Chapter 4 – Descriptive Spatial Statistics Scott Kilker Geog Advanced Geographic Statistics

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Learning Objectives Explain central tendency as applied in a spatial context Define spatial measures of dispersion and recognize possible applications Identify potential limitations and locational issues associated with applied descriptive spatial statistics

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Descriptive Spatial Statistics Descriptive Spatial Statistics, also referred to as Geostatistics, are the spatial equivalent to the basic descriptive statistics. They can be used to summarize point patterns and the dispersion of some phenomena.

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Central Tendency in a Spatial Context Mean Center Mean center represents an average center of a number of coordinates. This is calculated by averaging the X coordinates and Y coordinates separately and using the average for the Mean Center coordinate.

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Central Tendency in a Spatial Context Mean Center Considered the Center of Gravity Can be strongly affected by outliers Most well know use is the U.S. Bureau of Census geographic “center of population” calculation that shows the mean center of the U.S. population

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Central Tendency in a Spatial Context Mean Center in Action

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Central Tendency in a Spatial Context Weighted Mean Center Points can be weighted meaning they can be given more or less influence on the calculation of the mean center Points could represent cities, frequencies, volume of sales or some other value that will affect the points influence. Analogous to frequencies in the calculation of grouped statistics like the weighted mean Influenced by large frequencies of a point

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Central Tendency in a Spatial Context Least Squares Property Analogous to the least squares for a mean – Sum of squared deviations about mean is zero – Sum of squared deviations about a mean is less than the sum of squared deviations about any other number Deviations are distances – Calculated as the Euclidean distance

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Central Tendency in a Spatial Context Euclidean Median Considered the Median Center Often more useful than the Mean Center. Used when determining the central location that minimizes the unsquared rather than the squared Can be weighted

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Central Tendency in a Spatial Context Euclidean Median Used in economic geography to solve the “Weber” problem which searches for the “best” location for an industry. The best location will result in – Minimized transportation costs of raw material to factory – Minimized transportation costs of finished products to the market

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Central Tendency in a Spatial Context Euclidean Median Heavily used in public and private facility location Used to minimize the average distance a person must travel to reach a destination. – Useful in location of fire stations, police stations, hospitals and care centers – Used in conjunction with demographics to select store locations that will target the desired consumers

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Spatial measures of dispersion Standard Distance Analogous to the Standard Deviation in descriptive statistics Measures the amount of absolute dispersion in a point pattern Uses the straight-line Euclidean distance of each point from the mean center

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Spatial measures of dispersion Standard Distance Like Standard Deviation, strongly influenced by extreme locations Weighted standard distance can be used for problems that use the weighted mean center

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Manhattan Distance & Median Not all analysis would benefit from the use of straight line distances Manhattan distance is represented by a grid like city blocks in Manhanttan Manhattan Median is the center point in Manhattan space Manhattan Median cannot be found for a spatial pattern having an even number of points

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Spatial measures of dispersion Coefficient of Variation Calculated by dividing the standard deviation by the mean Measures the relative dispersion of values No analogous methods exists for measuring spatial dispersion Dividing the standard distance by the mean center does not provide meaningful results

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Spatial measures of dispersion Relative Distance To obtain a measure of relative dispersion, the standard distance must be divided by some measure of regional magnitude Region magnitude cannot be mean center Radius of a circle the same size that is being evaluated can be appropriate

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Spatial measures of dispersion Relative Distance Using a circle may not always be valid. For instance, if the region is wider than tall, it will have a strong influence on the dispersion A measure of relative dispersion is influenced by the boundary of the region being studied Region is not always a Circle Radius may not be the right choice

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Descriptive spatial statistics Limitations and Locational issues Geographers should look at geostatistics very carefully Interpretation can be difficult – The mean center for a high income area could be in a low income area Should view geostatistics as general indicators of location instead of precise measurements Point pattern analysis an benefit from consideration of other possible pattern characteristics – Using the knowledge of descriptive statistics like skewness and kurtosis can offer insights about the symmetry of the pattern that geographers could find useful when comparing point patterns – Value in comparing degrees of clustering and dispersal in different point patterns thought measuring spatial kurtosis levels

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe More Resources Wikipedia - Arthur J. Lembo at CrimeStat III Application – Stats in Action ESRI Spatial Statistics toolbox (ArcGIS 9.2) - overview_of_the_Spatial_Statistics_toolbox

31 October 2015 From 'An Introduction to Statistical Problem Solving in Geography' by McGrew & Monroe Summary Descriptive Spatial Statistics have many similarities with the descriptive statics – mean, median, standard deviation, weighted mean, measures of dispersion Care needs to be taken when evaluating geostatistics because results sometimes will not be meaningful – methods must be understood Methods applied with a GIS can be very powerful in their application to determine where industries, business, public and private facilities are located so they provide the greatest values to the owners and public