Loading in 5 sec....

Descriptive Statistics for Spatial DistributionsPowerPoint Presentation

Descriptive Statistics for Spatial Distributions

- By
**bona** - Follow User

- 294 Views
- Updated On :

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.

Related searches for Descriptive Statistics for Spatial Distributions

Download Presentation
## PowerPoint Slideshow about 'Descriptive Statistics for Spatial Distributions' - bona

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

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) For example: the center of a circle or square

- 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.

Mean Center (Centroid) For example: the center of a population

- 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

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:

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)

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 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

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)

Dispersion: Standard Distance

- Standard distance
- Analogous to standard deviation
- Represented graphically as circles on a 2-D scatter plot

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

- Bins typically assigned to standard directions
- 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 Problem

- Also 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

Download Presentation

Connecting to Server..