R: Addresses NLS problems with the Levenberg-Marquardt algorithm
nls.lm
R Documentation
Addresses NLS problems with the Levenberg-Marquardt algorithm
Description
The purpose of nls.lm is to minimize the sum square of the
vector returned by the function fn, by a modification of the
Levenberg-Marquardt algorithm. The user may also provide a
function jac which calculates the Jacobian.
A list or numeric vector of starting estimates. If
par is a list, then each element must be of length 1.
lower
A numeric vector of lower bounds on each parameter. If
not given, the default lower bound for each parameter is set to
-Inf.
upper
A numeric vector of upper bounds on each parameter. If
not given, the default upper bound for each parameter is set to
Inf.
fn
A function that returns a vector of residuals, the sum square
of which is to be minimized. The first argument of fn must be
par.
jac
A function to return the Jacobian for the fn function.
control
An optional list of control settings. See nls.lm.control for
the names of the settable control values and their effect.
...
Further arguments to be passed to fn and jac.
Details
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.
Value
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 fn evaluation; that is, the
residuals.
info
info is an integer code indicating
the reason for termination.
0
Improper input parameters.
1
Both actual and predicted relative reductions in the
sum of squares are at most ftol.
2
Relative error between two consecutive iterates is
at most ptol.
3
Conditions for info = 1 and info = 2 both hold.
4
The cosine of the angle between fvec and any column
of the Jacobian is at most gtol in absolute value.
5
Number of calls to fn has reached maxfev.
6
ftol is too small. No further reduction in the sum
of squares is possible.
7
ptol is too small. No further improvement in the
approximate solution par is possible.
8
gtol is too small. fvec is orthogonal to the
columns of the Jacobian to machine precision.
9
The number of iterations has reached maxiter.
message
character string indicating reason for termination
.
diag
The result list of diag. See Details.
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.
###### 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)