13th International Conference on Membrane Computing - MTA Sztaki

An analysis of correlative and quantitative causality in P systems
centrations, propagate through the networks and generate emergent patterns of
correlations. It combines several correlation coefficients to develop similarity indexes
which can be interpreted as fingerprint of underlying cause-effect events in
biological pathways. In particular, since experiments conducted under identical
conditions do not necessarily lead to identical results, we also focused, in Section
4, on different factors causing this variability. In fact, the computation of correlation
indexes from experimental data is necessarily complicate by uncertainty
due to measurements errors, natural fluctuations, noise, artifacts, unexpected external
variations effecting the experiment and missing data. As a consequence,
noise can affect the correlative signals, by making it weak. Therefore, the correlative
analysis that we described should take these uncertainties into account
as it could influence the correlation estimates and the predictive accuracy of the
resulting P system model.
As an ulterior step, to fix this problem, an initial phase of data preparation
and preprocessing could be applied [4]. It has to involve the elimination of
both noise and artifacts from experimental data. Let us consider a set of experimental
data obtained by sampling, possibly at a constant rate τ, substance
concentrations and chemo-physical parameter values of a certain biochemical
system. To remove artifacts from substance and parameter time-series, we can
consider curve fitting methods 4 which are often employed to find a smooth curve
which fits noisy data by reducing their random component while preserving the
main trend of the dynamics under investigation. Of course, if data are affected
by other kinds of errors regarding, for instance, consistency, integrity, or outliers,
then ad hoc techniques must be used [16], but it is out of the scope of this paper
to consider particular methods to process raw data. After such a preprocessing
of experimental data, we assume that fluctuations and measurement errors are
normally distributed around the average trend of the system dynamics, therefore
each observed substance and parameter time-series is fitted by a smooth
function using least-squares theory.
The transition P system Π which is defined based on the correlative causality
relations provides a correspondence between quantitative and correlative notions
of causality. When considering a time-series as an object of the transition P
system, its causes can either be nonexistent, which shows that the time-series is
not correlated to any other or it can be a single rule, which serves to pinpoint
the set of directly correlated or directly caused time-series. It remains to be
seen how these results presented in Section 4 can be extended to a more varied
combination of time-series (which corresponds to a generic multiset in Π). The
case studies in Section 5 present the quantitative-correlative correspondence for
more general cases.
Finally, we would like to point out that when causality is extracted by means
of correlative relations between time-series, then it has to be present between
4 Curve fitting is the process of constructing a curve which has the best fit to a series
of data points. Curve fitting can involve either interpolation, where an exact fit to
the data is required, or smoothing, in which a “smooth” function is constructed that
approximately fits the data.
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