nls.lm  R Documentation 
The purpose of nls.lm
is to minimize the sum square of the
vector returned by the function fn
, by a modification of the
LevenbergMarquardt algorithm. The user may also provide a
function jac
which calculates the Jacobian.
nls.lm(par, lower=NULL, upper=NULL, fn, jac = NULL, control = nls.lm.control(), ...)
par 
A list or numeric vector of starting estimates. If

lower 
A numeric vector of lower bounds on each parameter. If
not given, the default lower bound for each parameter is set to

upper 
A numeric vector of upper bounds on each parameter. If
not given, the default upper bound for each parameter is set to

fn 
A function that returns a vector of residuals, the sum square
of which is to be minimized. The first argument of 
jac 
A function to return the Jacobian for the 
control 
An optional list of control settings. See 
... 
Further arguments to be passed to 
Both functions fn
and jac
(if provided) must return
numeric vectors. Length of the vector returned by fn
must
not be lower than the length of par
. The vector returned by
jac
must have length equal to
length(\code{fn}(\code{par}, …)) * length(\code{par}).
The control
argument is a list; see nls.lm.control
for
details.
Successful completion.
The accuracy of nls.lm
is controlled by the convergence
parameters ftol
, ptol
, and gtol
. These
parameters are used in tests which make three types of comparisons
between the approximation par and a solution
par0. nls.lm
terminates when any of the tests
is satisfied. If any of the convergence parameters is less than
the machine precision, then nls.lm
only attempts to satisfy
the test defined by the machine precision. Further progress is not
usually possible.
The tests assume that fn
as well as jac
are
reasonably well behaved. If this condition is not satisfied, then
nls.lm
may incorrectly indicate convergence. The validity
of the answer can be checked, for example, by rerunning
nls.lm
with tighter tolerances.
First convergence test.
If z denotes the Euclidean norm of a vector z, then
this test attempts to guarantee that
fvec < (1 + \code{ftol})fvec0,
where fvec0 denotes the result of fn
function
evaluated at par0. If this condition is satisfied
with ftol
~ 10^(k), then the final
residual norm fvec has k significant decimal digits
and info
is set to 1 (or to 3 if the second test is also
satisfied). Unless high precision solutions are required, the
recommended value for ftol
is the square root of the machine
precision.
Second convergence test.
If D is the diagonal matrix whose entries are defined by the
array diag
, then this test attempt to guarantee that
D*(par  par0) < \code{ptol}D*par0,
If this condition is satisfied with ptol
~ 10^(k), then the larger components of
D*par have k significant decimal digits and
info
is set to 2 (or to 3 if the first test is also
satisfied). There is a danger that the smaller components of
D*par may have large relative errors, but if
diag
is internally set, then the accuracy of the components
of par is usually related to their sensitivity. Unless high
precision solutions are required, the recommended value for
ptol
is the square root of the machine precision.
Third convergence test.
This test is satisfied when the cosine of the angle between the
result of fn
evaluation fvec and any column of the
Jacobian at par is at most gtol
in absolute value.
There is no clear relationship between this test and the accuracy
of nls.lm
, and furthermore, the test is equally well
satisfied at other critical points, namely maximizers and saddle
points. Therefore, termination caused by this test (info
=
4) should be examined carefully. The recommended value for
gtol
is zero.
Unsuccessful completion.
Unsuccessful termination of nls.lm
can be due to improper
input parameters, arithmetic interrupts, an excessive number of
function evaluations, or an excessive number of iterations.
Improper input parameters.
info
is set to 0 if length(\code{par}) = 0, or
length(fvec) < length(\code{par}), or ftol
< 0,
or ptol
< 0, or gtol
< 0, or maxfev
<= 0, or factor
<= 0.
Arithmetic interrupts.
If these interrupts occur in the fn
function during an
early stage of the computation, they may be caused by an
unacceptable choice of par by nls.lm
. In this case,
it may be possible to remedy the situation by rerunning
nls.lm
with a smaller value of factor
.
Excessive number of function evaluations.
A reasonable value for maxfev
is 100*(length(\code{par}) + 1). If the
number of calls to fn
reaches maxfev
, then this
indicates that the routine is converging very slowly as measured
by the progress of fvec and info
is set to 5. In this
case, it may be helpful to force diag
to be internally set.
Excessive number of function iterations.
The allowed number of iterations defaults to 50, can be increased if
desired.
The list returned by nls.lm
has methods
for the generic functions coef
,
deviance
, df.residual
,
print
, residuals
, summary
,
confint
,
and vcov
.
A list with components:
par 
The best set of parameters found. 
hessian 
A symmetric matrix giving an estimate of the Hessian at the solution found. 
fvec 
The result of the last 
info 

message 
character string indicating reason for termination 
.
diag 
The result list of 
niter 
The number of iterations completed before termination. 
rsstrace 
The residual sum of squares at each iteration. Can be used to check the progress each iteration. 
deviance 
The sum of the squared residual vector. 
The public domain FORTRAN sources of MINPACK package by J.J. Moré, implementing the LevenbergMarquardt algorithm were downloaded from https://netlib.org/minpack/, and left unchanged. The contents of this manual page are largely extracted from the comments of MINPACK sources.
J.J. Moré, "The LevenbergMarquardt algorithm: implementation and theory," in Lecture Notes in Mathematics 630: Numerical Analysis, G.A. Watson (Ed.), SpringerVerlag: Berlin, 1978, pp. 105116.
optim
, nls
, nls.lm.control
###### example 1 ## values over which to simulate data x < seq(0,5,length=100) ## model based on a list of parameters getPred < function(parS, xx) parS$a * exp(xx * parS$b) + parS$c ## parameter values used to simulate data pp < list(a=9,b=1, c=6) ## simulated data, with noise simDNoisy < getPred(pp,x) + rnorm(length(x),sd=.1) ## plot data plot(x,simDNoisy, main="data") ## residual function residFun < function(p, observed, xx) observed  getPred(p,xx) ## starting values for parameters parStart < list(a=3,b=.001, c=1) ## perform fit nls.out < nls.lm(par=parStart, fn = residFun, observed = simDNoisy, xx = x, control = nls.lm.control(nprint=1)) ## plot model evaluated at final parameter estimates lines(x,getPred(as.list(coef(nls.out)), x), col=2, lwd=2) ## summary information on parameter estimates summary(nls.out) ###### example 2 ## function to simulate data f < function(TT, tau, N0, a, f0) { expr < expression(N0*exp(TT/tau)*(1 + a*cos(f0*TT))) eval(expr) } ## helper function for an analytical gradient j < function(TT, tau, N0, a, f0) { expr < expression(N0*exp(TT/tau)*(1 + a*cos(f0*TT))) c(eval(D(expr, "tau")), eval(D(expr, "N0" )), eval(D(expr, "a" )), eval(D(expr, "f0" ))) } ## values over which to simulate data TT < seq(0, 8, length=501) ## parameter values underlying simulated data p < c(tau = 2.2, N0 = 1000, a = 0.25, f0 = 8) ## get data Ndet < do.call("f", c(list(TT = TT), as.list(p))) ## with noise N < Ndet + rnorm(length(Ndet), mean=Ndet, sd=.01*max(Ndet)) ## plot the data to fit par(mfrow=c(2,1), mar = c(3,5,2,1)) plot(TT, N, bg = "black", cex = 0.5, main="data") ## define a residual function fcn < function(p, TT, N, fcall, jcall) (N  do.call("fcall", c(list(TT = TT), as.list(p)))) ## define analytical expression for the gradient fcn.jac < function(p, TT, N, fcall, jcall) do.call("jcall", c(list(TT = TT), as.list(p))) ## starting values guess < c(tau = 2.2, N0 = 1500, a = 0.25, f0 = 10) ## to use an analytical expression for the gradient found in fcn.jac ## uncomment jac = fcn.jac out < nls.lm(par = guess, fn = fcn, jac = fcn.jac, fcall = f, jcall = j, TT = TT, N = N, control = nls.lm.control(nprint=1)) ## get the fitted values N1 < do.call("f", c(list(TT = TT), out$par)) ## add a blue line representing the fitting values to the plot of data lines(TT, N1, col="blue", lwd=2) ## add a plot of the log residual sum of squares as it is made to ## decrease each iteration; note that the RSS at the starting parameter ## values is also stored plot(1:(out$niter+1), log(out$rsstrace), type="b", main="log residual sum of squares vs. iteration number", xlab="iteration", ylab="log residual sum of squares", pch=21,bg=2) ## get information regarding standard errors summary(out)
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