Krig                 package:fields                 R Documentation

_K_r_i_g_i_n_g _s_u_r_f_a_c_e _e_s_t_i_m_a_t_e

_D_e_s_c_r_i_p_t_i_o_n:

     Fits a surface to irregularly spaced data. The Kriging model
     assumes   that the unknown  function is a realization  of a
     Gaussian  random spatial processes. The assumed model is additive 
     Y = P(x) +  Z(X) + e, where P is a low order polynomial and Z is a
      mean zero,  Gaussian stochastic process with a   covariance that
     is unknown up to a scale constant. The main advantages of this
     function are the flexibility in specifying the covariance as an R
     language function and also the supporting functions plot, predict,
     predict.se, surface for subsequent analysis. Krig also supports a
     correlation model where the mean and marginal variances are
     supplied.

_U_s_a_g_e:

     Krig(
     x, Y, cov.function = "stationary.cov", lambda = NA, df
                      = NA, GCV=FALSE, Z = NULL, cost = 1, knots = NA, weights = NULL,
                      m = 2, nstep.cv = 80, scale.type = "user", x.center =
                      rep(0, ncol(x)), x.scale = rep(1, ncol(x)), rho = NA,
                      sigma2 = NA, method = "GCV", verbose = FALSE, mean.obj
                      = NA, sd.obj = NA, null.function =
                      "Krig.null.function", wght.function = NULL, offset =
                      0, outputcall = NULL, na.rm = TRUE, cov.args = NULL,
                      chol.args = NULL, null.args = NULL, wght.args = NULL,
                      W = NULL, give.warnings = TRUE, ...)

     ## S3 method for class 'Krig':
     fitted(object,...)

     ## S3 method for class 'Krig':
     coef(object,...)

     resid.Krig(object,...)

_A_r_g_u_m_e_n_t_s:

       x: Matrix of independent variables. These could the locations
          for spatial data or the indepedent variables in a regression.            

       Y: Vector of dependent variables. These are the values of the
          surface (perhaps with measurement error) at the locations or
          the dependent response in a regression. 

cov.function: Covariance function for data in the form of an R function
          (see  Exp.simple.cov as an example).   Default assumes that
          correlation is an exponential function of distance. See also
          'stationary.cov' for more general choice of covariance 
          shapes. 'exponential.cov' will be faster if only the
          exponential  covariance form is needed.  

       Z: A vector of matrix of covariates to be include in the fixed
          part of the  model. If NULL (default) no addtional covariates
          are included.

  lambda: Smoothing parameter that is the ratio of the error variance
          (sigma**2)  to the scale parameter of the   covariance
          function (rho). If omitted this is estimated by GCV ( see
          method below).  

      df: The effective number of parameters for the fitted surface.
          Conversely,  N- df, where N is the total number of
          observations is the degrees of freedom associated with the
          residuals.  This is an alternative to specifying lambda and
          much more interpretable. NOTE: GCV argument defaults to TRUE
          if this argument is used. 

     GCV: If TRUE matrix decompositions are done to allow estimating 
          lambda by GCV or REML and specifying smoothness by the
          effective degrees of freedom. So the GCV switch does more
          than just supply a GCV estimate. Also if lambda or df are
          passed the estimate will be evaluated at those values,  not
          at the GCV/REML estimates of lambda.  If FALSE Kriging 
          estimate is found under a fixed lambda model. 

    cost: Cost value used in GCV criterion. Corresponds to a penalty
          for   increased number of parameters. The default is 1.0 and
          corresponds to the usual GCV function.  

   knots: A matrix of locations similar to x. These can define an
          alternative set of basis functions for representing the
          estimate. One choice may be a space-filling subset of the
          original x locations, thinning out the design where locations
          cluster. The  default is to put a "knot" at all unique
          locations. (See details.) 

 weights: Weights are proportional to the reciprocal variance of the
          measurement   error. The default is equal weighting i.e.
          vector of unit weights.  

       m: A polynomial function of degree (m-1) will be   included in
          the model as the drift (or spatial trend) component.  The "m"
          notation is from thin-plate splines where m is the 
          derivative in the penalty function. With m=2 as the default a
          linear  model in the locations  will be fit a fixed part of
          the model.  

nstep.cv: Number of grid points for the coarse grid search  to 
          minimize the GCV  RMLE and other related criterian for
          finding lambda.  

scale.type: This is a character string among: "range", "unit.sd",
          "user", "unscaled". The independent variables and knots are
          scaled to the specified scale.type.  By default no scaling is
          done. This usuall makes sense for spatial locations. Scale
          type of "range" scales the data to the interval (0,1) by
          forming  (x-min(x))/range(x) for each x. Scale type of
          "unit.sd"  Scale type of "user" allows specification of an
          x.center and x.scale by the  user. The default for "user" is
          mean 0 and standard deviation 1. Scale  type of "unscaled"
          does not scale the data.  

x.center: Centering values to be subtracted from each column of the x
          matrix.  

 x.scale: Scale values that are divided into each column after
          centering.  

     rho: Scale factor for covariance.  

  sigma2: Variance of the errors, often called the nugget variance. If
          weights are specified then the error variance is sigma2
          divided by weights.  Note that lambda is defined as the ratio
           sigma2/rho. 

  method: Determines what "smoothing" parameter should be used. The
          default  is to estimate standard GCV  Other choices are:
          GCV.model, GCV.one, RMSE, pure error and REML. The 
          differences are explained below.   

 verbose: If true will print out all kinds of intermediate stuff.
          Default is false, of course as this is used mainly for
          debugging. 

mean.obj: Object to predict the mean of the spatial process. This used
          in when fitting a correlation model with varying spatial
          means and varying marginal variances. (See details.) 

  sd.obj: Object to predict the marginal standard deviation of the
          spatial process.  

null.function: An R function that creates the matrices for the null
          space model.   The default is fields.mkpoly, an R function
          that creates a polynomial  regression matrix with all terms
          up to degree m-1. (See Details)   

wght.function: An R function that creates a weights matrix to the
          observations.   This is only needed if the weight matirx has
          off diagonal elements.  The default is NULL indicating that
          the weight matrix is a diagonal, based on the weights
          argument. (See details) 

  offset: The offset to be used in the GCV criterion. Default is 0.
          This would be  used when Krig is part of a backfitting
          algorithm and the offset is  other model degrees of freedom
          from other regression components.  

outputcall: If NULL the output object will have a $call argument based
          on this call.  If no NULL the output call will have whatever
          is passed. This is kludge for the Tps function so that it
          return a Krig object but have the right call argument. Sorry
          no one promised that fields would be pretty.  

cov.args: A list with the arguments to call the covariance function.
          (in addition to the locations) 

   na.rm: If TRUE NAs will be removed from the 'y' vector and the 
          corresponding rows of 'x' - with a warning.  If FALSE Krig
          will just stop with a message. Once NAs are removed all 
          subsequent analysis in fields does not use those data. 

chol.args: Arguments to be passed to the cholesky decomposition in
          Krig.engine.fixed.  The default if NULL, assigned at the top
          level of this function, is  list( pivot=FALSE). This argument
          is useful when working with  the sparse matrix package. 

wght.args: Optional arguments to be passed to the weight function 
          (wght.function)  used to create the observation weight
          matrix.

       W: The explicit observatoin weight matrix.

null.args: Extra arguments for the null space function 
          'null.function'.  If 'fields.mkpoly' is passed as 
          'null.function' then this is set to a list with the value of
          'm'. So the default is use a polynomial of degree m-1 for the
          null space (fixed part) of the model.  

give.warnings: If TRUE warnings are given in gcv grid search limits. If
          FALSE warnings are not given. Best to leave this TRUE!

     ...: Optional arguments that appear are assumed to be additional
          arguments to the covariance function. Or are included in
          methods functions (resid, fitted, coef) as a required
          argument.

  object: A Krig object

_D_e_t_a_i_l_s:

     This function produces a object of class Krig. With this object it
     is easy to subsequently predict with this fitted surface, find
     standard errors, alter the y data ( but not x), etc. 

     The Kriging model is: Y.k= P(x.k) + Z(x.k) + e.k

     where ".k" means subscripted by k, Y is the dependent variable
     observed at location x.k, P is a low order polynomial, Z is a mean
     zero, Gaussian field with covariance function K and e is assumed
     to be independent normal errors. The estimated surface is the best
     linear unbiased estimate (BLUE) of P(x) + Z(x) given the observed
     data. For this estimate K, is taken to be rho*cov.function and the
     errors have variance sigma**2. In more conventional geostatistical
     terms rho is the "sill" if the covariance function is actually a
     correlation function and sigma**2 is the nugget variance or
     measure error variance (the two are confounded in this model.)  If
     the weights are given then the variance of e.k is sigma**2/
     weights.k . In the case that the weights are specified as a
     matrix, W,  using the wght.function option then the assumed
     covariance matrix for the errors is sigma**2 Wi, where Wi is the
     inverse of W. 

     If these parameters rho and sigma2 are omitted in the call, then
     they are estimated in the following way. If lambda is given, then
     sigma2 is estimated from the residual sum of squares divided by
     the degrees of freedom associated with the residuals.  Rho is
     found as the difference between the sums of squares of the
     predicted values having subtracted off the polynomial part and
     sigma2.  

     A useful extension of a stationary correlation to a nonstationary
     covariance is what we term a correlation model.  If mean and
     marginal standard deviation objects are included in the call.  
     Then the observed data is standardized based on these functions. 
     The spatial process is then estimated with respect to the
     standardized scale. However for predictions and standard errors
     the mean and standard deviation surfaces are used to produce
     results in the original scale of the observations.

     The GCV function has several alternative definitions when
     replicate observations are present or if one uses a reduced set
     knots.  Here are the choices based on the method argument:  

     GCV: leave-one-out GCV. But if there are replicates it is leave
     one group out. (Wendy and Doug prefer this one.)  

     GCV.one: Really leave-one-out GCV even if there are replicate
     points.  This what the old tps function used in FUNFITS.

     rmse: Match the estimate of sigma**2 to a external value ( called
     rmse)  

     pure error: Match the estimate of sigma**2 to the estimate based
     on replicated data (pure error estimate in ANOVA language).  

     GCV.model: Only considers the residual sums of squares explained
     by the basis functions.  

     WARNING: The covariance functions often have a nonlinear
     parameter(s) that often control the strength of the correlations
     as a function of separation, usually referred to as the range
     parameter. This parameter must be specified in the call to Krig
     and will not be estimated.

_V_a_l_u_e:

     A object of class Krig. This includes the predicted values in  
     fitted.values and the residuals in residuals. The results of the
     grid  search to minimize the generalized cross validation function
     are  returned in gcv.grid. 

     The coef.Krig function only returns the coefficients, "d",
     associated with the  fixed part of the model (also known as the
     null space or spatial drift).

    call: Call to the function  

       y: Vector of dependent variables.  

       x: Matrix of independent variables.  

 weights: Vector of weights.  

   knots: Locations used to define the basis functions.   

transform: List of components used in centering and scaling data.  

      np: Total number of parameters in the model.  

      nt: Number of parameters in the null space.  

matrices: List of matrices from the decompositions (D, G, u, X, qr.T).  

gcv.grid: Matrix of values from the GCV grid search. The first column 
          is the grid of lambda values used in the search, the second
          column   is the trace of the A matrix, the third column is
          the GCV values and  the fourth column is the estimated value
          of sigma conditional on the vlaue of lambda.   

lambda.est: A table of estimated smoothing parameters with
          corresponding degrees  of freedom and estimates of sigma
          found by different methods.   

    cost: Cost value used in GCV criterion.  

       m: Order of the polynomial space: highest degree polynomial is
          (m-1).  This is a fixed part of the surface often referred to
          as the drift  or spatial trend.   

  eff.df: Effective degrees of freedom of the model.  

fitted.values: Predicted values from the fit.  

residuals: Residuals from the fit.  

  lambda: Value of the smoothing parameter used in the fit.  

   yname: Name of the response.  

cov.function: Covariance function of the model.  

    beta: Estimated coefficients in the ridge regression format  

       d: Estimated coefficients for the polynomial basis functions
          that span the  null space  

fitted.values.null: Fitted values for just the polynomial part of the
          estimate  

   trace: Effective number of parameters in model.  

       c: Estimated coefficients for the basis functions derived from
          the  covariance.  

coefficients: Same as the beta vector.  

just.solve: Logical describing if the data has been interpolated using
          the basis   functions.   

    shat: Estimated standard deviation of the measurement error (nugget
          effect).  

  sigma2: Estimated variance of the measurement error (shat**2).  

     rho: Scale factor for covariance.  COV(h(x),h(x')) =
          rho*cov.function(x,x')  If the covariance is actually a 
          correlation function then rho is also the "sill".  

mean.var: Normalization of the covariance function used to find rho.  

best.model: Vector containing the value of lambda, the estimated
          variance of the   measurement error and the scale factor for
          covariance used in the fit.  

_R_e_f_e_r_e_n_c_e_s:

     See "Additive Models" by Hastie and Tibshirani, "Spatial
     Statistics" by     Cressie and the FIELDS manual.

_S_e_e _A_l_s_o:

     summary.Krig, predict.Krig, predict.se.Krig, predict.surface.se,
     predict.surface, plot.Krig, surface.Krig

_E_x_a_m_p_l_e_s:

     # a 2-d example 
     # fitting a surface to ozone  
     # measurements. Exponential covariance, range parameter is 20 (in miles) 

     fit <- Krig(ozone$x, ozone$y, theta=20)  
      
     summary( fit) # summary of fit 
     set.panel( 2,2) 
     plot(fit) # four diagnostic plots of fit  
     set.panel()
     surface( fit, type="C") # look at the surface 

     # predict at data
     predict( fit)

     # predict using 7.5 effective degrees of freedom:
     predict( fit, df=7.5)

     # predict on a grid ( grid chosen here by defaults)
      out<- predict.surface( fit)
      surface( out, type="C") # option "C" our favorite

     # predict at arbitrary points (10,-10) and (20, 15)
      xnew<- rbind( c( 10, -10), c( 20, 15))
      predict( fit, xnew)

     # standard errors of prediction based on covariance model.  
      predict.se( fit, xnew)

     # surface of standard errors on a default grid
      predict.surface.se( fit)-> out.p # this takes some time!
      surface( out.p, type="C")
      points( fit$x)

     # Using anohter stationary covariance. 
     # smoothness is the shape parameter for the Matern. 

     fit <- Krig(ozone$x, ozone$y, Covariance="Matern", theta=10, smoothness=1.0)  
     summary( fit)

     #
     # Roll your own: creating very simple user defined Gaussian covariance 
     #

     test.cov <- function(x1,x2,theta,marginal=FALSE,C=NA){
        # return marginal variance
          if( marginal) { return(rep( 1, nrow( x1)))}

         # find cross covariance matrix     
           temp<- exp(-(rdist(x1,x2)/theta)**2)
           if( is.na(C[1])){
               return( temp)}
           else{
               return( temp%*%C)}
           } 
     #
     # use this and put in quadratic polynomial fixed function 

      fit.flame<- Krig(flame$x, flame$y, cov.function="test.cov", m=3, theta=.5)

     #
     # note how range parameter is passed to Krig.   
     # BTW:  GCV indicates an interpolating model (nugget variance is zero) 
     #

     # take a look ...
      surface(fit.flame, type="I") 

     # 
     # Thin plate spline fit to ozone data using the radial 
     # basis function as a generalized covariance function 
     #
     # p=2 is the power in the radial basis function (with a log term added for 
     # even dimensions)
     # If m is the degree of derivative in penalty then p=2m-d 
     # where d is the dimension of x. p must be greater than 0. 
     #  In the example below p = 2*2 - 2 = 2  
     #

      out<- Krig( ozone$x, ozone$y,cov.function="Rad.cov", 
                            m=2,p=2,scale.type="range") 

     # See also the Fields function Tps
     # out  should be identical to  Tps( ozone$x, ozone$y)
     # 

     # A Knot example

      data(ozone2)
      y16<- ozone2$y[16,] 

     # there are some missing values -- remove them 
      good<- !is.na( y16)
      y<- y16[good] 
      x<- ozone2$lon.lat[ good,]

     #
     # the knots can be arbitrary but just for fun find them with a space 
     # filling design. Here we select  50 from the full set of 147 points
     #
      xknots<- cover.design( x, 50, num.nn= 75)$design  # select 50 knot points

      out<- Krig( x, y, knots=xknots,  cov.function="Exp.cov", theta=300)  
      summary( out)
     # note that that trA found by GCV is around 17 so 50>17  knots may be a 
     # reasonable approximation to the full estimator. 
     #

     # the plot 
      surface( out, type="C")
      US( add=TRUE)
      points( x, col=2)
      points( xknots, cex=2, pch="O")


     # A correlation model example

     # fit krig surface using a mean and sd function to standardize 
     # first get stats from 1987 summer Midwest O3 data set 
     # Compare the function Tps to the call to Krig given above 
     # fit tps surfaces to the mean and sd  points.  
     # (a shortcut is being taken here just using the lon/lat coordinates) 

      data(ozone2)
      stats.o3<- stats( ozone2$y)
      mean.o3<- Tps( ozone2$lon.lat, c( stats.o3[2,]))
      sd.o3<- Tps(  ozone2$lon.lat, c( stats.o3[3,]))

     #
     # Now use these to fit particular day ( day 16) 
     # and use great circle distance
     #NOTE: there are some missing values for day 16. 

      fit<- Krig( ozone2$lon.lat, y16, 
                 theta=350, mean.obj=mean.o3, sd.obj=sd.o3, 
                 Covariance="Matern", Distance="rdist.earth",
                 smoothness=1.0,
                 na.rm=TRUE) #

     # the finale
      surface( fit, type="I")
      US( add=TRUE)
      points( fit$x)
      title("Estimated ozone surface")

     #
     #
     # explore some different values for the range and lambda using REML
      theta <- seq( 300,400,,10)
      PLL<- matrix( NA, 10,80)
     # the loop 
      for( k in 1:10){

     # call to Krig with different ranges
     # also turn off warnings for GCV search 
     # to avoid lots of messages. (not recommended in general!)

       PLL[k,]<- Krig( ozone2$lon.lat[good,], y16[good],
                  cov.function="stationary.cov", 
                  theta=theta[k], mean.obj=mean.o3, sd.obj=sd.o3, 
                  Covariance="Matern",smoothness=.5, 
                  Distance="rdist.earth", nstep.cv=80,
                  give.warnings=FALSE)$gcv.grid[,7]
       
     #
     # gcv.grid is the grid search output from 
     # the optimization for estimating different estimates for lambda including 
     # REML
     # default grid is equally spaced in eff.df scale ( and should the same across theta)
     #  here 

      }

     # see the 2 column of $gcv.grid to get the effective degress of freedom. 

      cat( "all done", fill=TRUE)
      contour( theta, 1:80, PLL)
      

