Robust Fitting of Nonlinear Regression Models

nlrob fits a nonlinear regression model by robust methods. Per default, by an M-estimator, using iterated reweighted least squares (called “IRLS” or also “IWLS”).

robust, regression, nonlinear
nlrob(formula, data, start, lower, upper,
      weights = NULL, na.action =,
      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, model = FALSE,
      control = if(method == "M") nls.control() else
		nlrob.control(method, optArgs = list(trace=trace), ...),
      trace = FALSE, ...)

# S3 method for nlrob fitted(object, ...) # S3 method for nlrob residuals(object, type = , ...) # S3 method for nlrob predict(object, newdata, ...)


a nonlinear formula including variables and parameters of the model, such as y ~ f(x, theta) (cf. nls). (For some checks: if \(f(.)\) is linear, then we need parentheses, e.g., y ~ (a + b * x); (note that ._nlrob.w is not allowed as variable or parameter name))


an optional data frame containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which nlrob is called.


a named numeric vector of starting parameters estimates, only for method = "M".

lower, upper

numeric vectors of lower and upper bounds; if needed, will be replicated to be as long as the longest of start, lower or upper. For (the default) method = "M", if the bounds are unspecified all parameters are assumed to be unconstrained; also, for method "M", bounds can only be used with the "port" algorithm. They are ignored, with a warning, in cases they have no effect.

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

For methods "CM" and "mtl", the bounds must additionally have an entry named "sigma" as that is determined simultaneously in the same optimization, and hence its lower bound must not be negative.


an optional vector of weights to be used in the fitting process (for intrinsic weights, not the weights w used in the iterative (robust) fit). I.e., sum(w * e^2) is minimized with e = residuals, \(e_i = y_i - f(xreg_i, \theta)\), where \(f(x,\theta)\) is the nonlinear function, and w are the robust weights from resid * weights.


a function which indicates what should happen when the data contain NAs. The default action is for the procedure to fail. If NAs are present, use na.exclude to have residuals with length == nrow(data) == length(w), where w are the weights used in the iterative robust loop. This is better if the explanatory variables in formula are time series (and so the NA location is important). For this reason, 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 nrow(data) == length(w).


a character string specifying which method to use. The default is "M", for historical and back-compatibility reasons. For the other methods, primarily see nlrob.algorithms.


Computes an M-estimator, using nls(*, weights=*) iteratively (hence, IRLS) with weights equal to \(\psi(r_i) / r_i\), where \(r_i\) is the i-the residual from the previous fit.


Computes an MM-estimator, starting from init, either "S" or "lts".

% more: FIXME

Computes a Tau-estimator.


Computes a “Constrained M” (=: CM) estimator.


Compute as “Maximum Trimmed Likelihood” (=: MTL) estimator.

Note that all methods but "M" are “random”, hence typically to be preceded by set.seed() in usage, see also nlrob.algorithms.


a function (possibly by name) of the form g(x, 'tuning constant(s)', deriv) that for deriv=0 returns \(\psi(x)/x\) and for deriv=1 returns \(\psi'(x)\). Note that tuning constants can not be passed separately, but directly via the specification of psi, typically via a simple .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).


when not NULL (default), a positive number specifying a scale kept fixed during the iterations (and returned as Scale component).


character string specifying the convergence criterion. The relative change is tested for residuals with a value of "resid" (the default), for coefficients with "coef", and for weights with "w".


maximum number of iterations in the robust loop.


non-negative convergence tolerance for the robust fit.


previous name for tol, now deprecated.


character string specifying the algorithm to use for nls, see there, only when method = "M". The default algorithm is a Gauss-Newton algorithm.


a logical specifying if nlrob() should compute the asymptotic variance-covariance matrix (see vcov) already. This used to be hard-wired to TRUE; however, the default has been set to FALSE, as vcov(obj) and summary(obj) can easily compute it when needed.


a logical indicating if the model.frame should be returned as well.


an optional list of control settings.

for method = "M":

settings for nls(). See nls.control for the names of the settable control values and their effect.

for all methods but "M":

a list, typically resulting from nlrob.control(method, *).


logical value indicating if a “trace” of the nls iteration progress should be printed. Default is FALSE. If TRUE, in each robust iteration, the residual sum-of-squares and the parameter values are printed at the conclusion of each nls iteration. When the "plinear" algorithm is used, the conditional estimates of the linear parameters are printed after the nonlinear parameters.


an R object of class "nlrob", typically resulting from nlrob(..).

for nlrob: only when method is not "M", optional arguments for nlrob.control; for other functions: potentially optional arguments passed to the extractor methods.


a string specifying the type of residuals desired. Currently, "response" and "working" are supported.


a data frame (or list) with the same names as the original data, see e.g., predict.nls.


For method = "M", iterated reweighted least squares (“IRLS” or “IWLS”) is used, calling nls(*, weights= .) where weights \(w_i\) are proportional to \(\psi(r_i/ \hat{\sigma})\).

All other methods minimize differently, and work without nls. See nlrob.algorithms for details.


nlrob() returns an object of S3 class "nlrob", for method = "M" also inheriting from class "nls", (see nls).

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

residuals(.), by default type = "response", returns the residuals \(e_i\), defined above as \(e_i = Y_i - f_(x_i, \hat\theta)\). 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 working.residuals component.


This function (with the only method "M") used to be named rnls and has been in package sfsmisc in the past, but been dropped there.

See Also

nls, rlm.

  • nlrob
  • fitted.nlrob
  • residuals.nlrob
  • predict.nlrob
  • vcov.nlrob
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 ))
summary(Rm3DN2) # -> robustness weight of obs. 10 ~= 0.037
confint(Rm3DN2, method = "Wald")
                    c(density = "numeric", conc = "numeric")))

## 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
# }
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))
summary(gCM)# did fail; note it has  NA NA NA (std.err, t val, P val)
stopifnot(identical(Rm3DN2$dataClasses, gMM$dataClasses),
          identical(   gCM$dataClasses, gMM$dataClasses))
# }
<!-- %not (always) tested -->
# }
Documentation reproduced from package robustbase, version 0.93-4, License: GPL (>= 2)

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