Predictive maintenance solution for industrial systems -- an unsupervised approach based on log periodic power law

TL;DR

The paper presents an unsupervised predictive maintenance method based on Log-Periodic Power Law (LPPL) analysis grounded in renormalization-group theory to detect critical points in univariate time series and forecast failures in reciprocating compressors. It defines an initial breakdown (IB) point via LPPL fitting to a transformed variable $W(t)=\log(p(t))$ with a finite-time critical time $t_c$, and then estimates a future failure window using system dynamics. Applied to daily OSV data from a compressor, the method backtests IB detection, classifies failure risk with a fit-based threshold, and compares its performance to online changepoint methods, showing fewer alerts with competitive accuracy (e.g., $Precision=0.67$, $Recall=0.80$). The approach benefits from requiring no labeled data and working on short time series, offering actionable failure windows and root-cause insights (valve vs. discharge-leakage) with potential applicability to other industrial IoT domains. The LPPL model used takes the form $W(t) \approx A + |t_c - t|^{m}\left[ B + C_1 \cos(\omega \log|t_c - t|) + C_2 \sin(\omega \log|t_c - t|) \right]$, enabling detection of critical transitions and probabilistic forecasting in a physically interpretable framework.

Abstract

A new unsupervised predictive maintenance analysis method based on the renormalization group approach used to discover critical behavior in complex systems has been proposed. The algorithm analyzes univariate time series and detects critical points based on a newly proposed theorem that identifies critical points using a Log Periodic Power Law function fits. Application of a new algorithm for predictive maintenance analysis of industrial data collected from reciprocating compressor systems is presented. Based on the knowledge of the dynamics of the analyzed compressor system, the proposed algorithm predicts valve and piston rod seal failures well in advance.

Figures (5)

• Figure 1: Examples of LPPL function (\ref{['final_fit_1']}) fits for cases with positive trends (a) and with negative trends (b) for selected current times (dates in proleptic Gregorian ordinal units). Fit parameters calculated for both trend categories: (a) positive trend: current date: $t_{c}=2020-08-28$, length of the time series: $l_{max}=79$, mean squared error: $mse = 2.6\cdot 10^{-5}$; (b) negative trend: current date: $t_{c}=2020-09-23$, length of the time series: $l_{max}=55$, mean squared error: $mse = 5.7\cdot 10^{-5}$.
• Figure 2: Calculated IB points for the analyzed input time series (marked in figure by a solid line with measurement points). The vertical lines of different colors indicate the diagnosed by the algorithm IB points. For each group of the IB points, the mean value of the matching error ($mse$) of the LPPL function (\ref{['final_fit_1']}) is annotated. The black vertical lines indicate the dates of compressor repairs, which typically took place a few to several weeks after the algorithm detected the fault (IB points). The correlation with the diagnosed breakpoints is clearly visible, except for the prediction determined with the largest error (mse=0.000249) for 2020-12-14.
• Figure 3: Redrawn prediction of failures by the algorithm in comparison with reparation times and problems detected by experts. Figure shows representations of groups of discovered IB points (colored and continuous single vertical lines) and predicted failure time periods corresponding to them according to Definition (\ref{['bl:definition2']}) (corresponding colored areas bounded by vertical dashed lines). The categorization of the criticality of the forecasted problems (Definition (\ref{['bl:definition1']})) is represented by the splitting into 3 separate figures, each for a separate criticality category. The Monitoring event class remains without any marked vertical lines. The algorithm found no prediction for this category. The reparation (maintenance) dates are indicated in figures by black vertical lines. Time periods of abnormal compressor behavior requiring monitoring and not qualified for repair by experts are indicated by the gray colour without additional delimiting lines. The intersection of the areas determined by the algorithm as a period of failure occurrence with periods of abnormal behavior of the compressor can be seen for periods starting from 2021-07-15 in the Critical event category. The raw data of the input time series $t_{inp}$ are indicated by the solid blue line with the measurement points ($o$).
• Figure 4: The same graphs as in Figure (\ref{['fig:final_detection_examples']}) with added annotations describing the failures predicted by the algorithm - colored texts (different from gray and black), reasons for compressor reparations - black text and diagnosed abnormal compressor behavior requiring monitoring - gray text. For better visualization, areas of the diagnosed abnormal behaviour of the compressor that require monitoring are displayed in the "Monitored Event" category.
• Figure 5: Comparison of the results of the presented method (LPPL method - top figure) with the statistical method (changepoint_online - middle and bottom figures) at the level of determining IB points and trend change points. The analyzed time series $t_{inp}$ is shown by the solid blue line with the measurement points ($o$). The black vertical lines indicate the compressor repair dates. Time periods when abnormal compressor behavior requiring monitoring was not qualified for repair by experts are indicated by the gray colour without additional delimiting lines. Top figure: for the LPPL method, the vertical lines in different colors indicate the IB points diagnosed by the algorithm for the LPPL method (as in Figure \ref{['fig:detection_examples']}). Middle and bottom figure: the found points of upward (middle figure) and downward trend change (bottom figure). Calculations of trend change points are shown for the threshold value of the 75th percentile.

Theorems & Definitions (3)

• Theorem 1
• Definition 1
• Definition 2