Obtaining a Dynamic Chain Event Graph

Analogous to a CEG, a DCEG is constructed by merging all situations in the same position into a single vertex and then gathering all situations that represent terminated processes into a single position w∞. Every staged tree then spans a

Note that the definition of position is more restrictive in a DCEG than in a CEG because of the condition2associated with the bijection in Equation 6.4 (see Def- inition 31). So, whilst in a CEG any two situations are at the same position when their unfolding sequences of events and colour are equivalent, in a DCEG these equivalences of events and colours have additionally to hold in each time-slice. This is important to avoid topological ambiguities during the final construction of a DCEG graph since the set of vertices in a DCEG graph corresponds exactly to the set of positions yielded by its supporting stage tree.

Also observe that two processes unfolding from situations in the same position must be equivalent for all subsequent developments described by the staged tree. However processes evolving from situations in the same stage only have to be identified across the next step in their evolutions. Therefore, as in a CEG the set of positions in a DCEG constitutes a finer partition of its corresponding staged structure: if two situations are in the same position, then they must also be in the same stage but the converse is not always valid.

As observed in Barclay et al. (2015) a DCEG may have directed loops that allow it to have a finite number of vertices. Theorem 7 below tells us that this is always the case when a DCEG is based on aT-Periodic Staged Tree. Theorem 8 asserts that time-homogeneity suffices to satisfy this condition.

Theorem 7. A DCEG is finite if and only if its corresponding staged tree is T-Periodic after some time T.

Proof. Assume that a finite DCEG is supported by a staged tree that it is non- periodic for every T, T = 0,1, . . .. It will then follow that for every time-slice T,

T = 1,2, . . ., there must be at least one situation sa(T) that is at position w,

such that w does not merge any situationsb(t), t= 0, . . . , T −1. Therefore the

number of positions in this staged tree is infinite. This contradicts the hypothesis. It follows that the supporting staged tree of a finite DCEG must therefore be

T-Periodic Staged Tree for some T.

ta =T + 1, T + 2, . . ., there is some situation s(tb) at time tb, tb, t = 0, . . . , T,

such that sa and sb are at the same position. So the number of positions in the

corresponding DCEG does not need to be greater than the number of situations in _TT. This DCEG must therefore be finite.

Theorem 8. Every time-homogeneous staged tree after timeT has an associated DCEG with a finite graph.

Proof. In any event tree, letξ(sa, sb)be the sequence of events that happen along

a finite path between the situations sa and sb, where sb is down stream of sa.

Let S(t+ 1) be the set of all situations in times-slice t+ 1 whose parent are in time-slice t. Also let a(si, t) be the antecedent situation of a situation si such

that a(si, t)∈ S(t).

By assumption, the event tree is strong T-periodic. Initially assume that _T−1 = ∅. So for every situation si ∈ S(2T + 2) there is a situation s(t), t≤T, such

that the event trees _T(a(si, T + 1)) and T(s(t)) are graphically isomorphic. It

follows that there is a situations(t∗_{)∈ T(s(t)),t}∗ _{=t+T + 1, such thats(t}∗_)∈

S(t∗_{),ξ(s(t), s(t}∗_{)) =ξ(a(s}

i, T+1), si)and the event treesT(si)andT(s(t∗))are

graphical isomorphic. Time-homogeneity after time T then tell us that there is also a probabilistic isomorphism between the primitive probabilities associated with

T(si) and T(s(t∗)). Therefore, si and s(t∗), where t∗ ≤2T + 1, are in the same

position. So the number of positions in the corresponding DCEG does not need to be greater than the number of situations inT2T+1. This DCEG must therefore

be finite.

If there are time-invariant events (_T−1 6=∅), the result still holds for each subtree T∞(s), s ∈ l(T−1), because of the strong periodicity embedded into the time-

homogeneity condition. Therefore the DCEG is also finite in this case.

Figure 6.7 shows the finite DCEG associated with Example 3. Note that to draw an uncluttered graph without any loss, some of its edges are dashed and grey and the sink position w∞ is represented by two receiving vertices. In the next chapter

Figure 6.7: The 2T-DCEG associated with Example 3. The stage structure is given by the following partition: u0 = {w0}, u1 = {w1}, u2 = {w2}, u3 = {w3},

u4 ={w4, w5, w6},u5={w7, w8, w9},u6={w10, w13},u7={w11, w14},u8={w12, w15},

u9={w16},u10={w17},u11={w18}, u12 ={w19, w20, w21},u13={w22, w23, w24},

u14={w25, w26, w27}.

I will define a useful family of finite DCEGs named N Time-Slice Dynamic Chain Event Graph (NT-DCEG). It will become apparent later that the DCEG given in Figure 6.7 is indeed a 2T-DCEG. At that point I will explain how to read the conditional independences embedded in its topology.