
(no longer active)
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34. Next time: Inferential Statistics for Point Pattern Analysis
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35. Briggs Henan University 2010

36. Software for Spatial Statistics: Examples
Planned as a separate lecture
…but we couldn’t meet last Friday
…so I will look as some examples after today’s lecture, and again after the next lecture
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37. 1. Using ArcGIS to find the Population Centroid of China
Open ArcGIS
Add data files: China.shp and ChinaProvinceData.xls
Join ChinaProvinceData.xlx to China,shp
Right click China and select Joins ..
Use GMI_Admin as join field
Open ArcToolbox by clicking on
Go to Spatial Statistics Tools>Measuring Geographic Distribution>Mean Center
Input Feature Class: China
Output: China_MeanCenter.shp
Weight Field: Population 2008
Note the warning: we should have projected data first!
WARNING : The input feature class does not appear to contain projected data.
It is in south Henan province!
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38. Briggs Henan University 2010
2. Calculate Population Centroid using a Spreadsheet Program (e.g. Excel)
Make a copy of ChinaProvinceData.xls and open this copy ChinaProvinceData Copy.xls
It contains Centroids for each province obtained from GeoDA.
(You need the very expensive ArcInfo version to get centroids for all polygons from ArcGIS and I do not have it!)
Calculate: XCentroid * Weight (Population 2008), and then Sum
YCentroid * Weight (Population 2008), and then Sum
Divide each sum by the Sum of the Weights (Total Population 2008).
These are the X and Y coordinates for the China Population Centroid
Copy these values into a new worksheet, and create a very simple data table
ID X Y
Save spreadsheet and close Excel.
Read this table into ArcGIS
Right click on table name and select Display XY Data
This displays X, Y coordinates from a table on the map.
The results are very similar to the value calculated by ArcGIS itself!
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39. Briggs Henan University 2010
3. Use ArcGIS to Calculate Standard Deviation Ellipse for Population and for Illiterate Population
SDE for Population
Go to Spatial Statistics Tools>Measuring Geographic Distribution>
Directional Distribution
Input Feature Class: China
Output: SDE_Population.shp
Weight Field: Data$.Pop2008
Mean Center for Illiterate Percent
Go to Spatial Statistics Tools>Measuring Geographic Distribution>Mean Center
Output: MC_Illit_PerCent.shp
Weight Field: Data$.Illiterate_Prcnt
SDE for Illiterate Percent
Output: SDE_Illit_PerCent.shp
Weight Field: Data$.Illiterate_Prcnt.
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40. 4. Use GeoDA to find the Centroids of the Provinces of China
(Need ArcInfo to do this in ArcGIS, which is expensive. GeoDA is free. )
--The GeoDA program is on my Web site at: or go to
--download, unzip, and click the file OpenGeoDA.exe to start the software
--it does have some “bugs” so some things may not work or it may crash!
--Input the provinces shapefile: File>Open Shape File China.shp
--Open the data table: Table>Promotion to see what is there
--Create centroids for each province: Options> Add Centroids to Table
Place check mark in X coordinates and Y coordinates box, click OK
--Go to Table>Promotion to open the table—it has the X and Y centroid coordinates
--Save as a new shapefile: Table> Save to Shapefile as China_Centroids.shp
I then opened the China_centroids.dbf (part of the shapefile) file with Excel and copied the centroid values into the ChinaProvincesData.xls spreadsheet.
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