krig                 package:funfits                 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:

     The kriging model is Y(x)= P(x) + Z(x) + e where Y is the
     dependent variable observed at location x, 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. If
     these parameters 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. 

     WARNING: The covariance functions often have a nonlinear parameter
     that  controls the strength of the correlations as a function of
     separation,  usually refered to as the range parameter. This
     parameter must be  specified in the call to krig and will not be
     estimated.

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

     krig(x, Y, cov.function, lambda=NA, cost=1, knots, 
     weights=rep(1, length(Y)), m=2, return.matrices=T, 
     nstep.cv=50, scale.type="user", x.center=rep(0, ncol(x)), 
     x.scale=rep(1, ncol(x)), rho=NA, sigma2=NA, ...)

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

       x: Matrix of independent variables. 

       Y: Vector of dependent variables. 

cov.function: Covariance function for data in the form of an S-PLUS
          function (see exp.cov).  

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

    cost: Cost value used in GCV criterion. Corresponds to a penalty
          for  increased number of parameters. 

   knots: Subset of data used in the fit. 

 weights: Weights are proportional to the reciprocal variance of the
          measurement  error. The default is no 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. 

return.matrices: Matrices from the decompositions are returned. The
          default is T.  

nstep.cv: Number of grid points for minimum GCV search. 

scale.type: The independent variables and knots are scaled to the
          specified scale.type. By default the scale type is "unit.sd",
          whereby the data is scaled to have mean 0 and standard
          deviation 1. 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 "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 are subtracted from each column of the x
          matrix. 

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

     rho: Scale factor for covariance. 

  sigma2: Variance of e. 

     ...: Optional arguments. Theta can be specified. If the
          cov.parameters are  specified this list is assumed to be
          arguments to the covariance  function. 

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

     A list of class krig. This includes the predicted surface of
     fitted.values and the residuals. The results of the grid search to
     minimize the generalized cross validation function is returned in
     gcv.grid.

    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 used in 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
          variance. 

    cost: Cost value used in GCV criterion. 

       m: Order of the polynomial space: highest degree polynomial is
          (m-1). 

  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: Esimated 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') 

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 FUNFITS manual.

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

     summary.krig, predict.krig, predict.se.krig, plot.krig,
     surface.krig

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

     #2-d example
     krig(ozone$x, ozone$y, exp.cov) -> fit # fitting a surface to ozone 
     # measurements.
     plot(fit) # plots fit and residuals
     # data using a Gaussian covariance
     # first make up covariance function
     test.cov <- function(x1,x2){exp(-(rdist(x1,x2)/.5)**2)}
     krig(flame$x, flame$y, test.cov) -> fit.flame
     surface(fit.flame)

