13th International Conference on Membrane Computing - MTA Sztaki

An analysis of correlative and quantitative causality in P systems
correlation between x i and x j is defined as:
ρ C1 (x i , x j ) = min
k≠i,j |ρ(x i, x j | x k )| (2)
where
ρ(x i , x j | x k ) =
ρ(x i , x j ) − ρ(x i , x k )ρ(x j , x k )
√
(1 − ρ(xi , x k ) 2 )(1 − ρ(x j , x k ) 2 )) . (3)
If there is x k ≠ x i , x j which explains all the correlation between x i and x j ,
then ρ C1 (x i , x j ) ∼ = 0 and the pair (x i , x j ) is conditionally independent given x k .
In this case, we say that on an undirected graph x i and x j are not adjacent but
separated by x k . Therefore, if ρ C1 (x i , x j ) is smaller than a given threshold, then
we consider that there isn’t a significant interaction between x i and x j . From
the definition, we have that if x i [t] = x k [t] or x j [t] = x k [t], for all t, then (3) is
not defined. In this case, we set ρ(x i , x j | x k ) = 0.
The first order partial correlation allows us to remove many false positives
computed by Pearson correlation alone. However, low values of the coefficients
(1) and (2) guarantee that an interaction between two time-series is missing,
while high values of (2) do not guarantee that two time-series interact. Therefore,
we consider ρ Call (x i , x j ), which describes the partial correlation between x i and
x j conditioned on all the other n − 2 species. We follow this strategy because it
is possible that the correlation conditioned to a single species is high, but the
correlation conditioned to all the other species is low. Let Ω be the correlation
matrix of the n species of X, that is the n × n matrix whose (i, j)-th entry is
ρ(x i , x j ). A very powerful result allows us to compute ρ Call (x i , x j ) by using Ω −1
[13]. In fact, we have
ω i,j
ρ Call (x i , x j ) = −√ (4)
ωi,i ω j,j
where ω i,j is the (i, j)-th entry of Ω −1 . The critical step in the application of
(4) is the reliable estimation of the inverse of the correlation matrix when Ω is
either singular or else numerically very close to singularity. We apply the spectral
decomposition, which is based on the use of eigenvalues and eigenvectors, to
compute Ω −1 . According to the spectral decomposition, a rank-deficient matrix
can be decomposed into a smaller number of factors than the original matrix
and still preserve all of the information in the matrix.
The following definition provides the rules to infer direct interactions among
species and represents the first step in order to study correlative cause-effect
relationships among them.
Definition 2 (Directed Correlation) We say that two time-series x i and x j
are directly correlated if indexes |ρ(x i , x j )| and |ρ Call (x i , x j )| are above two fixed
thresholds.
Although a combination in the use of Pearson and partial correlation can
be viewed as a technique to develop new hypothesis of interactions among biochemical
components [7], we point out that the study of time-shifts in biological
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