On mining frequent chronicles for machine failure prediction

Abstract

In industry 4.0, machines generate a lot of data about several kinds of events that occur in the production process. This huge quantity of information contains valuable patterns that allow prediction of important events in the appropriate instant. In this paper, we are interested in mining frequent chronicles in the context of industrial data. We introduce a general approach to preprocess, mine, and use frequent chronicles to predict a special event; the failure of a machine. Our approach aims not only to predict the failure, but also the time of its appearance. Our approach is validated through a set of experiments performed on the chronicle mining phase as well as the prediction phase. Experiments were achieved on synthetic data in addition to a real industrial data set.

Appendices

Appendix A Proofs

Proof of Lemma 1

Let us prove ${{\,\mathrm{supp}\,}}(s') \le {{\,\mathrm{supp}\,}}(s)$ and ${{\,\mathrm{supp}\,}}(s') \ge {{\,\mathrm{supp}\,}}(s)$:

• Since s is contained within $s'$, any sequence containing $s'$ also contains s. Thus, ${{\,\mathrm{supp}\,}}(s') \le {{\,\mathrm{supp}\,}}(s)$.

• We are to prove that concatenation with $\omega$ as a suffix does not reduce the support of a given sequence in $FS_\omega$, i.e. ${{\,\mathrm{supp}\,}}(s') \ge {{\,\mathrm{supp}\,}}(s)$.

  Let $\gamma = \left\{ \gamma _1, \gamma _2, \ldots , \gamma _{{{\,\mathrm{supp}\,}}(s)} \right\}$ be the set of sequences of $D_\omega$ containing s. Then, we can write any $\gamma _i$ as follows:

  $$\begin{aligned} \gamma _k = \alpha _1 \mathbin {\Diamond }_i \langle s_1 \rangle \mathbin {\Diamond }_i \alpha _2 \mathbin {\Diamond }_i \cdots \mathbin {\Diamond }_i \langle s_p \rangle \mathbin {\Diamond }_i \alpha _{p+1} \mathbin {\Diamond }_i \omega \end{aligned}$$

  where $\alpha _i, i \in \llbracket 1, p+1 \rrbracket$ are the remaining sequences needed to build $\gamma _k$.

  One can easily notice that $s'$ is contained within $\gamma _k$. Thus, $\forall k \in \llbracket 1, {{\,\mathrm{supp}\,}}(s) \rrbracket , s' \subseteq \gamma _k$. Ultimately, ${{\,\mathrm{supp}\,}}(s') \ge {{\,\mathrm{supp}\,}}(s)$.

$\square$

Proof of proposition 1

Let us prove $\mathcal {CS}_\omega \subseteq \left\{ s \mathbin {\Diamond }_s \omega \vert s \in \mathcal {CS}\right\}$ and $\left\{ s \mathbin {\Diamond }_s \omega \vert s \in \mathcal {CS}\right\} \subseteq \mathcal {CS}_\omega$.

• Let us use a reduction ad absurd argument.

  Assume $\exists s_\omega \in \mathcal {CS}_\omega , \not \exists s \in \mathcal {CS}, s_\omega = s \mathbin {\Diamond }\omega$. Let $s_\omega = \langle s_{\omega ,1}, s_{\omega ,2}, \ldots , s_{\omega ,p} \rangle$. Let us consider $\gamma = \left\{ \gamma _i \right\} _{i=1}^{{{\,\mathrm{supp}\,}}(s_\omega )}$ the set of sequences of $D_\omega$ containing $s_\omega$. Then, as $\forall i \in \llbracket 1, {{\,\mathrm{supp}\,}}(s_\omega ) \rrbracket , \gamma _i \in D_\omega$, we can write $\gamma _i$ as follows:

  $$\begin{aligned} \gamma _i = \alpha _1 \mathbin {\Diamond }\langle s_1 \rangle \mathbin {\Diamond }\alpha _2 \mathbin {\Diamond }\ldots \mathbin {\Diamond }\langle s_p \rangle \mathbin {\Diamond }\alpha _{p+1} \mathbin {\Diamond }\omega \end{aligned}$$

  where $\alpha _i, i \in \llbracket 1, p+1 \rrbracket$ are the remaining sequences needed to build $\gamma _i$.

  Then, let $s_\omega ' = s_\omega \mathbin {\Diamond }\omega$. This sequence contains $s_\omega$, i.e. $s_\omega '$ is a super-sequence of $s_\omega$. Moreover, as per Lemma 1, ${{\,\mathrm{supp}\,}}(s_\omega ) = {{\,\mathrm{supp}\,}}(s_\omega \mathbin {\Diamond }\omega )$. Thus, $s_\omega$ is not closed, so $s_\omega \not \in \mathcal {CS}_\omega$, which is absurd. Therefore, $\forall s_\omega \in \mathcal {CS}_\omega , \exists s \in \mathcal {CS}, s_\omega = s \mathbin {\Diamond }\omega$.

• Let us consider $s_\omega = s \mathbin {\Diamond }\omega$ with $s \in \mathcal {CS}$. Let us show that $s_\omega \in \mathcal {CS}_\omega$.

  • $s \in \mathcal {CS}\Rightarrow s \in \mathcal {FS}$, so from lemma 1, ${{\,\mathrm{supp}\,}}(s_\omega ) = {{\,\mathrm{supp}\,}}(s)$. Moreover, let $\{ \gamma _i \}$ be the sequences of D where s occurs, then $s_\omega$ occurs in every sequence of $\{ \gamma _i \mathbin {\Diamond }\omega \} \subseteq D_\omega$. So $s_\omega \in \mathcal {FS}_\omega$.

  • Let us use a reductio ad absurdum argument. Let us assume $\exists \beta _\omega \in \mathcal {FS}_\omega , {{\,\mathrm{supp}\,}}(s_\omega ) = {{\,\mathrm{supp}\,}}(\beta _\omega ) \wedge s_\omega \subseteq \beta _\omega$. Let $\beta _\omega$ be of the form $\beta \mathbin {\Diamond }_s \omega$, so $\beta \in \mathcal {FS}$, without loss of generality. Lemma 1 gives us that ${{\,\mathrm{supp}\,}}(s_\omega ) = {{\,\mathrm{supp}\,}}(s)$ and ${{\,\mathrm{supp}\,}}(\beta _\omega ) = {{\,\mathrm{supp}\,}}(\beta )$, so ${{\,\mathrm{supp}\,}}(s) = {{\,\mathrm{supp}\,}}(\beta )$. In addition, we have $s \subseteq \beta$. Therefore, $s \not \in \mathcal {CS}$, which is absurd. Hence, there exist no $\beta _\omega$ in $\mathcal {FS}_\omega$ with same support as and which contains $s_\omega$. $\square$

Appendix B Suffix database with i-concatenation

In Table 15, we consider a version of the suffix database presented in Table 3 but with i-concatenation.

Table 15 Sample suffix database with i-concatenation: P is the failure event, s-append at the end of each sequence

Let us recall the motivation to build a suffix database is to always extract the last event of a sequence (corresponding to a failure in our application), so that we do not have to re-check the database to extract such event or do case-based approaches for each sequence. Moreover, we would like to keep the same number of closed frequent sequences, as this number is an important part of their discrimination role.

The problem with i-concatenation is that the resulting suffix database can produce a closed frequent sequences set with P missing from some patterns, and can eventually produce more sequences. Using the i-concatenation suffix database, and still considering a relative threshold of 0.8, we can see that $\langle A, A, A, B \rangle$ is closed (support of 0.8). It is contained within $\langle A, A, A, \{ B, P \} \rangle$ (support of 0.6) and $\langle A, A, A, \{ A, P \} \rangle$ (support of 0.2), but support of both of these sequences fall short to the one of $\langle A, A, A, B \rangle$. Thus, equality of Proposition 1 does not hold, and one cannot ensure we will have the same number of closed frequent sequences, nor that the failure event will be contained within every extracted pattern.