Some useful parameter transformations.
logit(p)expit(x)
log_barycentric(X)
inv_log_barycentric(Y)
numeric; a quantity in [0,1].
numeric; the log odds ratio.
numeric; a vector containing the quantities to be transformed according to the log-barycentric transformation.
numeric; a vector containing the log fractions.
Parameter transformations can be used in many cases to recast constrained optimization problems as unconstrained problems.
Although there are no limits to the transformations one can implement using the parameter_trans facilty, pomp provides a few ready-built functions to implement some very commonly useful ones.
The logit transformation takes a probability \(p\) to its log odds, \(\log\frac{p}{1-p}\). It maps the unit interval \([0,1]\) into the extended real line \([-\infty,\infty]\).
The inverse of the logit transformation is the expit transformation.
The log-barycentric transformation takes a vector \(X_i\), \(i=1,\dots,n\), to a vector \(Y_i\), where $$Y_i = \log\frac{X_i}{\sum_j X_j}.$$ If \(X\) is an \(n\)-vector, it takes every simplex defined by \(\sum_i X_i = c\), \(c\) constant, to n-dimensional Euclidean space \(R^n\).
The inverse of the log-barycentric transformation is implemented as inv_log_barycentric.
Note that it is not a true inverse, in the sense that it takes \(R^n\) to the unit simplex, \(\sum_i X_i = 1\).
Thus,
log_barycentric(inv_log_barycentric(Y)) == Y,
but
inv_log_barycentric(log_barycentric(X)) == X
only if sum(X) == 1.
More on implementing POMP models:
Csnippet,
accumulators,
basic_components,
covariate_table(),
distributions,
dmeasure_spec,
dprocess_spec,
parameter_trans(),
pomp-package,
prior_spec,
rinit_spec,
rmeasure_spec,
rprocess_spec,
skeleton_spec,
userdata