Statistical Surfaces Any geographic entity that can be

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)

Surfaces • Statistical surface • Continuous • Discrete

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)

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

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

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

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

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

Statistical Surfaces Estimating a point here: interpolation Sample data

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

Statistical Surfaces • Interpolation: Linear interpolation If Sample elevation data A = 8 feet and B = 4 feet A then C C = (8 + 4) / 2 = 6 feet B Elevation profile

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

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

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

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 3 200 2 160

Statistical Surfaces • Interpolation: IDW Weights 1 / (42) =. 0625 1 / (32) =. 1111 1 / (22) =. 2500 Adjusted Weights. 0625 /. 0625 = 1. 1111 /. 0625 = 1. 8. 2500 /. 0625 = 4 100 x 1 = 100 160 x 1. 8 = 288 200 x 4 = 800 100 +288 + 800 = 1188 4 1188 / 6. 8 = 175 3 200 2 160

Statistical Surfaces • Interpolation: 1 st 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

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

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

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

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