notLog2 and notExp2 are alternatives to log
and exp or notLog and notExp for
re-parameterization of variance parameters. They are used by the
pdTens and pdIdnot classes which in turn implement
smooths for gamm.
The functions are typically used to ensure that smoothing parameters are
positive, but the notExp2 is not monotonic: rather it cycles between
`effective zero' and `effective infinity' as its argument changes. The
notLog2 is the inverse function of the notExp2 only over an
interval centered on zero.
Parameterizations using these functions ensure that estimated smoothing
parameters remain positive, but also help to ensure that the likelihood is
never indefinite: once a working parameter pushes a smoothing parameter below
`effetive zero' or above `effective infinity' the cyclic nature of the
notExp2 causes the likelihood to decrease, where otherwise it might
simply have flattened.
This parameterization is really just a numerical trick, in order to get
lme to fit gamm models, without failing due to indefiniteness.
Note in particular that asymptotic results on the likelihood/REML criterion are
not invalidated by the trick,
unless parameter estimates end up close to the effective zero or effective
infinity: but if this is the case then the asymptotics would also have been invalid
for a conventional monotonic parameterization.
This reparameterization was made necessary by some modifications to the
underlying optimization method in lme introduced in nlme 3.1-62. It is
possible that future releases will return to the notExp parameterization.
Note that you can reset `effective zero' and `effective infinity': see below.