Online apnea–bradycardia detection based on hidden semi-Markov models

Abstract

In this paper, we propose a new online apnea–bradycardia detection scheme that takes into account not only the instantaneous values of time series, but also their temporal evolution. The detector is based on a set of hidden semi-Markov models, representing the temporal evolution of beat-to-beat interval (RR interval) time series. A preprocessing step, including quantization and delayed version of the observation vector, is also proposed to maximize detection performance. This approach is quantitatively evaluated through simulated and real signals, the latter being acquired in neonatal intensive care units (NICU). Compared to two conventional detectors used in NICU, our best detector shows an improvement on average of around 15 % in sensitivity and 7 % in specificity. Furthermore, a reduced detection delay of approximately 2 s is also observed with respect to conventional detectors.

Appendices

Appendix 1: The viterbi algorithm

The Viterbi algorithm has first appeared in the literature of coding in 1967 [43]. This algorithm selects the states individually most likely, in order to find the states sequence most likely associated with the observed sequence. The algorithm includes a step of recursion which travels the signal and records the states that maximize ${\mathcal {L}}(O_{1:T}|\lambda )$, and a step of back propagation that travels the signal in the opposite direction, starting from the state for which the likelihood at time $T$ is maximum, and taking into account the previous states stored in the recursion step.

To find the best state sequence $S_{1:T} = S_1, \ldots , S_T$ for the sequence of observation $O_{T} \triangleq O_1, \ldots , O_T$,

$$\begin{aligned} \delta _t(i) \triangleq \underset{s_{1:t-1}}{\max }\big ( P(s_1, s_2,\ldots ,s_t = i, o_{1:t}|\lambda ) \big ) \end{aligned}$$

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represents the maximum likelihood of a single path until time $t$, which takes into account the first $t$ observations and stops in state $i$. By induction we have

$$\begin{aligned} \delta _{t+1}(j) = \underset{i \in \mathcal {S}}{\max } \big ( \delta _{t}(i) a_{ij} \big ) b_j(o_{t+1}). \end{aligned}$$

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The argument that maximizes Eq. 15 for each $t$ and $j$, is recorded through the array $\varPsi _t(j)$. The full procedure to find the best state sequence is:

• Initialization:

  $$\begin{aligned} \delta _1(i)&= \pi (i) b_i(o_1), \quad i \in \mathcal {S} \\ \varPsi _1(i)&= 0. \end{aligned}$$

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• Recursion:

  $$\begin{aligned} \delta _t(j)&= \underset{i \in \mathcal {S}}{\max } \big ( \delta _{t-1}(i) a_{ij} \big ) b_j(o_t), \quad 2 \le t \le T, j \in \mathcal {S} \\ \varPsi _t(j)&= \underset{i \in \mathcal {S}}{\mathrm{arg\,max }} \big ( \delta _{t-1}(i) a_{ij} \big ), \quad 2 \le t \le T, j \in \mathcal {S} \end{aligned}$$

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• Termination:

  $$\begin{aligned} P^*&= \underset{i \in \mathcal {S}}{\max } \big ( \delta _T(i)\big ) \\ i_T^*&= \underset{i \in \mathcal {S}}{\mathrm{arg\,max }} \big ( \delta _T(i) \big ). \end{aligned}$$

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• backpropagation (path obtained):

  $$\begin{aligned} i_t^* = \varPsi _{t+1}(i_{t+1}^*), \quad t = T-1,T-2,\ldots ,,1. \end{aligned}$$

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Appendix 2: The viterbi algorithm extended to HSMM

The quantity

$$\begin{aligned} \delta _t(i,d) \triangleq \underset{s_{1:t-d}}{\max }\big ( P(s_{1:t-d}, S_{t-d+1:t} = i, o_{1:t}|\lambda ) \big ) \end{aligned}$$

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represents the maximum likelihood of a single path until time $t$, which takes into account the first $t$ observations and stops at the state $i$ of length $d$. By induction we have:

$$\begin{aligned} \delta _{t+d}(j,d) = \underset{i \in \mathcal {S} \setminus \{j\}, d' \in \mathcal {D}}{\max } \big ( \delta _{t}(i,d') a_{ij} \big ) p_j(d) b_j(o_{t+1:t+d}). \end{aligned}$$

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The array $\varPsi _t(j,d)$ is used to record the states sequence and times that maximize 21. The complete procedure is as follows:

• Initialization:

  $$\begin{aligned} \delta _1(i)&= \pi (i) b_i(o_1), \quad i \in \mathcal {S} \\ \varPsi _1(i)&= 0. \end{aligned}$$

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• Recursion:

  $$\begin{aligned} \delta _t(j,d)&= \underset{i \in \mathcal {S} \setminus \{j\}, d' \in \mathcal {D}}{\max } \big ( \delta _{t-d}(i,d') a_{ij} \big ) p_j(d) b_j(o_{t-d+1:t}), \\&\quad 2 \le t \le T, j \in \mathcal {S}, d \in \mathcal {D} \\ \varPsi _t(j,d)&= \underset{i \in \mathcal {S} \setminus \{j\}, d' \in \mathcal {D}}{\mathrm{arg\,max }} \big ( \delta _{t-d}(i,d') a_{ij} \big ), \\&\quad 2 \le t \le T, j \in \mathcal {S} , d \in \mathcal {D}. \end{aligned}$$

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• Termination:

  $$\begin{aligned} P^*&= \underset{i \in \mathcal {S}, d \in \mathcal {D}}{\max } \big ( \delta _T(i,d)\big ) \\ (i_T^*,d_T^*)&= \underset{i \in \mathcal {S},d \in \mathcal {D}}{\mathrm{arg\,max }} \big ( \delta _T(i,d) \big ). \end{aligned}$$

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• backpropagation (path obtained):

  $$\begin{aligned} (i_t^*,d_t^*) = \varPsi _{t+1}(i_{t+1}^*,d_{t+1}^*), \quad t = T-d_t^*,\ldots ,1. \end{aligned}$$

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