13th International Conference on Membrane Computing - MTA Sztaki

An analysis of correlative and quantitative causality in P systems
3 Correlative Causality
Associations among time-series of biological entities represent at least the
strength of relation between two species x i and x j . They can be measured by
several coefficient types, which can be classified into similarity and dissimilarity
measures. The first ones reflect the extent of similarity between species. The
larger the similarity between x i and x j , the more they are similar. In contrast,
dissimilarity measures reflect dissimilarities between x i and x j .
Correlation coefficients belong to the group of similarity measures and describe
at least the magnitude of the relation between two species. As a corollary
of the Cauchy-Schwarz inequality, the absolute value of each correlation coefficient
cannot exceed 1. However, these coefficients can be extended in order to
describe both magnitude and direction. Magnitude of a correlation represents
the strength of the relation: the strength of the tendency of variables to move
in the same or the opposite direction or how strong they covary across the set
of observations. The larger the absolute correlation, the stronger the variables
are associated. The direction of a correlation coefficient describes how two variables
are associated. If such a coefficient is positive, then the two variables move
in the same direction. Differently, if it is negative, then they move in opposite
directions.
Consider m to be the length of time-series available for each species of a set
X = {x 1 , x 2 , . . . , x n }. The time-series of x i and x j can be correlated directly to
compute the pairwise Pearson correlation coefficient given by:
∑ m
t=0
ρ(x i , x j ) =
((x i[t] − ¯x i )(x j [t] − ¯x j ))
√ ∑ m (
t=0 (x i[t] − ¯x i ) 2 )( ∑ m
t=0 (x (1)
j[t] − ¯x j ) 2 )
where ¯x i and ¯x j are the averages of x i [t] and x j [t], respectively. However, let
us suppose that at least one between x i and x j is in a stable-state, and then
ρ(x i , x j ) can not be defined. In this case, we assume that ρ(x i , x j ) = 0 because
no interesting relationships can be found between the two species.
A high Pearson correlation is an indication of coordinate and concurrent
behaviours, and can be used to gain knowledge about the regulative mechanisms
and then regarding cause-effect relationships. Pearson correlation values close to
1 indicate positive linear relationships between x i and x j , correlations equal to
0 indicate no linear associations, while correlations near to −1 indicate negative
linear relationships. Namely, the closer the coefficient is to either −1 or 1, the
stronger is the correlation between the variables (Figure 1).
In particular, a high correlation between two time-series may indicate a direct
interaction, an indirect interaction, or a joint regulation by a common unknown
regulator (Figure 2). However, only the direct interactions are of interest to infer
the regulatory mechanisms of a biological network.
For a better illustration, let us consider a simple example consisting of three
species: x 1 , x 2 and x 3 . Let us assume that, as represented in Figure 3 (a),
the reactions x 1 → x 2 and x 1 → x 3 exist, and that these reactions induce
strong correlations for the pairs (x 1 , x 2 ) and (x 1 , x 3 ). Therefore, we also observe
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