1. Spatial Statistics
Applied to point data

2. Centrographic Statistics
Most basic type of descriptor for spatial distributions, includes:
Mean Center
Median Center
Standard Deviation
Standard Distance
Standard Deviational Ellipse
Two dimensional correlates to basic statistical moments of a single-variable distribution
Modified from one dimensional to two dimensional

3. Mean Center Simply the mean of X and Y Also called center of gravity
Sum of differences between the mean X and all other X is zero (same for Y)

4. Weighted Mean Center
Produced by weighting each coordinate by another variable (e.g., population)
Points associated with areas can have the characteristics of those areas included

5. Standard Deviation of X and Y
A measure of dispersion
Does not provide a single summary statistic of the dispersion

6. Standard Distance Deviation
Represents the standard deviation of the distance of each point from the mean center
Is the two dimensional equivalent of standard deviation
Where:

7. Standard Distance Deviation
Because it is an average distance from the mean center, it is represented as a single vector

8. Standard Deviation Ellipse
While the standard distance deviation is a good single measure of the dispersion of the incidents around the mean center, it does not show the potential skewed nature of the data (anisotropy).
The standard deviation ellipse gives dispersion in two dimensions

9. Standard Deviational Ellipse

10. Testing the Differences

11. Crime Analysis with Centrographic Statistics
A good “free” software product for doing some basic spatial statistics is Crimestat
Review of Crimestat Figures 4.19 – 4.26
Seeing the relationship between mean center, standard distance, and standard deviational ellipse
Centrographic Statistics in Monroe County

12. Point Pattern Analysis
The spatial pattern of the distribution of a set of point features.
Spatial properties of the entire body of points are studied rather than the individual entities
Points are 0 dimensional objects, the only valid measures of distributions are the number of occurrences in the pattern and respective geographic locations

13. Descriptive Statistics of Point Features
Frequency: number of point features occurring on a map

14. Types of Distribution Three general patterns
Random any point is equally likely to occur at any location and the position of any point is not affected by the position of any other point. There is no apparent ordering of the distribution
Uniform every point is as far from all of its neighbors as possible
Clustered many points are concentrated close together, and large areas that contain very few, if any, points

15. Quadrat Analysis
Based on a measure derived from data obtained after a uniform grid network is drawn over a map of the distribution of interest
The frequency count, the number of points occurring within each quadrat is recorded first
These data are then used to compute a measure called the variance
The variance compares the number of points in each grid cell with the average number of points over all of the cells
The variance of the distribution is compared to the characteristics of a random distribution

16. RANDOM
UNIFORM
CLUSTERED

18. Quadrat Analysis
A random distribution would indicate that that the variance and mean are the same. Therefore, we would expect a variance-mean ratio around 1
Values other than 1 would indicate a non- random distribution.

19. Weakness of Quadrat Analysis
Quadrat size and orientation
If the quadrats are too small, they may contain only a couple of points. If they are too large, they may contain too many points
Some have suggested that quadrat size should be twice the size of the mean area per point
Or, test different sizes (or orientations) to determine the effects of each test on the results

20. Weakness of Quadrat Method
Actually a measure of dispersion, and not really pattern, because it is based primarily on the density of points, and not their arrangement in relation to one another
Results in a single measure for the entire distribution, so variations within the region are not recognized

21. Nearest-Neighbour Analysis
Unlike quadrat analysis uses distances between points as its basis.
The mean of the distance observed between each point and its nearest neighbour is compared with the expected mean distance that would occur if the distribution were random

22. RANDOM
UNIFORM
CLUSTERED

24. Advantages of Nearest Neighbor over Quadrat Analysis
No quadrat size problem to be concerned with
Takes distance into account
Problems
Related to the entire boundary size
Must consider how to measure the boundary
Arbitrary or some natural boundary
May not consider a possible adjacent boundary