A.2 Convergence citeria

WOMBAT calculates four different criteria to determine whether an analysis has converged. The first two are simple changes, available for all maximisation algorithms, the other two are based on the derivatives of the log likelihood function, i.e. can only be obtained for the AI algorithm. The criteria are :

1.
The increase in log likelihood values between subsequent iterates, i.e.     t      t−1
log ℒ − logℒ  , with     t
logℒ  the log likelihood for iterate t  .
2.
The change in the vector of parameter estimates from the last iterate. This is evaluated as
┌ ------------------------
││ ∑p (        )2 ∑p (  )2
∘     ˆθti − ˆθti−1 ∕    θˆti
  i=1            i=1
(A.1)

where θˆt
 i  denotes the estimate of the i− th parameter from iterate t  , and p  is the number of parameters.

3.
The norm of the gradient vector, i.e., for git= ∂log ℒt∕∂θi  ,
┌ --------
││ ∑p
∘    (gti)2
  i=1
(A.2)

4.
The ‘Newton decrement’, i.e.
  ∑p ∑p
−       gtigtjHtij
  i=1j=1
(A.3)

where Ht
 ij  is the ij− the element of the inverse of the average information matrix for iterate t  . This gives a measure of the expected difference of     t
log ℒ  from the maximum, and has been suggested as an alternative convergence criterion [2].


Table A.1: Default thresholds for convergence criteria
Algorithm
Criterion AI (PX-) EM Powell
Change in logℒ < 5 ×10− 4  < 10−5  < 10−4
Change in parameters < 10− 8  < 10−8  < 10−8
Norm of gradient vector < 10− 3
Newton decrement not used

Default values for the thresholds for these criteria used in WOMBAT are summarised in Table A.1.

N.B.: Current values used are rather stringent; ‘softer’ limits combining several criteria may be more appropriate for practical estimation.