1. URBDP 422 URBAN AND REGIONAL GEO-SPATIAL ANALYSIS
Lecture 13: Surface Analysis and Interpolation
March 4, 2014
URBDP 422 Urban and Regional Spatial Anaysis

2. Lecture’s Objectives Characterize statistical surfaces
Understand spatial autocorrelation
Examine implications of spatial autocorrelation for scientific inference
Learn methods of interpolation

3. Statistical Surfaces
Any geographic entity that can be thought of as containing a Z value for each X,Y location
topographic elevation being the most obvious example
but can be any numerically measurable attribute that varies over space

4. Statistical Surfaces Two types of surfaces:
Data that are continuous by nature and measured as fields (e.g. temperature and elevation)
Punctiform: data are composed of individuals whose distribution can be modeled as a field (population density)

5. Continuous Statistical Surfaces
What are some examples

6. Continuous Statistical Surfaces
Puget Sound DEM
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

7. Statistical Surfaces from Punctiform Data
What are some examples?
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

8. Statistical Surfaces from Punctiform Data
Point data
Density surface 2 4 3 6 1
Distribution of trees
# of trees w/in a specified radius

9. Punctiform Statistical Surfaces
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

10. URBDP 422 Urban and Regional Spatial Anaysis - Alberti
Data Models for Statistical Surfaces
- Isolines: contour lines describing the curve connecting
all locations of the same elevation.
- Lattice: regularly spaced network of equidistant points
with elevation values assigned
- Grid: raster representation, a generalized form of lattice
where each lattice point becomes a cell center.
A commonly used format for Digital Elevation Model (DEM).
- TIN (triangulated irregular network): an irregularly spaced
sample of points that can be adopted to the terrain, with
more points in areas of rough terrain and fewer in smooth
terrain. In TIN the sample points are connected by lines
to form triangles.
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

11. URBDP 422 Urban and Regional Spatial Anaysis - Alberti
Statistical Surfaces
A contour line (also isoline, isopleth, or isarithm) is a curve along which the function has a constant value. In cartography, a contour line (often just called a "contour") joins points of equal elevation (height) above a given level, such as mean sea level.
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

12. URBDP 422 Urban and Regional Spatial Anaysis - Alberti
Statistical Surfaces
Contour lines 10 20 30 40 50 80 70 60 60
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

13. URBDP 422 Urban and Regional Spatial Anaysis - Alberti
Surface in Raster Data Model
Lattice
A lattice is a surface interpretation of a grid,
represented by equally spaced sample points
referenced to a common origin and a constant
sampling distance in the x y direction.
The surface value is recorded for each cell and
stored as center point of the cell; it does not imply an area of constant value.
The lattice support accurate surface calculations
including slope, aspect, and contour interpolation.
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

14. URBDP 422 Urban and Regional Spatial Anaysis - Alberti
Lattice versus Grid
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

15. URBDP 422 Urban and Regional Spatial Anaysis - Alberti
Statistical Surfaces
Sets of points with associated Z values
Regular Lattice
Irregular Points
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

16. Triangulated Irregular Network (TIN)
Model:
Useful for terrain (surface) modeling and analysis – an alternative to grid DEM
Irregularly spaced sample points
Adaptive: More points in rough terrain
Sample points are triangulated and each triangle represented by a simple surface (usually a plane)
The corner points of each triangle can match the given data exactly.
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

17. URBDP 422 Urban and Regional Spatial Anaysis - Alberti
TIN Data Structure
The tin data structure is based on two basic
elements: - points with x,y,z values
- edges joining these points to form triangles
node (x,y,z)
triangle
edge
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

18. URBDP 422 Urban and Regional Spatial Anaysis - Alberti
Spatial Autocorrelation
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

19. URBDP 422 Urban and Regional Spatial Anaysis - Alberti
Spatial Autocorrelation
Points that are closer together in space are more likely to have similar properties than those farther apart.
The first law of geography (Tobler's Law) "Everything is related to everything else, but near things are more related than distant things.”
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

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Why is spatial autocorrelation important?
the bad news
Spatial patterns indicate that data are not independent of one another, violating the assumption of independence for some statistical tests. Inflate the degree of freedom.
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

23. URBDP 422 Urban and Regional Spatial Anaysis - Alberti
Why is spatial autocorrelation important?
the good news
It provides the basis for interpolation. We can determine values of a variable at unobserved points based on the spatial autocorrelation among known data points.
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

24. Understanding spatial autocorrelation The Semivariogram
semivariance
Distance between points
(due to error,
spatial variation at
distances smaller
than sampling interval
Nugget
Range
sill
Binning (pairs of points are grouped based on distance between them, and their average distance and semivariance are plotted as one point)
(distance that is the limit
of spatial autocorrelation)
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

25. An Example: Assessed value of Seattle residential land
Do you think the distribution of land value (per parcel) is spatially correlated?
If yes, is it negatively or positively correlated in space?
Mean land value = $1.9 million
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

26. Positive spatial correlation No spatial correlation

27. Measuring spatial autocorrelation Moran’s I
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

28. Measuring spatial autocorrelation Moran’s I
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

29. URBDP 422 Urban and Regional Spatial Anaysis - Alberti
Seattle housing sales
Hedonic Pricing Theory - value is a function of
Structural characteristics of home
Characteristics of neighborhood
Environmental characteristics
Are these characteristics randomly distributed across the landscape?
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

30. What other phenomenon do you expect to be spatially correlated?1. 2. 3.
Why do you expect these are spatially correlated?
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

31. Issues with spatial autocorrelation
Dependence of samples – means degrees of freedom (your count of observations -1) is overestimated
Influences statistical tests:
Are average housing sales in Ballard more or less than sales in West Seattle?
Is housing price correlated with views of Mt Rainier?
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

32. Solutions (to conduct statistical tests)
Filter – as long as your samples are separated by the distance of influence (range), they are independent
Detrend your data – in this case it looks like distance from downtown might explain the spatial correlation
Geographically weighted regression
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

33. Solution – detrend data
Linear regression model where distance from downtown is the independent variable explaining land value
To get an independent sample, we can select parcels >10,000 feet from each other (and control for dist from DT)

34. Solution – detrend data
Map of residuals
What other factors might explain prices? We can use these to further detrend data.
Waterfront properties
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

35. Applications of spatial autocorrelation
Can use the spatial structure to estimate values at unsampled locations (interpolation) via kriging
Can also identify clustering of phenomenon; to make inference about source. Pollution hotspots, disease/health epidemics, etc.
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

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Interpolation
Estimate values of properties of interest at unsampled sites using information from sampled sites
Task: chose the best model fitting the data so that value of points among those sampled can be predicted
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

37. Interpolation (Bacastow 2001)
Estimating a point here: interpolation
Sample data
Bacastow 2001
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

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39. Interpolation vs. Extrapolation
estimating the values of locations for which there is no data using the known data values of nearby locations
Extrapolation
estimating the values of locations outside the range of available data using the values of known data
Interpolation vs. Extrapolation
Interpolation is prediction within the range of our data
E.g., having temperature values for a bunch of locations all throughout PA, predict the temperature values at all other locations within PA
Note that the methods we are talking about are strictly those of interpolation, and not extrapolation
Extrapolation is prediction outside the range of our data
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

40. Interpolation vs. Extrapolation
Estimating a point here: interpolation
Estimating a point here: extrapolation
Bacastow 2001
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

41. Interpolation Techniques
Deterministic
Directly based on the surrounding measured values or on specified mathematical formulas that determine the smoothness of the resulting surface.
Grouped into two groups: global and local
Global techniques use the entire dataset
Local techniques used points within a distance or neighborhood
Geo-Statistical
based on statistical models that include autocorrelation (statistical relationships among the measured points).
Not only do these techniques have the capability of producing a prediction surface, but they can also provide some measure of the certainty or accuracy of the predictions.
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

42. URBDP 422 Urban and Regional Spatial Anaysis - Alberti
Linear Interpolation If
A = 8 feet and
B = 4 feet
then
C = (8 + 4) / 2 = 6 feet
Sample elevation data A C B
Elevation profile
Bacastow 2001
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

43. Nonlinear Interpolation
Sample elevation data
Often results in a more realistic interpolation but estimating missing data values is more complex A C B
Elevation profile
Bacastow 2001
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

44. Interpolation: Global
use all known sample points to estimate a value at an unsampled location
Use entire data set to estimate value
Bacastow 2001
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

45. URBDP 422 Urban and Regional Spatial Anaysis - Alberti
Interpolation: Local
use a neighborhood of sample points to estimate a value at an unsampled location
Use local neighborhood data to estimate value, i.e. closest n number of points, or within a given search radius
Bacastow 2001
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

46. Interpolation: Distance Weighted
Inverse Distance Weighted - IDW
the weight (influence) of a neighboring data value is inversely proportional to the square of its distance from the location of the estimated value
100 4
160 3 2
200
Bacastow 2001
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

47. URBDP 422 Urban and Regional Spatial Anaysis - Alberti
Interpolation: IDW
Weights
Adjusted Weights
1 / (42) = / (32) = / (22) = .2500
.0625 / = / = / = 4
100 x 1 = x 1.8 = x 4 = 800
100
= 1188 4
1188 / 6.8 = 175
160 3 2
200
Bacastow 2001
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

48. Interpolation: 1st degree Trend Surface
global method
multiple regression (predicting z elevation with x and y location
conceptually a plane of best fit passing through a cloud of sample data points
does not necessarily pass through each original sample data point
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

49. Interpolation: 1st degree Trend Surface
In two dimensions
In three dimensions z y y x x
Bacastow 2001
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

50. Interpolation: Spline and higher degree trend surface
local
fits a mathematical function to a neighborhood of sample data points
a ‘curved’ surface
surface passes through all original sample data points
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

51. Interpolation: Spline and higher degree trend surface
In two dimensions
In three dimensions z y y x x
Bacastow 2001
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

52. Interpolation: kriging
common for geologic applications
addresses both global variation (i.e. the drift or trend present in the entire sample data set) and local variation (over what distance do sample data points ‘influence’ one another)
provides a measure of error
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

53. URBDP 422 Urban and Regional Spatial Anaysis - Alberti
Kriging methods rely on defining autocorrelation
Autocorrelation:
statistical correlation in variables between points dependent on distance and direction separating the points; close points are more similar
Correlation
Distance
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

54. URBDP 422 Urban and Regional Spatial Anaysis - Alberti
Fitting a Variogram Model
Spheric Model
semivariance
Distance between points
Nugget
Range
sill
Exponential Model
URBDP 422 Urban and Regional Spatial Anaysis - Alberti

55. URBDP 422 Urban and Regional Spatial Anaysis - Alberti
Take-home messages
Statistical Surfaces: Geographic entity containing a Z value for each X,Y location
Spatial Autocorrelation: points that are closer together in space are more likely to have similar properties than those farther apart
Spatial autocorrelation violates assumption of independence in sampling without correcting for spatial autocorrelation.
Interpolation: best model fitting the data so that value of un-sampled points among those sampled can be predicted
URBDP 422 Urban and Regional Spatial Anaysis - Alberti