mla: An Algorithm for Least-squares Curve Fitting

In ellessenne/mla: An Algorithm for Least-squares Curve Fitting

Description

This algorithm provides a numerical solution to the problem of minimizing a function. This is more efficient than the Gauss-Newton-like algorithm when starting from points very far from the final minimum. A new convergence test is implemented (RDM) in addition to the usual stopping criterion: stopping rule is when the gradients are small enough in the parameters metric (GH-1G).

This algorithm provides a numerical solution to the problem of minimizing a function. This is more efficient than the Gauss-Newton-like algorithm when starting from points very far from the final minimum. A new convergence test is implemented (RDM) in addition to the usual stopping criterion: stopping rule is when the gradients are small enough in the parameters metric (GH-1G).

Usage

mla(
  par,
  fn,
  gr = NULL,
  ...,
  hessian = FALSE,
  control = list(),
  verbose = FALSE
)

Arguments

par	A vector containing the initial values for the parameters.
fn	The function to be minimized (or maximized), with argument the vector of parameters over which minimization is to take place. It should return a scalar result.
gr	A function to return the gradient value for a specific point. If missing, finite-difference approximation will be used.
...	Further arguments to be passed to fn and gr.
hessian	A function to return the hessian matrix for a specific point. If missing, finite-difference approximation will be used.
control	A list of control parameters. See ‘Details’.
verbose	Equals to TRUE if report (parameters at iteration, function value, convergence criterion ...) at each iteration is requested. Default value is FALSE.

Details

Convergence criteria are very strict as they are based on derivatives of the log-likelihood in addition to the parameter and log-likelihood stability. In some cases, the program may not converge and reach the maximum number of iterations fixed at 500. In this case, the user should check that parameter estimates at the last iteration are not on the boundaries of the parameter space. If the parameters are on the boundaries of the parameter space, the identifiability of the model should be assessed. If not, the program should be run again with other initial values, with a higher maximum number of iterations or less strict convergence tolerances. The control argument is a list that can supply any of the following components:

• maxiter Optional maximum number of iterations for the iterative algorithm. Default is 500.

• eps Not documented (yet). Default is 1e-6.

• epsa Optional threshold for the convergence criterion based on the parameter stability. Default is 1e-6.

• epsb Optional threshold for the convergence criterion based on the log-likelihood stability. Default is 1e-6.

• epsd Optional threshold for the relative distance to minimum. This criterion has the nice interpretation of estimating the ratio of the approximation error over the statistical error, thus it can be used for stopping the iterative process whathever the problem. Default is 1e-6.

• digits Number of digits to print in outputs. Default value is 8.

• blinding Equals to TRUE if the algorithm is allowed to go on in case of an infinite or not definite value of function. Default value is FALSE.

• multipleTry Integer, different from 1 if the algorithm is allowed to go for the first iteration in case of an infinite or not definite value of gradients or hessian. This account for a starting point to far from the definition set. As many tries as requested in multipleTry will be done by changing the starting point of the algorithm. Default value is 25.

Value

A list with the following elements:

• par, stopping point value;

• value, function evaluation at the stopping point;

• ni, number of iterations before reaching stopping criterion;

• convergence, status of convergence: =1 if the convergence criteria were satisfied, =2 if the maximum number of iterations was reached, =4 if the algorithm encountered a problem in the function computation;

• hessian, a symmetric matrix giving an estimate of the Hessian at the solution found;

• ca, convergence criteria for parameters stabilisation;

• cb, convergence criteria for function stabilisation;

• rdm, convergence criteria on the relative distance to minimum.

Author(s)

Alessandro Gasparini ([e-mail])

Daniel Commenges

Melanie Prague

Amadou Diakite

Alessandro Gasparini

References

Donald W. Marquardt (1963). An algorithm for least-squares estimation of nonlinear parameters. Journal of the Society for Industrial and Applied Mathematics, 11(2):431–441

Daniel Commenges, H Jacqmin-Gadda, C. Proust, J. Guedj (2006). A Newton-like algorithm for likelihood maximization the robust-variance scoring algorithm. arxiv:math/0610402v2

Donald W. Marquardt (1963). An algorithm for least-squares estimation of nonlinear parameters. Journal of the Society for Industrial and Applied Mathematics, 11(2):431–441

Daniel Commenges, H Jacqmin-Gadda, C. Proust, J. Guedj (2006). A Newton-like algorithm for likelihood maximization the robust-variance scoring algorithm. arxiv:math/0610402v2

Examples

### 1
### initial values
par <- c(8, 9)
### your function
fn <- function(b) {
  return(4 * (b[1] - 5)^2 + (b[2] - 6)^2)
}
## Call
test1 <- mla(par = par, fn = fn)
test1

### 2
### initial values
b <- c(3, -1, 0, 1)
### your function
f2 <- function(b) {
  return((b[1] + 10 * b[2])^2 + 5 * (b[3] - b[4])^2 + (b[2] - 2 * b[3])^4 + 10 * (b[1] - b[4])^4)
}

## Call
test2 <- mla(par = b, fn = f2)
test2

ellessenne/mla documentation built on March 29, 2020, 12:20 p.m.