1. Spatial Statistics II
RESM 575
Spring 2010
Lecture 8

2. Last time Part A: Spatial statistics
What are spatial measurements and statistics?
Part B:
Measuring geographic distributions
Testing statistical significance
Identifying patterns

3. Review
Used in lab7

4. What we will use spatial stats to find
How features are distributed
Where are the clusters
What is the pattern created by the features
What are the relationships between sets of features or values

5. Identifying patterns Part A:
Measuring the pattern of discrete or noncontinuous features (points, lines, polygons)
Part B:
Measuring the spatial pattern of feature values

6. Measuring the pattern of discrete or noncontinuous features (points, lines, polygons)
Options:
Overlaying areas of equal size
Calculating the average distance between features
Counting the number of features within defined distances

7. Overlaying areas of equal size
Termed Quadrat analysis
GIS overlays areas of equal size on the study area and counts the number of features in each
Calculates the expected counts for a random distribution
If fewer areas than expected contain most of the features and more areas than expected contain few or no features, the features form a clustered pattern . . . .

8. Notes Traditionally, squares are used for the quadrats
The quadrat size will affect whether any patterns are identified!
What size to use?
Originally done with archeological digs where squares were easiest to lay out with stakes and a string

9. Quadrat size
Quadrat size = twice the size of the mean area per feature
Length of a side of quadrat =
sqrt[{2*(Area of the extent / # of features)}]

10. Example
Area = 6,000m * 6,000m
= 36,000,000m2
n = 36
Length of a side of quadrat = sqrt[{2*(36,000,000 / 36)}] m

11. Hawth’s spatial ecology tools
Already loaded in 317

12. Use Hawth’s tools to create quadrat
ArcMap ->View -> Toolbars ->
1412

13. Quadrat analysis
Because the method counts the number of features per unit area, it measures the density of features
It doesn’t take into account the proximity of features to each other or their arrangement

14. Calculating the quadrat analysis stat
GIS can be used to count the number of features in each quadrat and then summarize it to create a frequency table
We want a table listing the number of quadrats containing 0, 1, 2, etc features
How do we create this in GIS??????

15. Creating a frequency table
Spatially join the point features to the quadrat polygons (right click on the quadrat polygons ->joins and relates -> join…)
Make sure it is on spatial location join
The Point features
Sum because we want total # of points per quadrat
Creates a new output shapefile to be used in step 2

16. Creating a frequency table
Now summarize the count field from the output shapefile (right click Count field)
Will create final frequency table shown on next slide

17. Finished frequency table
# of points per quad
# of quads that had that many points in them
Right clicking on Options will allow us to export dbf file for stat tests in Excel

18. Calculating the quadrat analysis stat
We then need a frequency table for the expected distribution usually based on a Poisson distribution
Calculating the probability that a given number of features will occur in a quadrat
λ = n / k
Where λ = the ave # of features per quadrat
n = the # of features
k = the # of quadrats

19. Finding the prob of a # of features occurring in any given quadrat P(x)
P(x) = e-λλx / x!
e is Euler’s constant, with a value of
X! is x factorial (if x is 3 then 3! is 3*2*1= 6
Excel can be used to easily calculate this equation and create a frequency table
NOTE: After calculating P(x), or the frequency table, we then multiply the results by the total # of features to get the number of quadrats expected to contain that number of features

21. Analysis
By comparing the two frequency tables, you can see whether the features create a pattern
If the observed distribution table has more features than the table for the random distribution, then the features create a clustered pattern

22. Kolmogorov-Smirnov test
Stat tests to find how much the frequency tables (and thus the patterns) differ
Kolmogorov-Smirnov test
Compares the difference in frequencies between the tables
Chi-Square test
Compares the cumulative difference in frequencies

23. Kolmogorov-Smirnov test
Calculates the proportion of quads in each class for each line in the frequency table (the number of quads in the class divided by the total number of quads)
Then creates a running cumulative total of proportions from top of the table to the bottom – the total of all classes equals 1 (100% of the quads)

24. Kolmogorov-Smirnov test
Calculated difference in proportions
Compares this value to the critical value for a confidence level you specify

25. Critical value of Kolmogorov-Smirnov test
Critical value = K / sqrt(m)
Where
m = number of quads
k = is the constant for a confidence interval you’re using
For a CI of .20 (80%), K is 1.07
If 63 quads, Critical value = .135
If the calculated difference in proportions between the observed and random distribution is greater than the critical value, the difference can be considered statistically significant.

26. Chi-Square Also based on the frequency table
Test to find if two sets of frequencies (or two distributions) are significantly different

27. Chi-Square test
You then compare the test value with the critical value
If the X2 value is greater than the critical value, the distributions are significantly different, and you can consider the observed distribution to not be random, but either clustered or dispersed

28. Factors influencing the results of quadrat analysis
Works best with small, tight clusters (earthquakes, bird studies)
Since it does not measure the relationship or distance between features (just if they fall in the quad or not) it may not recognize certain patterns . . . . . . .

29. Measuring the pattern of discrete or noncontinuous features (points, lines, polygons)
Options:
Overlaying areas of equal size
Calculating the average distance between features
Counting the number of features within defined distances

30. Calculating the Average Distance Between Features
Referred to as the Nearest Neighbor index
Measures the distance between each feature and its nearest neighbor an then calculates the average
Distributions that have a smaller ave distance between features than a random distribution would have are considered clustered . . . . .

31. Notes on Nearest Neighbor
Use if there is a direct interaction between features
Good for data analyzed along a line (ie plants or wildlife observations collected along a transect)
Stat is calculated using the distance between features

32. Calculating the index in GIS

33. Interpretation of results

34. Testing the results statistically of NNI
The null hypoth is that the features are randomly distributed
A spatially random distribution would result in a normal distribution of the NNI
Z score

35. Testing the results statistically of NNI
A positive Z score indicates a dispersed pattern, negative Z score is clustered
At 95% CI the z would have to be > 1.96 or < to be significantly significant
Can compare distributions (the higher the z the more clustered)
Make sure study areas are the same size

36. Factors influencing the NNI
For lines, NNI could be misleading if endpoints are close and midpoints aren’t
If single continuous lines are really several joined lines
Centroids of large, convoluted polygons or of different sizes, area boundaries may be close while the centroids are far apart

37. Side note: finding points of lines, or points of polygons (centroids) in ArcGIS
Select source tab

38. Factors influencing the NNI
Extent of the study area

39. Measuring the pattern of discrete or noncontinuous features (points, lines, polygons)
Options:
Overlaying areas of equal size
Calculating the average distance between features
Counting the number of features within defined distances

40. Counting the number of features within defined distances
Known as the K-function, or Ripley’s K-function
Here you specify a distance interval and use GIS to calculate the average number of neighboring features within the distance of each feature
If the ave # of features found at a distance is greater than the ave concentration of features throughout the study area, then the distribution is considered clustered at that distance . . . . . .
K is the distance bands

41. Notes on K-function
Good for if you are interested in how the pattern changes at different scales of analysis
Includes all distances to neighbors within a given distance
Emergency calls, bird nests

42. K function in GIS
Finds the distance from each point to every other point
Then, for each point, counts up the number of surrounding points within a given distance

43. Interpreting the K stat

44. Paper on class website for today.

45. Factors influencing the K function
Points near the edge of a study area are likely to have less neighboring points
At larger distances its more likely that fewer neighboring points will be found than actually exist

46. Comparing methods for measuring the pattern of feature locations