Correction to: Discrete Event Dyn Syst (2022) 32: 65
In (Lefebvre et al. 2022), there was a flaw in the proof of Lemma 2. The statement of Lemma 2 is correct by itself but its proof requires a slightly different definition of the etransition probability matrix given in Definition 5. This note provides the corrections and adjusts Examples 3 and 4 accordingly. The other results, proofs, and examples in (Lefebvre et al. 2022) remain unchanged.
Correction to Definition 5 in (Lefebvre et al. 2022): Given an LCTMM G = (X, E,Λ, π_{0}), for each event e ∈ E its etransition probability matrix \(Q_{e} = (q_{e,i,j}) \in \mathbb{R} _{\geq 0}^{n \times n}\) (where q_{e, i, j} is the element of matrix Q_{e} in row i and column j) is defined by q_{e, i, j} = μ(x_{i}, e, x_{j}), where for x_{i} ∈ X, e ∈ E, and x_{j} ∈ Post(x_{i}), μ(x_{i}, e, x_{j}) is the sum of the firing rates of the etransitions from state x_{i} to x_{j} (μ(x_{i}, e, x_{j}) = 0 if no etransition exists from state x_{i} to x_{j}).
Correction to Example 3 in (Lefebvre et al. 2022): The atransition and btransition probability matrices of the LCTMM in Fig. 1 with alphabet E = {a, b} are the matrices Q_{a} and Q_{b} detailed below: ◇
Correction to Lemma 2 in (Lefebvre et al. 2022): Consider an LCTMM G = (X, E,Λ,π_{0}) and its etransition probability matrices as in the revised Definition 5 above. Given an observation σ = (e, t) with e ∈ E, it holds that:
where 1_{n×1} is the all ones column vector of dimension n.
For each state x_{j} of the LCTMM it holds that
The numerator and denominator of the previous expression are reformulated.

Given an infinitesimal interval dt, the quantity q_{e, i, j} ⋅ dt represents the probability that a transition to x(t) = x_{j} occurs when event e is observed in interval (t − dt, t] given that x(t − dt) = x_{i}. More formally, Pr(x(t) = x_{j} ∩ (e,(t − dt, t])∣x(t − dt) = x_{i}) = q_{e, i, j} ⋅ dt.

On the other hand,
$$\begin{array}{lll} Pr((e,(tdt, t]))&=&\displaystyle \sum\limits_{i=1}^{n} Pr((e,(tdt, t]) \mid x(tdt)=x_{i}) \cdot Pr(x(tdt)=x_{i})\\ &=&\displaystyle \sum\limits_{i=1}^{n} \left( \sum\limits_{j=1}^{n}q_{e,i,j} .dt \right) \cdot Pr(x(tdt)=x_{i}) \end{array} $$
Considering that \(\lim \limits _{dt \rightarrow 0} Pr(x(tdt)=x_{i}) = \pi _{i}(t^{} \mid \boldsymbol {\pi }_{0}), \) we have
or eq. (1) in matrix form. Observe that the denominator in eq. (1) is nonzero because the event e has been observed at time t, i.e., there must exist a state x_{i} from which a transition labeled e may occur and such that \(\pi _{i}(t^{} \mid \boldsymbol {\pi }_{0}) > 0\). □
Correction to Example 4 in (Lefebvre et al. 2022): Consider the LCTMM in Fig. 1 with firing rates μ_{1,1} = 2, μ_{3,1} = 3, all other rates being equal to 1, and sequence of observations σ = (a,1)(b,3)(a,4)(a,5) within the time interval [0,7]. The state probabilities are reported in Fig. 2.
In order to illustrate how the time stamps of the observations influence the probabilities of the states, consider also the sequence of observations σ = (a, t_{1}) with several values of t_{1} within the time interval [0,4]. Observe in Fig. 3 that the probability of x_{3} at time t = 4 changes depending on the value of t_{1}.
Let us consider some basic cases that explain and illustrate Definition 5 and Lemma 2.
Consider the LCTMM in Fig. 4(a) with π_{0} = [1 0 0] where a and b are two observable labels. No label is observed up to time t, we have \(\boldsymbol {\pi }(t^{} \mid \boldsymbol {\pi }_{0}) = [ 1~0~0]\) because there exists no silent evolution from state x_{1}. When a label a is observed at t we will obtain π(t∣π_{0}, σ) = [0 1 0] with σ = (a, t). According to Eq. (1) this can be written as
Note that the probability Pr((a,(t − dt, t]) to observe a within (t − dt, t] assuming that nothing was observed before time t − dt (and consequently that the system stays at x_{1} before t − dt) is equal to the probability that the delay of a is smaller than dt (which is μdt) and that the delay of b is greater than dt (which is \(1  \mu ^{\prime } dt\)). Since the events a and b are independent and dt is an infinitesimal duration, we have: \(Pr(a,dt) = (\mu dt) \cdot (1  \mu ^{\prime } dt) = \mu dt  \mu \mu ^{\prime } dt^{2} \approx \mu dt\).
Consider the LCTMM in Fig. 4(b) with π_{0} = [1 0 0] where a is the single observable label. No label a is observed, we have \(\boldsymbol {\pi }(t^{} \mid \boldsymbol {\pi }_{0}) = [ 1~0~0]\). When a label a is observed at t we will obtain π(t∣π_{0}, σ) with σ = (a, t) according to Equation (1):
with Δ = μ + μ.^{′}
Consider finally the LCTMM in Fig. 4(c) with π_{0} = [1 0 0]. This example evolves exactly as the example in Fig. 4(b) up to the first observation of the label a at time t. From that time, and despite the fact that no silent transition exists in this system, the probability of the states x_{2} and x_{3} will change depending on the values of μ and \(\mu ^{\prime }\) and according to the extended ε subchain of the system (Definition 4 in (Lefebvre et al. 2022)). In particular, for a given value of time \(t^{\prime } \geq t\), there exists \(\alpha _{t^{\prime }} \in [0, 1]\) such that \(\boldsymbol {\pi }(t^{\prime } \mid \boldsymbol {\pi }_{0}, (a,t)) =[0~\alpha _{t^{\prime }}~1\alpha _{t^{\prime }}]\). When a second label a is observed at \(t^{\prime }\) we will obtain \(\boldsymbol {\pi }(t^{\prime } \mid \boldsymbol {\pi }_{0}, \sigma ) \) with \(\sigma = (a,t) (a,t^{\prime })\) that can be written as
with \({\Delta }^{\prime } = \mu \alpha _{t^{\prime }} +\mu ^{\prime } (1\alpha _{t^{\prime }})\).
References
Lefebvre D., Seatzu C., Hadjicostis C. N., Giua A. (2022) Probabilistic state estimation for labeled continuous time Markov models with applications to attack detection Journal of Discrete Event Systems
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Lefebvre, D., Seatzu, C., Hadjicostis, C.N. et al. Correction to: Probabilistic state estimation for labeled continuous time Markov models with applications to attack detection. Discrete Event Dyn Syst 32, 539–544 (2022). https://doi.org/10.1007/s10626022003646
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DOI: https://doi.org/10.1007/s10626022003646