13th International Conference on Membrane Computing - MTA Sztaki

R. Pagliarini, O. Agrigoroaiei, G. Ciobanu, V. Manca
s ∈ G. Since G has at least two elements there exists some s ′ ∈ R, not necessarily
different from s, such that G ≥ s+s ′ , which is equivalent to G−s ′ ≥ s. Therefore
rhs(G) ∩ x i = x i = rhs(G − s ′ ) ∩ x i which contradicts G being a cause for x i .
We show that each possible cause of x i corresponds to a certain correlation
of x i as a time-series with the other time-series in X. In Proposition 2 we show
that for a time-series x i not to be cross-correlated with any other time-series nor
to cause any other time-series is equivalent to the object x i having the empty
multiset of rules as cause in Π.
Proposition 2. The empty multiset of rules 0 R is a cause for x i in Π if and
only if both C xi and D xi are empty.
Proof. If 0 R is a cause for x i then it follows that there is no rule r ∈ R such that
lhs(r) ≤ x i , which is equivalent to saying that x i cannot be the left hand side
of any rule, therefore both C xi and D xi are empty.
If both C xi and D xi are empty then there is no rule r ∈ R such that lhs(r) ≤
x i \rhs(0 R ) = x i therefore the first condition of Definition 1 is fulfilled. The
second condition follows immediately since there is no rule r such that r ∈ 0 R .
In the next proposition we show that having certain rules which correspond
to a time-series x j as causes for x i is equivalent to x i i) being directly correlated
with x j , ii) being directly caused by x j or iii) being directly correlated with a
time-series directly caused by x j .
Proposition 3. Consider x i ∈ X. Then the following hold:
1. r j is a cause for x i ⇔ x i ∈ C xj ⇔ r i is a cause for x j ;
2. r j is a cause for x i ⇔ i = j and C xi ≠ ∅;
3. r jk is a cause for x i ⇔ x i ∈ C xk , or i = k and x i ∈ D xj .
Proof. We start by showing that for any rule s, the multiset G = s is a cause
for x i in Π if and only if x i ∈ rhs(s).
If x i ∈ rhs(s) then the first condition of Definition 1 is always fulfilled, since
x i \rhs(G) = 0 and therefore there exists no rule r such that lhs(r) ≤ x i \rhs(G).
The second condition follows from r ∈ G implies r = s, thus rhs(G − r) ∩ x i =
0 < rhs(G) ∩ x i = x i .
If G = s is a cause for x i then rhs(s) ∩ x i > 0 (supposing otherwise would
contradict the second condition of Definition 1), i.e., x i ∈ rhs(s).
Therefore r j is a cause for x i iff x i ∈ α j , which is equivalent to x i ∈ C xj .
However direct correlation is symmetrical therefore it is equivalent to x j ∈ C xi .
The latter is equivalent to x j ∈ α i = rhs(r i ), which is equivalent to r i is a cause
for x j .
We have that r j is a cause for x i iff x i = x j and C xj = C xi ≠ ∅. We have
that r jk is a cause for x i iff x i ∈ {x k } ∩ C xk , which amounts to x i ∈ C xk , or
i = k and x i ∈ D xj .
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