Spatial Analysis

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

Videos http://vid01.esri.com/winmmedia/avflu.wmv http://www.youtube.com/user/stchome#p/a/u/1/pM-z2GXwdsc

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.

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

Interpolation

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

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

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

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

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

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

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)

3. Distance Weighted Methods

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

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

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

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?

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

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)

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

IDW Wizard in Geostatistical Analyst define data source

Further define interpolation Power of distance 4 sectors

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

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

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

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

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.

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.

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)

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

Analysis of Variogram

Fitting a Model, Directional Effects

How Many Neighbors?

Cross Validation

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

Slightly Better Cross Validation

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

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

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

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

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

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

Gateway to the Literature Lam, N.S.-N., Spatial interpolation methods: A review, Am. Cartogr., 10 (2), 129-149, 1983. Gold, C.M., Surface interpolation, spatial adjacency, and GIS, in Three Dimensional Applications in Geographic Information Systems, edited by J. Raper, pp. 21-35, 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), 143-162, 1994. Davies, C., and D. Medyckyj-Scott, GIS usability: Recommendations based on the user's view, Int. J. Geographical Info. Sys., 8 (2), 175-189, 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. 2000 Quantitative Geography: Perspectives on Spatial Data Analysis. London: Sage. Fotheringham A.S. and O’Kelly M.E. 1989 Spatial Interaction Models: Formulations and Applications. Dordrecht: Kluwer.

More Resources ... a link to a USDA geostatistical workshop   http://www.ars.usda.gov/News/docs.htm?docid=12555 ... an EPA workshop with presentations on geostatistical applications for stream networks: http://oregonstate.edu/dept/statistics/epa_program/sac2005js.htm