13th International Conference on Membrane Computing - MTA Sztaki

An analysis of correlative and quantitative causality in P systems
directionality: what happens first ought to be upstream of what happens next
[20].
Cross-correlation can be applied to infer causal-effect relationships among
time-series. It extends Pearson correlation [6] by determining the best correlations
among variables shifted in time. For time-series x i and x j of length m, the
cross-correlation at lag τ is defined as:
φ(x i , x j , τ) =
∑ m−τ
t=0 ((x i[t] − ¯x i )(x j [t + τ] − ¯x j ))
√
(
∑ m
t=0 (x i[t] − ¯x i ) 2 )( ∑ m
t=0 (x j[t] − ¯x j ) 2 ) . (5)
In particular, if at least one between x i and x j is in a stable-state, then we
set φ(x i , x j , τ) = 0 because stable-state is an indication that a species is not
involved in a cause-effect relationships.
By using the cross-correlation we introduce a definition of cause-effect between
time-series which extends that of directed correlation. This concept of
causality rests on the fact that predictability can be tested by determining if
one time-series is related to past or current values of another time-series.
Definition 3 (Cross-Correlation Causality) We say that a time-series x i
causes another time-series x j with lag τ if
max {|φ(x i , x j , θ)|} = |φ(x i , x j , τ)| . (6)
θ
τ
Let us assume that x i causes x j with a lag τ 1 , x 1→
i xj , but that there is
x z such that an indirect causal relationship x i → x z → x j exists. Given τ 1 , we
consider the first order partial cross-correlation to correct for the delayed effect
of x z on the cross-correlation between x i and x j :
where
with
φ C1 (x i , x j ) =
min |ψ(x i , x j , τ 1 | x z , τ 2 )| (7)
0≤τ 2