Descriptive Statistics for Spatial Distributions

Chapter 3 of the textbook

Pages 76-115

Descriptive Statistics for Point Data

Also called geostatistics

Used to describe point data including:– The center of the points– The dispersion of the points

Descriptive spatial statistics:Centrality

Assume point data.

Example types of geographic centers:– U.S. physical center– U.S. population center

Mean center

Median center

Mean Center (Centroid)

A centroid is the arithmetic mean (a.k.a. the “center of mass”) of a spatial data object or set of objects, which is calculated mathematically

In the simplest case the centroid is the geographic mean of a single object

• I.e., imagine taking all the points making up the outer edge of of a polygon, adding up all the X values and all the Y values, and dividing each sum by the number of points. The resulting mean X and Y coordinate pair is the centroid.

For example: the center of a circle or square

Mean Center (Centroid)

A more complicated case is when a centroid is the geographic mean of many spatial objects

This type of centroid would be calculated using the geographic mean of all the objects in one or more GIS layer

• I.e., the coordinates of each point and/or of each individual polygon centroid are used to calculate an overall mean

For example: the center of a population

Mean Center (Centroid) in Irregular Polygons

Where is the centroid for the following shapes?

In these cases the true centroid is outside of the polygons

Measures of Central Tendency – Arithmetic Mean

A standard geographic application of the mean is to locate the center (centroid) of a spatial distribution

Assign to each member a gridded coordinate and calculating the mean value in each coordinate direction --> Bivariate mean or mean center

This measure minimizes the squared distances

For a set of (x, y) coordinates, the mean center is calculated as:

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Weighted Mean Center

Calculated the same as the normal mean center, but with an additional Z value multiplied by the X and Y coordinates

This would be used if, for example, the points indicated unequal amounts (e.g., cities with populations)

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Manhattan Median

The point for which half of the distribution is to the left, half to the right, half above and half below

For an even number of points there is no exact solution

For an odd number of points the is an exact solution

The solution can change if we rotate the axes

May also called the bivariate median

Manhattan Median Equation

The book describes this as something created graphically (e.g., drawing lines between points)

However it can be calculated by using the median X and Y values

If there are an even number of points the Manhattan median is actually a range

Euclidian Median

The point that minimizes aggregate distance to the center

For example: if the points were people and they all traveled to the a single point (the Euclidian Median), the total distance traveled would be minimum

May also called the point of Minimum Aggregate Travel (MAT) or the median center

Euclidian Median

Point that minimizes the sum of distancesMust be calculated iterativelyIterative calculations:– When mathematical solutions don’t exist.– Result from one calculation serves as input into next

calculation. – Must determine:

• Starting point• Stopping point• Threshold used to stop iterating

This may also be weighted in the same way we weight values for the mean center

Euclidian Median Equations

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Measures of Central TendencyHow do they differ?

Mean center: – Minimizes squared distances– Easy to calculate– Affected by all points

Manhattan Median:– Minimizes absolute deviations– Shortest distances when traveling only N-S and/or E-W– Easy to calculate– No exact solution for an even number of points

Euclidian Median:– True shortest path– Harder to calculate (and no exact solution)

Dispersion: Standard Distance

Standard distance– Analogous to standard deviation

– Represented graphically as circles on a 2-D scatter plot

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Dispersion (not discussed in textbook)

Average distance– Often more interesting– Distances are always positive, so average distance from a center

point is not 0.

Relative distance– Standard distance is measured in units (i.e. meters, miles).– The same standard distance has very different meanings when the

study area is one U.S. state vs. the whole U.S.– Relative distance relates the standard distance to the size of the

study area.

Dispersion: Quartilides

Quartilides are determined like the Manhattan median, but for only X or Y, not both

Similar to quantiles (e.g., percentiles and quartiles) from chapter 2, but in 2-D

Examples: Northern, Southern, Eastern, Western

Pattern Analysis

This will be discussed in greater detail later in the class, but some of these measures start hinting at things like clustering

Directional Statistics

Directional statistics are concerned with…

Characterizing and quantifying direction is challenging, in part, because 359 and 0 degrees are only one degree apart

To deal with this we often use trigonometry to make measurements easier to use

For example, taking the cosine of a slope aspect measurement provides an indication of north or south facing

Directional Graphics

Circular histogram– Bins typically assigned to standard directions

• 4 – N, S, E, W• 8 – N, NE, E, SE, S, SW, W, NW• 16 – N, NNE, NE, ENE, E, ESE, SE, SSE, S, SSW, SW,

WSW, W, WNW, NW, NNW

Rose diagram– May used radius length or area (using radius ^0.5) to

indicate frequency

Directional Statistics

Directional Mean– Assumes all distances are equal– Calculates a final direction angle– An additional equation is required to determine the quadrant – Derived using trigonometry

Unstandardized variance– Tells the final distance, but not the direction

Circular Variance – Based on the unstandardized variance– Gives a standardized measure of variance– Values range from 0 to 1, with 1 equaling a final distance of zero

Problems Associated With Spatial Data

Boundary Problem

Scale Problem

Modifiable Units Problem

Problems of Pattern

Boundary Problem

Can someone give me a concise definition of the boundary problem?

Which of these boundaries are “correct” and why?

How can we improve the boundaries?

Scale ProblemAlso referred to as the aggregation problem

When scaling up, detail is lost

Scaling down creates an ecological fallacy

Modifiable Units Problem

Also called the Modifiable Area Units Problem (MAUP)

Similar to scaling problems because they also involve aggregation

The take home message is that how we aggregate the input units will impact the values of the output units

A real world example of this is Gerrymandering voting districts

Problems of Pattern

This “problem” relates to the limitations of some statistics (e.g., LQ, CL, Lorenz Curves)

Fortunately there are many other types of statistics that can be used in addition to or instead of these limited measured (e.g., pattern metrics)

For Monday

Read pages 145-164