Inferential Spatial Statistics:
Introduction to Concepts
Infer
Population
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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Spatial Analysis: successive levels of sophistication
1. Spatial data description: classic GIS capabilities
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–

Spatial queries & measurement,
buffering, map layer overlay
2. Exploratory Spatial Data Analysis (ESDA):
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–
–

searching for patterns and possible explanations
GeoVisualization through data graphing and mapping
Descriptive spatial statistics
3. 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
4. Spatial modeling or prediction
–
Constructing models (of processes) to predict spatial
outcomes (patterns)
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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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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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Populations and Samples
Population: all occurrences
of a particular phenomena
Sample: a part (subset) of
the population for which we
have data. You are a sample of
the population of all
people in the world.
The sample is used to make
inferences about the population.
Infer
We draw conclusions about the
population from the sample.
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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.
Infer
Population
Processes
Sample
Create
Patterns
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The Importance of the Sample
How “good “ (or “accurate”
or “true”) are our
inferences or conclusions?
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.
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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
Some Definitions
• Population
– All occurences
• Parameters
– Numbers
calculated from
the population
• Sample
− Subset of population for
which we have data
• Statistics
– Numbers calculated
from the sample
statistics are estimates of
parameters
We can calculate the statistic because we have data for
samples. We cannot calculate the parameter because we
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do not have data for entire population.
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Example:
Are girls more intelligent than boys?
• Sample of boys
• Sample of girls
– IQ* = 115
– 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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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
 g  b
This is called the Null Hypothesis:  g  b
--there is no difference between girls and boys
in the population
The Alternative Hypothesis:  g  b
--in the population, girls are smarter than boys
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Choosing between Null and Alternative
• In our two samples:
X g  X b  130  115  15
– 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 12
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How do I calculate the probability
of getting a difference of 15?
We use the sampling distribution.
What is this?
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All girls
All boys
(the population of girls)
(the population of boys)
Random
samples
Random
samples
X g  Xb
For every pair of samples,
calculate the mean of each, and
then the difference between
these means.
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The Sampling Distribution
If we have a thousand sample pairs, we have a
thousand values for X g  X b
We can draw a frequency distribution showing how
often or frequently different values occur
2.5%
2.5%
-1.96
X g  Xb
0
1.96
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
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Using the Sampling Distribution
2.5%
2.5%
-1.96
X g  Xb
0
Here, a sample difference of 15
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is quite likely:
Conclusion: Accept the Null.
Boys and Girls are the same
The probability should be less than
5% (.05) to reject the null hypothesis.
This probability is called the statistical
significance of the test.
1.96
15
Here, a sample difference of 15
is very unlikely:
Conclusion:
Reject the Null
Accept the Alternative
Girls are smarter than boys
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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:
It is a Normal Frequency
X g  Xb
Distribution if the sample sizes
z
are greater than 30.
2
2
s g  sb
S2g =variance for girls
n
n
g
b
S2b =variance for boys
• Note: test statistics always have “degrees of freedom”
which are calculated from the sample size (N)
Test Statistic for Normal Frequency Distribution
2.5%
2.5%
X g  Xb
-1.96
0
1.96
To reject the Null Hypothesis, the Z test
statistic should have a value greater than
1.96 (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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Standard Error:
Standard Deviation of the Sampling Distribution
Smaller standard error
Test statistic for the difference between
two means:
X g  Xb
z
s 2 g  s 2b
ng
nb
Larger standard error
2.5%
-1.96
2.5%
-1.96
2.5% 2.5%
X g  Xb
0
1.96 1.96
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
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Null Hypothesis
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•
•
•
•
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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The Normal or Gaussian Probability Distribution.
2.5%
2.5%
-1.96
X g  Xb
0
This is the sampling
distribution for tests
involving differences
between means.
Why is it this shape?
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
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• Symmetrical—same on each side of the mean
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How do we find the Sampling
Distribution and Test Statistic?
Two methods:
1. By mathematical theory:
•
•
test statistics and sampling distributions already
known through theory
common distributions are Z (Normal), Chi-square, and
F distributions
2. 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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Spatial Statistical Inference
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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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What do we mean by spatially random?
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”
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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High Peak district biomass index:
ratio of remotely sensed data spectral bands
B3 and B4
Spatially clustered
Geographically random
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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Bank Robberies—First Order or Second Order effect?
– Bank robberies are clustered
Bank robbery
– First order--because banks are
Banks
clustered
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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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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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
Empirical frequency
is used to measure the probability
distribution from 500
of obtaining our actual
random patterns
(“samples”)
observed spatial statistic
Our observed value:
--highly unlikely to have occurred if the process was random
--conclude that process is not random
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
http://www.icpsr.umich.edu/NACJD/crimestat.html
– 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
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–
–
–
Download from: http://geodacenter.asu.edu/
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,
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–
–
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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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– www.insightful.com
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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. 2001 (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: http://www.icpsr.umich.edu/NACJD/crimestat.html
• Arthur J. Lembo at
http://www.css.cornell.edu/courses/620/css620.html
(no longer active)
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Next time: Inferential Statistics
for Point Pattern Analysis
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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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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 000916: The input feature class does not appear to contain projected data.
It is in south Henan province!
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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
113.4696704 32.3797596
Copy these values into a new worksheet, and create a very simple data table
ID
X
Y
1
113.4697 32.3798
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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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$.Pop2008
Mean Center for Illiterate Percent
Go to Spatial Statistics Tools>Measuring Geographic Distribution>Mean Center
Input Feature Class: China
Output: MC_Illit_PerCent.shp
Weight Field: Data$.Illiterate_Prcnt
SDE for Illiterate Percent
Go to Spatial Statistics Tools>Measuring Geographic Distribution>
Directional Distribution
Input Feature Class: China
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Output: SDE_Illit_PerCent.shp
Weight Field: Data$.Illiterate_Prcnt.
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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: www.utdallas.edu/~briggs or go to
http://geodacenter.asu.edu/
--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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