1. Spatial Analysis

2. Early Spatial Analysis
John Snow, 1854
Cholera via polluted water, not air
“John Snow’s pump” X X X X X X X

3. Videos http://vid01.esri.com/winmmedia/avflu.wmv

4. Categories of Spatial Analysis (Longley et al.)
Hypothesis testing
Queries and reasoning
Map & database/catalog queries, buffer, polygon overlay
Measurements
Aspects of geographic data, length, area, etc.

5. Categories of Spatial Analysis (Longley et al.)
Transformations
New data, raster to vector, geometric rules
Buffer, polygon overlay
Interpolation, Density Estimation, Terrain Analysis (Lab 6)
Descriptive summaries
Essence of data in 1 or 2 parameters
Spatial statistics (including fragmentation statistics)
Optimization - ideal locations, routes
Network analysis (Lab 5), Routing

6. Interpolation

8. Nonlinear Interpolation
When things aren't or shouldn’t be so simple
Basic types: 1. Trend surface analysis / Polynomial
2. Minimum Curvature Spline
3. Inverse Distance Weighted 4. Kriging

9. Fitting Continuous Surfaces to Data
(1) FLAT plane
(2) flat but TILTED to fit data better
(3) tilted but WARPED to fit data even better

11. 1. Trend Surface/Polynomial
point-based
Fits a polynomial to input points
When calculating function that will describe surface, uses least-square regression fit
approximate interpolater
Resulting surface doesn’t pass through all data points
global trend in data, varying slowly overlain by local but rapid fluctuations

12. 1. Trend Surface cont.
flat but TILTED plane to fit data
surface is approximated by linear equation (polynomial degree 1)
z = a + bx + cy
tilted but WARPED plane to fit data
surface is approximated by quadratic equation (polynomial degree 2)
z = a + bx + cy + dx2 + exy + fy2

13. Trend Surfaces
Simplifies the surface representation to allow visualization of general trends.
Polynomials of higher order

14. Windows (not Microsoft’s)
generates estimates based on existing data in the “region”
“region” = “roving window”
moves about study area
summarizes data it encounters
reach (search radius)
number of samples
Direction
WHERE might you find unusual responses?
results extend non-spatial concept of central tendency

15. 2. Minimum Curvature Splines
Fits a minimum-curvature surface through input points
Like bending a sheet of rubber to pass through points
While minimizing curvature of that sheet
repeatedly applies a smoothing equation (piecewise polynomial) to the surface
Resulting surface passes through all points
best for gently varying surfaces, not for rugged ones (can overshoot data values)

17. 3. Distance Weighted Methods

18. 3. Inverse Distance Weighted
Each input point has local influence that diminishes with distance
estimates are averages of values at n known points within window
where w is some function of distance

19. The estimate is a weighted average Weights decline with distance
point i
known value zi
location xi
weight wi distance di
unknown value (to be interpolated)
location x
The estimate is a weighted average
Weights decline with distance

20. IDW (cont.)
an almost infinite variety of algorithms may be used, variations include:
the nature of the distance function (w)
varying the number of points used
the direction from which they are selected

21. IDW (cont.) IDW is popular, easy, but not panacea
interpolated values limited by the range of the data
no interpolated value will be outside the observed range of z values
how many points should be included in the averaging?
what to do about irregularly spaced points?

22. A potentially undesirable characteristic of IDW interpolation
This set of six data points clearly suggests a hill profile. But in areas where there is little or no data the interpolator will move towards the overall mean. Blue line shows the profile interpolated by IDW

23. IDW Example ozone concentrations at CA measurement stations
1. estimate a complete field, make a map
2. estimate ozone concentrations at specific locations (e.g., Los Angeles)

24. Data for IDW Example measuring stations and concentrations
(point shapefile)
CA cities (point shapefile)
CA outline (polygon shapefile)
DEM (raster)

25. IDW Wizard in Geostatistical Analyst define data source

26. Further define interpolation
Power of distance
4 sectors

27. Cross validation
removing one of the n observation points and using the remaining n-1 points to predict its value.
Error = observed - predicted

28. Results amount of detail where there is no data
generally smooth surface
highs in LA, S central valley

29. 4. Kriging
Assumes distance or direction betw. sample points shows a spatial correlation that help describe the surface
Fits function to
Specified number of points OR
All points within a window of specified radius
based on an analysis of the data, then an application of the results of this analysis to interpolation
Most appropriate when you already know about spatially correlated distance or directional bias in data
Kriging
developed by Georges Matheron, as the "theory of regionalized variables", and D.G. Krige as an optimal method of interpolation for use in the mining industry
the basis of this technique is the rate at which the variance between
points changes over space
this is expressed in the variogram which shows how the average difference between values at points changes with distance between points
Kriging is based on an analysis of the data, then an application of the results of this analysis to interpolation

30. Kriging (cont.) Involves several steps
Exploratory statistical analysis of data
Variogram modeling
Creating the surface based on variogram
Variograms
vertical axis is E(zi - zj)2, i.e. "expectation" of the difference
i.e. the average difference in elevation of any two points
distance d apart
d (horizontal axis) is distance between i and j

31. Explore with Trend analysis
You may wish to remove a trend from the dataset before using kriging. The Trend Analysis tool can help identify global trends in the input dataset.

32. SemiVariogram in Kriging
how avg. difference between values at points changes with distance between points
Range – no more surprises
sill
the upper limit (asymptote) is called the sill
the distance at which this limit is reached is called the range
The intersection with the y axis is called the nugget
a non-zero nugget indicates that repeated measurements at the same point yield different values
in developing the variogram it is necessary to make some assumptions about the nature of the observed variation on the surface:
simple Kriging assumes that the surface has a constant mean, no underlying trend and that all variation is statistical
universal Kriging assumes that there is a deterministic trend in the surface that underlies the statistical variation
in either case, once trends have been accounted for (or assumed not to exist), all other variation is assumed to be a function of distance
nugget
A semivariogram. Each cross represents a pair of points. The solid circles are obtained by averaging within the ranges or bins of the distance axis. The solid line represents the best fit to these five points, using one of a small number of standard mathematical functions.

33. Kriging Results
once the variogram has been developed, it is used to estimate distance weights for interpolation
computationally very intensive w/ lots of data points
estimation of the variogram complex
No one method is absolute best
Results never absolute, assumptions about distance, directional bias
Deriving the variogram
the input data for Kriging is usually an irregularly spaced sample of points
to compute a variogram we need to determine how variance increases with distance
begin by dividing the range of distance into a set of discrete intervals, e.g. 10 intervals between distance 0 and the maximum distance in the study area
for every pair of points, compute distance and the squared difference in z values
assign each pair to one of the distance ranges, and accumulate total variance in each range
after every pair has been used (or a sample of pairs in a large dataset) compute the average variance in each distance range
plot this value at the midpoint distance of each range
fit one of a standard set of curve shapes to the points
"model" the variogram
Computing the estimates:
interpolated values are the sum of the weighted values of some number of known points where weights depend on the distance between the interpolated and known points
weights are selected so that the estimates are:
unbiased (if used repeatedly, Kriging would give the correctresult on average)
minimum variance (variation between repeated estimates is minimum)

34. Kriging Example surface has a constant mean, no underlying trend
Surface has no constant mean
Maybe no underlying trend
surface has a constant mean,
no underlying trend
allows for a trend
binary data

35. Analysis of Variogram

36. Fitting a Model, Directional Effects

37. How Many Neighbors?

38. Cross Validation

39. Kriging Result similar pattern to IDW less detail in remote areas
smooth

40. Slightly Better Cross Validation

41. IDW vs. Kriging Kriging IDW
Kriging appears to give a more “natural” look to the data
Kriging avoids the “bulls eye” effect
Kriging gives us a standard error
IDW

42. Which Method to Use? Trend - rarely goes through your original points
Spline - best for surfaces that are already smooth
Elevations, water table heights, etc.
IDW - assumes variable decreases in influence w/distance from sampled location
Interpolating a surface of consumer purchasing power for a retail store
Kriging - if you already know correlated distances or directional bias in data
Geology, soil science

43. Which to Use? cont.
Kriging - Allows user greater flexibility in defining the model to be used in the interpolation
Tracks changes in spatial dependence across study area (may not be linear)
Produces
a smooth, interpolated surface
variogram (how well pixel value fits overall model)
Diagnostic tool to refine model
Want to get variances close as possible to zero

44. Interpolation Software
ArcGIS with Geostatistical Analyst
Surfer (Golden Software)
Surface II package (Kansas Geological Survey)
GEOEAS (EPA)
Spherekit (NCGIA, UCSB)
Matlab

45. ArcInfo Workstation Interpolation Methods
TREND (Grid function)
SPLINE (Grid function, minimum curvature spline)
IDW (Grid function)
KRIGING (Arc command)

46. Research Issues... "easy to use" “effective"
choose correct technique w/o having a Ph.D. in math or stats
“effective"
techniques should be informative,
highlighting the essential nature of the data and/or surface
meet needs of the study
“natural language” interface
series of questions about the intentions, goals and aims of the user and about the nature of the data
articles on prototypes in the literature

47. Gateway to the Literature
Lam, N.S.-N., Spatial interpolation methods: A review, Am. Cartogr., 10 (2), , 1983.
Gold, C.M., Surface interpolation, spatial adjacency, and GIS, in Three Dimensional Applications in Geographic Information Systems, edited by J. Raper, pp , Taylor and Francis, Ltd., London, 1989.
Robeson, S.M., Spherical methods for spatial interpolation: Review and evaluation, Cartog. Geog. Inf. Sys., 24 (1), 3-20, 1997.
Mulugeta, G., The elusive nature of expertise in spatial interpolation, Cart. Geog. Inf. Sys., 25 (1), 33-41, 1999.
Wang, F., Towards a natural language user interface: An approach of fuzzy query, Int. J. Geog. Inf. Sys., 8 (2), , 1994.
Davies, C., and D. Medyckyj-Scott, GIS usability: Recommendations based on the user's view, Int. J. Geographical Info. Sys., 8 (2), , 1994.
Blaser, A.D., M. Sester, and M.J. Egenhofer, Visualization in an early stage of the problem-solving process in GIS, Comp. Geosci, 26, 57-66, 2000.
Fotheringham A.S., Brunsdon C., and Charlton M Quantitative Geography: Perspectives on Spatial Data Analysis. London: Sage.
Fotheringham A.S. and O’Kelly M.E Spatial Interaction Models: Formulations and Applications. Dordrecht: Kluwer.

48. More Resources ... a link to a USDA geostatistical workshop
... an EPA workshop with presentations on geostatistical applications for stream networks: