# "micro Krig" Spatial process estimate of a curve or surface, "kriging" with a known covariance function. This is a simple version of the Krig function that is optimized for large data sets, sparse linear algebra, and a clear exposition of the computations. Lambda, the smoothing parameter must be fixed. This function is called higher level functions for maximum likelihood estimates of covariance paramters. ```r mKrig(x, y, weights = rep(1, nrow(x)), Z = NULL, ZCommon = NULL, cov.function = "stationary.cov", cov.args = NULL, lambda = NA, m = 2, chol.args = NULL, find.trA = TRUE, NtrA = 20, iseed = NA, na.rm = FALSE, collapseFixedEffect = TRUE, tau = NA, sigma2 = NA, verbose = FALSE, simpleKriging = FALSE, ...) ## S3 method for class 'mKrig' predict( object, xnew=NULL,ynew=NULL, grid.list = NULL, derivative=0, Z=NULL,drop.Z=FALSE,just.fixed=FALSE, collapseFixedEffect = object$collapseFixedEffect, ...) ## S3 method for class 'mKrig' summary(object, ...) ## S3 method for class 'mKrig' print( x, digits=4,... ) ## S3 method for class 'mKrigSummary' print( x, digits=4,... ) mKrig.coef(object, y, collapseFixedEffect=TRUE) mKrig.trace( object, iseed, NtrA) mKrigCheckXY(x, y, weights, Z, ZCommon, na.rm) ``` ## Arguments - `collapseFixedEffect`: If replicated fields are given to mKrig (i.e. `y` has more than one column) there is the choice of estimating the fixed effect coefficients (`d` in the returned object) separately for each replicate or pooling across replicates and deriving a single estimate. If `collapseFixedEffect` is TRUE (default) the estimates are pooled. - `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.) - `cov.args`: A list of optional arguments that will be used in calls to the covariance function. - `cov.function`: The name, a text string of the covariance function. - `derivative`: If zero the surface will be evaluated. If not zero the matrix of partial derivatives will be computed. - `digits`: Number of significant digits used in printed output. - `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. - `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 NtrA solutions of the linear system but the cholesky decomposition is reused. - `grid.list`: A grid.list to evaluate the surface in place of specifying arbitrary locations. - `iseed`: Random seed ( using `set.seed(iseed)`) used to generate iid normals for Monte Carlo estimate of the trace. Set this to an integer to insure the same random draws are used to compute the approximate trace and be consistent across different values of the covariance parameters. - `just.fixed`: If TRUE only the predictions for the fixed part of the model will be evaluted. - `lambda`: Smoothing parameter or equivalently the ratio between the nugget and process varainces. - `m`: The degree of the polynomial used in the fixed part. This is follows the thin plate spline convention that the degree is m-1. Default is m=2 for a linear function, m=1 a constant and, as an quick logical switch, m=0 will result in no fixed part in the spatial model. - `na.rm`: If TRUE NAs in y are omitted along with corresonding rows of x. - `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. - `object`: Object returned by mKrig. (Same as "x" in the print function.) - `sigma2`: Value of the process variance. - `simpleKriging`: If TRUE there is no fixed linear part in the spatial model. This is a more rational way than setting `m=0`. - `tau`: Value of the measurment error (aka nugget) standard deviation. - `verbose`: If TRUE prints out intermediate values of objsects for debugging. - `weights`: Precision ( 1/variance) of each observation - `x`: Matrix of unique spatial locations (or in print or surface the returned mKrig object.) - `xnew`: Locations for predictions. - `y`: Vector or matrix of observations at spatial locations, missing values are not allowed! Or in mKrig.coef a new vector of observations. If y is a matrix the columns are assumed to be independent replicates of the spatial field. I.e. observation vectors generated from the same covariance and measurment error model but independent from each other. - `ynew`: New observation vector. `mKrig` will reuse matrix decompositions and find the new fit to these data. - `Z`: Linear covariates to be included in fixed part of the model that are distinct from the default low order polynomial in `x`. (NOTE the order of the polynomial determined by `m`) - `ZCommon`: Linear covariates to be included in fixed part of the model. This option is only relevant for independent realizations of the fields (i.e. y is a matrix). The number of rows of `ZCommon` should be equal to the number of observations times the number of realizations. - ``...``: In `mKrig` and `predict` additional arguments that will be passed to the covariance function. ## Details This function is an abridged version of Krig and uses a more direct computation for the linear algebra that faciliates compactly supported covariance functions. The m stand for micro and done for a fixed lambda parameter and other covariance parameters (e.g. aRange) and for unique spatial locations. This is also the basic computational element of the top level `spatialProcess` function for finding MLEs. See the source code for a commented version that described the computation. These restrictions simplify the code for reading. Note that also little checking is done and the spatial locations are not transformed before the estimation. Repeated locations To keep the coding simple `x` must be a set of unique locations. When there are multiple observations at a location -- also known as replicated locations -- one strategy is to use the average spatial value for this location in place of mulitple observations. One can also adjust the weights accordingly for this location if that makes sense. To do this use the function `Krig.replicates`. For example ``` out<- Krig.replicates( x=x, y=y) ``` and `out$xM, out$yM, out$weightsM` will be the unique locations with averaged values and the number of observations. ``` out<- Krig.replicates( x=x, y=y, weights= weights) ``` will also combine the weight vector based on the replications and be returned as `out$weightsM`. Multiple fields 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 `y` as a matrix and the computations are done efficiently for each set. The likelihood across all replicates is combined and denoted with `FULL` at the end. Also note the `collapseFixedEffects` switch to determine if the regression part is found seperately for each replicate or combined into a single model. 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 the trace of the smoothing matrix is (trace A(lambda)) is computationally expensive and a Monte Carlo approximation is supplied instead. Supporting functions See `predictSE.mKrig` for prediction standard errors and `sim.mKrig.approx` to quantify the uncertainty in the estimated function using conditional simulation. `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. `summary.mKrig` creates a list of class `mKrigSummary` along with a table of standard errors for the fixed linear parameters. `print.mKrigSummary` prints the `mKrigSummary` object and adds some more explanation about the model and results `print.mKrig` prints a summary for the `mKrig` object that the combines the summary and print methods. `mKrig.coef` finds "beta"" and "c" coefficients representing the solution using the previous cholesky decomposition but 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, fixed part, and the covariance matrix all remain the same . Sparse covariance functions 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 `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 `nzR` to 450000 one would use the following argument for chol.args in mKrig: `chol.args=list(pivot=TRUE,memory=list(nnzR= 450000))` Approximation trace estimate `mKrig.trace` This is an internal function called by `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 `NtrA` independent N(0,1) vectors and the mean and standard deviation are reported in the `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 (`np`) then the algorithm switches to finding the trace exactly based on applying A to `np` unit vectors. ## Returns - **summary**: A named array with the values of covariance parameters and log likelihoods. - **beta**: 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. - **c.coef**: Coefficients of the nonparametric part. - **nt**: Dimension of fixed part. - **np**: Dimension of c.coef. - **nZ**: Number of columns of Z covariate matrix (can be zero). - **ind.drift**: Logical vector that indicates polynomial coefficients in the `d` coefficients vector. This is helpful to distguish between polynomial part and the extra covariates coefficients associated with Z. - **lambda.fixed**: The fixed lambda value - **x**: Spatial locations used for fitting. - **knots**: The same as x - **cov.function.name**: Name of covariance function used. - **args**: A list with all the covariance arguments that were specified in the call. - **m**: Order of fixed part polynomial. - **chol.args**: A list with all the cholesky arguments that were specified in the call. - **call**: A copy of the call to mKrig. - **non.zero.entries**: Number of nonzero entries in the covariance matrix for the process at the observation locations. - **lnDetCov**: Log determinant of the covariance matrix for the observations having factored out sigma. - **Omega**: GLS covariance for the estimated parameters in the fixed part of the model (d coefficients0. - **qr.VT, Mc**: QR and cholesky matrix decompositions needed to recompute the estimate for new observation vectors. - **fitted.values, residuals**: Usual predictions from fit. - **eff.df**: Estimate of effective degrees of freedom. Either the mean of the Monte Carlo sample or the exact value. - **trA.info**: If NtrA ids less than `np` then the individual members of the Monte Carlo sample and `sd(trA.info)/ sqrt(NtrA)` is an estimate of the standard error. If NtrA is greater than or equal to `np` then these are the diagonal elements of A(lamdba). - **GCV**: Estimated value of the GCV function. - **GCV.info**: Monte Carlo sample of GCV functions ## Author(s) Doug Nychka, Reinhard Furrer, John Paige ## See Also Krig, surface.mKrig, Tps, fastTps, predictSurface, predictSE.mKrig, sim.mKrig.approx, `mKrig.grid` ## Examples ```r # # 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 mKrigMLEGrid to choose lambda by maxmimum likelihood out<- mKrig( x,y, aRange = 2.0, lambda=.01) out.p<- predictSurface( out) surface( out.p) # # NOTE this should be identical to # Krig( x,y, aRange=2.0, lambda=.01) ############################################################################## # an example using a "Z" covariate and the Matern family # again see mKrigMLEGrid 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", aRange=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() ######################################################################### # 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, aRange=.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 aRange # are less than the default maximum allocated size of the # sparse covariance matrix. # see options() for the default values. The names follow the convention # spam.arg where arg is the name of the spam component # e.g. spam.nearestdistnnz # The following will give a warning for aRange=.9 because # allocation for the covariance matirx storage is too small. # Here aRange controls the support of the covariance and so # indirectly the number of nonzero elements in the sparse matrix ## Not run: look2<- mKrig( x,y, cov.function="wendland.cov",k=2, aRange=.9, lambda=.1) ## End(Not run) # The warning resets the memory allocation for the covariance matrix # according the to values options(spam.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. options( spam.nearestdistnnz=c(416052,400)) look2<- mKrig( x,y, cov.function="wendland.cov",k=2, aRange=.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 ## Not run: # 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 # look2<- mKrig( x,y, cov.function="wendland.cov",k=2, aRange=.1, lambda=1e2 ) ## End(Not run) # to avoid the warning look2<-mKrig( x,y, cov.function="wendland.cov", k=2, aRange=.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, aRange=.1, lambda=1e2 ) # note slight difference in summary because two data sets have been fit. print( look3) #} ## Not run: ################################################################## # finding a good choice for aRange 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 # and saving computation time by not looking at the complete set # of points # NOTE: This could also be done directly using the FNN package for finding nearest # neighbors set.seed(223) ind<- sample( 1:N,100) hold<- rdist( x[ind,], x) dd<- apply( hold, 1, quantile, p= 50/N ) 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 hold2<- nearest.dist( x, x, delta= dguess ) # here is trick to find the number of nonsero rows for a matrix in spam format. hold3<- diff( hold2@rowpointers) # min( hold3) = 43 which we declare close enough. This also counts the diagonal # So there are a minimum of 42 nearest neighbors ( median is 136) # see table( hold3) for the distribution # now the following will use no less than 43 - 1 nearest neighbors # due to the tapering. mKrig( x,y, cov.function="wendland.cov",k=2, aRange=dguess, lambda=1e2) -> look2 ## End(Not run) ############################################################################### # use precomputed distance matrix # ## Not run: 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) #precompute distance matrix in compact form distMat = rdist(x, compact=TRUE) look3<- mKrig( x,Y,cov.function="stationary.cov", aRange=.1, lambda=1e2, distMat=distMat ) #precompute distance matrix in standard form distMat = rdist(x) look3<- mKrig( x,Y,cov.function="stationary.cov", aRange=.1, lambda=1e2, distMat=distMat ) ## End(Not run) ``` ## References [https://github.com/dnychka/fieldsRPackage](https://github.com/dnychka/fieldsRPackage)

mKrig function