Probabilistic Performance Analysis of Fault Diagnosis Schemes

Since relabeling the states of a Markov chain does not alter its tractability, the next step
is to seek conditions under which the states can be permuted to achieve upper-triangular
transition probability matrices. The following lemmas show that the existence of such
permutations can be related to the presence of cycles in the graph (Θ, A).
Lemma 4.22. Let Θ = {0,1,...,m}. Given a matrix Π ∈ R (m+1)×(m+1) , define the matrix A as in
Theorem 4.11. Then, there exists a permutation σ: Θ → Θ, such that the matrix
ˆΠ(i , j ) := Π ( σ(i ),σ(j ) ) , i , j ∈ Θ (4.8)
is upper-triangular if and only if the directed graph with vertices Θ and adjacency matrix A
is acyclic.
Proof. Suppose that the permutation σ makes ˆΠ upper triangular. Let n > 0 and let
v 0 → v 1 ,→ ··· → v n
be a path on the graph (Θ, A). For i = 1,...,n, the existence of the edge v i−1 → v i implies
that v i−1 ≠ v i and
Π(v i−1 , v i ) = ˆΠ ( σ −1 (v i−1 ),σ −1 (v i ) ) ≠ 0.
Since ˆΠ is upper-triangular,
σ −1 (v 0 ) < σ −1 (v 1 ) < ··· < σ −1 (v n ).
Therefore, no path can never visit the same vertex twice, i.e., the graph (Θ, A) is acyclic.
Now, suppose that the graph (Θ, A) is acyclic. The Depth-First Search (dfs) algorithm
generates a pre-visit and post-visit number for each vertex v ∈ Θ as it begins exploring v
and finishes exploring v, respectively. Because (Θ, A) is acyclic, A(u, v) = 1 if and only if
pre(u) < pre(v) < post(v) < post(u) (4.9)
(see [21] for details). Define the map ρ : v → pre(v), for all v ∈ Θ. Since each vertex has a
unique pre-visit number in Θ, the map ρ is a permutation. Let σ := ρ −1 and define ˆΠ as in
equation (4.8). Note that for any i , j ∈ Θ, such that i > j ,
A ( σ(i ),σ(j ) ) = 0 =⇒ Π ( σ(i ),σ(j ) ) = ˆΠ(i , j ) = 0.
Therefore, the permutation σ makes ˆΠ upper-triangular.
Lemma 4.23. Let Θ = {0,1,...,m}. Given a sequence of matrices {Π k } in R (m+1)×(m+1) , define
53