  The problem of statistical inference for open chaotic systems measured with
error is complicated by the interaction of the uncertainty introduced by chaos,
and the various sources of random or external variation. Here a method of
representing measured data from large open chaotic systems subject to error as
collections of threads of plausible pseudo future histories to enable
statistical analysis is described. This representation provides asymptotically
consistent predictive distributions, for use in developing predictive
likelihood methods which: 1. provide a framework for variable selection, 2.
provide a framework for Bayesian updating, so for example 4 season ahead
predictions learn naturally as the 3rd season ahead is measured. 3. allows
examination of conditional scenarios along the future histories for planning
purposes. 4. allows the ranking of variable, delay combinations with higher
signal to noise ratio.
  The method is tested for learning and variable selection by examining its
behavior in predicting 9 years across 4 seasons of climate variables, including
local temperature and rainfall measurements at two locations, predicting up to
4 seasons ahead.
