Hi all,
I am wondering if anyone has implemented (or at least tried to) an automatic
reparametrization in order to satisfy "trivial" constraints (in the sense of
Dennis & Schnabel, 1983) in optimization problems.
To be perhaps clearer let us consider a simple bi-exponential model for some
recorded signal (sorry for the LaTex notations I hope they aren't too
confusing):
$s(t) = A (p_{fast} \exp(-\frac{t}{t_{fast}}) + p_{slow}
\exp(-\frac{t}{t_{slow}}) $
where we would have: $t_{fast} > 0$ and $t_{slow} > 0$ , $0 < p_{fast},
p_{slow} < 1$ and $p_{fast}+p_{slow}=1$. In addition we could want to
enforce $A>0$, say, because our signal corresponds physically to a
concentration and for ease of interpretation having $t_{fast} < t_{slow}$
would not hurt. We could then reparametrize our problem with:
$A = \exp (\alpha)$
$t_{fast} = \exp (\tau_{fast})$
$t_{slow} = t_{fast} + \exp (\tau_{slow})$
$p_{fast} = 0.5 + 0.5 \frac{\pi_{fast}}{\sqrt{1+\pi_{fast}^2}}$
$p_{slow} = 1 - p_{fast}$
and the four parameters: $\alpha, \tau_{fast}, \tau_{slow}, \pi_{fast}$
would be unconstrained.
One could then think of allowing "optim" users to specify more general
boundaries ("lower", "upper") which would be used to generate a parameter
transformation function. This function, say u2c (for "unconstrained to
constrained"), would take a vector of parameters as one of its arguments
and would return another vector (perhaps of a longer length). Then optim
would internaly optimize f(u2c(p)) instead of f. Clearly more sophisticated
constraint specifications (than "lower" and "upper") would be required, like
a way to declare that some parameters form a simplex or that others are
encoding a variance-covariance matrix (in the latter case the pdLogChol
function of nlme already provides the reparametrization if I'm not wrong).
Assuming that such a functionality does not exist yet in R (if it does,
sorry to have missed it) do you guys think that:
1) it's not necessary because users can take care of it for themselves
2) it would be complicated and not general enough
3) it would be worth trying
Thanks for your opinions,
Christophe
--
A Master Carpenter has many tools and is expert with most of them. If you
only know how to use a hammer, every problem starts to look like a nail.
Stay away from that trap.
Richard B Johnson.
--
Christophe Pouzat
Laboratoire de Physiologie Cerebrale
CNRS UMR 8118
UFR biomedicale de l'Universite Paris V
45, rue des Saints Peres
75006 PARIS
France
tel: +33 (0)1 42 86 38 28
fax: +33 (0)1 42 86 38 30
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