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Statistical Surfaces

Discover the different types of statistical surfaces, such as continuous and discrete surfaces, and learn about interpolation and estimation methods used in Geographic Information Systems (GIS). Explore linear and nonlinear interpolation, global and local methods, kriging, and trend surfaces.

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Statistical Surfaces

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  1. 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 measureble attribute that varies continuously over space, such as temperature and population density (interval/ratio data)

  2. Surfaces • Statistical surface • Continuous • Discrete

  3. Statistical Surfaces • Two types of surfaces: • data are not countable (i.e. temperature) and geographic entity is conceptualized as a field • punctiform: data are composed of individuals whose distribution can be modeled as a field (population density)

  4. 2 2 3 2 2 3 3 6 3 3 3 4 4 2 2 3 4 4 3 1 2 2 3 2 1 Statistical Surfaces • Surface from punctiform data Point data Density surface Distribution of trees Find # of trees w/in the neighborhood of each grid cell

  5. Statistical Surfaces • Storage of surface data in GIS • raster grid • TIN • isarithms (e.g. contours for topographic elevation) • lattice

  6. Statistical Surfaces • Isarithm 10 20 30 40 50 80 70 60 60

  7. Statistical Surfaces • Lattice: a set of points with associated Z values Regular Irregular

  8. Statistical Surfaces • Interpolation • 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 We will be talking about point interpolation

  9. Statistical Surfaces Estimating a point here: interpolation Sample data

  10. Statistical Surfaces Estimating a point here: interpolation Estimating a point here: extrapolation

  11. Statistical Surfaces • Interpolation: 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

  12. Statistical Surfaces • Interpolation: 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

  13. Statistical Surfaces • Interpolation: Global • use all known sample points to estimate a value at an unsampled location Use entire data set to estimate value

  14. Statistical Surfaces • 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

  15. Statistical Surfaces • 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

  16. Statistical Surfaces • Interpolation: IDW Weights Adjusted Weights 1 / (42) = .0625 1 / (32) = .1111 1 / (22) = .2500 .0625 / .0625 = 1 .1111 / .0625 = 1.8 .2500 / .0625 = 4 100 x 1 = 100 160 x 1.8 = 288 200 x 4 = 800 100 100 +288 + 800 = 1188 4 1188 / 6.8 = 175 160 3 2 200

  17. Statistical Surfaces • 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

  18. Statistical Surfaces • Interpolation: 1st degree Trend Surface In two dimensions In three dimensions z y y x x

  19. Statistical Surfaces • 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

  20. Statistical Surfaces • Interpolation: Spline and higher degree trend surface In two dimensions In three dimensions z y y x x

  21. Statistical Surfaces • 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

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