fields - tools for spatial data

fields is a collection of functions for curve and function fitting with an emphasis on spatial data and spatial statistics. It was developed over 20+ years to provide easy to use but sophisticated tools for analyzing spatial data, particularly that encountered in the environmental sciences. For the impatient users, jump to the examples below to see how easy this is use. Please send bugs and questions to Doug Nychka, nychka@mines.edu. Positive comments are also welcome!
The major methods implemented include cubic and thin plate splines, universal Kriging and Kriging for large data sets. A more modern framework for Kriging is spatial process estimation with covariance parameters determined by maximum likelihood and the uncertainty derived from assumptions of a Gaussian process. Throughout we try to include reasonable defaults in functions that reflect our experience with analyzing spatial data. For example, the Matern covariance function is the default choice for the main spatial method.
A key feature of this package is any covariance function implemented in R code can be used for spatial prediction through the spatial functions. Another important feature is that fields will take advantage of compactly supported covariance functions in a seamless way through the spam package. See library( help=fields) for a listing of all the fields contents. We also recommend the thoughtful vignette created by Ashton Weins, Mitchell Krock, and Emma Lilly (fieldsVignette.pdf) in the fields github repository.
fields strives to have readable and tutorial code. Take a look at the source code for mKrig to see how things work "under the hood". E.g. how a linear algebra computation is overloaded to handle sparse matrices and how an output object is built up sequentially throughout a computation.
The fields source code is liberally commented. Unfortunately on loading this package, R will strip comments from the source for efficiency. You can go to CRAN fields page to download the latest "tarball" ( aka Package Source) and unzip to get code with comments. We also keep the most recent version of this package at the fields github repository. and for commented source go to the the subdirectory fields/R. This may be a more recent version, however, than what is posted to CRAN.
Details

Major methods
spatialProcess An easy to use method that fits a spatial process model ( e.g. Kriging) but also estimates the key spatial parameters: nugget variance, sill variance and range parameter by maximum likelihood. The default covariance model is a Matern covariance. This function and related functions called by this are the core methods in fields and have much flexibility.

spatialProcess allows one to supply a covariance function that is written in native R code. See (stationary.cov) that includes several families of covariances including the Matern and several distance metrics including great circle distance . sim.spatialProcess and simLocal.spatialProcess provide "one liners" for conditional simulation of the fitted surface.
Tps Thin Plate spline regression including GCV and REML estimates for the smoothing parameter. For moderate size data sets as a first look we use Tps all the time. See also fastTps for an approximate method to handle very large numbers of spatial locations. Also see the help file for spatialProcess to see how to fit a thin plate plate using the more extensive set of spatial stats functions.
sreg , splint Fast 1-D cubic smoothing splines and interpolating splines, a workhorse algorithm for more EDA and more complicated methods.
mKrig (micro Krig) Efficient Universal Kriging and Gaussian process function, that can take advantage of sparse covariance functions and is the core algorithm called by optimization functions and for spatial predictio.
QTps A easy to use extension of thin plate splines for quantile and robust surface fitting.
mKrigMLEGrid and mKrigMLEJoint for maximum likelihood estimates of covariance parameters. These functions also handle replicate fields, assumed to be independent realizations, at the same locations and can also take any covariate function function written in R following the fields format
Other noteworthy functions
vgram and vgram.matrix find variograms for spatial data (and with temporal replications.
cover.design Generates space-filling designs where the distance function is expresed in R code.
There are many convenient functions for working with image data and rationally (well, maybe reasonably) creating and placing a color scale on plots. This suite of tools are for the users who want to extend the "base R grahics" and retain control over details. See the last amusing example in help(imagePlot) for an example. as.image, imagePlot, bubblePlot, drape.plot, quilt.plot add.image, crop.image, half.image, average.image, designer.colors, color.scale, in.poly See also grid.list for how fields works with grids and US and world for adding a map quickly.
Generic functions that support the methods
plot - diagnostic plots of fit
summary- statistical summary of fit
print- shorter version of summary
surface- graphical display of fitted surface
predict- evaluation fit at arbitrary points
predictSE- prediction standard errors at arbitrary points.
sim.rf- Simulate a random fields on a 2-d grid.
Getting Started

Try some of the examples from help files for spatialProcess or Tps.
Some fields datasets

CO2 Global satelite CO2 concentrations (simulated field)
COmonthlyMet Monthly mean temperatures and precip for Colorado
glacier An elevation dataset of a glacier also used by the applied math community to test interpolation methods.
lennon Image of John Lennon
NorthAmericanRainfall 50+ year average and trend for summer rainfall at 1700+ stations.
ozone2 Daily max 8 hour ozone concentrations for the US midwest for summer 1987.
PRISMelevation Digital elevations for the continental US at approximately 4km resolution
rat.diet Small paired study on rat food intake over time.
RCMexample Regional climate model output
RMelevation Digital elevations for the Rocky Mountain Empire
WorldBankCO2 Demographic and carbon emission data for 75 countries and for 1999.
DISCLAIMER:

The authors can not guarantee the correctness of any function or program in this package.
Examples

# some air quality data, daily surface ozone measurements for 
# the Midwest:

data(ozone2)
s<-ozone2$lon.lat
y<- ozone2$y[16,] # June 18, 1987 

# quick plot of spatial data with map
bubblePlot( s,y)
US( add=TRUE) # add US map


# fitting a thin plate spline surface (always a good place to 
# start). Here the  default smoothing (aka lambda) found by cross-validation
  fit0<- Tps(s,y)
# fits a GCV thin plate smoothing spline surface to ozone measurements.
# Hey, it does not get any easier than this!

  summary(fit0) #diagnostic summary of the fit 
#> CALL:
#> Tps(x = s, Y = y)
#>                                                  
#>  Number of Observations:                147      
#>  Number of unique points:               147      
#>  Number of parameters in the null space 3        
#>  Parameters for fixed spatial drift     3        
#>  Effective degrees of freedom:          74.6     
#>  Residual degrees of freedom:           72.4     
#>  MLE tau                                8.307    
#>  GCV tau                                9.276    
#>  MLE sigma                              958000   
#>  Scale passed for covariance (sigma)    <NA>     
#>  Scale passed for nugget (tau^2)        <NA>     
#>  Smoothing parameter lambda             7.204e-05
#> 
#> Residual Summary:
#>       min     1st Q    median     3rd Q       max 
#> -30.38000  -3.07900   0.04153   3.35100  20.65000 
#> 
#> Covariance Model: Rad.cov
#>   Names of non-default covariance arguments: 
#>        p
#> 
#> DETAILS ON SMOOTHING PARAMETER:
#>  Method used:   GCV    Cost:  1
#>    lambda       trA       GCV   GCV.one GCV.model    tauHat 
#> 7.204e-05 7.461e+01 1.747e+02 1.747e+02        NA 9.276e+00 
#> 
#>  Summary of all estimates found for lambda
#>               lambda   trA   GCV tauHat -lnLike Prof converge
#> GCV        7.204e-05 74.61 174.7  9.276        609.3        6
#> GCV.model         NA    NA    NA     NA           NA       NA
#> GCV.one    7.204e-05 74.61 174.7  9.276           NA        6
#> RMSE              NA    NA    NA     NA           NA       NA
#> pure error        NA    NA    NA     NA           NA       NA
#> REML       5.653e-04 41.20 188.3 11.643        600.9        5
  set.panel(2,2)
#> plot window will lay out plots in a 2 by 2 matrix 
  plot(fit0) # four diagnostic plots of fit and residuals.


# quick plot of predicted surface
  set.panel()
#> plot window will lay out plots in a 1 by 1 matrix 
  surface(fit0) # contour/image plot of the fitted surface
# see also predictSurface for more control over the evaluation grid
#
  US( add=TRUE, col="magenta", lwd=2) # US map overlaid
  title("Daily max 8 hour ozone in PPB,  June 18th, 1987")

####
  fit2<- spatialProcess( s,y)
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0.148289775837323, 0.0422438333108693, 0.0385124069595538, 0.208412708836621, 0.0234169767334396, 0.0181120659191808, 0.0221831515820746, 0.0214673906685065, 0.0120466817104608, 0.0127421734611369, 0.0124585020466162, 0.023155289594295, 0.0241842124642385, 0.0190285645816875, 0.0214687009057733, 0.0222164419408822, 0.0155296783032353, 0.0145212946265917, 0.0141770019645178, 0.0212528698037939, 0.0118884767194995, 0.0922515367232717, 0.0604189945113337, 0.0530197146375534, 0.0997339071883292, 0.211909464981588, 0.183440300695817, 0.0353276331296414, 0.312399920820965, 0.293340392806194, 0.0393281076653689, 0.0534087986165907, 0.0538728549367662, 0.0380900384269337, 0.036382232071515, 0.0346604304029978, 0.0447208172616078, 0.00542183201956858, 0.0424512666088325, 0.0329018726864422, 0.033768001491719, 0.0355760920052331, 0.0390829831290969, 0.0373533867140323, 0.0339253863083627, 0.0355606839804911, 0.0362614186668707, 0.0373253050630728, 0.0250041141463037, 0.0258195158797121, 0.0190601527075838, 0.0189092059995773, 0.0164482937193614, 0.0158047410190547, 0.0163336345003018, 0.0231314061643122, 0.0259292201583745, 0.0232755433143924, 0.0400414191654425, 0.0395430695174553, 0.0279578609527963, 0.0363375935990314, 0.0231495250953575, 0.697403351176605, 0.614443756850939, 0.652188514621153, 0.84242280111772, 0.934978805863661, 0.667292869606824, 0.492541793032429, 0.678444595513967, 0.263616092337356, 0.756539278702474, 0.640142121641741, 0.666664967096088, 0.627334404250739, 0.673530863944187, 0.589516637189421, 0.950885009232243, 0.720849307436018, 0.528959405072991, 0.496996439181866, 0.582379407957787, 0.745260575197119, 0.782765171342818, 0.667035826388711, 1.0001), dim = c(147L, 147L)), pivot = FALSE): the leading minor of order 39 is not positive
# a "Kriging" model. The covariance defaults to a Matern 
# with smoothness 1.0.
# the nugget, sill and range parameters are found by maximum likelihood
# summary, plot, and surface also work for \code{fit2} !

  surface(fit2) # contour/image plot of the fitted surface
#> Error: object 'fit2' not found
  US( add=TRUE, col="magenta", lwd=2) # US map overlaid
  title("Daily max 8 hour ozone in PPB,  June 18th, 1987")

if (FALSE) { # \dontrun{
# And 20 approximate conditional draws of the spatial field on a grid
# with uncertainty in the 120PPB contour 
   look<- simLocal.spatialProcess(fit2, M=20)
for( k in 1:20){
contour( look$x, look$y, look$z[,,k], add=TRUE, level=c(120),
  col="white",  drawlabels=FALSE)
}

} # }