Statistical Surfaces
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Transcript of Statistical Surfaces
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
Distribution of trees
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Find # of trees w/in the neighborhood of each grid cell
Point data Density surface
Statistical Surfaces
• Storage of surface data in GIS
– raster grid– TIN– isarithms (e.g. contours for topographic
elevation)– lattice
Statistical Surfaces
• Isarithm
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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
Elevation profile
Sample elevation data
A
B
If
A = 8 feet and
B = 4 feet
then
C = (8 + 4) / 2 = 6 feetC
Statistical Surfaces• Interpolation: Nonlinear interpolation
Elevation profile
Sample elevation data
A
B
C
Often results in a more realistic interpolation but estimating missing data values is more complex
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
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Statistical Surfaces• Interpolation: IDW
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100 x 1 = 100 160 x 1.8 = 288 200 x 4 = 800
1 / (42) = .0625 1 / (32) = .1111 1 / (22) = .2500
.0625 / .0625 = 1 .1111 / .0625 = 1.8 .2500 / .0625 = 4
Weights Adjusted Weights
100 +288 + 800 = 1188
1188 / 6.8 = 175
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
Statistical Surfaces• Interpolation: 1st degree Trend Surface
x
yz
x
y
In two dimensions In three dimensions
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
x
yz
x
y
In two dimensions In three dimensions
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