Probabilistic Performance Analysis of Fault Diagnosis Schemes

of storage. In the simplest case, where Θ = {0,1}, there are 2 k+1 terms to store. For example,
at k = 36, the total storage needed is
12 × 2 36+1 ≈ 1.65 × 10 12 bytes > 1terabyte!
Since physical systems are often sampled at twice their bandwidth or more, the amount of time
represented by 36 discrete samples is small compared to the time-scale of the system.
4.2.1 Limiting Complexity with Structured Markov Chains
Although the number of paths in Θ k grows exponentially with k, not all of the paths need
to be considered in computing equation (4.3), because some paths have zero probability
of occurring. That is, some sequences of faults cannot occur under the given model. This
section explores, from a theoretical perspective, what properties the Markov chain must
have in order to reduce the number of terms in equation (4.3) to a tractable number.
Terminology
Definition 4.3. Given a Markov chain θ taking values in Θ, let l ≥ 0 and ϑ 0:l ∈ Θ l+1 . If the
event {θ 0:l = ϑ 0:l } has nonzero probability, then ϑ 0:l is said to be a possible path of {θ k }.
Otherwise, ϑ 0:l is said to be an impossible path.
Definition 4.4. A Markov chain is said to be tractable if the number of possible paths of
length l is O(l c ), for some constant c.
Definition 4.5. Let θ be a Markov chain taking values in Θ. A state ϑ ∈ Θ is said to be
degenerate if P(θ k = ϑ) = 0, for all k (i.e., no possible path ever visits ϑ). A Markov chain with
one or more degenerate states is said to be degenerate.
Remark 4.6. Our definition of a tractable Markov chain is based on the conventional notion
that polynomial-time algorithms are tractable, whereas algorithms requiring superpolynomial
time are intractable [19]. This idea is known as Cobham’s Thesis or the Cobham–
Edmonds Thesis [16, 29].
Remark 4.7. Suppose that θ ∼ ( Θ,{Π k },π 0
)
is a Markov chain with a nonempty set of degenerate
states ¯Θ ⊂ Θ. Let ˆθ be the Markov chain formed by removing the degenerate states
from Θ and trimming the matrices {Π k } and the pmf π 0 accordingly. Clearly, any possible
path of θ is a possible path of ˆθ, so the tractability of θ can be determined by analyzing the
non-degenerate Markov chain ˆθ.
Since the goal is to relate the tractability of Markov chains to properties of directed
graphs, we must first establish some definitions from graph theory.
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