idw {phylin}

Inverse Distance Weighting interpolation

Description

This function interpolates a list of samples with location and a value to a table of coordinates, that generally represent a spatial grid. The interpolation is based on inverse distance weighting algoritm with three different methods available for weight calculation.

Usage

idw(values, coords, grid, method = "Shepard", p = 2, R = 2, N = 15,
    distFUN = geo.dist, ...)

Arguments

values

A table of points to be interpolated. Table must contain x and y locations, and a column of values to be interpolated.

coords

A table wit x and y coordinates of the samples.

grid

Coordinates of locations to interpolate. It is assumed to be in the same order as 'values' table.

method

Method to calculate weights for idw. Should be "Shepard" (default), "Modified", "Neighbours", or distinctive abreviations of each. See details section for additional help on each method.

p

The power to use in weight calculation.

R

Radius to use with Modified Shepard method.

N

Maximum number of neighbours to use with Shepard with neighbours.

distFUN

Distance function used to calculate distances between locations. The default is 'geo.dist' which calculates simple euclidean distances between the locations. This function must have a 'from' and a 'to' arguments to specify, respectively, the source and destination localities.

...

Other arguments to be passed to distFUN.

Details

The IDW interpolation algorithm is commonly used to interpolate genetic data over a spatial grid. This function provides a simple interface to interpolate such data with three methods:

1. Shepard: weights are the inverse of the distance between the interpolation location x and the sample points x_i, raised to the power p

   w(x) = \frac{1}{d(x, x_i)^p}

2. Modified Shepard: distances are weighted with a search radius r to calculate the interpolation weights

   w(x) = \left(\frac{r-d(x, x_i)}{r.d(x, xi)}\right)^p

3. Shepard with neighbours: A maximum ammount of N neighbours is allowed to the weight calculation following Shepard method.

Value

It return a vector for each row of the 'coords' table with the respective interpolated value.

Author(s)

Pedro Tarroso <[e-mail]>

References

Fortin, M. -J. and Dale, M. (2006) Spatial Analysis: A guide for Ecologists. Cambridge: Cambridge University Press.

Isaaks, E. H. and Srivastava, R. M. (1989) An Introduction to applied geostatistics. New York: Oxford University Press.

Legendre, P. and Legendre, L. (1998) Numerical ecology. 2nd english edition. Amesterdam: Elsevier

Vandergast, A. G.,Hathaway, S. A., Fisher, R. N., Boys, J., Bohonak, A. J., (2008) Are hotspots evolutionary potential adequately protected in southern California? Biological Conservation, 141, 1648-1664.

See Also

intgen.idw krig

Examples

data(vipers)
data(d.gen)
data(grid)

# interpolate and plot the genetic distances for sample s2 in the d.gen
int <- idw(d.gen[,2], vipers[,1:2], grid)

grid.image(int, grid, main='IDW interpolation', xlab='Longitude', 
           ylab='Latitude', sclab="Genetic distance to sample s2")

points(vipers[,1:2], cex=d.gen[,2]*15+0.2)

# change idw power (i.e. points will have a larger influence in the 
# surroundings)
int <- idw(d.gen[,2], vipers[,1:2], grid, p=5)

result <- data.frame(grid, int)
grid.image(int, grid, main='IDW interpolation', xlab='Longitude', 
           ylab='Latitude', sclab="Genetic distance to sample s2")

points(vipers[,1:2], cex=d.gen[,2]*15+0.2)


# change idw method to "Modified Shepard" and define a maximum 
# neighbour distance
int <- idw(d.gen[,2], vipers[,1:2], grid, 'Modified', R=10)

grid.image(int, grid, main='IDW interpolation', xlab='Longitude', 
           ylab='Latitude', sclab="Genetic distance to sample s2")

points(vipers[,1:2], cex=d.gen[,2]*15+0.2)

#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#
#        Example following methods in Vandergast et al. 2008        #
#            Fit a linear model and recover the residuals           #
# ATENTION:                                                         #
#    1- Vandergast et al. (2008) suggests a RMA instead of a        # 
#       ordinary linear regression as in this example. Try package  # 
#       'lmodel2' or or other similar for RMA linear regression.    #
#    2- This example tests if the package 'geometry' is installed   #
#       to compute midpoints. If TRUE, a Delaunay triangulation is  # 
#       used, similarly to Vandergast et al. (2008). Otherwise,     #
#       midpoints are computed for the combination of all pairs of  #
#       samples.                                                    #
#                                                                   #
# the d.gen and d.real matrices in this example have the same       # 
# column and row order!                                             #
#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#

if (is.element('geometry', installed.packages()[,1])) 
    all=FALSE else 
    all=TRUE

mp <- midpoints(vipers[,1:2], all=all)
d.real <- as.matrix(dist(vipers[,1:2]))

fit <- lm(as.vector(d.gen) ~ as.vector(d.real))
resid <- matrix(fit$residuals, nrow(vipers), nrow(vipers))
dimnames(resid) <- dimnames(d.gen)
mp$z <- extract.val(resid, mp[,1:2])

int <- idw(mp[,5], mp[,3:4], grid)

grid.image(int, grid, main='IDW interpolation', 
           xlab='Longitude', ylab='Latitude', 
           sclab="Residuals of genetic vs. real distances")

# plot samples connecting lines
for (i in 1:nrow(mp))
{
    pair <- as.character(unlist(mp[i,1:2]))
    x <- c(vipers[pair[1],1], vipers[pair[2],1])
    y <- c(vipers[pair[1],2], vipers[pair[2],2])
    lines(x, y, lty=2)
}
points(vipers[,1:2], pch=16) # plot samples points in black
points(mp[,3:4], pch=16, col='gray') # plot midpoints in gray