Given a nonlinear model expressed as an expression of the form
lhs ~ formula_for_rhs
and a start vector where parameters used in the model formula are named,
attempts to find the minimum of the residual sum of squares using the
Nash variant (Nash, 1979) of the Marquardt algorithm, where the linear
sub-problem is solved by a qr method. This is a restructured version
of a function by the same name from package nlmrt which is now
deprecated.
nlxb(formula, start, trace=FALSE, data, lower=-Inf, upper=Inf,
masked=NULL, weights=NULL, control)A list of the following items
A named vector giving the parameter values at the supposed solution.
The sum of squared residuals at this set of parameters.
The residual vector at the returned parameters.
The jacobian matrix (partial derivatives of residuals w.r.t. the parameters) at the returned parameters.
The number of residual evaluations (sum of squares computations) used.
The number of Jacobian evaluations used.
This is a modeling formula of the form (as in nls)
lhsvar ~ rhsexpression
for example,
y ~ b1/(1+b2*exp(-b3*tt))
You may also give this as a string.
Note that the residuals are computed within this code using
residual <- rhsexpression - lhsvar
which is the negative of the usual choice, but the sum of squares is the same.
A named parameter vector. For our example, we could use
start=c(b1=1, b2=2.345, b3=0.123)
Logical TRUE if we want intermediate progress to be reported. Default is FALSE.
A data frame containing the data of the variables in the formula. This data may, however, be supplied directly in the parent frame.
Lower bounds on the parameters. If a single number, this will be applied to all
parameters. Default -Inf.
Upper bounds on the parameters. If a single number, this will be applied to all
parameters. Default Inf.
Character vector of quoted parameter names. These parameters will NOT be altered by the algorithm. Masks may also be defined by setting lower and upper bounds equal for the parameters to be fixed. Note that the starting parameter value must also be the same as the lower and upper bound value.
A vector of fixed weights. The objective function that will be minimized is the
sum of squares where each residual is multiplied by the square root of the
corresponding weight. Default NULL implies unit weights.
A list of controls for the algorithm. These are:
watchMonitor progress if TRUE. Default is FALSE.
phiDefault is phi=1, which adds phi*Identity to
Jacobian inner product.
lamdaInitial Marquardt adjustment (Default 0.0001).
Odd spelling is deliberate.
offsetShift to test for floating-point equality.
Default is 100.
lamincFactor to use to increase lamda. Default is 10.
lamdecFactor to use to decrease lamda is lamdec/laminc.
Default lamdec=4.
femaxMaximum function (sum of squares) evaluations.
Default is 10000, which is extremely aggressive.
jemaxMaximum number of Jacobian evaluations.
Default is 5000.
rofftestDefault is TRUE. Use a termination test of the relative offset
orthogonality type. Useful for nonlinear regression problems.
smallsstestDefault is TRUE. Exit the function if the sum of squares
falls below (100 * .Machine$double.eps)^4 times the initial sumsquares. This
is a test for a ``small'' sum of squares, but there are problems which are
very extreme for which this control needs to be set FALSE.
John C Nash <nashjc@uottawa.ca>
nlxb attempts to solve the nonlinear sum of squares problem by using
a variant of Marquardt's approach to stabilizing the Gauss-Newton method using
the Levenberg-Marquardt adjustment. This is explained in Nash (1979 or 1990) in
the sections that discuss Algorithm 23.
In this code, we solve the (adjusted) Marquardt equations by use of the
qr.solve(). Rather than forming the J'J + lambda*D matrix, we augment
the J matrix with extra rows and the y vector with null elements.
Nash, J. C. (1979, 1990) _Compact Numerical Methods for Computers. Linear Algebra and Function Minimisation._ Adam Hilger./Institute of Physics Publications
others!!
cat("See examples in nlsr-package.Rd\n")
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