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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.

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Descriptive statistics for spatial distributions l.jpg

Descriptive Statistics for Spatial Distributions

Chapter 3 of the textbook

Pages 76-115

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Descriptive Statistics for Point Data

  • Also called geostatistics

  • Used to describe point data including:

    • The center of the points

    • The dispersion of the points

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Descriptive spatial statistics:Centrality

  • Assume point data.

  • Example types of geographic centers:

    • U.S. physical center

    • U.S. population center

  • Mean center

  • Median center

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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

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    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

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    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

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    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

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    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

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    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

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    Euclidian Median

    • Point that minimizes the sum of distances

    • Must be calculated iteratively

    • Iterative 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

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    Measures of Central Tendency

    • How 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)

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    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.

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    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

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    Pattern Analysis

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

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    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

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    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

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    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

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    Problems Associated With Spatial Data

    • Boundary Problem

    • Scale Problem

    • Modifiable Units Problem

    • Problems of Pattern

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    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?

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    Scale Problem

    • Also referred to as the aggregation problem

    • When scaling up, detail is lost

    • Scaling down creates an ecological fallacy

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    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

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    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)

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    For Monday

    • Read pages 145-164