Revision c103e17c562b32dcb179c0853306f28a0e456d50 authored by Martin Schlather on 11 April 2005, 00:00:00 UTC, committed by Gabor Csardi on 11 April 2005, 00:00:00 UTC
1 parent e7b694a
fitvario.Rd
\name{fitvario}
\alias{fitvario}
\alias{mleRF}% obsolete
\alias{fitvario.default}
\title{LSQ and Maximum Likelihood Estimation
of Random Field Parameters}
\description{
This function estimates arbitrary parameters of
a random field specification.
}
\usage{
fitvario(x, y=NULL, z=NULL, T=NULL, data, model, param,
lower=NULL, upper=NULL, sill=NA, ...)
fitvario.default(x, y=NULL, z=NULL, T=NULL, data, model, param,
lower=NULL, upper=NULL, sill=NA, trend,
use.naturalscaling=TRUE, PrintLevel=RFparameters()$Print,
optim.control=NULL, bins=20, nphi=1, ntheta=1, ntime=20,
distance.factor=0.5,
upperbound.scale.factor=10, lowerbound.scale.factor=20,
lowerbound.scale.LS.factor=5,
upperbound.var.factor=10, lowerbound.var.factor=100,
lowerbound.sill=1E-10, scale.max.relative.factor=1000,
minbounddistance=0.001, minboundreldist=0.02,
approximate.functioncalls=50, refine.onborder=TRUE,
pch=RFparameters()$pch,
var.name="X",
time.name="T", transform=NULL, standard.style=NULL,
lsq.methods=c("self", "plain", "sqrt.nr", "sd.inv", "internal"),
mle.methods=c("ml", "reml"),
cross.methods=c("cross.sq", "cross.abs", "cross.ign", "cross.crps"),
users.guess=NULL, users.beta=NULL, table.format=FALSE)
}
\arguments{
\item{x}{(\eqn{n\times2}{n x 2})-matrix of coordinates, or vector of
x-coordinates}
\item{y}{vector of y coordinates}
\item{z}{vector of z coordinates}
\item{T}{vector of T coordinates; these coordinates are given in
triple notation, see \code{\link{GaussRF}}}
\item{data}{vector or matrix of values measured at \code{coord};
If also a time component is given, the in the data the indices for
the spatial components run the fastest.
}
\item{model}{string or list;
covariance model, see \code{\link{CovarianceFct}}, or
type \code{\link{PrintModelList}()} to get all options.
% See also t
If \code{model} is a list, then the parameters with value \code{NA}
are estimated. Parameters that have value \code{NaN} should be
explicitely be defined by the function \code{transform}.
An alternative to define \code{NaN} values and
the function \code{transform}, is to replace the \code{NaN}
by a real-valued function with solely parameter a list defining
a covariance model. In case of the anisotropy matrix, the matrix
must be replaced by a list if functions are introduced.
Only the list elements
variance, scale or anisotropy, and kappas can be used, and
not the mean or the trend.
Further, the mean or the trend cannot be set by such a function.
See also \code{transform} below.
}
\item{param}{
vector or matrix or NULL.
If vector then
\code{param=c(mean, variance, nugget, scale,...)};
the parameters must be given
in this order. Further parameters are to be added in case of a
parametrised class of covariance functions, see
\code{\link{CovarianceFct}}.
\emph{Any} components set to \code{NA} are estimated; the others
are kept fix.
See also \code{model} above.
}
\item{lower}{list or vector. Lower bounds for the parameters.
If \code{lower} and \code{param} are vectors and \code{length(lower) <
length(param)}
then \code{lower} must match the number of additional parameters
\eqn{\kappa}{a,b,c,...}.
If \code{param} is matrix the length of \code{lower} must match the
number columns of param or being 2 elements smaller (then
\code{lower} is filled with \code{NA} from the left.
The bounds are equally applied to all rows.
If \code{lower} is a list, then elements that are not given are
considered as \code{NA}.
If \code{lower} is not given, or \code{lower} contains \code{NA}
then the missing bounds are generated automatically.
}
\item{upper}{list or vector. Upper bounds for the parameters.
See also lower.
}
\item{sill}{If not \code{NA} the sill is kept fix. Only used if the
standard format for the covariance model is given. See Details.}
\item{trend}{
Not programmed yet.
May only be set if \code{missing(param)};
linear formula : uses \code{X1}, \code{X2},... and \code{T} as internal
parameters for the coordinates; all parameters are estimated\cr
matrix : must have the same number of rows as \code{x}\cr
\emph{fixed} mean + matrix or linear formula : not possible within
this function (just subtract the mean from your data before calling
this function)
}
\item{...}{arguments as given in \code{fitvario.default} and listed in the
following.}
\item{use.naturalscaling}{
logical. Only used if model is given in standard (simple) way.
If \code{TRUE} then \emph{internally}, rescaled
covariance functions will be used for which
cov(1)\eqn{\approx}{~=}0.05.
\code{use.naturalscaling} has the advantage that \code{scale}
and the form parameters of the model get \sQuote{orthogonal},
but \code{use.naturalscaling} does not work for all models.
See Details.}
\item{PrintLevel}{level to which messages are shown. See Details.}
\item{optim.control}{control list for \code{\link{optim}}, which uses
'L-BFGS-B'. However 'parscale' may not be given.}
\item{bins}{number of bins of the empirical variogram. See Details.}
\item{nphi}{scalar or vector of 2 components.
If it is a vector then the first component gives the first angle
of the xy plane
and the second one gives the number of directions on the half circle.
If scalar then the first angle is assumed to be zero}
\item{ntheta}{scalar or vector of 2 components.
If it is a vector then the first component gives the first angle
in the third direction
and the second one gives the number of directions on the half circle.
If scalar then the first angle is assumed to be zero.}
\item{ntime}{scalar or vector of 2 components.
if \code{ntimes} is a vector, then the first component are the
maximum time distance (in units of the grid length \code{T[3]}) and the
second component gives the step size (in units of the grid length
\code{T[3]}). If scalar then the step size is assumed to 1 (in units
of the grid length \code{T[3]}.
}
\item{distance.factor}{relative right bound for the bins. See Details.}
\item{upperbound.scale.factor}{relative upper bound for scale in LSQ
and MLE.
See Details.}
\item{lowerbound.scale.factor}{relative lower bound for scale in MLE.
See Details.}
\item{lowerbound.scale.LS.factor}{relative lower bound for scale
in LSQ. See Details.}
\item{upperbound.var.factor}{relative upper bound for variance
and nugget. See Details.}
\item{lowerbound.var.factor}{relative lower bound for variance.
See Details.}
\item{lowerbound.sill}{absolute lower bound for variance
and nugget. See Details.}
\item{scale.max.relative.factor}{relative lower bound
for scale below
which an additional nugget effect is detected. See Details.}
\item{minbounddistance}{absolute distance to the bounds
below which a part of the algorithm is considered as
having failed. See Details.}
\item{minboundreldist}{relative distance to the bounds
below which a part of the algorithm is considered as
having failed. See Details.}
\item{approximate.functioncalls}{approximate evaluations
of the ML target function on a grid. See Details.}
\item{refine.onborder}{logical.
If \code{refine.onborder=TRUE} and if the result of
any maximum likelihood method or cross validation method
is on a borderline, then the optimisation is redone
in a modified way (which takes about double extra time)
}
\item{pch}{character shown before evaluating any method;
if \code{pch!=""} then one or two
additional steps in the MLE methods are
marked by \dQuote{+}.
Default: "*".}
\item{var.name}{basic name for the coordinates in the formula of the
trend. Default: \sQuote{X}}
\item{time.name}{basic name for the time component in the formula of the
trend. Default: \sQuote{X}}
\item{transform}{vector of strings.
Essentially, \code{transform} allows for the definition of a parameter as a
function of other estimated or fixed parameters.
All the parameters are supposed to be in a vector called \sQuote{param}
where the positions are given by \code{\link{parampositions}}.
An example of \code{transform} is
\code{function(param) \{param[3] <- 5 - param[1]; param\}}.
Any parameter that is set by \code{transform}, should be \code{NaN}
in the model definition. If it is \code{NA} a warning is given.
Note that the mean and the trend of the model can be neither set nor used in
\code{transform}. See also \code{standard.style}.
Instead of giving \code{transform}, in the model definition,
all NaN values are replaced by functions whose only parameter
is a bare model list, i.e., only the list elements
variance, scale or anisotropy, and kappas can be used, and
not the mean or the trend.
Further, the mean or the trend cannot be set by such a function.
Default: \code{NULL}}
\item{standard.style}{logical or \code{NULL}. This variable should only be
set by the advanced user. If \code{NULL} \code{standard.style} will be
\code{TRUE} if the covariance model allows for a \sQuote{standard}
definition (see \command{\link{convert.to.readable}} and
\command{\link{CovarianceFct}}) and \code{transform} is \code{NULL}.
Otherwise \code{standard.style} will be \code{FALSE}.
If a \\sQuote{standard} definition is given and both the variance and the
nugget are either not estimated or do not appear on the right hand
side of the \code{transform}, then \code{standard.style} might be set
to \code{TRUE} by the user. This accelerates the MLE algorithm.
The responsibility is completely left to the user, then.
Currently \command{fitvario} is only implemented for the
\\sQuote{standard} definition of the covariance model.
Hence \code{standard.style} must always be \code{TRUE} and consequently,
neither the variance nor the nugget may appear on either side
of the \code{transform}
}
\item{lsq.methods}{
variants of the least squares fit of the variogram. See Details.
}
\item{mle.methods}{
variants of the maximum likelihood fit of the covariance function.
See Details.
}
\item{cross.methods}{
Not implemented yet.
%variants of the cross validation fit of the covariance function.
%See Details.
}
\item{users.guess}{
User's guess of the parameters. All the parameters must be given
using the same rules as for either \code{param} (except that no NA's should
be contained) or \code{model}.
}
\item{users.beta}{
User's guess of the trend.
}
\item{table.format}{logical.
The parameters controls the output. See Value.
}
}
\details{
The optimisations are performed using \code{\link{optimise}} if one
parameter has to be estimated only and \code{\link{optim}}, otherwise.
First, by means of various control parameters, see below, the algorithm
first tries to estimate the bounds for the parameters to be estimated,
if the bounds for the parameters are not given.
Independently whether \code{users.guess} is given,
the algorithm guesses initial values for the parameters.
The automatic guess and the user's guess will be called
primitive methods in the following.
Second, the variogram model is fitted by various least squares
methods (according to the value of \code{lsq.methods}) using
the best parameter set among the primitive methods as initial value
if the effective number of parameters is greater than 1.
[Remarks: (i) \dQuote{best} with respect to the target value of
the respective lsq method;
(ii) the effective number of parameters in the optimisation algorithm
can be smaller than the number of estimated parameters, since in some
cases, some parameters can be calculated explicitely;
relevant for the choice between
\code{\link{optimise}} and \code{\link{optim}} is the
effective number of parameters.]
Third, the model is fitted by various maximum likelihood methods
(according to the value of \code{mle.methods}) using
the best parameter set among the primitive methods and the lsq methods
as initial value (if the effective number of parameters is greater
than 1).
% Forth, the model is fitted by various cross validation methods
% (according to the value of \code{mle.methods}) using
% the best parameter set among the primitive, lsq and mle methods
% as initial value (if the effective number of parameters is greater than 1).
Comments on specific parameters:
\itemize{
\item \code{lower}\cr
The lower bounds are technical bounds that should not
restrict really restrict the domaine of the value.
However, if these values are too small the optimisation
algorithm will frequently run into local minima or get
stuck close the border of the parameter domain.
It is advised to limit seriously the domain of the additional
parameters of the covariance model and/or the total number of
parameters to be estimated, if \dQuote{many} parameters
of the covariance model are estimated.
If the model is given in standard form, the user may supply
the lower bounds for the whole parameter vector, or only for
the additional form parameters of the model.
The lower bound for the mean will be ignored.
\code{lower} may contain \code{NA}s, then these values
are generated by the
If a nested model is given, the bounds may again be supplied
for all parameters or only for the additional form parameters
of the model. The bounds given apply uniformely to all
submodels of the nested model.
If the \code{model} is given in list format, then
\code{lower} is a list, where components may be missing
or \code{NA}. These are generated by the algorithm, then.
If \code{lower} is \code{NULL} all lower bounds are generated
automatically.
\item \code{upper.kappa}\cr
See \code{lower.kappa}.
\item \code{sill}\cr
Additionally to estimating \code{nugget} and \code{variance}
separately, they may also be estimated together under the
condition that \code{nugget} + \code{variance} = \code{sill}.
For the latter a finite value for \code{sill} has to be supplied,
and \code{nugget} and \code{variance} are set to \code{NA}.
\code{sill} is only used for the standard model.
\item \code{use.naturalscaling}\cr
logical. If \code{TRUE} then internally, rescaled
covariance functions will be used for which
cov(1)\eqn{\approx}{~=}0.05. However this parameter
does not influence
the output of \code{fitvario}: the parameter vector
returned by \code{fitvario} refers
\emph{always} to the standard covariance model as given in
\code{\link{CovarianceFct}}. (In contrast to \code{PracticalRange}
in \code{\link{RFparameters}}.)\cr
Advantages if \code{use.naturalscaling==TRUE}:
\itemize{
\item \code{scale} and the shape parameter of a parameterised
covariance model can be estimated better if they are estimated
simultaneously.
\item The estimated bounds calculated by means of
\code{upperbound.scale.factor} and \code{lowerbound.scale.factor},
etc. might be more realistic.
\item in case of anisotropic models, the inverse of the elements
of the anisotropy matrix should be in the above bounds.
}
Disadvantages if \code{use.naturalscaling==TRUE}:
\itemize{
\item For some covariance models with additional parameters, the
rescaling factor has to be determined numerically.
Then, more time is needed to perform \code{fitvario}.
}
Default: \code{TRUE}.
\item \code{PrintLevel}\cr
0 : no message\cr
1 : error messages\cr
2 : warnings\cr
3 : minimum debugging information\cr
5 : extended debugging information, including graphics\cr
Default: \code{0}.
\item \code{trace.optim}\cr
see control parameter \code{trace} of
\code{\link{optim}}. Default: \code{0}.
\item \code{bins}\cr
vector of explicit boundaries for the bins or the
number of bins for the empirical variogram (used in the
LSQ target function, which is described at the beginning
of the Details). Default: \code{20}.
\item \code{distance.factor}\cr
right boundary of the last
bin is calculated as \code{distance.factor} * (maximum distance
between all pairs of points). Only used if \code{bins} is a scalar.
Default: \code{0.5}.
\item \code{upperbound.scale.factor}\cr
The upper bound for the scale is determined
as \code{upperbound.scale.factor} * (maximum distance
between all pairs of points).
Default: \code{10}.
\item \code{lowerbound.scale.factor}\cr
The lower bound for the scale is determined
as
\eqn{\hbox{(minimum distance
between different pairs of points)} /
\code{lowerbound.scale.factor}}{(minimum distance
between different pairs of points) /
\code{lowerbound.scale.factor}}.
Default: \code{20}.\cr
\item \code{lowerbound.scale.LS.factor}\cr
For the LSQ target function a different lower bound
for the scale is used. It is determined as
\eqn{\hbox{(minimum distance
between different pairs of points)} /
\code{lowerbound.scale.LS.factor}}{(minimum distance
between different pairs of points) /
\code{lowerbound.scale.LS.factor}}.
Default: \code{5}.\cr
\item \code{upperbound.var.factor}\cr
The upper bound for the variance and the nugget is determined
as
\deqn{\code{upperbound.var.factor} *
{\hbox{var}}(\code{data}).}{\code{upperbound.var.factor} *
var(\code{data}).}
Default: \code{10}.
\item \code{lowerbound.var.factor}\cr
The lower bound for the variance and the nugget is determined
as
\deqn{\hbox{var}(\code{data}) /
\code{lowerbound.var.factor}.}{var(\code{data}) /
\code{lowerbound.var.factor}.}
If a standard model definition is given and
either the nugget or the variance is fixed,
the parameter to be estimated
must also be greater than \code{lowerbound.sill}.
If a non-standard model definition is given
then \code{lowerbound.var.factor} is only used
for the first model; the other lower bounds for the
variance are zero.
Default: \code{100}.
\item \code{lowerbound.sill}\cr
See \code{lowerbound.var.factor}.
Default: \code{1E-10}.
\item \code{scale.max.relative.factor}\cr
If the initial scale value for the ML estimation
obtained by the LSQ target function is
less than
\eqn{\hbox{(minimum distance
between different pairs of points)} /
\code{scale.max.relative.factor}}{ (minimum distance
between different pairs of points) /
\code{scale.max.relative.factor}}
a warning is given that probably a nugget effect
is present.
Note: if \code{scale.max.relative.factor} is greater
than \code{lowerbound.scale.LS.factor} then
no warning is given as
the scale has the lower bound (minimum distance
between different pairs of points) /
\code{lowerbound.scale.LS.factor}.
Default: \code{1000}.
\item \code{minbounddistance}\cr
If any value of the parameter vector
returned from the ML estimation
is closer than \code{minbounddistance}
to any of the bounds or if any value
has a relative distance smaller than
\code{minboundreldist}, then it is assumed that
the MLE algorithm has dropped into a local minimum,
and it will be continued with evaluating the
ML target function on a grid, cf. the beginning paragraphs
of the Details.
Default: \code{0.001}.
\item \code{minboundreldist}\cr
See \code{minbounddistance}.
Default: \code{0.02}.
\item \code{approximate.functioncalls}\cr
In case the parameter vector is too close to the given
bounds, the ML target function is evaluated on a grid
to get a new initial value for the ML estimation.
The number of points of the grid is approximately
\code{approximate.functioncalls}.
Default: \code{50}.
\item \code{lsq.methods}\cr
Variogram fit by least squares methods; first, a preliminary
trend is estimated
by a simple regression; second, the variogram is fitted; third,
the trend is fitted using the estimated covariance structure.
\itemize{
\item \code{"self"} weighted lsq. Weights are the values of the
fitted variogram to the power of -2
\item \code{"plain"} model fitted by least squares; trends are never taken
into account
\item \code{"sqrt.nr"} weighted lsq. Weight is the square root
of the number
of points in the bin
\item \code{"sd.inv"} weighted lsq. Weight is the inverse the
standard deviation of the
variogram cloud within the bin
}
\item \code{mle.methods}\cr
Model fit by various maximum likelihood methods
(according to the value of \code{mle.methods}) using
the best parameter set among the primitive methods and the lsq methods
as initial value (if the effective number of parameters is greater
than 1). If the best parameter vector of the MLE found so far is too close
to some given bounds, see the specific parameters above, it is
assumed that
\code{\link{optim}} ran into a local minimum because of a bad starting
value.
In this case and if \code{refine.onborder=TRUE}
the MLE target function is calculated on a grid, the
best parameter vector is taken, and the optimisation is restarted with
this parameter vector.
\itemize{
\item \code{"ml"} maximum likelihood;
since ML and REML give the same result if there are not any
covariates, ML is performed in that case, independently whether
it is given or not.
\item \code{"reml"} restricted maximum likelihood
}
% \item \code{cross.methods}\cr
% Here, the parameters are chosen so that
% the sum of the distances between the measured and estimated
% values in the cross validation are minimised.
% The various methods differ in the distance definition.
%
% The best parameter set among the primitive, lsq and mle methods
% are used as initial parameter value in the optimisation
% (if the effective number of parameters is greater
% than 1). If the best parameter vector of the cross validation
% algorithm
% found so far is too close
% to some given bounds, see the specific parameters above, it is
% assumed that
% \code{\link{optim}} ran into a local minimum because of a bad starting
% value.
% In this case and if \code{refine.onborder=TRUE}
% the target function is calculated on a grid, the
% best parameter vector is taken, and the optimisation is restarted with
% this parameter vector.
%
% \itemize{
% \item \code{"cross.sq"} squared distance
% \item \code{"cross.abs"} absolute value
% \item \code{"cross.ign"} \sQuote{ignorance} distance, see
% Gneiting et al. (2005)
% \item \code{"cross.crps"} \sQuote{xxxxxxx} distance, see
% Gneiting et al. (2005)
% }
}
}
\value{
If \code{table.format=TRUE} then a matrix is returned where the
first rows contain the estimated parameters, followed by
the target values of all methods for the given set of parameters;
the last rows give the lower and upper bounds used in the estimations.
The columns correspond to the various estimation methods for the parameters.
If \code{table.format=FALSE} then
the function returns a list with the following elements
\item{ev}{list returned by \code{\link{EmpiricalVariogram}}}
\item{variogram}{list of fitted models; they include the fixed and the
estimated parameters; each list element gives the estimated
covariance model}
\item{values}{list of optimal target values}
}
\references{
\itemize{
\item Least squares and mle methods
Cressie, N.A.C. (1993)
\emph{Statistics for Spatial Data.}
New York: Wiley.
% \item Cross validation methods
% Gneiting, T., M. Schlather, B. Huwe (2005)
% \emph{}
% In preparation.
\item Related software\cr
Ribeiro, P. and Diggle, P. (2001) Software for geostatistical analysis
using R and S-PLUS: geoR and geoS, version 0.6.15.
\url{http://www.maths.lancs.ac.uk/~ribeiro/geoR.html}.
}
}
\author{Martin Schlather, \email{schlath@hsu-hh.de}
\url{http://www.unibw-hamburg.de/WWEB/math/schlath/schlather.html}}
\note{This function does not depend on the value of
\code{\link{RFparameters}()$PracticalRange}.
The function \code{fitvario} always uses the standard specification
of the covariance model as given in \code{\link{CovarianceFct}}.
% The function is still in an experimental stage (and will probably
% never leave it). The function works well if
% only a few components of the parameter vector are to be estimated.
}
\section{Acknowledgement}{
Thanks to Paulo Ribeiro for hints and comparing the first versions
of \code{fitvario} to
\code{\link[geoR]{likfit}} of the package \code{geoR} whose homepage is at
\url{http://www.est.ufpr.br/geoR/}.
}
\seealso{
\code{\link{CovarianceFct}},
\code{\link{GetPracticalRange}},
\command{\link{parampositions}}
\code{\link{RandomFields}},
% \code{\link{winddata}}.
}
\examples{
% library(RandomFields, lib="~/TMP"); source("~/article/R/NEW.RF/RandomFields/R/mle.old.R");
model <-"gencauchy"
param <- c(0, 1, 0, 1, 1, 2)
estparam <- c(0, NA, 0, NA, NA, 2) ## NA means: "to be estimated"
## sequence in `estparam' is
## mean, variance, nugget, scale, (+ further model parameters)
## So, mean, variance, and scale will be estimated here.
## Nugget is fixed and equals zero.
points <- 100
x <- runif(points,0,3)
y <- runif(points,0,3) ## 100 random points in square [0, 3]^2
d <- GaussRF(x=x, y=y, grid=FALSE, model=model, param=param, n=10)
str(fitvario(x=cbind(x,y), data=d, model=model, param=estparam,
lower=c(0.1, 0.1), upper=c(1.9, 5), cross.me=NULL))
## The next two estimations give about the same result.
## For the first the sill is fixed to 1.5. For the second the sill
## is reached if the estimated variance is smaller than 1.5
estparam <- c(0, NA, NA, NA, NA, NA)
str(fitvario(x=cbind(x,y), data=d, model=model, param=estparam,
sill=1.5, cross.me=NULL))
estparam <- c(0, NA, NaN, NA, NA, NA)
parampositions(model=model, param=estparam)
f <- function(param) {
param[5] <- max(0, 1.5 - param[1])
return(param)
}
str(fitvario(x=cbind(x,y), data=d, model=model, param=estparam,
sill=1, transform=f, cross.me=NULL))
## the next call gives a warning, since the user may programme
## strange things in this setup, and the program cannot check it.
estparam <- c(0, NA, NA, NA, NA, NaN)
parampositions(model=model, param=estparam)
f <- function(param) {param[3] <- param[2]; param}
unix.time(str(fitvario(x=cbind(x,y), data=d, model=model,
param=estparam, transform=f, standard.style=TRUE,
cross.me=NULL), vec.len=6))
## much better programmed, but also much slower:
estmodel <- list(list(model="gencauchy", var=NA, scale=NA,
kappa=list(NA, function(m) m[[1]]$kappa[1])))
unix.time(str(fitvario(x=cbind(x,y), data=d, model=estmodel,
cross.me=NULL), vec.len=6))
%#################################################################
%## ##
%## for a further, more complicated example, see data(winddata) ##
%## ##
%#################################################################
%
% check space time extension in MLE\n add examples\n add test
}
\keyword{spatial}
% LocalWords: fitvario mleRF LSQ param NA naturalscaling PrintLevel optim pch
% LocalWords: RFparameters nphi ntheta ntime lowerbound minbounddistance var
% LocalWords: minboundreldist functioncalls eqn coord CovarianceFct emph cr
% LocalWords: PrintModelList cov variogram uote parampositions NaN kappas MLE
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