% fields, Tools for spatial data
% Copyright 2004-2013, Institute for Mathematics Applied Geosciences
% University Corporation for Atmospheric Research
% Licensed under the GPL -- www.gpl.org/licenses/gpl.html

\name{mKrig}
\alias{mKrig}
\alias{predict.mKrig}
\alias{mKrig.coef}
\alias{mKrig.trace}
\alias{print.mKrig}
\alias{summary.mKrig}

\title{"micro Krig"  Spatial process estimate of a curve or surface, 
"kriging" with a known covariance function. }
\description{
This is a simple version of the Krig function that is 
optimized for large data sets and a clear exposition of the
computations. Lambda, the smoothing parameter must be fixed. 
}
\usage{
mKrig(x, y, weights = rep(1, nrow(x)),  Z=NULL,
  lambda = 0, cov.function = "stationary.cov", 
    m = 2, chol.args=NULL,cov.args=NULL, find.trA = TRUE, NtrA = 20,
                 iseed = 123, llambda=NULL,  ...)

\method{predict}{mKrig}( object, xnew=NULL,ynew=NULL, grid.list = NULL, derivative=0, Z=NULL,
                           drop.Z=FALSE,just.fixed=FALSE, ...)
\method{summary}{mKrig}(object, ...)
\method{print}{mKrig}( x, digits=4,... )
 mKrig.coef(object, y)

mKrig.trace( object, iseed, NtrA)
}

\arguments{
  \item{x}{Matrix of unique spatial locations (or in print or surface
       the returned mKrig object.) }

  \item{y}{ Vector or matrix of observations at spatial locations,
          missing values are not allowed! Or in \code{mKrig.coef} a
          new vector of observations. If y is a matrix the columns are
          assumed to be independent observations vectors generated
          from the same covariance and measurment error model. }

  \item{weights}{Precision ( 1/variance) of each observation }

  \item{Z}{Linear covariates to be included in fixed part of the model
  that are distinct from the default low order polynomial in \code{x}}

  \item{drop.Z}{If true the fixed part will only be evaluated at the
polynomial part of the fixed model. The contribution from the other
covariates will be omitted.}

  \item{lambda}{ Smoothing parameter or equivalently the ratio between
the nugget and process varainces.}

  \item{llambda}{If not \code{NULL} then \code{lambda = exp(
  llambda)}}

  \item{cov.function}{The name, a text string of the covariance
  function.}

  \item{m}{The degree of the polynomial used in teh fixed part is
(m-1) }

  \item{chol.args}{A list of optional arguments (pivot, nnzR) that
 will be used with the call to the cholesky decomposition. Pivoting is
 done by default to make use of sparse matrices when they are
 generated.  This argument is useful in some cases for sparse
 covariance functions to reset the memory parameter nnzR.  (See
 example below.) }

 \item{cov.args}{A list of optional arguments that will be used in
calls to the covariance function.}

 \item{find.trA}{If TRUE will estimate the effective degrees of
freedom using a simple Monte Carlo method. This will add to the
computational burden by approximately \code{NtrA} solutions of the
linear system but the cholesky decomposition is reused.}
 \item{grid.list}{A grid.list to evaluate the surface in place of specifying 
            arbitrary locations.}
\item{NtrA}{Number of Monte Carlo samples for the trace. But if NtrA
is greater than or equal to the number of observations the trace is
computed exactly.}

\item{iseed}{ Random seed ( using \code{set.seed(iseed)}) used to
generate iid normals for Monte Carlo estimate of the trace.}

  \item{\dots}{In \code{mKrig} and \code{predict} additional arguments
                 that will be passed to the covariance function. }
                 \item{object}{Object returned by mKrig. (Same as "x"
                 in the print function.)}

  \item{xnew}{Locations for predictions.}

   \item{ynew}{New observation vector. \code{mKrig} will reuse matrix
decompositions and find the new fit to these data.}

  \item{derivative}{If zero the surface will be evaluated. If not zero
    the matrix of partial derivatives will be computed.}

\item{just.fixed}{If TRUE only the predictions for the fixed part of
the model will be evaluted.}

\item{digits}{Number of significant digits used in printed output.}

}
\details{
 This function is an abridged version of Krig. The m stand for micro
and this function focuses on the computations in Krig.engine.fixed
done for a fixed lambda parameter, for unique spatial locations and
for data without missing values.

These restrictions simplify the code for reading. Note that also
little checking is done and the spatial locations are not transformed
before the estimation.  Because most of the operations are linear
algebra this code has been written to handle multiple data
sets. Specifically if the spatial model is the same except for
different observed values (the y's), one can pass \code{y} as a matrix
and the computations are done efficiently for each set.  Note that
this is not a multivariate spatial model just an efficient computation
over several data vectors without explicit looping.A big difference in
the computations is that an exact expression for thetrace of the
smoothing matrix is (trace A(lambda)) is computationally expensive and
a Monte Carlo approximation is supplied instead.

See \code{predictSE.mKrig} for prediction standard errors and 
\code{sim.mKrig.approx} to quantify the uncertainty in the estimated function using conditional 
simulation.  

\code{predict.mKrig} will evaluate the derivatives of the estimated
function if derivatives are supported in the covariance function.  For
example the wendland.cov function supports derivatives.

\code{print.mKrig} is a simple summary function for the object.

\code{mKrig.coef} finds the "d" and "c" coefficients represent the
solution using the previous cholesky decomposition for a new data
vector. This is used in computing the prediction standard error in
predictSE.mKrig and can also be used to evalute the estimate
efficiently at new vectors of observations provided the locations and
covariance remain fixed.

Sparse matrix methods are handled through overloading the usual linear
algebra functions with sparse versions. But to take advantage of some
additional options in the sparse methods the list argument chol.args
is a device for changing some default values. The most important of
these is \code{nnzR}, the number of nonzero elements anticipated in
the Cholesky factorization of the postive definite linear system used
to solve for the basis coefficients. The sparse of this system is
essentially the same as the covariance matrix evalauted at the
observed locations.  As an example of resetting \code{nzR} to 450000
one would use the following argument for chol.args in mKrig:

\code{ chol.args=list(pivot=TRUE,memory=list(nnzR= 450000))}

\code{mKrig.trace} This is an internal function called by \code{mKrig}
to estimate the effective degrees of freedom.  The Kriging surface
estimate at the data locations is a linear function of the data and
can be represented as A(lambda)y.  The trace of A is one useful
measure of the effective degrees of freedom used in the surface
representation. In particular this figures into the GCV estimate of
the smoothing parameter.  It is computationally intensive to find the
trace explicitly but there is a simple Monte Carlo estimate that is
often very useful.  If E is a vector of iid N(0,1) random variables
then the trace of A is the expected value of t(E)AE. Note that AE is
simply predicting a surface at the data location using the synthetic
observation vector E. This is done for \code{NtrA} independent N(0,1)
vectors and the mean and standard deviation are reported in the
\code{mKrig} summary.  Typically as the number of observations is
increased this estimate becomse more accurate. If NtrA is as large as
the number of observations (\code{np}) then the algorithm switches to
finding the trace exactly based on applying A to \code{np} unit
vectors.
}
\value{
  \item{d}{Coefficients of the polynomial fixed part and if present
  the covariates (Z).To determine which is which the logical vector
  ind.drift also part of this object is TRUE for the polynomial
  part. }

  \item{c}{ Coefficients of the nonparametric part.}

  \item{nt}{ Dimension of fixed part.}

  \item{np}{ Dimension of c.}

  \item{nZ}{Number of columns of Z covariate matrix (can be zero).}

   \item{ind.drift}{Logical vector that indicates polynomial
   coefficients in the \code{d} coefficients vector. This is helpful
   to distguish between polynomial part and the extra covariates
   coefficients associated with Z. }
 
 \item{lambda.fixed}{The fixed lambda value}

  \item{x}{Spatial locations used for fitting.}

  \item{knots}{The same as x}

  \item{cov.function.name}{Name of covariance function used.}

 \item{args}{ A list with all the covariance arguments that were
 specified in the call.}

 \item{m}{Order of fixed part polynomial.}

  \item{chol.args}{ A list with all the cholesky arguments that were
         specified in the call.}

 \item{call}{ A copy of the call to mKrig.}

 \item{non.zero.entries}{ Number of nonzero entries in the covariance
matrix for the process at the observation locations.}

 \item{shat.MLE}{MLE of sigma.}

 \item{rho.MLE}{MLE or rho.}

 \item{rhohat}{Estimate for rho adjusted for fixed model degrees of
 freedom (ala REML).}

 \item{lnProfileLike}{log Profile likelihood for lambda}

 \item{lnDetCov}{Log determinant of the covariance matrix for the
  observations having factored out rho.}

 \item{Omega}{GLS covariance for the estimated parameters in the fixed
part of the model (d coefficients0.}

 \item{qr.VT, Mc}{QR and cholesky matrix decompositions needed to
recompute the estimate for new observation vectors.}

 \item{fitted.values, residuals}{Usual predictions from fit.}

 \item{eff.df}{Estimate of effective degrees of freedom. Either the
mean of the Monte Carlo sample or the exact value. }

 \item{trA.info}{If NtrA ids less than \code{np} then the individual
members of the Monte Carlo sample and \code{sd(trA.info)/ sqrt(NtrA)}
is an estimate of the standard error. If NtrA is greater than or equal
to \code{np} then these are the diagonal elements of A(lamdba).}

 \item{GCV}{Estimated value of the GCV function.}

 \item{GCV.info}{Monte Carlo sample of GCV functions}

}
\author{Doug Nychka, Reinhard Furrer }
\seealso{ Krig, surface.mKrig, Tps, fastTps, predictSurface, predictSE.mKrig, sim.mKrig.approx, 
  \code{ \link{mKrig.grid}}}
\examples{
#
# Midwest ozone data  'day 16' stripped of missings 
  data( ozone2)
  y<- ozone2$y[16,]
  good<- !is.na( y)
  y<-y[good]
  x<- ozone2$lon.lat[good,]
# nearly interpolate using defaults (Exponential covariance range = 2.0)
# see also mKrig.MLE to choose lambda by maxmimum likelihood
  out<- mKrig( x,y, theta = 2.0, lambda=.01)
  out.p<- predictSurface( out)
  surface( out.p)
#
# NOTE this should be identical to 
# Krig( x,y, theta=2.0, lambda=.01) 

##############################################################################
# an example using a "Z" covariate and the Matern family
#  again see mKrig.MLE to choose parameters by MLE.
data(COmonthlyMet)
yCO<- CO.tmin.MAM.climate
good<- !is.na( yCO)
yCO<-yCO[good]
xCO<- CO.loc[good,]
Z<- CO.elev[good]
out<- mKrig( xCO,yCO, Z=Z, cov.function="stationary.cov", Covariance="Matern",
                    theta=4.0, smoothness=1.0, lambda=.1)
set.panel(2,1)
# quilt.plot with elevations
quilt.plot( xCO, predict(out))
# Smooth surface without elevation linear term included
surface( out)
set.panel()

#########################################################################
# Interpolate using tapered version of the exponential, 
# the taper scale is set to 1.5 default taper covariance is the Wendland.
# Tapering will done at a scale of 1.5 relative to the scaling 
# done through the theta  passed to the covariance function.
data( ozone2)
  y<- ozone2$y[16,]
  good<- !is.na( y)
  y<-y[good]
  x<- ozone2$lon.lat[good,]
  mKrig( x,y,cov.function="stationary.taper.cov",
       theta = 2.0, lambda=.01, 
       Taper="Wendland",  Taper.args=list(theta = 1.5, k=2, dimension=2)
           ) -> out2

# Try out GCV on a grid of lambda's.
# For this small data set 
# one should really just use Krig or Tps but this is an example of
# approximate GCV that will work for much larger data sets using sparse 
# covariances and the Monte Carlo trace estimate
#
# a grid of lambdas:
  lgrid<- 10**seq(-1,1,,15) 
  GCV<- matrix( NA, 15,20)
  trA<-  matrix( NA, 15,20)
  GCV.est<- rep( NA, 15)
  eff.df<- rep( NA, 15)
  logPL<- rep( NA, 15) 
# loop over lambda's
  for(  k in 1:15){
      out<- mKrig( x,y,cov.function="stationary.taper.cov",
                    theta = 2.0, lambda=lgrid[k],
          Taper="Wendland",  Taper.args=list(theta = 1.5, k=2, dimension=2)  ) 
      GCV[k,]<- out$GCV.info
      trA[k,]<- out$trA.info
      eff.df[k]<- out$eff.df
      GCV.est[k]<- out$GCV
      logPL[k]<- out$lnProfileLike
  }
#
# plot the results different curves are for individual estimates  
# the two lines are whether one averages first the traces or the GCV criterion.
#
  par( mar=c(5,4,4,6))
  matplot( trA, GCV, type="l", col=1, lty=2,
            xlab="effective degrees of freedom", ylab="GCV")
  lines( eff.df, GCV.est, lwd=2, col=2)
  lines( eff.df, rowMeans(GCV), lwd=2)
# add exact GCV computed by Krig 
  out0<-  Krig( x,y,cov.function="stationary.taper.cov",
              theta = 2.0, 
              Taper="Wendland", Taper.args=list(theta = 1.5, k=2, dimension=2),
              spam.format=FALSE)  
  lines( out0$gcv.grid[,2:3], lwd=4, col="darkgreen")

# add profile likelihood 
  utemp<- par()$usr
  utemp[3:4] <- range( -logPL)
  par( usr=utemp)
  lines( eff.df, -logPL, lwd=2, col="blue", lty=2)
  axis( 4)
  mtext( side=4,line=3, "-ln profile likelihood", col="blue")
  title( "GCV ( green = exact) and  -ln profile likelihood", cex=2)

#########################################################################
# here is a series of examples with bigger datasets  
# using a compactly supported covariance directly

set.seed( 334)
N<- 1000
x<- matrix( 2*(runif(2*N)-.5),ncol=2)
y<- sin( 1.8*pi*x[,1])*sin( 2.5*pi*x[,2]) + rnorm( 1000)*.1
  
look2<-mKrig( x,y, cov.function="wendland.cov",k=2, theta=.2, 
            lambda=.1)

# take a look at fitted surface
predictSurface(look2)-> out.p
surface( out.p)

# this works because the number of nonzero elements within distance theta
# are less than the default maximum allocated size of the 
# sparse covariance matrix. 
#  see  spam.options() for the default values 

# The following will give a warning for theta=.9 because 
# allocation for the  covariance matirx storage is too small. 
# Here theta controls the support of the covariance and so 
# indirectly the  number of nonzero elements in the sparse matrix

\dontrun{
 look2<- mKrig( x,y, cov.function="wendland.cov",k=2, theta=.9, lambda=.1)
}

# The warning resets the memory allocation  for the covariance matrix
# according the to values   'spam.options(nearestdistnnz=c(416052,400))'
# this is inefficient becuase the preliminary pass failed. 

# the following call completes the computation in "one pass"
# without a warning and without having to reallocate more memory. 

spam.options(nearestdistnnz=c(416052,400))
  look2<- mKrig( x,y, cov.function="wendland.cov",k=2,
                    theta=.9, lambda=1e-2)
# as a check notice that 
#   print( look2)
# reports the number of nonzero elements consistent with the specifc allocation
# increase in spam.options


# new data set of 1500 locations
  set.seed( 234)
  N<- 1500
  x<- matrix( 2*(runif(2*N)-.5),ncol=2)
  y<- sin( 1.8*pi*x[,1])*sin( 2.5*pi*x[,2]) + rnorm( N)*.01

\dontrun{  
# the following is an example of where the allocation  (for nnzR) 
# for the cholesky factor is too small. A warning is issued and 
# the allocation is increased by 25% in this example
#
 look2<- mKrig( x,y, 
            cov.function="wendland.cov",k=2, theta=.1, lambda=1e2  )
}
# to avoid the warning 
 look2<-mKrig( x,y, 
            cov.function="wendland.cov", k=2, theta=.1, 
            lambda=1e2, chol.args=list(pivot=TRUE, memory=list(nnzR= 450000)))

###############################################################################
# fiting multiple data sets
#
#\dontrun{ 
  y1<- sin( 1.8*pi*x[,1])*sin( 2.5*pi*x[,2]) + rnorm( N)*.01
  y2<- sin( 1.8*pi*x[,1])*sin( 2.5*pi*x[,2]) + rnorm( N)*.01
  Y<- cbind(y1,y2)
  look3<- mKrig( x,Y,cov.function="wendland.cov",k=2, theta=.1, 
            lambda=1e2  )
# note slight difference in summary because two data sets have been fit.
  print( look3)
#}

##################################################################
# finding a good choice for theta as a taper 

# Suppose the target is a spatial prediction using roughly 50 nearest neighbors
# (tapering covariances is effective for roughly 20 or more in the situation of 
#  interpolation) see Furrer, Genton and Nychka (2006).

# take a look at a random set of 100 points to get idea of scale

  set.seed(223)
  ind<- sample( 1:N,100)
  hold<- rdist( x[ind,], x)
  dd<- (apply( hold, 1, sort))[65,]
  dguess<- max(dd)
# dguess is now a reasonable guess at finding cutoff distance for
# 50 or so neighbors

# full distance matrix excluding distances greater than dguess
# but omit the diagonal elements -- we know these are zero!
  hold<- nearest.dist( x, delta= dguess,upper=TRUE)
# exploit spam format to get quick of number of nonzero elements in each row
  hold2<-  diff( hold@rowpointers)
#  min( hold2) = 55   which we declare close enough 
# now the following will use no less than 55 nearest neighbors 
# due to the tapering. 
\dontrun{
  mKrig( x,y, cov.function="wendland.cov",k=2, theta=dguess, 
            lambda=1e2) ->  look2
}

}
% Add one or more standard keywords, see file 'KEYWORDS' in the
% R documentation directory.
\keyword{spatial }