1. Chapter 10 Problems
Even problems are at end of text. 19. What is a kernel in a moving window operation? Does the kernel size or shape change for different portions of the data set? Why or why not?
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2. 23. Calculate the cost of travel between A and B, and A and C.
A to B
A to C
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3. Chapter 11 Problems
Answers to even numbered problems are in the back of the text. 3. Define slope and aspect and give the mathematical formulas used to derive them from elevation data. Slope is the change in elevation compared to a change in the horizontal position.
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4. Aspect is the direction of the slope.
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5. Lab 5 – Spatial Analysis Will be up on the website later today.
You will be creating your own Report Sheet.
10 screen shots plus a layout.
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6. Uncheck all layers except for the one you are screen printing.
Crop the image.
Set the size to 3”
Total pages I am willing to print per this lab is 4.
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7. Spatial Estimation Chapter 12 – Part 1
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8. Introduction
Spatial prediction methods are used to estimate values at un-sampled locations.
Not all locations can be sampled:
Time
Money
An infinite number of samples or a large finite number of samples.
Locations are difficult or impossible to get to.
Missing or suspect data.
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9. Spatial Interpolation
The prediction of variables at unmeasured locations based on the sampling of the same variables at known location.
Soil temperature
Elevation
Ocean productivity
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10. Spatial Prediction
The estimation of variables at un-sampled locations based, at least in part, on other variables, and often on a total set of measurements.
Elevation as a proxy for temperature.
Soil nitrogen content as a proxy for crop yield.
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11. Spatial Prediction (cont’d)
Usually translates to same or higher dimension:
Point data to point or line data
In the opposite direction we have the MAU problem.
250
deer
1000 deer
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12. Core Area
A high use, density, intensity or probability of the occurence of a variable or an event.
A core area is defined from a set of samples and are used to predict the frequency or likelihood of the occurence of an object or event.
Location of traffic accidents
High crime areas
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13. Sampling Sampling Common sampling patterns in spatial analysis:
A shortcut method for investigating a whole population
Data is gathered on a small part of the whole parent population or sampling frame, and used to inform what the whole picture is like
We control the number of samples and the pattern
Common sampling patterns in spatial analysis:
Systematic
Random
Cluster
Adaptive
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14. Difficult to stay on lines
Systematic sampling pattern
Easy
Samples spaced uniformly at fixed X, Y intervals
Parallel lines
Advantages
Easy to understand
Disadvantages
All receive same attention
Difficult to stay on lines
May be biases
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15. Advantages Disadvantages Random Sampling
Select point based on random number process
Plot on map
Visit sample
Advantages
Less biased (unlikely to match pattern in landscape)
Disadvantages
Does nothing to distribute samples in areas of high
Difficult to explain, location of points may be a problem
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16. Advantages Cluster Sampling
Cluster centers are established (random or systematic)
Samples arranged around each center
Plot on map
Visit sample
(e.g. US Forest Service, Forest Inventory Analysis (FIA)
Clusters located at random then systematic pattern of samples at that location)
Advantages
Reduced travel time
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17. Advantages Disadvantages Adaptive sampling
More sampling where there is more variability.
Need prior knowledge of variability, e.g. two stage sampling
Advantages
More efficient, homogeneous areas have few samples, better representation of variable areas.
Disadvantages
Need prior information on variability through space
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18. INTERPOLATION
Many methods - All combine information about the sample coordinates with the magnitude of the measurement variable to estimate the variable of interest at the unmeasured location
Methods differ in weighting and number of observations used
Different methods produce different results
No single method has been shown to be more accurate in every application
Accuracy is judged by withheld sample points
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19. INTERPOLATION Raster surface
Outputs typically:
Raster surface
Values are measured at a set of sample points
Raster layer boundaries and cell dimensions established
Interpolation method estimate the value for the center of each unmeasured grid cell
Contour Lines
Iterative process
From the sample points estimate points of a value Connect these points to form a line
Estimate the next value, creating another line with the restriction that lines of different values do not cross.
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20. Lecture 12

21. Sampled locations and values
Example Base
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Elevation contours
Sampled locations and values

22. INTERPOLATION 1st Method - Thiessen Polygon
Assigns interpolated value equal to the value found at the nearest sample location
Conceptually simplest method
Only one point used (nearest)
Often called nearest sample or nearest neighbor
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23. INTERPOLATION Thiessen Polygon Advantage: Ease of application
Accuracy depends largely on sampling density
Boundaries often odd shaped as transitions between polygons
are often abrupt
Continuous variables often not well represented
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24. Thiessen Polygon
Draw lines connecting the points to their nearest neighbors.
Find the bisectors of each line.
Connect the bisectors of the lines and assign the resulting polygon the value of the center point
Source:
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25. Thiessen Polygon
Start: 1) 2) 3)
Draw lines connecting the points to their nearest neighbors.
Find the bisectors of each line.
Connect the bisectors of the lines and assign the resulting polygon the value of the center point 1 2 3 5 4
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26. Sampled locations and values Thiessen polygons
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27. INTERPOLATION Fixed-Radius – Local Averaging
More complex than nearest sample
Cell values estimated based on the average of nearby samples
Samples used depend on search radius
(any sample found inside the circle is used in average, outside ignored)
Specify output raster grid
Fixed-radius circle is centered over a raster cell
Circle radius typically equals several raster cell widths
(causes neighboring cell values to be similar)
Several sample points used
Some circles many contain no points
Search radius important; too large may smooth the data too much
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28. Fixed-Radius – Local Averaging
INTERPOLATION
Fixed-Radius – Local Averaging
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29. Fixed-Radius – Local Averaging
INTERPOLATION
Fixed-Radius – Local Averaging
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30. Fixed-Radius – Local Averaging
INTERPOLATION
Fixed-Radius – Local Averaging
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31. INTERPOLATION Inverse Distance Weighted (IDW)
Estimates the values at unknown points using the distance and values to nearby know points (IDW reduces the contribution of a known point to the interpolated value)
Weight of each sample point is an inverse proportion to the distance.
The further away the point, the less the weight in helping define the unsampled location
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32. INTERPOLATION Inverse Distance Weighted (IDW)
Zi is value of known point
Dij is distance to known point
Zj is the unknown point
n is a user selected exponent
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33. INTERPOLATION
Inverse Distance Weighted (IDW)
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34. INTERPOLATION Inverse Distance Weighted (IDW)
Factors affecting interpolated surface:
Size of exponent, n affects the shape of the surface
larger n means the closer points are more influential
A larger number of sample points results in a smoother surface
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35. INTERPOLATION
Inverse Distance Weighted (IDW)
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36. INTERPOLATION
Inverse Distance Weighted (IDW)
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37. INTERPOLATION Splines
Name derived from the drafting tool, a flexible ruler, that helps create smooth curves through several points
Spline functions are use to interpolate along a smooth curve.
Force a smooth line to pass through a desired set of points
Constructed from a set of joined polynomial functions
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38. Spline Surface created with Spline interpolation
Passes through each sample point
May exceed the value range of the sample point set
                                                                                                                                                                                                                             
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39. Lecture 12

40. INTERPOLATION : Splines
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41. Measuring Interpolation Accuracy
Hold back 10% of the sample points.
Create your interpolated surface.
Add the withheld points, how well did the surface predict the values of the withheld points.
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42. Interpolation vs. Prediction
Spatial prediction is more general than spatial interpolation.
Both are used to estimate values of a variable at unknown locations.
Interpolation use only the measured target variable and sample coordinates to estimate the variable at unknown locations.
Prediction methods address the presence of spatial autocorrelation.
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43. Tobler’s Law Tobler's First Law of Geography.
A formulation of the concept of spatial autocorrelation by the geographer Waldo Tobler (1930-), which states:
"Everything is related to everything else, but near things are more related than distant things."
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44. Spatial Prediction
Based on mathematical models often built by a statistical process.
Distinction between interpolation and prediction may be viewed as artificial, but it can be more general than interpolation.
In addition to autocorrelation, variables may show cross-correlation, the tendency for two variables to change together.
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45. Spatial autocorrelation:
Higher autocorrelations, points near each other are alike.
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46. Cross-correlation – two variables change in concert (positive or negative)
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47. Moran’s I – A Measure of Spatial Correlation
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48. Moran’s I (cont’d)
Moran’s I approaches a value of +1 in areas of positive spatial correlation:
Large values tend to be clumped together.
Small values tend to be clumped together.
Values approach 0 in areas of low spatial correlation.
A value of -1 are anti-correlated, a large value is next to a small value.
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49. Spatial Prediction Trend Surface
Fitting a statistical model, a trend surface, through the measured points. (typically polynomial)
Where Z is the value at any point x
Where ais are coefficients estimated in a regression model
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50. Spatial Prediction
Trend Surface
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51. Global Mathematical Functions Polynomial Trend Surface
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52. Global Mathematical Functions Polynomial Trend Surface
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53. Kriging
A surface created with Kriging can exceed the value range of the sample points but will not pass through the points.

55. INTERPOLATION (cont.) Exact/Non Exact methods
Exact – predicted values equal observed
Theissen
IDW
Spline
Non Exact-predicted values might not equal observed
Fixed-Radius
Trend surface
Kriging