# A RUL Estimation System from Clustered Run-to-Failure Degradation Signals

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## Abstract

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## 1. Introduction

- We made improvements in cleaning spikes or possible outlines and smoothing time-series in the pre-processing data step in the fault detection framework developed in [15] to reduce the remaining noise level while maintaining its relevant characteristics such as trends and stationarity.
- We show that the fault detection framework in [15], together with our pre-processing method, improves the robustness of the framework and can be transferable to another problem with similar degradation, although with different statistical characteristics.
- We built a strategy using clustering run-to-failure critical segments to define an appropriate failure threshold that improves the RUL estimation. Moreover, using this strategy, we predict the RUL of another problem with similar degradation.

## 2. Background

#### 2.1. Fault Detection

#### 2.2. Prognostic

#### 2.3. Recurrent Neural Networks (RNNs)

#### 2.4. Prophet Model

## 3. Methodology

#### 3.1. Pre-Processing Data

**Spikes cleaning**: it consists of clearing possible outliers and spikes points by comparing time series values with the values of their preceding time window, identifying a time point as anomalous if the change of value from its preceding average or median is anomalously large.An advantage of this outlier reduction strategy is that it considers the local dynamics of the signal with time windows. Therefore, managing to identify as outliers the samples that are outside the local range and thus reduce the number of samples that are normal but that were identified as outliers, as could happen with traditional methods that depend on the global mean and standard deviation. This method is implemented in the ADTK library [59].**Double exponential smoothing**: this filter [26,60,61,62,63,64] is commonly used for forecasting in time series, but it can also be used for noise reduction. This method is particularly useful in time series to smooth its behavior, preserving the trend and without losing almost any information in the dynamics of the series. Also, the model is simple to implement, depending on two main parameters. For more details, see [15].**Convolutional smoothing**: this consist of applying the Fourier transform with a fixed window size to smooth the signal maintaining the trend. In other words, this method applies a central weighted moving average to the signal allowing short-term fluctuations to be reduced and long-term trends to be highlighted. It is implemented in the TSmoothie library [65].

#### 3.2. Run-to-Failures Critical Segments Clustering

#### 3.3. Prognostic Method

#### 3.3.1. Strategy A

#### 3.3.2. Strategy B

**Cluster-Model stage**: it consists of the usage of clustering described in Section 3.2, so that, for each cluster we can fit a regression model. The train data is defined by the critical signals limited by a defined failure threshold in the cluster with its residual RUL, i.e., for each critical signal S with length $l\left(S\right)$ in cluster C and ${S}^{\prime}\subset S$ such that ${S}_{0}^{\prime}={S}_{0}$, and ${S}_{l\left({S}^{\prime}\right)}^{\prime}\approx failure\_threshold$. Then, each sample ${S}_{i}^{\prime}\in {S}^{\prime}$ has a residual RUL$${r}_{i}:=Normalize\left({S}_{i}^{\prime}\right)\xb7l\left({S}^{\prime}\right),$$$$Normalize\left({S}_{i}\right)=\frac{{S}_{i}-min\left(S\right)}{max\left(S\right)-min\left(S\right)},$$**Prediction stage**: it consists mainly in predicting the RUL of a component in the signal that has been diagnosed as a fault, which means a degradation behavior has started. In this step, we took a segment of the signal after a fault has been detected; it is pre-processed and submitted to a classifier to identify to which cluster it belongs and select the related regression model, already fitted in the Cluster-Model stage, to predict the RUL. This procedure is executed when new samples are available.The classifier works in matching segments to all run-to-failure critical segments using Minimum Variance Matching (MVM) [68,69,70], which is a popular method for elastic matching of two sequences of different lengths by mapping the problem of the best matching subsequence to the problem of the shortest path in a directed acyclic graph providing the minimum distance. The classification scope provides the assignment by a voting criterion, i.e., the maximum number of signals of a cluster closer to a given segment will be taken. A flow chart of this prognostic process is shown in Figure 7.

## 4. Application Setting

#### 4.1. Crack Growth

#### 4.1.1. Problem Description

#### 4.1.2. Prognostic

- Strategy A: following the methodology in Section 3.3.1, we estimate RUL shifting the time window by 15 days in every iteration, 1 year size of time-window, and 2 years of forecast.The results are shown in Figure 9. In the prognostic horizon, Figure 9b, we can see that all the models underestimate RUL, with some exceptions like the Dense neural network model. Neural network models had poor performances of RUL estimation and mostly fall outside of the confidence interval. Only the Prophet model is relatively close to the ground truth RUL. Concerning the $\alpha \u2013\lambda $ accuracy, only Prophet has a segment close to the ground truth but then falls outside of the confidence interval, underestimating the RUL.
- Strategy B: using the technique proposed in Section 3.3.2 in this problem, we will simplify some steps of this process. Given that all the degradation trajectories are similar, we can assume only one cluster and the classifier will assign to it every time. Hence, the Cluster-Model stage has only one model, which is used to predict the RUL. Basically, this scheme becomes a simple regression model where it is fitted with all the historical-critical segment trajectories limited by its failure threshold and its residual RUL. We use 100 trajectories as run-to-failure signals generated from Equation (2) to fit the model.The performances can be seen in Figure 9d,e. All the models fall inside the confidence interval in the prognostic horizon and are getting closer to the ground truth as they reach the EoL, as illustrated in Figure 9d. Similar behavior is obtained for $\alpha \u2013\lambda $ accuracy, as shown in Figure 9d. Only a few times, some methods go out and then go back into the confidence interval, e.g., LSTM and GRU, but these behaviors are acceptable.

#### 4.2. Intermediate Frequency Processor Degradation Problem

#### 4.2.1. Problem Description

#### 4.2.2. Prognostic

- Strategy A: the performances of this method are illustrated in Figure 10b,c, in which we can see that none of these models give good predictions of RUL, nor when it approaches the EoL.
- Strategy B: from the historical run-to-failure signals, different degradation levels appears in each voltage’s current of the IFP. In this application, each voltage’s signals are clustered into a few clusters so that signals in each cluster have similar degradation levels making it easier to define an appropriate failure threshold in each cluster, just as described in Section 3.2, defining a total of 5 clusters for this problem: 2 cluster for 6.5 volts, 1 cluster for 8 volts, and 2 clusters for 10 volts; they are shown in Figure 11, in which, for each cluster has its corresponded failure threshold value, i.e., 0.566 is the failure threshold for cluster 1, 0.2 for cluster 2, 0.127 for cluster 3, 0.246 for cluster 4, and 0.275 for cluster 5; or it can be explained as 5.7%, 2%, 36%, 18%, and 8.3% of degradation levels for each cluster, respectively. These clusters are used to classify the new arriving pre-processed signal to select the appropriate failure threshold and predict the RUL.The cluster generation criterion focuses mainly on the Minimum Variance Matching (MVM) similarity metric, which is obtained by solving a shortest path (SP) problem that measures the distance between pairs of signals. The principle is to fix a signal as a centroid and compute the distances with the other signals; these distances are ordered, and using the same fundamentals of the elbow method, a group of signals is selected to form a cluster ${C}_{1}$ and the rest in another group ${C}_{2}$. This process is repeated for the cluster ${C}_{2}$ to verify if the signals are similar or if another cluster is generated, and so on. Repeated runs were made, resulting in most cases with 5 clusters being enough to separate these signals.The performances under both metrics, Figure 10d,e, show that almost all models have relatively good predictions of RUL falling inside of the confidence interval. Only ESN has some irregularities, but these underestimations are acceptable. The Dense neural network model outperforms the others slightly when it gets close to the EoL.Analyzing the results, the models that used strategy A showed a problem similar to what occurs in the application of the Crack Growth in Section 4.1.2, in which the models remain sensitive to small variations, generating a great variability in the estimation of EoL and therefore, affects the prediction of the RUL.Taking into account these effects that it could have on the models, if strategy B is used and a set of historical run-to-failure signals is considered that have great variability in the degradation behavior, different from that used in Section 4.1.2 in which the signals are quite similar, could affect the models in predicting the RUL due to these variations in the level of degradation of the historical signals.To avoid this, it was decided to group the signals into groups that are similar in degradation level and address them separately. As a consequence, the performance in different models manages to predict the RUL close to the real value.

#### 4.3. Validation in a Different Setting

#### 4.3.1. Problem Description

#### 4.3.2. Fault Detection

#### 4.3.3. Prognostic

- Strategy A: applying this method, we can see Figure 15b,c, that neural networks have a poor quality of predictions, whereas the Prophet model has some segments that fall inside the confidence interval, but it is not good enough because of its irregular behaviour.
- Strategy B: in this problem, there are no historical run-to-failure signals. So, clustering over this component is not possible. However, given that the degradation behavior present in this component is similar to the IFP of ALMA, we can use these clusters and try to transfer to this problem. To achieve this, it is necessary to transform the new arriving pre-processed signal Q and scale it to every cluster described in Section 3.2, this means, for each cluster, we define a transformed signal of Q as follows$$\begin{array}{ccc}\hfill S& =& \kappa \xb7Q,\hfill \end{array}$$$$\begin{array}{ccc}\hfill {S}^{\prime}& =& S-{S}_{0}+{k}_{i}\hfill \end{array}$$$$\kappa =\frac{{k}_{i}-{k}_{i}^{*}}{{Q}_{0}-{q}^{*}}$$The classifier result gives the final scope, which is used for model selection in the prediction of RUL. In the prognostic horizon metric, Figure 15d, the GRU model outperforms the other models. However, the other models fall inside the confidence interval after 200 days. So, all the models in this metric are acceptable. From the $\alpha \u2013\lambda $ accuracy side, most of the time, these models are not inside the confidence interval, underestimating the RUL on the first 300 days $(\lambda =3/4)$. After that, they are around the ground truth up to the EoL. In this case, the GRU model is close to the frontier of the confidence interval, which is not as bad as an instance for RUL computation by using a similar degradation signal developed from another system or component like the IFP Problem.

## 5. Discussion

- Strategy A: time-window size was 365 days, 2 years of forecasting, a lookback of 19 samples format (e.g., samples from time $t-19$ until time t with a total of 20 samples) as input, and 20 epochs for neural networks adjustments. For simplicity, we assume for this method that new data is available every 15 days to update RUL estimation. The model hyperparameters used for prognostics are summarized in Table 1.
- Strategy B: a lookback of 9 samples format (e.g., samples from time $t-9$ until time t with a total of 10 samples) as input, and 15 epochs for neural networks adjustments. The model hyperparameters used for prognostics are summarized in Table 2.

- Crack Growth in Section 4.1.2: is a classical problem in the literature in which the degradation is a monotonical non-decreasing trajectory. The worst performances are given by strategy A, where only the Prophet model was relatively close to the ground truth RUL. Whereas, the strategy B, all prediction models are significantly well performed on both metrics.
- IFP Degradation in Section 4.2.2: the historical degradation signals are not totally monotonous with different degradation levels and speeds, resulting in different failure threshold values for a set of signals. With this insight, defining a unique failure threshold for all the signals and forecasting the dynamic of the signal until reaching the failure threshold as described by strategy A does not work well. Therefore, clustering signals by degradation levels helps to define appropriately the failure threshold given the characteristic of similarity to a set of historical run-to-failure signals from a cluster. Therefore, using strategy B improves the prediction of RULs, in which ESN is the less accurate model than the other models tested.
- Camera Resolution Degradation in Section 4.3.3: the degradation trajectory showed irregularities similar to the IFP signals, in which there is some segment increase and then decrease, and vice versa. Therefore, the degradation trajectory is also not completely monotonous. Addressing this problem with strategy A showed some difficulties, particularly trying to forecast the dynamic or trend of the signal when the trend of the segment changes in the opposite sense to the degradation, obtaining an overestimation of the RUL. Working with this strategy showed that only the Prophet approximates the ground truth, but it is still not good enough and acceptable. From the strategy B perspective and using the RUL predictive model transferred from the IFP setting provided better results compared to the previous strategy, converging to the ground truth as it reaches the EoL with a few minor exceptions.

## 6. Conclusions

## 7. Future Work

- Improving the computation of uncertainty measurements of RUL predictions. This computation will help develop new prescriptive maintenance approaches that help in the decision-making process of maintenance procedures.
- Test this approach on other problems with similar degradation faults to continue evaluating the robustness of this run-to-failure critical segment clustering approach to predict a component’s RUL value.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RUL | Remaining Useful Life |

RNN | Recurrent Neural Network |

ALMA | Atacama Large Millimeter Array |

CBM | Condition-Based Maintenance |

PHM | Prognostic and Health Management |

PH | Prognostic Horizon |

EoL | End-of-Life |

ESN | Echo State Network |

LSTM | Long-Short Term Memory |

GRU | Gated Recurrent Unit |

ADTK | Anomaly Detection Toolkit |

MVM | Minimum Variance Matching |

IFP | Intermediate Frecuency Processor |

LSB | Lower Sideband |

USB | Upper Sideband |

SW | Switch matrix current |

UT | Unit Telescope |

CCD | Charge-Coupled Device |

EoP | End-of-Prediction |

ANN | Artificial Neural Network |

SP | Shortest Path |

## Appendix A. Evaluation Metrics

**Pronostic Horizon (PH)**: it identifies whether a method predicts within specified limits around the ground truth End-of-Life (EoL) so that the predictions are considered trustworthy. If it does, how much time does it allow for any maintenance action to be taken. The longer PH better the model and more time to act based on the prediction with some desired credibility. This metric is defined as:$$PH={t}_{EoL}-{t}_{i},$$$$i=min\left\{j|(j\in \mathcal{J})\wedge \left[{\varrho}_{\alpha}^{-}\le r\left(j\right)\le {\varrho}_{\alpha}^{+}\right]\right\},$$$$\begin{array}{ccc}\hfill {\varrho}_{\alpha}^{-}& =& {r}_{*}-\alpha \xb7{t}_{EoL},\hfill \\ \hfill {\varrho}_{\alpha}^{+}& =& {r}_{*}+\alpha \xb7{t}_{EoL}.\hfill \end{array}$$**$\mathit{\alpha}\mathbf{\u2013}\mathit{\lambda}$ Accuracy**: this metric quantifies the prediction quality by identifying whether the prediction falls within specified limits at a particular time; this is a more stringent requirement as compared to PH since it requires predictions to stay within a cone of accuracy. Its output is binary since we need to evaluate whether the following condition is met,$$(1-\alpha )\xb7{r}_{*}\left(t\right)\le r\left({t}_{\lambda}\right)\le (1+\alpha )\xb7{r}_{*}\left(t\right),$$$${t}_{\lambda}={t}_{P}+\lambda \xb7({t}_{EoL}-{t}_{P}).$$**Relative Accuracy**: a similar notion as $\alpha \u2013\lambda $ accuracy where, instead of finding out whether the predictions fall within given accuracy levels at a given time ${t}_{\lambda}$, we also quantitatively measure the accuracy by the following$$R{A}_{\lambda}=1-\frac{{r}_{*}\left({t}_{\lambda}\right)-{r}_{{t}_{\lambda}}}{{r}_{*}\left({t}_{\lambda}\right)},$$$$CR{A}_{\lambda}=\frac{1}{\left|{\mathcal{J}}_{\lambda}\right|}\sum _{i=1}^{{\mathcal{J}}_{\lambda}}w\left(r\right)R{A}_{\lambda}$$**Convergence**: it is a useful metric since we expect a prognostics algorithm to converge to the true value as more information accumulates over time. Besides, it shows that the distance between the origin and the centroid of the area under the curve for a metric quantifies convergence, and a faster convergence is desired to achieve high confidence in keeping the prediction horizon as large as possible. Lower distance means a faster convergence. The computation of this metric is defined as, let $({x}_{c},{y}_{c})$ be the center of mass of the area under the curve $M\left(i\right)$. Then, the convergence ${C}_{M}$ can be represented by the Euclidean distance between the center of mass and $({t}_{P},0)$, where$${C}_{M}=\sqrt{{({x}_{c}-{t}_{P})}^{2}+{y}_{c}^{2}},$$$${x}_{c}=\frac{1}{2}\frac{{\sum}_{i=P}^{EoP}\left({t}_{i+1}^{2}-{t}_{i}^{2}\right)M\left(i\right)}{{\sum}_{i=P}^{EoP}\left({t}_{i+1}-{t}_{i}\right)M\left(i\right)},$$$${y}_{c}=\frac{1}{2}\frac{{\sum}_{i=P}^{EoP}\left({t}_{i+1}-{t}_{i}\right){M}^{2}\left(i\right)}{{\sum}_{i=P}^{EoP}\left({t}_{i+1}-{t}_{i}\right)M\left(i\right)},$$$M\left(i\right)$ is a non-negative prediction error accuracy or precision metric. In other words, this metric measures the fastness of convergence of a method.

## Appendix B. Recurrent Neural Networks

#### Appendix B.1. Echo State Networks (ESNs)

#### Appendix B.2. Long-Short Term Memory (LSTM)

**Block input**: it consists of combining the input $\mathbf{u}\left(t\right)$ and the previous output of LSTM units $\mathbf{h}(t-1)$ for each time step t, and it is defined as$$\mathbf{z}\left(t\right)=\varphi \left({\mathbf{W}}_{z}\mathbf{u}\left(t\right)+{\mathbf{R}}_{z}\mathbf{h}(t-1)+{\mathbf{b}}_{z}\right).$$**Input gate**: this gate decides which values needs to be updated with new information to the cell state. It is computed as a combination of the input $\mathbf{u}\left(t\right)$, the previous output of LSTM units $\mathbf{h}(t-1)$, and the previous cell state $\mathbf{c}(t-1)$ for each time step t,$$\begin{array}{ccc}\hfill \mathbf{i}\left(t\right)& =& \sigma \left({\mathbf{W}}_{i}\mathbf{u}\left(t\right)+{\mathbf{R}}_{i}\mathbf{h}(t-1)\right.\hfill \\ & & \phantom{\rule{1.em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left.+{\mathbf{p}}_{i}\odot \mathbf{c}(t-1)+{\mathbf{b}}_{i}\right).\hfill \end{array}$$**Forget gate**: it makes the decision of what information needs to be removed from the LSTM memory, and it is calculated similarly to the input gate.$$\begin{array}{ccc}\hfill \mathbf{f}\left(t\right)& =& \sigma \left({\mathbf{W}}_{f}\mathbf{u}\left(t\right)+{\mathbf{R}}_{f}\mathbf{h}(t-1)\right.\hfill \\ & & \phantom{\rule{1.em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left.+{\mathbf{p}}_{f}\odot \mathbf{c}(t-1)+{\mathbf{b}}_{f}\right).\hfill \end{array}$$**Cell state**: this step provides an update for the LSTM memory in which the current value is given by the combination of block input $\mathbf{z}\left(t\right)$, input gate $\mathbf{i}\left(t\right)$, forget gate $\mathbf{f}\left(t\right)$ and the previous cell state $\mathbf{c}(t-1)$.$$\mathbf{c}\left(t\right)=\mathbf{z}\left(t\right)\odot \mathbf{i}\left(t\right)+\mathbf{c}(t-1)\odot \mathbf{f}\left(t\right).$$**Output gate**: this gate makes the decision of what part of the LSTM memory contributes to the output and it is related to the current input vector $\mathbf{u}\left(t\right)$, the previous output $\mathbf{h}(t-1)$, and the current cell state $\mathbf{c}\left(t\right)$.$$\begin{array}{ccc}\hfill \mathbf{o}\left(t\right)& =& \sigma \left({\mathbf{W}}_{o}\mathbf{u}\left(t\right)+{\mathbf{R}}_{o}\mathbf{h}(t-1)\right.\hfill \\ & & \phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}\left.+{\mathbf{p}}_{o}\odot \mathbf{c}\left(t\right)+{\mathbf{b}}_{o}\right).\hfill \end{array}$$**Block output**: finally, this step computes the output $\mathbf{h}\left(t\right)$, which combines the current cell state $\mathbf{c}\left(t\right)$ and the current output gate $\mathbf{o}\left(t\right)$.$$\mathbf{h}\left(t\right)=\psi \left(\mathbf{c}\right(t\left)\right)\odot \mathbf{o}\left(t\right)$$

#### Appendix B.3. Gated Recurrent Unit (GRU)

**Update gate**: this gate determines how much previously learned information should be passed on to the future,$$\mathbf{z}\left(t\right)=\sigma \left({\mathbf{W}}_{z}\mathbf{u}\left(t\right)+{\mathbf{R}}_{z}\mathbf{h}(t-1)+{\mathbf{b}}_{z}\right).$$**Reset gate**: this gate decides how much previously learned information to forget.$$\mathbf{r}\left(t\right)=\sigma \left({\mathbf{W}}_{r}\mathbf{u}\left(t\right)+{\mathbf{R}}_{r}\mathbf{h}(t-1)+{\mathbf{b}}_{r}\right).$$**Cell state**: it consists of storing the relevant information from the past, using the reset gate to affect the memory content.$$\begin{array}{ccc}\hfill \mathbf{c}\left(t\right)& =& tanh\left({\mathbf{W}}_{c}\mathbf{u}\left(t\right)\right.\hfill \\ & & \phantom{\rule{1.em}{0ex}}\left.+{\mathbf{R}}_{c}\mathbf{h}(t-1)\odot \mathbf{r}\left(t\right)+{\mathbf{b}}_{c}\right).\hfill \end{array}$$**Block output**: finally, compute the output $\mathbf{y}\left(t\right)$$$\mathbf{h}\left(t\right)=\mathbf{c}\left(t\right)\odot \mathbf{z}\left(t\right)+\mathbf{h}(t-1)\odot \left(1-\mathbf{z}\left(t\right)\right)$$

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**Figure 2.**Application of each method separately to the raw signal. (

**a**) Outliers and spikes cleaning. (

**b**) Double Exponential Smoothing. (

**c**) Convolutional smoothing. (

**d**) Proposed pre-processing method.

**Figure 3.**Time-window examples. (

**a**) Time-window with missing values. (

**b**) Time-window without missing values.

**Figure 5.**An example of RUL estimation using a time-window size of 365 days. (

**a**) Time-window samples until fault date ${t}_{P}$. (

**b**) Time-window shifted by 300 days.

**Figure 9.**The crack growth prognostic. (

**a**) Testing: a crack growth trajectory. (

**b**) Strategy A: the prognostic horizon metric. (

**c**) Strategy A: the $\alpha \u2013\lambda $ accuracy metric. (

**d**) Strategy B: the prognostic horizon metric. (

**e**) Strategy B: the $\alpha \u2013\lambda $ accuracy metric.

**Figure 10.**The IFP prognostic. (

**a**) Testing: a signal from an IFP. (

**b**) Strategy A: the prognostic horizon metric. (

**c**) Strategy A: the $\alpha \u2013\lambda $ accuracy metric. (

**d**) Strategy B: the prognostic horizon metric. (

**e**) Strategy B: the $\alpha \u2013\lambda $ accuracy metric.

**Figure 11.**The IFP signals clustering, the red dashed lines represent the failure threshold defined for each cluster, and continuous lines are the critical segments segmented from the run-to-failure IFP signals (

**a**) Class 1: 6.5 Volts (Degradation type 1). (

**b**) Class 2: 6.5 Volts (Degradation type 2). (

**c**) Class 3: 8 Volts. (

**d**) Class 4: 10 Volts (Degradation type 1). (

**e**) Class 5: 10 Volts (Degradation type 2).

**Figure 15.**The Camera Resolution prognostic. (

**a**) Testing: Resolution media trajectory. (

**b**) Strategy A: The prognostic horizon metric. (

**c**) Strategy A: The $\alpha \u2013\lambda $ accuracy metric. (

**d**) Strategy B: The prognostic horizon metric. (

**e**) Strategy B: The $\alpha \u2013\lambda $ accuracy metric.

Model | ||||||

ESN | GRU | LSTM | ||||

Hyperparameter | input_size: | 20 | input_shape: | (20, 1) | input_shape: | (20, 1) |

output_size: | 1 | units (GRU): | 20 | units (LSTM): | 20 | |

reservoir_size: | 100 | activation (GRU): | reLU | activation (LSTM): | reLU | |

spectralRadius: | 0.75 | units (Dense): | 20 | units (Dense): | 20 | |

noise_scale: | 0.001 | activation (Dense): | reLU | activation (Dense): | reLU | |

leaking_rate: | 0.5 | units (Dense): | 1 | units (Dense): | 1 | |

sparsity: | 0.3 | activation (Dense): | linear | activation (Dense): | linear | |

activation: | tanh | optimizer: | adam | optimizer: | adam | |

feedback: | True | |||||

regularizationType: | Ridge | |||||

regularizationParam: | auto | |||||

Prophet | ||||||

changepoint_prior_scale: | 0.05 | |||||

seasonality_prior_scale | 0.01 | |||||

daily_seasonality: | False |

Model | ||||||
---|---|---|---|---|---|---|

ESN | GRU | Dense | ||||

Hyperparameter | input_size: | 10 | input_shape: | (10, 1) | input_shape: | 10 |

output_size: | 1 | units (GRU): | 15 | units (Dense): | 50 | |

reservoir_size: | 250 | activation (GRU): | reLU | activation (Dense): | reLU | |

spectralRadius: | 1.0 | recurrent_dropout (GRU): | 0.5 | dropout: | 0.5 | |

noise_scale: | 0.001 | units (GRU) | 15 | units (Dense): | 25 | |

leaking_rate: | 0.7 | activation (GRU): | reLU | activation (Dense): | reLU | |

sparsity: | 0.2 | recurrent_dropout (GRU): | 0.5 | dropout: | 0.5 | |

activation: | tanh | units (Dense): | 1 | units (Dense): | 1 | |

feedback: | False | activation (Dense): | linear | activation (Dense): | linear | |

regularizationType: | Ridge | optimizer: | adam | optimizer: | adam | |

regularizationParam: | 0.01 |

Problem | Prophet | ESN | LSTM | GRU |
---|---|---|---|---|

Crack growth | 252.40 | 109.49 | 2170.89 | 2197.84 |

Resolution Degradation | 193.41 | 31.60 | 1995.64 | 1997.99 |

IFP Degradation | 82.28 | 38.20 | 892.36 | 890.27 |

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**MDPI and ACS Style**

Cho, A.D.; Carrasco, R.A.; Ruz, G.A. A RUL Estimation System from Clustered Run-to-Failure Degradation Signals. *Sensors* **2022**, *22*, 5323.
https://doi.org/10.3390/s22145323

**AMA Style**

Cho AD, Carrasco RA, Ruz GA. A RUL Estimation System from Clustered Run-to-Failure Degradation Signals. *Sensors*. 2022; 22(14):5323.
https://doi.org/10.3390/s22145323

**Chicago/Turabian Style**

Cho, Anthony D., Rodrigo A. Carrasco, and Gonzalo A. Ruz. 2022. "A RUL Estimation System from Clustered Run-to-Failure Degradation Signals" *Sensors* 22, no. 14: 5323.
https://doi.org/10.3390/s22145323