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Spatial Analysis. Longley et al., Ch 14,15. Transformations . Buffering (Point, Line, Area) Point-in-polygon Polygon Overlay Spatial Interpolation Theissen polygons Inverse-distance weighting Kriging Density estimation. Basic Approach. Map. New map. Transformation. Point-in-polygon.

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spatial analysis

Spatial Analysis

Longley et al., Ch 14,15

transformations
Transformations
  • Buffering (Point, Line, Area)
  • Point-in-polygon
  • Polygon Overlay
  • Spatial Interpolation
    • Theissen polygons
    • Inverse-distance weighting
    • Kriging
    • Density estimation
basic approach
Basic Approach

Map

New map

Transformation

point in polygon
Point-in-polygon

Select point known to be outside

Select point to be tested

Create line segment

Intersect with all boundary

segments

Count intersections

EVEN=OUTSIDE

ODD=INSIDE

combining maps
Combining maps
  • RASTER
  • As long as maps have same extent, resolution, etc, overlay is direct (pixel-to-pixel)
  • Otherwise, needs interpolation
  • Use map algebra (Tomlin)
  • Tomlin’s operators
    • Focal, Local, Zonal
combining maps8
Combining Maps
  • VECTOR
  • A problem
creating new zones
Creating new zones

Town buffer

River buffer

other spatial analysis methods
Other spatial analysis methods
  • Centrographic analysis (mean center)
  • Dispersion measures (stand. Dist)
  • Point clustering measures (NNS)
  • Moran’s I: Spatial autocorrelation (Clustering of neighboring values)
  • Fragmentation and fractional dimension
  • Spatial optimization
    • Point
    • Route
  • Spatial interpolation
moran s i
Moran’s I

http://gis.esri.com/library/userconf/proc02/pap1064/p106413.gif

spatial autocorrelation
Spatial autocorrelation
  • Correlation of a field with itself

Low

High

Maximum

spatial optimization
Spatial optimization

www.giscenter.net/eng/work_03_e.html

linear interpolation
Linear interpolation

C

B

Half way from A to B,

Value is (A + B) / 2

A

nonlinear interpolation
Nonlinear Interpolation
  • When things aren\'t or shouldn’t be so simple
  • Values computed by piecewise “moving window”
  • Basic types:1. Trend surface analysis / Polynomial

2. Minimum Curvature Spline

3. Inverse Distance Weighted 4. Kriging

1 trend surface polynomial
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 interpolator
    • 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
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
2 minimum curvature splines
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 inverse distance weighted
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 R

where w is some function of distance (e.g., w = 1/dk)

slide24
IDW
  • IDW is popular, easy, but problematic
  • 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 about irregularly distributed points?
  • What about the map edges?
idw example
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)

ozone in s cal text example
Ozone in S. Cal: Text Example

measuring stations and concentrations

(point shapefile)

CA cities (point shapefile)

CA outline (polygon shapefile)

DEM (raster)

cross validation
Cross validation
  • removing one of the n observation points and using the remaining n-1 points to predict its value.
  • Error = observed - predicted
4 kriging
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
  • Involves several steps
    • Exploratory statistical analysis of data
    • Variogram modeling
    • Creating the surface based on variogram
kriging
Kriging
  • Breaks up topography into 3 elements:Drift (general trend), small deviations from the drift and random noise.

To be stepped over

explore with trend analysis
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.
kriging results
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
kriging example
Kriging Example

Surface has no constant mean

Maybe no underlying trend

surface has a constant mean,

no underlying trend

allows for a trend

binary data

kriging result
Kriging Result
  • similar pattern to IDW
  • less detail in remote areas
  • smooth
idw vs kriging
IDW vs. Kriging
  • Kriging appears to give a more “smooth” look to the data
  • Kriging avoids the “bulls eye” effect
  • Kriging gives us a standard error

Kriging

IDW

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