Tps                  package:fields                  R Documentation

_T_h_i_n _p_l_a_t_e _s_p_l_i_n_e _r_e_g_r_e_s_s_i_o_n

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

     Fits a thin plate spline surface to irregularly spaced data. The 
     smoothing parameter is chosen by generalized cross-validation. The
     assumed  model is additive  Y = f(X) +e  where f(X) is a d
     dimensional surface.  This is a special case of the spatial
     process estimate.  A "fast" version of this function uses a
     compactly supported Wendland covariance and computes the estimate
     for a fixed smoothing pararmeter.

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

     Tps(x, Y, m = NULL, p = NULL, scale.type = "range", ...)

     fastTps(x, Y, m = NULL, p = NULL,  theta,  ...)

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

     To be helpful, a more complete list of arguments are described
     that are the  same as those for the Krig function.  

       x: Matrix of independent variables. Each row is a location or a
          set of  independent covariates.  

       Y: Vector of dependent variables.  

       m: A polynomial function of degree (m-1) will be   included in
          the model as the drift (or spatial trend) component.  Default
          is the value such that 2m-d is greater than zero where d is
          the  dimension of x.  

       p: Exponent for radial basis functions. Default is 2m-d.  

scale.type: The independent variables and knots are scaled to the
          specified  scale.type.  By default the scale type is "range",
          whereby  the locations are transformed   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.   

   theta: The tapering range that is passed to the Wendlend compactly 
          supported covariance. The covariace (i.e. the radial bais
          function) is  zero beyond range theta.

     ...: Any argument that is valid for the Krig function. Some of the
          main ones are listed below. 



          _l_a_m_b_d_a 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. 


          _d_f 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.


          _c_o_s_t 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.


          _w_e_i_g_h_t_s Weights are proportional to the reciprocal variance
               of the measurement   error. The default is no weighting
               i.e. vector of unit weights. 

          _n_s_t_e_p._c_v Number of grid points for minimum GCV search. 

          _x._c_e_n_t_e_r Centering values are subtracted from each column of
               the x matrix.  Must  have scale.type="user".

          _x._s_c_a_l_e Scale values that divided into each column after
               centering.  Must  have scale.type="user".

          _r_h_o Scale factor for covariance. 

          _s_i_g_m_a_2 Variance of errors or if weights are not equal to 1
               the variance is sigma**2/weight. 


          _m_e_t_h_o_d 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.

          _v_e_r_b_o_s_e If true will print out all kinds of intermediate
               stuff.  

          _m_e_a_n._o_b_j Object to predict the mean of the spatial process. 

          _s_d._o_b_j Object to predict the marginal standard deviation of
               the spatial process. 


          _n_u_l_l._f_u_n_c_t_i_o_n 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)


          _o_f_f_s_e_t The offset to be used in the GCV criterion. Default is
               0. This would be  used when Krig/Tps is part of a
               backfitting algorithm and the offset has to be included 
               to reflect other model degrees of freedom. 


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

     A thin plate spline is result of minimizing the residual sum of 
     squares subject to a constraint that the function have a certain 
     level of smoothness (or roughness penalty). Roughness is 
     quantified by the integral of squared m-th order derivatives. For
     one  dimension and m=2 the roughness penalty is the integrated
     square of  the second derivative of the function. For two
     dimensions the  roughness penalty is the integral of  

     (Dxx(f))**22 + 2(Dxy(f))**2 + (Dyy(f))**22 

     (where Duv denotes the second partial derivative with respect to u
      and v.) Besides controlling the order of the derivatives, the
     value of  m also determines the base polynomial that is fit to the
     data.  The degree of this polynomial is (m-1). 

     The smoothing parameter controls the amount that the data is 
     smoothed. In the usual form this is denoted by lambda, the
     Lagrange  multiplier of the minimization problem. Although this is
     an awkward  scale, lambda =0 corresponds to no smoothness
     constraints and the data  is interpolated.  lambda=infinity
     corresponds to just fitting the  polynomial base model by ordinary
     least squares.  

     This estimator is implemented by passing the right generalized
     covariance function based on radial basis functions to the more
     general function Krig.  One advantage of this implementation is
     that once a Tps/Krig object is created the estimator can be found
     rapidly for other data and smoothing parameters provided the
     locations remain unchanged. This makes simulation within R
     efficient (see example below). Tps does not currenty support the
     knots argument where one can use a reduced set of basis functions.
     This is mainly to simplify and a good alternative using knots
     would be to use a valid covariance from the Matern family and a
     large range parameter. 

     See also the mKrig function for handling larger data sets and also
     for an example of combining Tps and mKrig for evaluation on a
     larger grid. 

     The function 'fastTps' is really a convenient wrapper function
     that calls 'mKrig' with the Wendland covariance function. This is
     experimental  and some care needs to exercised in specifying the
     taper range and power ( 'p') which describes the behavior of the
     Wendland at the  origin. Note that unlike Tps the locations are
     not scaled to unit range and this can cause havoc in smoothing
     problems with variables in  very different units. So  ' x<- 
     scale(x)' might be a good idea for putting the  varaibles on a
     common sacel for smoothing.  This function does have the potential
     to approximate estimates of  Tps for very large spatial data sets.
     See 'wendland.cov' and  help on the SPAM package for more
     background.

_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
     minimizing the generalized cross validation function is  returned
     in gcv.grid. Note that the GCV/REML optimization is  done even if
     lambda or df is given.  Please see the documentation on Krig for
     details of the returned  arguments.

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

     See "Nonparametric Regression and Generalized Linear Models"   by
     Green and Silverman.  See "Additive Models" by Hastie and
     Tibshirani.

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

     Krig, summary.Krig, predict.Krig, predict.se.Krig, plot.Krig,
     mKrig   'surface.Krig',  'sreg'

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

     #2-d example 

     fit<- Tps(ozone$x, ozone$y)  # fits a surface to ozone measurements. 

     set.panel(2,2)
     plot(fit) # four diagnostic plots of  fit and residuals. 
     set.panel()

     summary(fit)

     # NOTE: the predict function is quite flexible:

          look<- predict( fit, lambda=2.0)
     #  evaluates the estimate at lambda =2.0  _not_ the GCV estimate
     #  it does so very efficiently from the Krig fit object.

          look<- predict( fit, df=7.5)
     #  evaluates the estimate at the lambda values such that 
     #  the effective degrees of freedom is 7.5
      

     # compare this to fitting a thin plate spline with 
     # lambda chosen so that there are 7.5 effective 
     # degrees of freedom in estimate
     # Note that the GCV function is still computed and minimized
     # but the spline estimate is uses 7.5 df.

     fit1<- Tps(ozone$x, ozone$y,df=7.5)

     set.panel(2,2)
     plot(fit1) # four diagnostic plots of  fit and residuals.
               # GCV function (lower left) has vertical line at 7.5 df.
     set.panel()

     # The basic matrix decompositions are the same for 
     # both fit and fit1 objects. 

     # predict( fit1) is the same as predict( fit, df=7.5)
     # predict( fit1, lambda= fit$lambda) is the same as predict(fit) 

     # predict onto a grid that matches the ranges of the data.  

     out.p<-predict.surface( fit)
     image( out.p) 
     surface(out.p) # perspective and contour plots of GCV spline fit 
     # predict at different effective 
     # number of parameters 
     out.p<-predict.surface( fit,df=10)


     #1-d example 
     out<-Tps( rat.diet$t, rat.diet$trt) # lambda found by GCV 
     plot( out$x, out$y)
     lines( out$x, out$fitted.values)

     # 
     # compare to the ( much faster) one spline algorithm 
     #  sreg(rat.diet$t, rat.diet$trt) 
     # 

     # Adding a covariate to the fixed part of model
     # Note: this is a fairly big problem numerically (850+ locations)

     Tps( RMprecip$x,RMprecip$y, Z=RMprecip$elev)-> out

     surface( out, drop.Z=TRUE)
     US( add=TRUE, col="grey")

     ### 
     ### fast Tps

     # m=2   p= 2m-d= 2
     #
     # Note: theta =3 degrees is a very generous taper range. 
     # Use trials with rdist.nearest to sort an efficient tapre range 
     # for large spatial problems 

     fastTps( RMprecip$x,RMprecip$y,m=2,lambda= 1e-2, p=2, theta=3.0) -> out2



     #
     # simulation reusing Tps/Krig object
     #
     fit<- Tps( rat.diet$t, rat.diet$trt)
     true<- fit$fitted.values
     N<-  length( fit$y)
     temp<- matrix(  NA, ncol=50, nrow=N)
     sigma<- fit$shat.GCV
     for (  k in 1:50){
     ysim<- true + sigma* rnorm(N) 
     temp[,k]<- predict(fit, y= ysim)
     }
     matplot( fit$x, temp, type="l")

     # 
     #4-d example 
     fit<- Tps(BD[,1:4],BD$lnya,scale.type="range") 

     # plots fitted surface and contours 
     # default is to hold 3rd and 4th fixed at median values 

     surface(fit)   

