idw: Inverse Distance Weighting interpolation
In phylin: Spatial Interpolation of Genetic Data

Description Usage Arguments Details Value Author(s) References See Also Examples

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.

1 2	idw(values, coords, grid, method = "Shepard", p = 2, R = 2, N = 15, distFUN = geo.dist, ...)

`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.

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:

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) = 1/d(x, xi)^p
Modified Shepard: distances are weighted with a search radius r to calculate the interpolation weights

w(x) = ((r-d(x, xi)) / (r*d(x, xi)))^p
Shepard with neighbours: A maximum ammount of N neighbours is allowed to the weight calculation following Shepard method.

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

Pedro Tarroso <ptarroso@cibio.up.pt>

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.

intgen.idw krig

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