\name{nlrob}
\title{Robust Fitting of Nonlinear Regression Models}
\alias{nlrob}
\alias{fitted.nlrob}
\alias{residuals.nlrob}
\alias{predict.nlrob}
\alias{vcov.nlrob}
\description{
  \code{nlrob} fits a nonlinear regression model by robust methods.
  Per default, by an M-estimator, using iterated reweighted least
  squares (called \dQuote{IRLS} or also \dQuote{IWLS}).
}
\usage{
nlrob(formula, data, start, lower, upper,
      weights = NULL, na.action = na.fail,
      method = c("M", "MM", "tau", "CM", "mtl"),
      psi = .Mwgt.psi1("huber", cc=1.345), scale = NULL,
      test.vec = c("resid", "coef", "w"), maxit = 20,
      tol = 1e-06, acc, algorithm = "default", doCov = FALSE,
      control = if(method == "M") nls.control() else
		nlrob.control(method, optArgs = list(trace=trace), ...),
      trace = FALSE, ...)

\method{fitted}{nlrob}(object, ...)
\method{residuals}{nlrob}(object, type = , ...)% FIXME: more 'type's + DOCU
\method{predict}{nlrob}(object, newdata, ...)
}
\arguments{
  \item{formula}{a nonlinear \code{\link{formula}} including variables
    and parameters of the model, such as \code{y ~ f(x, theta)} (cf. \code{\link{nls}}).
    (For some checks: if \eqn{f(.)} is linear, then we need
    parentheses, e.g., \code{y ~ (a + b * x)};
    (note that \code{._nlrob.w} is not allowed as variable or parameter name))
    %% FIXME in code -- long overdue, as nls() is more flexible *SINCE* R 2.2.1
    %% Do not use \code{w} as variable or parameter name!
    %% FIXME: this should really no longer be needed ==> add a check
  }
  \item{data}{an optional data frame containing the variables
    in the model.  If not found in \code{data}, the variables are taken
    from \code{environment(formula)}, typically the environment from
    which \code{nlrob} is called.}
  \item{start}{a named numeric vector of starting parameters estimates,
    only for \code{method = "M"}.}
  \item{lower, upper}{numeric vectors of lower and upper bounds; if
    needed, will be replicated to be as long as the longest of \code{start},
    \code{lower} or \code{upper}.  For (the default) \code{method = "M"},
    if the bounds are unspecified all parameters are assumed to be
    unconstrained; also, for method \code{"M"}, bounds can only be used
    with the \code{"port"} algorithm.  They are ignored, with a warning,
    in cases they have no effect.

    For all other methods, currently these bounds \emph{must} be
    specified as finite values, and one of them must have
    \code{\link{names}} matching the parameter names in \code{formula}.

    For methods \code{"CM"} and \code{"mtl"}, the bounds must
    \emph{additionally} have an entry named \code{"sigma"} as that is
    determined simultaneously in the same optimization, and hence its
    \code{lower} bound must not be negative.
  }
  \item{weights}{an optional vector of weights to be used in the fitting
    process (for intrinsic weights, not the weights \code{w} used in the
    iterative (robust) fit). I.e.,
    \code{sum(w * e^2)} is minimized with \code{e} = residuals,
    \eqn{e_i = y_i - f(xreg_i, \theta)}{e[i] = y[i] - f(xreg[i], theta)},
    where \eqn{f(x,\theta)}{f(x, theta)} is the nonlinear function,
    and \code{w} are the robust weights from \code{resid * weights}.}
  \item{na.action}{a function which indicates what should happen when the data
    contain \code{NA}s.  The default action is for the procedure to
    fail.  If NAs are present, use \code{na.exclude} to have residuals with
    \code{length == nrow(data) == length(w)}, where \code{w} are the
    weights used in the iterative robust loop.  This is better if the
    explanatory variables in
    \code{formula} are time series (and so the NA location is important).
    For this reason, \code{na.omit}, which leads to omission of cases
    with missing values on any required variable, is not suitable
    here since the residuals length is different from
    \code{nrow(data) == length(w)}.
  }
  \item{method}{a character string specifying which method to use.  The
    default is \code{"M"}, for historical and back-compatibility
    reasons.  For the other methods, primarily see
    \code{\link{nlrob.algorithms}}. % nlrob-algos.Rd

    \describe{
      \item{"M"}{Computes an M-estimator, using \code{\link{nls}(*,
	  weights=*)} iteratively (hence, IRLS) with weights equal to
	\eqn{\psi(r_i) / r_i}, where \eqn{r_i} is the i-the residual
	from the previous fit.}
      \item{"MM"}{Computes an MM-estimator, starting from \code{init},
	either "S" or "lts".}% more: FIXME
      \item{"tau"}{Computes a Tau-estimator.}
      \item{"CM"}{Computes a \dQuote{Constrained M} (=: CM) estimator.}
      \item{"mtl"}{Compute as \dQuote{Maximum Trimmed Likelihood} (=: MTL)
	estimator.}
    }
    Note that all methods but \code{"M"} are \dQuote{random}, hence
    typically to be preceded by \code{\link{set.seed}()} in usage, see
    also \code{\link{nlrob.algorithms}}. % nlrob-algos.Rd
  }
  \item{psi}{a function (possibly by name) of the form \code{g(x, 'tuning
      constant(s)', deriv)} that for \code{deriv=0} returns
    \eqn{\psi(x)/x}{psi(x)/x} and for \code{deriv=1} returns
    \eqn{\psi'(x)}{psi'(x)}.  Note that tuning constants can \emph{not}
    be passed separately, but directly via the specification of \code{psi},
    typically via a simple \code{\link{.Mwgt.psi1}()} call as per
    default.

    Note that this has been a deliberately non-backcompatible change
    for robustbase version 0.90-0 (summer 2013 -- early 2014).
  }
  \item{scale}{when not \code{NULL} (default), a positive number
    specifying a scale kept \emph{fixed} during the iterations (and
    returned as \code{Scale} component).}
  \item{test.vec}{character string specifying the convergence
    criterion. The relative change is tested for residuals with a value
    of \code{"resid"} (the default), for coefficients with
    \code{"coef"}, and for weights with \code{"w"}.}
  \item{maxit}{maximum number of iterations in the robust loop.}
  \item{tol}{non-negative convergence tolerance for the robust fit.}
  \item{acc}{previous name for \code{tol}, now deprecated.}
  \item{algorithm}{character string specifying the algorithm to use for
    \code{\link{nls}}, see there, only when \code{method = "M"}.  The
    default algorithm is a Gauss-Newton algorithm.}
  \item{doCov}{a logical specifying if \code{nlrob()} should compute the
    asymptotic variance-covariance matrix (see \code{\link{vcov}})
    already.  This used to be hard-wired to \code{TRUE}; however, the
    default has been set to \code{FALSE}, as \code{\link{vcov}(obj)} and
    \code{\link{summary}(obj)} can easily compute it when needed.}
  \item{control}{an optional list of control settings.
    \describe{
      \item{for \code{method = "M"}:}{settings for \code{\link{nls}()}.
	See \code{\link{nls.control}} for the names of the settable control
	values and their effect.}
      \item{for all \code{method}s but \code{"M"}:}{a list, typically
	resulting from \code{\link{nlrob.control}(method, *)}.}
    }
  }
  \item{trace}{logical value indicating if a \dQuote{trace} of
    the \code{nls} iteration progress should be printed.  Default is
    \code{FALSE}. \cr
    If \code{TRUE}, in each robust iteration, the residual
    sum-of-squares and the parameter values are printed at the
    conclusion of each \code{nls} iteration.
    When the \code{"plinear"} algorithm is used, the conditional
    estimates of the linear parameters are printed after the nonlinear
    parameters.}

  \item{object}{an \R object of class \code{"nlrob"}, typically
    resulting from \code{nlrob(..)}.}
  \item{\dots}{for \code{nlrob}: only when \code{method} is \emph{not} \code{"M"},
    optional arguments for \code{\link{nlrob.control}};
    \cr
    for other functions: potentially optional arguments passed to the
    extractor methods.}
  \item{type}{a string specifying the \emph{type} of residuals desired.
    Currently, \code{"response"} and \code{"working"} are supported.
    %% FIXME: 1. document these (here)  2. write and support more types
  }
  \item{newdata}{a data frame (or list) with the same names as the
    original \code{data}, see e.g., \code{\link{predict.nls}}.}
}
\details{
  For \code{method = "M"}, iterated reweighted least squares
  (\dQuote{IRLS} or \dQuote{IWLS}) is used, calling \code{\link{nls}(*,
    weights= .)} where \code{weights} \eqn{w_i} are proportional to
  \eqn{\psi(r_i/ \hat{\sigma})}{psi(r_i/ sig.)}.

  All other methods minimize differently, and work \bold{without}
  \code{\link{nls}}.  See \link{nlrob.algorithms} % -> nlrob-algos.Rd
  for details.
}
\value{
  \code{nlrob()} returns an object of S3 class \code{"nlrob"},
  for \code{method = "M"} also inheriting from class \code{"nls"}, (see
  \code{\link{nls}}).

  It is a list with several components; they are not documented yet,
  as some of them will probably change.
  Instead, rather use \dQuote{accessor} methods, where possible:
  There are methods (at least) for the generic accessor functions
  \code{\link{summary}()}, \code{\link{coefficients}()} (aka \code{coef()})
  \code{fitted.values()}, \code{residuals()}, \code{\link{sigma}()} and
  \code{\link{vcov}()}, the latter for the variance-covariance matrix of
  the estimated parameters, as returned by \code{coef()}, i.e., not
  including the variance of the errors.
  For \code{nlrob()} results, \code{\link{estimethod}()} returns the
  \dQuote{estimation method}, which coincides with the \code{method}
  argument used.

  \code{residuals(.)}, by default \code{type = "response"}, returns
  the residuals \eqn{e_i}, defined above as
  \eqn{e_i = Y_i - f_(x_i, \hat\theta)}{e[i] = Y[i] - f(x[i], theta^)}.
  These differ from the standardized or weighted residuals which, e.g.,
  are assumed to be normally distributed, and a version of which is
  returned in \code{working.residuals} component.
  %% and another is working.residuals/Scale
}

\author{
  Andreas Ruckstuhl (inspired by \code{\link[MASS]{rlm}}() and
  \code{\link{nls}}()), in July 1994 for S-plus.\cr
  Christian Sangiorgio did the update to \R and corrected some errors,
  from June 2002 to January 2005, and Andreas contributed slight changes
  and the first methods in August 2005.

  Since then, the help page, testing, more cleanup, new methods: Martin
  Maechler.
}
\note{
  This function (with the only method \code{"M"}) used to be named
  \code{rnls} and has been in package \pkg{sfsmisc} in the past, but
  been dropped there.
}
\seealso{ \code{\link{nls}}, \code{\link[MASS]{rlm}}.
}
\examples{
DNase1 <- DNase[ DNase$Run == 1, ]

## note that selfstarting models don't work yet % <<< FIXME !!!

##--- without conditional linearity ---

## classical
fmNase1 <- nls( density ~ Asym/(1 + exp(( xmid - log(conc) )/scal ) ),
                data = DNase1,
                start = list( Asym = 3, xmid = 0, scal = 1 ),
                trace = TRUE )
summary( fmNase1 )

## robust
RmN1  <- nlrob( density ~ Asym/(1 + exp(( xmid - log(conc) )/scal ) ),
                data = DNase1, trace = TRUE,
                start = list( Asym = 3, xmid = 0, scal = 1 ))
summary( RmN1 )

##--- using conditional linearity ---

## classical
fm2DNase1 <- nls( density ~ 1/(1 + exp(( xmid - log(conc) )/scal ) ),
                  data = DNase1,
                  start = c( xmid = 0, scal = 1 ),
                  alg = "plinear", trace = TRUE )
summary( fm2DNase1 )

## robust
frm2DNase1 <- nlrob(density ~ 1/(1 + exp(( xmid - log(conc) )/scal ) ),
                  data = DNase1, start = c( xmid = 0, scal = 1 ),
                  alg = "plinear", trace = TRUE )
summary( frm2DNase1 )
## Confidence for linear parameter is quite smaller than "Asym" above
c1 <- coef(summary(RmN1))
c2 <- coef(summary(frm2DNase1))
rownames(c2)[rownames(c2) == ".lin"] <- "Asym"
stopifnot(all.equal(c1[,1:2], c2[rownames(c1), 1:2], tol = 0.09)) # 0.07315

### -- new examples -- "moderate outlier":
DN2 <- DNase1
DN2[10,"density"] <- 2*DN2[10,"density"]

fm3DN2 <- nls(density ~  Asym/(1 + exp(( xmid - log(conc) )/scal ) ),
              data = DN2, trace = TRUE,
              start = list( Asym = 3, xmid = 0, scal = 1 ))

## robust
Rm3DN2 <- nlrob(density ~  Asym/(1 + exp(( xmid - log(conc) )/scal ) ),
                data = DN2, trace = TRUE,
                start = list( Asym = 3, xmid = 0, scal = 1 ))
Rm3DN2
summary(Rm3DN2) # -> robustness weight of obs. 10 ~= 0.037
confint(Rm3DN2, method = "Wald")

## utility function sfsmisc::lseq() :
lseq <- function (from, to, length)
  2^seq(log2(from), log2(to), length.out = length)
## predict() {and plot}:
h.x <- lseq(min(DN2$conc), max(DN2$conc), length = 100)
nDat <- data.frame(conc = h.x)

h.p  <- predict(fm3DN2, newdata = nDat)# classical
h.rp <- predict(Rm3DN2, newdata = nDat)# robust

plot(density ~ conc, data=DN2, log="x",
     main = format(formula(Rm3DN2)))
lines(h.x, h.p,  col="blue")
lines(h.x, h.rp, col="magenta")
legend("topleft", c("classical nls()", "robust nlrob()"),
       lwd = 1, col= c("blue", "magenta"), inset = 0.05)

## See  ?nlrob.algorithms for examples
\dontrun{
DNase1 <- DNase[DNase$Run == 1,]
form <- density ~ Asym/(1 + exp(( xmid -log(conc) )/scal ))
gMM  <- nlrob(form, data = DNase1, method = "MM",
              lower = c(Asym = 0, xmid = 0, scal = 0),
              upper = 3, trace = TRUE)

## "CM" (and "mtl") additionally need bounds for "sigma" :
gCM  <- nlrob(form, data = DNase1, method = "CM",
              lower = c(Asym = 0, xmid = 0, scal = 0, sigma = 0),
              upper = c(3,3,3, sigma = 0.8))
}%not run
}
\keyword{robust}
\keyword{regression}
\keyword{nonlinear}
