In [10] we considered an industrial process for extracting
aluminium by electrolysis from a solution of alumina.
Experiments are performed, under similar operating conditions,
to measure the percentage concentration of
alumina in solution every ten minutes, terminating when the concentration
falls to a prespecified level. The measurements are the
responses 
representing, for runs j,
the concentrations of alumina remaining in solution at thirteen time points
i timed from the end of the experiment (the alumina level
is essentially fixed at the end of the experiment).
A key property of the process is that the experiments are
exchangeable, and one aim of the analysis will be to assess the underlying mean values of the
process. Data is available for three runs of the process.
The following exchangeable regressions model (an adaptation of the localised regression model introduced in [11] and of the dynamic linear model discussed in [13]), which takes into account various physical determinants and complicating features, was suggested to represent the process.
The intercepts 
and the slopes 
are
second-order exchangeable over runs, 
, and the are uncorrelated with the
. We allow for the regression slopes to drift slowly over time, and so
introduce strong correlations between neighbouring slopes and
choose a correlation function so that the slopes
have the markov property in being mutually uncorrelated given their neighbours.
One way of achieving this aim is to use the following correlation
function:
where, for convenience, we define 
to be the
correlation between the most distant slopes 
and 
, so that values of
describe the degree of local stability. As we expect only minor
regression drift, we feel that values of about 
,
giving a correlation of about 0.9913 between neighbouring
slopes, might be appropriate. Smaller values of 
will still yield high neighbouring
correlations, but the dependencies between slopes will tail off much more quickly.
The error term 
is constructed from uncorrelated components
representing various error terms: simple noise, a random walk with drift,
and an autoregressive term. (Full details of the development of the error structure can be
found in [4].)
To assess the sensitivity of the model to changes in the stability
parameter, and so discover whether our actual choice needs careful
thought, we construct the model for a range of values of and
estimate the regression slopes. We assess the overall implications of
changing the stability parameter by evaluating the structure of the
resolution transform for each .
