Complexity and entropy representation for machine component diagnostics

The Complexity-entropy causality plane (CECP) is a parsimonious representation space for time series. It has only two dimensions: normalized permutation entropy (HS) and Jensen-Shannon complexity (CJS) of a time series. This two-dimensional representation allows for detection of slow or rapid drifts in the condition of mechanical components monitored through sensor measurements. The CECP representation can be used for both predictive analytics and visual monitoring of changes in component condition. This method requires minimal pre-processing of raw signals. Furthermore, it is insensitive to noise, stationarity, and trends. These desirable properties make CECP a good candidate for machine condition monitoring and fault diagnostics. In this work we study the effectiveness of CECP on three rotary component condition assessment applications. We use CECP representation of vibration signals to differentiate various machine component health conditions for rotary machine components, namely roller bearing and gears. The results confirm that the CECP representation is able to detect, with high accuracy, changes in underlying dynamics of machine component degradation states. From class separability perspective, the CECP representation is able to generate linearly separable classes for the classification of different fault states. This classification performance improves with increasing signal length. For signal length of 16,384 data points, the fault classification accuracy varies from 90% to 100% for bearing applications, and from 85% to 100% for gear applications. We observed that the optimum parameter for CECP representatino depends on the application. For bearing applications we found that embedding dimension D = 4, 5, 6, and embedding delay τ = 1, 2, 3 are suitable for good fault classification. For gear applications we find that embedding dimension D = 4, 5, and embedding delay τ = 1, 5 are suitable for fault classification.

classifier. Thr ROC and AUC computations was performed using perfcurve() function. The validation accuracy (ACC) was computed using kfoldLoss() function. For three class problem fitcecoc() function was used to model a linear SVM. The validation process and evaluation of the classifier performance remains the same as above.

CECP Plots for MFPT Ball Bearing Experiment
Vibration signals for inner race and outer race fault of roller bearings were generated. The details and the data source is given in the main text.

Sensitivity Analysis
For each load condition we vary the parameters as given below Signal Length n = 2048, 4096, 8192, 16384, 32768, Embedding Dimension D = 3, 4, 5, 6, and, Embedding Delay τ = 1, 2, 3, 4, 5. In the following sections we plot H S and C JS values for load conditions 25, 100 and 300 lb. We also plot the performance of SVM classifiers for the aforementioned three load conditions. For the SVM classification, we consider parameter values D = 6 and τ = 1. For SVM we consider signal lengths, n = 2048, 4096, 8192, 16384 for training. We did not consider signal length n = 32768 because it yields fewer data points not sufficient for training SVM.

Classifier Accuracy for all Operating Conditions
We have included the results of the sensitivity analysis and the SVM classification for three load conditions. We observe the same pattern for other load conditions where the classifier performance improves with respect to increase in the signal length. The plots for all conditions are not shown to conserve space. The SVM results for all the load conditions for D = 6 and τ = 1 are presented in Table S1. Since the number of data points are very less for signal length n = 32768, we did not include in SVM calculations.

CECP Plots for CWRU Bearing Experiments
Vibration signals were collected for ball fault, inner race fault and good working condition (baseline) bearing. The details of the experiment is given in the main text. Sensor signals were collected at a frequency of 12,000 Hz.
The length of each fault-related signals was varied between 120,000 and 130,000 data points while the length of the baseline signals was varied between 200,000 and 500,000 data points.

Operating Conditions
The experimental parameters are outlined in Table S2. For all the parameter variations the fault depth was maintained at 0.2794 mm (0.011 inches). In the following sections we include the plots for one case of drive end bearing and one case of fan end bearing. Table ST3 includes

Classifier Accuracy for all Operating Conditions
We have included the results of the sensitivity analysis and the SVM classification for two cases. We observe the same pattern for other operating conditions where the classifier performance improves with respect to increase in the signal length. The plots for all conditions are not shown to conserve space. The SVM results for fan-end and drive-end bearings under all operating conditions, and parameters, D = 6 and τ = 1 are presented in Table S3 and S4 respectively.

CECP Plots for PHM Gear Experiments
The details of the experiment is given in the main text. From the dataset we consider the case titled helical 1 (has no known gear defects were found) as the baseline case. We consider helical 2 (has a chipped tooth in helical gear with 24 teeth) as a chipped tooth gear category and helical 5 (has a broken tooth in helical gear with 24 teeth) as a broken tooth gear category. In all the three cases, we use the vibration signals recorded from accelerometer 2 (placed on the output side). The signals were recorded under two different load conditions (labeled as Low and High) and five different rotational speeds (i.e., 30 rps (1800 rpm), 35 rps (2100 rpm), 40 rps (2400 rpm), 45 rps (2700 rpm), and 50 rps (3000 rpm) ). For each of these settings, two signals were recorded for four seconds each. Thus for one fault signal, 533,312 data points were generated for eight-second recording.
In the sections below we show plots of the sensitivity analysis for operating conditions speed 1800 rpm and 3000 rpm under low and high loads. The plots showing the performance of the SVM classifier for the above mentioned cases are also included.    Table S5.