(co)kriging cross validation, n-fold or leave-one-out

Cross validation functions for simple, ordinary or universal point (co)kriging, kriging in a local neighbourhood.

Usage

gstat.cv(object, nfold, remove.all = FALSE, verbose = interactive(), 
  all.residuals = FALSE, ...)
krige.cv(formula, locations, ...)
krige.cv.locations(formula, locations, data, model = NULL, ..., beta = NULL, 
  nmax = Inf, nmin = 0, maxdist = Inf, nfold = nrow(data), 
  verbose = interactive(), debug.level = 0)
krige.cv.spatial(formula, locations, model = NULL, ..., beta = NULL, 
  nmax = Inf, nmin = 0, maxdist = Inf, nfold = nrow(locations), 
  verbose = interactive(), debug.level = 0)

Arguments

object: object of class gstat; see function gstat
nfold: integer; if larger than 1, then apply n-fold cross validation; if nfold equals nrow(data) (the default), apply leave-one-out cross validation; if set to e.g. 5, five-fold cross validation is done. To specify the folds, pass an integer vector of length nrow(data) with fold indexes.
remove.all: logical; if TRUE, remove observations at cross validation locations not only for the first, but for all subsequent variables as well
verbose: logical; if FALSE, progress bar is suppressed
all.residuals: logical; if TRUE, residuals for all variables are returned instead of for the first variable only
...: other arguments that will be passed to predict in case of gstat.cv, or to gstat in case of krige.cv
formula: formula that defines the dependent variable as a linear model of independent variables; suppose the dependent variable has name z, for ordinary and simple kriging use the formula z~1; for simple kriging also define beta (see below); for universal kriging, suppose z is linearly dependent on x and y, use the formula z~x+y
locations: data object deriving from class Spatial or sf
data: data frame (deprecated); should contain the dependent variable, independent variables, and coordinates; only to be provided if locations is a formula
model: variogram model of dependent variable (or its residuals), defined by a call to vgm or fit.variogram
beta: only for simple kriging (and simulation based on simple kriging); vector with the trend coefficients (including intercept); if no independent variables are defined the model only contains an intercept and this should be the simple kriging mean
nmax: for local kriging: the number of nearest observations that should be used for a kriging prediction or simulation, where nearest is defined in terms of the space of the spatial locations. By default, all observations are used
nmin: for local kriging: if the number of nearest observations within distance maxdist is less than nmin, a missing value will be generated; see maxdist
maxdist: for local kriging: only observations within a distance of maxdist from the prediction location are used for prediction or simulation; if combined with nmax, both criteria apply
debug.level: print debugging information; 0 suppresses debug information

Methods

formula = "formula", locations = "formula": locations specifies which coordinates in data refer to spatial coordinates
formula = "formula", locations = "Spatial": Object locations knows about its own spatial locations

Details

Leave-one-out cross validation (LOOCV) visits a data point, and predicts the value at that location by leaving out the observed value, and proceeds with the next data point. (The observed value is left out because kriging would otherwise predict the value itself.) N-fold cross validation makes a partitions the data set in N parts. For all observation in a part, predictions are made based on the remaining N-1 parts; this is repeated for each of the N parts. N-fold cross validation may be faster than LOOCV.

Value

data frame containing the coordinates of data or those of the first variable in object, and columns of prediction and prediction variance of cross validated data points, observed values, residuals, zscore (residual divided by kriging standard error), and fold.

If all.residuals is true, a data frame with residuals for all variables is returned, without coordinates.

References

http://www.gstat.org/

Author

Edzer Pebesma

Note

Leave-one-out cross validation seems to be much faster in plain (stand-alone) gstat, apparently quite a bit of the effort is spent moving data around from R to gstat.

Examples

library(sp)
data(meuse)
coordinates(meuse) <- ~x+y
m <- vgm(.59, "Sph", 874, .04)
# five-fold cross validation:
x <- krige.cv(log(zinc)~1, meuse, m, nmax = 40, nfold=5)
bubble(x, "residual", main = "log(zinc): 5-fold CV residuals")


# multivariable; thanks to M. Rufino:
meuse.g <- gstat(id = "zn", formula = log(zinc) ~ 1, data = meuse)
meuse.g <- gstat(meuse.g, "cu", log(copper) ~ 1, meuse)
meuse.g <- gstat(meuse.g, model = vgm(1, "Sph", 900, 1), fill.all = TRUE)
x <- variogram(meuse.g, cutoff = 1000)
meuse.fit = fit.lmc(x, meuse.g)
out = gstat.cv(meuse.fit, nmax = 40, nfold = 5) 
#> Linear Model of Coregionalization found. Good.
#> [using ordinary cokriging]
#> Linear Model of Coregionalization found. Good.
#> [using ordinary cokriging]
#> Linear Model of Coregionalization found. Good.
#> [using ordinary cokriging]
#> Linear Model of Coregionalization found. Good.
#> [using ordinary cokriging]
#> Linear Model of Coregionalization found. Good.
#> [using ordinary cokriging]
summary(out)
#> Object of class SpatialPointsDataFrame
#> Coordinates:
#>      min    max
#> x 178605 181390
#> y 329714 333611
#> Is projected: NA 
#> proj4string : [NA]
#> Number of points: 155
#> Data attributes:
#>     zn.pred          zn.var           observed        residual        
#>  Min.   :4.660   Min.   :0.04106   Min.   :4.727   Min.   :-1.046096  
#>  1st Qu.:5.314   1st Qu.:0.05054   1st Qu.:5.288   1st Qu.:-0.133537  
#>  Median :5.775   Median :0.05452   Median :5.787   Median : 0.006930  
#>  Mean   :5.882   Mean   :0.05632   Mean   :5.886   Mean   : 0.003514  
#>  3rd Qu.:6.403   3rd Qu.:0.06034   3rd Qu.:6.514   3rd Qu.: 0.160171  
#>  Max.   :7.687   Max.   :0.09869   Max.   :7.517   Max.   : 0.518525  
#>      zscore              fold      
#>  Min.   :-4.22112   Min.   :1.000  
#>  1st Qu.:-0.53431   1st Qu.:2.000  
#>  Median : 0.03024   Median :3.000  
#>  Mean   : 0.01530   Mean   :2.942  
#>  3rd Qu.: 0.67320   3rd Qu.:4.000  
#>  Max.   : 2.45696   Max.   :5.000  
out = gstat.cv(meuse.fit, nmax = 40, nfold = c(rep(1,100), rep(2,55))) 
#> Linear Model of Coregionalization found. Good.
#> [using ordinary cokriging]
#> Linear Model of Coregionalization found. Good.
#> [using ordinary cokriging]
summary(out)
#> Object of class SpatialPointsDataFrame
#> Coordinates:
#>      min    max
#> x 178605 181390
#> y 329714 333611
#> Is projected: NA 
#> proj4string : [NA]
#> Number of points: 155
#> Data attributes:
#>     zn.pred          zn.var           observed        residual       
#>  Min.   :4.856   Min.   :0.04393   Min.   :4.727   Min.   :-1.52437  
#>  1st Qu.:5.382   1st Qu.:0.05944   1st Qu.:5.288   1st Qu.:-0.27045  
#>  Median :5.903   Median :0.07107   Median :5.787   Median :-0.09095  
#>  Mean   :5.966   Mean   :0.07283   Mean   :5.886   Mean   :-0.07990  
#>  3rd Qu.:6.428   3rd Qu.:0.08454   3rd Qu.:6.514   3rd Qu.: 0.14914  
#>  Max.   :7.734   Max.   :0.10621   Max.   :7.517   Max.   : 0.55106  
#>      zscore             fold      
#>  Min.   :-4.7612   Min.   :1.000  
#>  1st Qu.:-0.9359   1st Qu.:1.000  
#>  Median :-0.3354   Median :1.000  
#>  Mean   :-0.2678   Mean   :1.355  
#>  3rd Qu.: 0.6001   3rd Qu.:2.000  
#>  Max.   : 2.3860   Max.   :2.000  
# mean error, ideally 0:
mean(out$residual)
#> [1] -0.07990434
# MSPE, ideally small
mean(out$residual^2)
#> [1] 0.1101475
# Mean square normalized error, ideally close to 1
mean(out$zscore^2)
#> [1] 1.487834
# correlation observed and predicted, ideally 1
cor(out$observed, out$observed - out$residual)
#> [1] 0.8955611
# correlation predicted and residual, ideally 0
cor(out$observed - out$residual, out$residual)
#> [1] -0.1103516