Pycnophylactic Interpolation

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.

Details

For use in conjunction with the sp package, computes pycnophylactic surfaces given a SpatialPolygonsDataFrame and a population for each poplygon. A pycnophylactic surface is smooth, but populations allocated to each pixel sum up to the initial polygon counts, when summed over the polygons contained in each pixel.

Author(s)

Chris Brunsdon

Maintainer: Francesca Bitonti francesca.bitonti@unict.it

References

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

Examples

library(sp)
# 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"))

nc.sids <- as(sf::st_read(system.file("shape/nc.shp", package="sf")), "Spatial")
row.names(nc.sids) <- as.character(nc.sids$FIPSNO)

# 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
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)

Estimate populations for a set of zones based on pycnophylactic interpolation

Description

Given a SpatialGridDataFrame of population estimates and a set of polygons, compute a population estimate based on Tobler's pycnophylactic interpolation algorithm for each zone. The result is a vector.

Usage

estimate.pycno(sgdf, spdf)

Arguments

Details

Takes the estimate of population density for each pixel, checks which polygon each pixel is in, and aggregates them. Accuracy depends on the scale of pixels in the initial interpolation.

Value

A vector in which each each pixel set at the estimated population aggregation to each zone in spdf.

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.

See Also

pycno

Examples

library(sp)
# 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"))

nc.sids <- as(sf::st_read(system.file("shape/nc.shp", package="sf")), "Spatial")
row.names(nc.sids) <- as.character(nc.sids$FIPSNO)

# 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 functions
births74 <- pycno(nc.sids,nc.sids$BIR74,0.05,converge=1)

# Create a new 'blocky' set of zones
#blocks <- gUnionCascaded(nc.sids,1*(coordinates(nc.sids)[,2] > 36) + 
#  2*(coordinates(nc.sids)[,1] > -80))

crds <- sf::st_coordinates(sf::st_centroid(sf::st_geometry(sf::st_as_sf(nc.sids)),
 of_largest_polygon = TRUE))
block_ID <- 1*(crds[,2] > 36) + 2*(crds[,1] > -80)
temp <- sf::st_as_sf(nc.sids)
temp$block_ID <- block_ID
blocks <- as(aggregate(temp, by=list(temp$block_ID), head, n=1), "Spatial")

# Plot the blocky zones
plot(blocks)
# Aggregate data to them
estimates <- estimate.pycno(births74,blocks)
# Write the estimates on to the map
text(coordinates(blocks),as.character(estimates))

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

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

Arguments

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

library(sp)
# 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"))

nc.sids <- as(sf::st_read(system.file("shape/nc.shp", package="sf")), "Spatial")
row.names(nc.sids) <- as.character(nc.sids$FIPSNO)

# 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)