R: Nonlinear Least Squares

nls {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 (and `formula` is not a self-starting model, see `selfStart`), 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. Can be abbreviated.
`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` | 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. To use these is deprecated, as they are available from `convInfo` now.

### 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, θ) + eps

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

y = f(x, θ)

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.

### Note

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.

### Author(s)

Douglas M. Bates and Saikat DebRoy: David M. Gay for the Fortran code used by `algorithm = "port"`.

### 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.

https://www.netlib.org/port/ for the Port library documentation.

### See Also

`summary.nls`, `predict.nls`, `profile.nls`.

Self starting models (with ‘automatic initial values’): `selfStart`.

### 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")
    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))
    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),
                     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))
    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))
    ## terminates in an error, because convergence cannot be confirmed:
    try(nls(y ~ a + b*x, start = list(a = 0.12345, b = 0.54321)))

    ## 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))

    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.

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

    ## 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]))
    ## IGNORE_RDIFF_BEGIN
    summary(musc.2)
    ## IGNORE_RDIFF_END

[Package stats version 4.0.5 Index]