maxLik (version 1.0-0)

maxBFGS: BFGS, SANN and Nelder-Mead Maximization

Description

These functions are wrappers for optim where the arguments are compatible with maxNR. Note that there is a maxNR-based BFGS implementation maxBFGSR.

Usage

maxBFGS(fn, grad = NULL, hess=NULL, start, fixed = NULL,
   print.level = 0, iterlim = 200, constraints = NULL,
   tol = 1e-08, reltol=tol,
   finalHessian=TRUE,
parscale=rep(1, length=length(start)), ... )
maxSANN(fn, grad = NULL, hess = NULL, start, fixed = NULL,
   print.level = 0, iterlim = 10000, constraints = NULL,
   tol = 1e-08, reltol=tol,
finalHessian=TRUE,
cand = NULL, temp = 10, tmax = 10,
   parscale = rep(1, length = length(start)),
   random.seed = 123, ... )
maxNM(fn, grad = NULL, hess = NULL, start, fixed = NULL,
   print.level = 0, iterlim = 500, constraints = NULL,
   tol = 1e-08, reltol=tol,
finalHessian=TRUE,
 parscale = rep(1, length = length(start)),
   alpha = 1, beta = 0.5, gamma = 2, ...)

Arguments

fn

function to be maximised. Must have the parameter vector as the first argument. In order to use numeric gradient and BHHH method, fn must return vector of observation-specific likelihood values. Those are summed by maxNR if

grad

gradient of the function. Must have the parameter vector as the first argument. If NULL, numeric gradient is used (only maxBFGS uses gradient). Gradient may return a matrix, where columns correspond to the parameters and rows t

hess

Hessian of the function. Not used by any of these methods, for compatibility with maxNR.

start

initial values for the parameters.

fixed

parameters that should be fixed at their starting values: either a logical vector of the same length as argument start, a numeric (index) vector indicating the positions of the fixed parameters, or a vector of character stri

print.level

a larger number prints more working information.

iterlim

maximum number of iterations.

constraints

either NULL for unconstrained optimization or a list with two components. The components may be either eqA and eqB for equality-constrained optimization $A \theta + B = 0$; or ineqA and

tol, reltol

the relative convergence tolerance (see optim). tol is for compatibility with maxNR.

finalHessian

how (and if) to calculate the final Hessian. Either FALSE (not calculate), TRUE (use analytic/numeric Hessian) or "bhhh"/"BHHH" for information equality approach. The latter approach is only suit

cand

a function used in the "SANN" algorithm to generate a new candidate point; if it is NULL, a default Gaussian Markov kernel is used (see argument gr of optim)

temp

controls the '"SANN"' method. It is the starting temperature for the cooling schedule. Defaults to '10'.

tmax

is the number of function evaluations at each temperature for the '"SANN"' method. Defaults to '10'. (see optim)

random.seed

an integer used to seed R's random number generator. This is to ensure replicability when the Simulated Annealing method is used. Defaults to 123.

parscale

A vector of scaling values for the parameters. Optimization is performed on 'par/parscale' and these should be comparable in the sense that a unit change in any element produces about a unit change in the scaled value. (see

alpha, beta, gamma

Scaling parameters for the '"Nelder-Mead"' method. 'alpha' is the reflection factor (default 1.0), 'beta' the contraction factor (0.5) and 'gamma' the expansion factor (2.0). (see optim

...

further arguments for fn and grad.

Value

• Object of class "maxim":
• maximum
  value of fn at maximum.
• estimate
  best set of parameters found.
• gradient
  vector, gradient at parameter value estimate.
• gradientObs
  matrix of gradients at parameter value estimate evaluated at each observation (only if grad returns a matrix or grad is not specified and fn returns a vector).
• hessian
  value of Hessian at optimum.
• code
  integer. Success code, 0 is success (see optim).
• message
  character string giving any additional information returned by the optimizer, or NULL.
• fixed
  logical vector indicating which parameters are treated as constants.
• iterations
  two-element integer vector giving the number of calls to fn and gr, respectively. This excludes those calls needed to compute the Hessian, if requested, and any calls to fn to compute a finite-difference approximation to the gradient.
• type
  character string "BFGS maximisation".
• constraints
  A list, describing the constrained optimization (NULL if unconstrained). Includes the following components:

  • type

  {type of constrained optimization} outer.iterations{number of iterations in the constraints step} barrier.value{value of the barrier function}

Details

The state (or seed) of R's random number generator is saved at the beginning of the maxSANN function and restored at the end of this function so that this function does not affect the generation of random numbers although the random seed is set to argument random.seed and the SANN algorithm uses random numbers.

See Also

optim, nlm, maxNR, maxBHHH, maxBFGSR.

Examples

# Maximum Likelihood estimation of the parameter of Poissonian distribution
n <- rpois(100, 3)
loglik <- function(l) n*log(l) - l - lfactorial(n)
# we use numeric gradient
summary(maxBFGS(loglik, start=1))
# you would probably prefer mean(n) instead of that ;-)
# Note also that maxLik is better suited for Maximum Likelihood
###
### Now an example of constrained optimization
###
f <- function(theta) {
  x <- theta[1]
  y <- theta[2]
  exp(-(x^2 + y^2))
  ## Note: you may want to use exp(- theta \%*\% theta) instead ;-)
}
## use constraints: x + y >= 1
A <- matrix(c(1, 1), 1, 2)
B <- -1
res <- maxNM(f, start=c(1,1), constraints=list(ineqA=A, ineqB=B),
print.level=1)
print(summary(res))