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Remaining useful life prediction using prognostic methodology based on logical analysis of data and Kaplan–Meier estimation

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Abstract

Most of the reported prognostic techniques use a small number of condition indicators and/or use a thresholding strategies in order to predict the remaining useful life (RUL). In this paper, we propose a reliability-based prognostic methodology that uses condition monitoring (CM) data which can deal with any number of condition indicators, without selecting the most significant ones, as many methods propose. Moreover, it does not depend on any thresholding strategies provided by the maintenance experts to separate normal and abnormal values of condition indicators. The proposed prognostic methodology uses both the age and CM data as inputs to estimate the RUL. The key idea behind this methodology is that, it uses Kaplan–Meier as a time-driven estimation technique, and logical analysis of data as an event-driven diagnostic technique to reflect the effect of the operating conditions on the age of the monitored equipment. The performance of the estimated RUL is measured in terms of the difference between the predicted and the actual RUL of the monitored equipment. A comparison between the proposed methodology and one of the common RUL prediction technique; Cox proportional hazard model, is given in this paper. A common dataset in the field of prognostics is employed to evaluate the proposed methodology.

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Abbreviations

RUL:

Remaining useful life

CM:

Condition monitoring

KM:

Kaplan–Meier

LAD:

Logical analysis of data

CBM:

Condition based maintenance

ANNs:

Artificial neural networks

PHM:

Proportional hazards model

LR:

Logistic regression

SVMs:

Support vector machines

RVM:

Relative vector machine

HMM:

Hidden Markov model

MILP:

Mixed integer linear programming

LASD:

Logical analysis of survival data

SL:

Short life

LL:

Long life

MTTF:

Mean time to failure

TTF:

Time to failure

MRUL:

Mean remaining useful life

RMSE:

Root mean squared error

\( Cov (p)\) :

The set of observations covered by the pattern \(p\)

\(\varOmega ^{SL}\) :

The set of SL equipment

\(\varOmega ^{LL}\) :

The set of LL equipment

\(\varOmega \) :

The training data set

\(O(i,t_{F_i},Z_{i,t_{Fi}})\) :

The failure observation collected from the \(i\)th equipment \((i=1,2,\ldots T)\), where \(Z_{i,t_{Fi}}\) is the corresponding vector of covariates, \(t_S\) is the separation time between the two classes (SL and LL), and \(t_{Fi}\) is the failure time of the \(i\)th equipment

\(O\left( {i,t_{Fi} \le t_S, Z_{i,t_{Fi} \le t_S}}\right) \) :

Set of positive (short life) observations in the training dataset

\(O\left( {i,t_{Fi} >t_S,Z_{i,t_{Fi} >t_S}}\right) \) :

Set of negative (long life) observations in the training dataset

\(O(u,t_k,Z_{u,t_k})\) :

The updating observation collected from the \(u\)th equipment \((u=1,2,\ldots U)\) at time \(t_k \;(t_k =1,2,\ldots t_{Fu})\), and \(Z_{u,t_k}\) is the vector of covariates at time \(t_k\)

\(S_b(t)\) :

The baseline survival function estimated by KM

\(d_{t_{Fi}}\) :

The number of equipment that failed at time \(t_{Fi}\)

\(Y_{t_{Fi}}\) :

The number of equipment which are at risk at time \(t_{Fi}\)

\(P\) :

The set of generated patterns

\(|P|\) :

The cardinality of the set of generated patterns \(P\) (i.e. the number of generated patterns)

\(p_{j}\) :

A pattern belongs to the set of generated patterns \(P\), where \(j=1,2,\ldots |P|\)

\(S_{O(u,t_k,Z_{u,t_k})} (t)\) :

The updated survival curve of the updating observation \(O(u,t_k,Z_{u,t_k})\)

\(S_f (t)\) :

The former updated survival curve obtained from the previous updating observation

\(S(\tau )\) :

The survival function, where \(\tau \) is a dummy variable

\(MRUL_u ({t_k})\) :

The mean remaining useful life of equipment \(u\), calculated at time \(t_k\)

\(\varDelta t_r\) :

Is the monitoring interval which is the difference between two consecutive inspection intervals i.e. \(\varDelta t_r = t_{r+1} -t_r\)

\(RMSE\left( u\right) \) :

The calculated RMSE for the MRUL estimation of the system \(u\) in the updating dataset

\(N_{Fu}\) :

The actual number of operational cycles until the failure of the equipment \(u\)

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Authors and Affiliations

  1. Industrial Engineering and Applied Mathematics Department, École Polytechnique de Montréal, Montréal, Canada

    Ahmed Ragab, Mohamed-Salah Ouali, Soumaya Yacout & Hany Osman

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  1. Ahmed Ragab
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  2. Mohamed-Salah Ouali
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  3. Soumaya Yacout
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  4. Hany Osman
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Correspondence to Soumaya Yacout.

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Ragab, A., Ouali, MS., Yacout, S. et al. Remaining useful life prediction using prognostic methodology based on logical analysis of data and Kaplan–Meier estimation. J Intell Manuf 27, 943–958 (2016). https://doi.org/10.1007/s10845-014-0926-3

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