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Spatial Statistics I

Spatial Statistics I. RESM 575 Spring 2011 Lecture 10. Spatial Statistics. Measuring geographic distributions Testing statistical significance Identifying patterns. Measuring geographic distributions. Identify spatial characteristics of a distribution Where is the center?

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Spatial Statistics I

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  1. Spatial Statistics I RESM 575 Spring 2011 Lecture 10

  2. Spatial Statistics Measuring geographic distributions Testing statistical significance Identifying patterns

  3. Measuring geographic distributions Identify spatial characteristics of a distribution • Where is the center? • What feature is most central? • How are features dispersed around the center?

  4. ArcTools in ArcGIS

  5. Where is the center? • Mean Center tool • Computes the average X and Y coordinates of all features • Creates a new point feature

  6. Example

  7. Example

  8. Mean center tool • More common use: • To compare distributions of different types of features or to find the center of features based on an attribute value

  9. Example

  10. Example

  11. Example

  12. What is the most central feature? • Central feature tool • Identifies the most centrally located feature • Feature having the lowest total distance to all other features

  13. Example

  14. Example

  15. What is the most central feature? • More interestingly is by adding a weight in the analysis such as population • We are now finding not just which site is most central but which is the most accessible to the greatest number of people.

  16. Example

  17. Measuring feature distribution • Standard distance tool • Measures distribution of features around the mean • Result is a summary statistic representing distance • If circle is large, incidents are widespread • If small, incidents are more localized

  18. Example

  19. Distributional trends • Directional distribution (standard ellipse) tool • Identify spatial trends in the distribution of features • Uses • Compare distributions • Examine different time periods • Show compactness and orientation

  20. Example

  21. Testing statistical significance • The next section of identifying patterns or later spatial relationships allows us to perform significance tests on the results before accepting them

  22. Using significance tests with spatial data • Spatial data contradicts some of the assumptions of inferential statistics • You need to be aware of these limitations!

  23. Assumptions • Testing a random sample • With GIS data in a database you may not know if the data was randomly sampled • How large the sample is in relation to the population? • Even with randomness assumed, spatial data often violates the independence of observations in a sample Spatial data is rarely evenly distributed across a region

  24. For spatial pattern analysis… • The null hypothesis is that features are evenly distributed across the study area • Hard to imagine this being true • You have to make one of two common sampling assumptions: randomization or normalization

  25. Identifying Patterns • Why study patterns? Range from completely clustered to completely dispersed

  26. Identifying patterns (applications) Forestry applications USFS may measure the pattern of clear cuts to ensure sufficient contiguous forest habitat remaining Agency may allow a level of clustering of clear cuts and then make sure it is not exceeded Wildlife studies if population is dispersed then species can live in a wide range of habitats, if clustered then it has very specific habitat requirements

  27. Goal for analyzing spatial patterns • Are there underlying spatial processes influencing the locations of our features? • Are our features randomly located throughout the study area, or are they displaying a clustering or dispersed pattern?

  28. Approaches for analyzing spatial patterns • Approach #1 Global calculations • Identifies overall patterns or trends in the data • Effective for complex messy data • Interested in broad overall results • Work by comparing feature locations and/or attributes to a theoretical random distribution to determine if you have statistically significant clustering or dispersion

  29. Analyzing spatial patterns • Can be found with the Average Nearest Neighbor which does not require specifying an attribute, or • If based on an attribute we can test for Spatial autocorrelation using the (Moran’s I) tool • Things that are closer are more alike than things that are not • Measures similarity of neighboring features • Identifies if features are clustered or dispersed

  30. After tool runs, go to Geoprocessing - > Results

  31. Double click to open the html report for your latest run

  32. After tool runs, go to Geoprocessing - > Results

  33. Global SA interpretation • I > 0 positive spatial autocorrelation • I < 0 negative spatial autocorrelation The more positive or more negative, the greater amount of spatial autocorrelation

  34. 2nd Approach • Local calculations • Identify the extent and locations of clustering • Answer where do we have spatial clustering • Process every feature within the context of its neighboring features in order to determine whether it represents a spatial outlier, or if part of a statistically significant spatial cluster

  35. Locate the hot spots • This is a local question that requires a hot spot analysis (Getis-Ord Gi*) tool • Indicates the extent to which each feature is surrounded by similarly high or low values • Where do features with similar attribute values cluster spatially together

  36. Getis-Ord Gi* tool • Identifies where clustering occurs in both high and low values • Calculates a Z score for each feature • High Z = hot spot (when a feature has a high value and it is surrounded by other features with high values) • Low Z = cold spot (when we have features with low values surrounded by other features with low values)

  37. Notes on the Z score • Z score is a measure of standard deviation • It is a reference value that’s associated with a standard normal distribution • A very high or low Z score would be found in the tails

  38. More notes on Z score • A very high or low Z score means that the pattern deviates significantly from a hypothetical random pattern • For example, when using a 95% CI, Z scores are -1.96 and +1.96 • If Z is between these -1.96 and +1.96 you can’t reject the null • You are seeing one version of a random pattern • If very high or low (ie -2.5 or +5.4) you have a pattern that’s too unusual to be a pattern of random choice so we reject the null hypoth REMEMBER: The null hypothesis is that features are evenly distributed across the study area

  39. Reference • Mitchell, A. 2005. ESRI Guide to GIS Analysis, Volume 2. ESRI press, Redlands, CA. • ArcGIS 10 Help

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