Perform Tobler's pycnophylactic algorithm.

Description

Given a SpatialPolygonsDataFrame and a set of populations for each polygon, compute a population density estimate based on Tobler's pycnophylactic interpolation algorithm. The result is a SpatialGridDataFrame.

Usage

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pycno(x, pops, celldim, r = 0.2, converge = 3, verbose=TRUE)

Arguments

x

A SpatialPolygonsDataFrame representing a set of zones for which total populations are known.

pops

A list of populations, one for each zone represented in x.

celldim

The size of the cell sides for the output SpatialGridDataFrame, or a SpatialGrid or SpatialGridDataFrame with the grid topology of the desired output.

r

A relaxation parameter for the iterative step in the pycnophylactic algorithm. Prevents over-compensation in the smoothing step. In practice the default value works well.

converge

A convergence parameter, informing the decision on when iterative improvements on the smooth surface have converged sufficiently - see details.

verbose

If set to TRUE (the default value) makes the function report the maximum change in any pixel value for each iterative step. If set to FALSE the function is silent.

Details

This method uses an iterative approach, and for each iteration notes the maximum change in a pixel. When this value falls below a certain level (actually 10^(-converge) times the largest initial pixel value) the iterations stop and the current surface is reported.

Value

A SpatialGridDataFrame with each pixel set at the estimated density computed with the Pycnophylactic algorithm, these values being stored in the data item dens. The extent of this object is set at by the bounding box of the input SpatialPolygonsDataFrame. Pixels outside of all of the polygons are set to NA.

Note

Pycnophylatic interpolation has the property that the sum of the estimated values associated with all of the pixels in any polygon equals the supplied population for that polygon. A further property is that all pixel values are greater than or equal to zero. The method is generally used to obtain pixel-based population estimates when total populations for a set of irregular polygons (eg. counties) are known.

Author(s)

Chris Brunsdon

References

Tobler, W.R. (1979) Smooth Pycnophylactic Interpolation for Geographical Regions. Journal of the American Statistical Association, v74(367) pp. 519-530.

Examples

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# Read in data for North Carolina as a SpatialPolygonsDataFrame
nc.sids <- readShapeSpatial(system.file("shapes/sids.shp", package="maptools")[1], 
   IDvar="FIPSNO", proj4string=CRS("+proj=longlat +ellps=clrk66"))

# Compute the pycnophylactic surface for 1974 births as a SpatialGridDataFrame
# Note probably shouldn't really base grid cells on Lat/Long coordinates
# This example just serves to illustrate the use of the function
# It is suggested to use a hihger value for 'converge' - this value justs speeds 
# things up for the example.
births74 <- pycno(nc.sids,nc.sids$BIR74,0.05,converge=1)

# Draw it
image(births74)

# Overlay North Carolina county boundaries for reference
plot(nc.sids,add=TRUE)

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