Chapter 4 – Descriptive Spatial Statistics Scott Kilker Geog 3000- Advanced Geographic Statistics

Chapter 4 – Descriptive Spatial Statistics

Scott Kilker Geog 3000- Advanced Geographic Statistics

April 20, 2023From 'An Introduction to Statistical Problem Solving in Geography'

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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


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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.


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Central Tendency in a Spatial Context

Mean CenterMean 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.


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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


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Mean Center in Action


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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


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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


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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


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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


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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


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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


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Standard Distance

Like Standard Deviation, strongly influenced by extreme locations

Weighted standard distance can be used for problems that use the weighted mean center


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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


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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


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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


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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 CircleRadius may not be the right choice


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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


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More Resources

Wikipedia - http://en.wikipedia.org/wiki/Spatial_descriptive_statistics

Arthur J. Lembo at http://www.css.cornell.edu/courses/620/css620.html

CrimeStat III Application – Stats in Action - http://www.icpsr.umich.edu/icpsrweb/CRIMESTAT/about.jsp

ESRI Spatial Statistics toolbox (ArcGIS 9.2) - http://webhelp.esri.com/arcgisdesktop/9.2/index.cfm?TopicName=An_overview_of_the_Spatial_Statistics_toolbox


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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

Documents

Chapter 4 – Descriptive Spatial Statistics Scott Kilker Geog 3000- Advanced Geographic Statistics