of those before the start of the work. Due to commercial competition, the original dataset cannot be shared, but the publication should assist to find evidence that with the correct approach a predictive maintenance system can be incredibility successful. This dataset is composed of 51 different bearing, and for each the channels of acceleration, envelope and velocity were considered. Data were collected for 24 months, with 1h-sampled data, providing a total of more than 850.000 data-points to analyse. 4.2 Health Indicator construction The process of developing a Health Index (HI) holds a crucial role in the field of machinery prognostics. An appropriate HI can significantly enhance the accuracy of prognostic modelling, thereby improving the precision of the prediction results. Health indexes can be classified into two categories based on their construction methods: physical HIs and virtual HIs. Physical HIs are associated with the physics of failure and are typically derived from monitoring signals using statistical or signal processing techniques. In contrast, virtual HIs are often created by merging multiple physical HIs or multi-sensor signals, resulting in a loss of physical significance and offering only a virtual representation of the machinery's degradation trends [27]. RMS is the most commonly employed physical Health Index in the prediction of remaining useful life for machinery. [28] predicted the remaining useful life of bearings using RMS applied to vibration data. [29] extracted RMS and peak values from the wavelet coefficients as a means of predicting the remaining useful life of bearings. Similarly, approaches set out it [30] [31] also applied RMS to predict bearing anomalies in machineries. Some researchers constructed new physical Health Indexes based on statistical characteristics of signals in time domain. To evaluate the evolution of cracks in gear teeth, [32] quantified the proportion of a residual error signal that exceeded a baseline threshold. [33] developed a Physical Health Index for bearings by computing the energy ratio between residual signals obtained through autoregressive (AR) filtering and the original signals. On the side of virtual Health Index instead, the most popular technique is certainly PCA [14]. [34] used PCA to reduce the dimension of feature sets and further calculated the deviations between unknown states and the healthy state as a virtual Health Index. [35] used PCA combined with isometric feature mapping to construct a VHI for cutting tools. When dimensionality is a challenge however, [36] innovated with the usage of self-organizing map techniques, and since then this technique has been widely used in the VHI construction [37] [38] [26]. When faced with limited features, it is very popular in the literature [39] [40] [41] [42] to use a statistical, linear or non-linear, data transformation approach to create a virtual Health Index by combining multiple features. This latter has a clear advantage of understandability, which makes it more reliable to validate. Given the practical objective of the health index proposed in this paper, we chose to adopt the aforementioned method. 4.3 Health Stage division The concept of Health Stage division is akin to the commonly used terms "fault detection" or "fault diagnosis" in the field of prognostics and health management [43]. However, their objectives differ. Fault diagnosis aims to identify the specific fault patterns and severity of a given machine at a single point in time. In contrast, Health Stage division seeks to partition the ongoing degradation process of a machine into distinct Health Stages based on the fluctuating trends of health indicators [14] A simple but empirically-solid strategy for health stage division is presented by Wang [44], who developed a two-stage division
system where the unhealthy stage begins when the health-index exceeds a pre-specified alarm threshold based on historical initial points of defects for bearings. More complex methods for a threshold-defined two-stage division are found in the literature (Chebyshev inequality function [45], 3 V Box-Cox transformation [46], Hotelling T 2 statistic [47], locality preserving projection [48]) but complexity does not come at benefits of precision, as no method outperforms the others. When variations in fault patterns or operational conditions occur, the degradation trends of machinery may change, leading to an additional layer of complexity. In such cases, it becomes challenging to accurately describe the degradation processes using a signal degradation model. Consequently, it is necessary to further subdivide the unhealthy stage into different stages based on the various degradation trends. Some researchers have addressed this issue by dividing the degradation processes into multiple stages through the analysis of change points in health indexes or spectra [14]. The number of identified levels and the methods used for their identification vary and sophistication raises compared to the two-stage case. From using confidence levels to build a four-stage system [49], to analyse changes of frequency amplitudes in the power spectral density to develop a five-stage model [50], and apply classification or clustering algorithms, such as K-nearest neighbour [51], fuzzy c- means [52] and K-means [53], to develop the multi-stage division of machinery, the available methodologies for researchers are numerous. For those industrial-oriented real applications though, semi-supervised learning has recently emerged as a prominent methodology [54] [55] [56], as in practical applications, obtaining accurate labels based on real-time bearing conditions can be far more challenging and semi-supervised approach allows for effective utilization of dataset when only a small subset of data have labels. In this paper, given the practical usage of the outcome, the latter approach is further developed and applied. 5. RESULTS The dataset used in this study consisted of the complete data records of approximately 50 bearings that were monitored over a period of 2 years across 3 distinct paper machines situated in the same mill. In order to establish the methodology, data from 23 bearings on a single machine were exclusively analysed. Subsequently, the developed methods were tested on the remaining machines to validate their applicability and generalizability. pp y g y
Figure 2 - Bearing vibration signal. (a) raw (b) (c) filtered To extract valuable insights from the vast amount of data in vibration-based Smart Maintenance, appropriate filtering techniques must be employed to mitigate the impact of noise on the analysis. Noise, which can arise from environmental factors or intentional sources, can interfere with the accuracy of the system,
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