Nash variant of Marquardt nonlinear least squares solution via qr linear solver.

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Description

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.

Usage

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   nlfb(start, resfn, jacfn=NULL, trace=FALSE, lower=-Inf, upper=Inf, 
         maskidx=NULL, data=NULL, control, ...)

Arguments

resfn

A function that evaluates the residual vector for computing the elements of the sum of squares function at the set of parameters start.

jacfn

A function that evaluates the Jacobian of the sum of squares function, that is, the matrix of partial derivatives of the residuals with respect to each of the parameters. If NULL (default), uses an approximation. ?? put in character form as in optimx??

The Jacobian must be returned as the attribute "gradient" of this function, allowing jacfn to have the same name and be the same code block as resfn, which may permit some efficiencies of computation.

start

A named parameter vector. For our example, we could use start=c(b1=1, b2=2.345, b3=0.123) nls() takes a list, and that is permitted here also.

trace

Logical TRUE if we want intermediate progress to be reported. Default is FALSE.

lower

Lower bounds on the parameters. If a single number, this will be applied to all parameters. Default -Inf.

upper

Upper bounds on the parameters. If a single number, this will be applied to all parameters. Default Inf.

maskidx

Vector if indices of the parameters to be masked. These parameters will NOT be altered by the algorithm. Note that the mechanism here is different from that in nlxb which uses the names of the parameters.

data

Data frame of variables used by resfn and jacfn to compute the required residuals and Jacobian.

control

A list of controls for the algorithm. These are:

watch

Monitor progress if TRUE. Default is FALSE.

phi

Default is phi=1, which adds phi*Identity to Jacobian inner product.

lamda

Initial Marquardt adjustment (Default 0.0001). Odd spelling is deliberate.

offset

Shift to test for floating-point equality. Default is 100.

laminc

Factor to use to increase lamda. Default is 10.

lamdec

Factor to use to decrease lamda is lamdec/laminc. Default lamdec=4.

femax

Maximum function (sum of squares) evaluations. Default is 10000, which is extremely aggressive.

jemax

Maximum number of Jacobian evaluations. Default is 5000.

ndstep

Stepsize to use to computer numerical Jacobian approximatin. Default is 1e-7.

rofftest

Default is TRUE. Use a termination test of the relative offset orthogonality type. Useful for nonlinear regression problems.

smallsstest

Default 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.

...

Any data needed for computation of the residual vector from the expression rhsexpression - lhsvar. Note that this is the negative of the usual residual, but the sum of squares is the same. It is not clear how the dot variables should be used, since data should be in 'data'.

Details

nlfb 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.

Value

A list of the following items

coefficients

A named vector giving the parameter values at the supposed solution.

ssquares

The sum of squared residuals at this set of parameters.

resid

The residual vector at the returned parameters.

jacobian

The jacobian matrix (partial derivatives of residuals w.r.t. the parameters) at the returned parameters.

feval

The number of residual evaluations (sum of squares computations) used.

jeval

The number of Jacobian evaluations used.

Note

Special notes, if any, will appear here.

Author(s)

John C Nash <nashjc@uottawa.ca>

References

Nash, J. C. (1979, 1990) _Compact Numerical Methods for Computers. Linear Algebra and Function Minimisation._ Adam Hilger./Institute of Physics Publications

others!!

See Also

Function nls(), packages optim and optimx.

Examples

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cat("See examples in nls14-package.Rd\n")