nls                  package:stats                  R Documentation

Nonlinear Least Squares

Description:

     Determine the nonlinear (weighted) least-squares estimates of the
     parameters of a nonlinear model.

Usage:

     nls(formula, data, start, control, algorithm,
         trace, subset, weights, na.action, model,
         lower, upper, ...)

Arguments:

 formula: a nonlinear model formula including variables and parameters.
           Will be coerced to a formula if necessary.

    data: an optional data frame in which to evaluate the variables in
          'formula' and 'weights'.  Can also be a list or an
          environment, but not a matrix.

   start: a named list or named numeric vector of starting estimates. 
          When 'start' is missing, a very cheap guess for 'start' is
          tried (if 'algorithm != "plinear"'). 

 control: an optional list of control settings.  See 'nls.control' for
          the names of the settable control values and their effect.

algorithm: character string specifying the algorithm to use. The
          default algorithm is a Gauss-Newton algorithm.  Other
          possible values are '"plinear"' for the Golub-Pereyra
          algorithm for partially linear least-squares models and
          '"port"' for the 'nl2sol' algorithm from the Port library -
          see the references.

   trace: logical value indicating if a trace of the iteration progress
          should be printed.  Default is 'FALSE'.  If 'TRUE' the
          residual (weighted) sum-of-squares and the parameter values
          are printed at the conclusion of each iteration. When the
          '"plinear"' algorithm is used, the conditional estimates of
          the linear parameters are printed after the nonlinear
          parameters.  When the '"port"' algorithm is used the
          objective function value printed is half the residual
          (weighted) sum-of-squares.

  subset: an optional vector specifying a subset of observations to be
          used in the fitting process.

 weights: an optional numeric vector of (fixed) weights.  When present,
          the objective function is weighted least squares.

na.action: a function which indicates what should happen when the data
          contain 'NA's.  The default is set by the 'na.action' setting
          of 'options', and is 'na.fail' if that is unset.  The
          'factory-fresh' default is 'na.omit'.  Value 'na.exclude' can
          be useful.

   model: logical.  If true, the model frame is returned as part of the
          object. Default is 'FALSE'.

lower, upper: vectors of lower and upper bounds, replicated to be as
          long as 'start'.  If unspecified, all parameters are assumed
          to be unconstrained.  Bounds can only be used with the
          '"port"' algorithm.  They are ignored, with a warning, if
          given for other algorithms.

     ...: Additional optional arguments.  None are used at present.

Details:

     An 'nls' object is a type of fitted model object.  It has methods
     for the generic functions 'anova', 'coef', 'confint', 'deviance',
     'df.residual', 'fitted', 'formula', 'logLik', 'predict', 'print',
     'profile', 'residuals', 'summary', 'vcov' and 'weights'.

     Variables in 'formula' (and 'weights' if not missing) are looked
     for first in 'data', then the environment of 'formula' and finally
     along the search path.  Functions in 'formula' are searched for
     first in the environment of 'formula' and then along the search
     path.

     Arguments 'subset' and 'na.action' are supported only when all the
     variables in the formula taken from 'data' are of the same length:
     other cases give a warning.

     Note that the 'anova' method does not check that the models are
     nested: this cannot easily be done automatically, so use with
     care.

Value:

     A list of 

       m: an 'nlsModel' object incorporating the model.

    data: the expression that was passed to 'nls' as the data argument.
           The actual data values are present in the environment of the
          'm' component.

    call: the matched call with several components, notably
          'algorithm'.

na.action: the '"na.action"' attribute (if any) of the model frame.

dataClasses: the '"dataClasses"' attribute (if any) of the '"terms"'
          attribute of the model frame.

   model: if 'model = TRUE', the model frame.

 weights: if 'weights' is supplied, the weights.

convInfo: when 'algorithm' is not "port", a list with convergence
          information.

 control: the control 'list' used, see the 'control' argument.

convergence, message: for an 'algorithm = "port"' fit only, a
          convergence code ('0' for convergence) and message. 


     *Note* that setting 'warnOnly = TRUE' in the 'control' argument
     (see 'nls.control') returns a non-converged object (since R
     version 2.5.0) which might be useful for further convergence
     analysis, _but *not* for inference_.

Warning:

     *Do not use 'nls' on artificial "zero-residual" data.*

     The 'nls' function uses a relative-offset convergence criterion
     that compares the numerical imprecision at the current parameter
     estimates to the residual sum-of-squares.  This performs well on
     data of the form 

                        y = f(x, theta) + eps

     (with 'var(eps) > 0').  It fails to indicate convergence on data
     of the form

                           y = f(x, theta)

     because the criterion amounts to comparing two components of the
     round-off error.  If you wish to test 'nls' on artificial data
     please add a noise component, as shown in the example below.

     The 'algorithm = "port"' code appears unfinished, and does not
     even check that the starting value is within the bounds. Use with
     caution, especially where bounds are supplied.

Author(s):

     Douglas M. Bates and Saikat DebRoy

References:

     Bates, D. M. and Watts, D. G. (1988) _Nonlinear Regression
     Analysis and Its Applications_, Wiley

     Bates, D. M. and Chambers, J. M. (1992) _Nonlinear models._
     Chapter 10 of _Statistical Models in S_ eds J. M. Chambers and T.
     J. Hastie, Wadsworth & Brooks/Cole.

     <URL: http://www.netlib.org/port/> for the Port library
     documentation.

See Also:

     'summary.nls', 'predict.nls', 'profile.nls'.

Examples:

     require(graphics)

     DNase1 <- subset(DNase, Run == 1)

     ## using a selfStart model
     fm1DNase1 <- nls(density ~ SSlogis(log(conc), Asym, xmid, scal), DNase1)
     summary(fm1DNase1)
     ## the coefficients only:
     coef(fm1DNase1)
     ## including their SE, etc:
     coef(summary(fm1DNase1))

     ## using conditional linearity
     fm2DNase1 <- nls(density ~ 1/(1 + exp((xmid - log(conc))/scal)),
                      data = DNase1,
                      start = list(xmid = 0, scal = 1),
                      algorithm = "plinear", trace = TRUE)
     summary(fm2DNase1)

     ## without conditional linearity
     fm3DNase1 <- nls(density ~ Asym/(1 + exp((xmid - log(conc))/scal)),
                      data = DNase1,
                      start = list(Asym = 3, xmid = 0, scal = 1),
                      trace = TRUE)
     summary(fm3DNase1)

     ## using Port's nl2sol algorithm
     fm4DNase1 <- nls(density ~ Asym/(1 + exp((xmid - log(conc))/scal)),
                      data = DNase1,
                      start = list(Asym = 3, xmid = 0, scal = 1),
                      trace = TRUE, algorithm = "port")
     summary(fm4DNase1)

     ## weighted nonlinear regression
     Treated <- Puromycin[Puromycin$state == "treated", ]
     weighted.MM <- function(resp, conc, Vm, K)
     {
         ## Purpose: exactly as white book p. 451 -- RHS for nls()
         ##  Weighted version of Michaelis-Menten model
         ## ----------------------------------------------------------
         ## Arguments: 'y', 'x' and the two parameters (see book)
         ## ----------------------------------------------------------
         ## Author: Martin Maechler, Date: 23 Mar 2001

         pred <- (Vm * conc)/(K + conc)
         (resp - pred) / sqrt(pred)
     }

     Pur.wt <- nls( ~ weighted.MM(rate, conc, Vm, K), data = Treated,
                   start = list(Vm = 200, K = 0.1),
                   trace = TRUE)
     summary(Pur.wt)

     ## Passing arguments using a list that can not be coerced to a data.frame
     lisTreat <- with(Treated,
                      list(conc1 = conc[1], conc.1 = conc[-1], rate = rate))

     weighted.MM1 <- function(resp, conc1, conc.1, Vm, K)
     {
          conc <- c(conc1, conc.1)
          pred <- (Vm * conc)/(K + conc)
         (resp - pred) / sqrt(pred)
     }
     Pur.wt1 <- nls( ~ weighted.MM1(rate, conc1, conc.1, Vm, K),
                    data = lisTreat, start = list(Vm = 200, K = 0.1))
     stopifnot(all.equal(coef(Pur.wt), coef(Pur.wt1)))

     ## Chambers and Hastie (1992) Statistical Models in S  (p. 537):
     ## If the value of the right side [of formula] has an attribute called
     ## 'gradient' this should be a matrix with the number of rows equal
     ## to the length of the response and one column for each parameter.

     weighted.MM.grad <- function(resp, conc1, conc.1, Vm, K)
     {
       conc <- c(conc1, conc.1)

       K.conc <- K+conc
       dy.dV <- conc/K.conc
       dy.dK <- -Vm*dy.dV/K.conc
       pred <- Vm*dy.dV
       pred.5 <- sqrt(pred)
       dev <- (resp - pred) / pred.5
       Ddev <- -0.5*(resp+pred)/(pred.5*pred)
       attr(dev, "gradient") <- Ddev * cbind(Vm = dy.dV, K = dy.dK)
       dev
     }

     Pur.wt.grad <- nls( ~ weighted.MM.grad(rate, conc1, conc.1, Vm, K),
                        data = lisTreat, start = list(Vm = 200, K = 0.1))

     rbind(coef(Pur.wt), coef(Pur.wt1), coef(Pur.wt.grad))

     ## In this example, there seems no advantage to providing the gradient.
     ## In other cases, there might be.

     ## The two examples below show that you can fit a model to
     ## artificial data with noise but not to artificial data
     ## without noise.
     x <- 1:10
     y <- 2*x + 3                            # perfect fit
     yeps <- y + rnorm(length(y), sd = 0.01) # added noise
     nls(yeps ~ a + b*x, start = list(a = 0.12345, b = 0.54321),
         trace = TRUE)
     ## Not run: 
     ## terminates in an error, because convergence cannot be confirmed:
     nls(y ~ a + b*x, start = list(a = 0.12345, b = 0.54321),
          trace = TRUE)
     ## End(Not run)

     ## the nls() internal cheap guess for starting values can be sufficient:

     x <- -(1:100)/10
     y <- 100 + 10 * exp(x / 2) + rnorm(x)/10
     nlmod <- nls(y ~  Const + A * exp(B * x), trace=TRUE)

     plot(x,y, main = "nls(*), data, true function and fit, n=100")
     curve(100 + 10 * exp(x / 2), col=4, add = TRUE)
     lines(x, predict(nlmod), col=2)

     ## The muscle dataset in MASS is from an experiment on muscle
     ## contraction on 21 animals.  The observed variables are Strip
     ## (identifier of muscle), Conc (Cacl concentration) and Length
     ## (resulting length of muscle section).
     utils::data(muscle, package = "MASS")

     ## The non linear model considered is
     ##       Length = alpha + beta*exp(-Conc/theta) + error
     ## where theta is constant but alpha and beta may vary with Strip.

     with(muscle, table(Strip)) # 2,3 or 4 obs per strip

     ## We first use the plinear algorithm to fit an overall model,
     ## ignoring that alpha and beta might vary with Strip.

     musc.1 <- nls(Length ~ cbind(1, exp(-Conc/th)), muscle,
                   start = list(th=1), algorithm="plinear")
     summary(musc.1)

     ## Then we use nls' indexing feature for parameters in non-linear
     ## models to use the conventional algorithm to fit a model in which
     ## alpha and beta vary with Strip.  The starting values are provided
     ## by the previously fitted model.
     ## Note that with indexed parameters, the starting values must be
     ## given in a list (with names):
     b <- coef(musc.1)
     musc.2 <- nls(Length ~ a[Strip] + b[Strip]*exp(-Conc/th),
                   muscle,
                   start = list(a=rep(b[2],21), b=rep(b[3],21), th=b[1]))
     summary(musc.2)

