Imputing missing values keeping circular trend in mind - r

Question

Think of a picture of Sunrise where a red circle is surrounded by yellow thick ring and then blue background. Take red as 3 then yellow as 2 and blue as 1.
11111111111
11111211111
11112221111
11222322211
22223332222
11222322221
11112221111
11111211111
This is the desired output. But, the record/file/data has missing values (30% of all elements are missing).
How can we impute missing values so as to get this desired output keeping the circular trend in mind.

Answer 1

This is how I would solve a problem of this sort in a very simple, straightforward way. Please note that I corrected your sample data above to be symmetric:
d <- read.csv(header=F, stringsAsFactors=F, text="
1,1,1,1,1,1,1,1,1,1,1
1,1,1,1,1,2,1,1,1,1,1
1,1,1,1,2,2,2,1,1,1,1
1,1,2,2,2,3,2,2,2,1,1
2,2,2,2,3,3,3,2,2,2,2
1,1,2,2,2,3,2,2,2,1,1
1,1,1,1,2,2,2,1,1,1,1
1,1,1,1,1,2,1,1,1,1,1
")
library(raster)
## Plot original data as raster:
d <- raster(as.matrix(d))
plot(d, col=colorRampPalette(c("blue","yellow","red"))(255))
## Simulate 30% missing data:
d_m <- d
d_m[ sample(1:length(d), length(d)/3) ] <- NA
plot(d_m, col=colorRampPalette(c("blue","yellow","red"))(255))
## Construct a 3x3 filter for mean filling of missing values:
filter <- matrix(1, nrow=3, ncol=3)
## Fill in only missing values with the mean of the values within
## the 3x3 moving window specified by the filter. Note that this
## could be replaced with a median/mode or some other whole-number
## generating summary statistic:
r <- focal(d_m, filter, mean, na.rm=T, NAonly=T, pad=T)
## Plot imputed data:
plot(r, col=colorRampPalette(c("blue","yellow","red"))(255), zlim=c(1,3))
This is an image of the original sample data:
With 30% missing values simulated:
And only those missing values interpolated with the mean of the 3x3 moving window:

Answer 2

Here I compare Forrest's approach with a thin plate spline (TPS). Their performance is about the same -- depending on the sample. The TPS could be preferable if the gaps were larger such that focal could not estimate anymore --- but in that case you could also use a a larger (and perhaps Gaussian, see ?focalWeight) filter.
d <- matrix(c(
1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,2,1,1,1,1,1,
1,1,1,1,2,2,2,1,1,1,1,
1,1,2,2,2,3,2,2,2,1,1,
2,2,2,2,3,3,3,2,2,2,2,
1,1,2,2,2,3,2,2,2,1,1,
1,1,1,1,2,2,2,1,1,1,1,
1,1,1,1,1,2,1,1,1,1,1), ncol=11, byrow=TRUE)
library(raster)
d <- raster(d)
plot(d, col=colorRampPalette(c("blue","yellow","red"))(255))
## Simulate 30% missing data:
set.seed(1)
d_m <- d
d_m[ sample(1:length(d), length(d)/3) ] <- NA
plot(d_m, col=colorRampPalette(c("blue","yellow","red"))(255))
# Forrest's solution:
filter <- matrix(1, nrow=3, ncol=3)
r <- focal(d_m, filter, mean, na.rm=T, NAonly=T, pad=T)
#an alterative:
rp <- rasterToPoints(d_m)
library(fields)
# thin plate spline interpolation
#(for a simple pattern like this, IDW might work, see ?interpolate)
tps <- Tps(rp[,1:2], rp[,3])
# predict
x <- interpolate(d_m, tps)
# use the orginal values where available
m <- cover(d_m, x)
i <- is.na(d_m)
cor(d[i], m[i])
## [1] 0.8846869
cor(d[i], r[i])
## [1] 0.8443165