# Prognostics of Electromechanical Actuator with Partial Time Scaling Invariant Temporal Alignment
## 4 Data Framework Surrounding the Prognostic Method  
One monitored industrial asset's RUL is estimated by comparing its operational behavior to a knowledge database. In general, this knowledge database contains a variety of realizations of similar assets in an industrial context. It could be either normal or degraded behaviors. It can also be filled with specific aging scenarios where the asset is accelerated to the limits. For the latter, in practice, the asset is placed inside an abusive environment, and we enable it to fulfill its function in a looping manner to simulate an accelerated aging pattern. We focus our work on the latter database content. Such a database is filled with time series representing physical measurements taken during the aging process of the same family of assets. We can model this database as a four-dimensional tensor \(\mathfrak{D}_{a}\in\mathfrak{A}_{n,p,q,r}\) with \((n,p,q,r)\in\mathbb{N}^{4}\), where \(n\) is the number of acquisitions made per monitored system, \(p\) is the number of different types of measurements (current, voltage, etc. ), \(q\) is the number of assets monitored (always of the same family), and \(r\) represents the number of points of acquisition. Note that the hypothesis is made that the data along all the assets share the same number of points \(r\). Hence, they are only one shared acquisition profile repeated for every measure. Instead of working directly with raw data measured from the monitored system, we provide several aggregation functions for each signal. One such function will compute a _feature_ (or a _descriptor_). Let \(\mathcal{X}\in\mathcal{M}_{(n,1)}(\mathbb{R})\) be a raw time series extracted from \(\mathfrak{D}\); one function \(f\) can be defined as \(f:\mathcal{M}_{(n,1)}(\mathbb{R})\rightarrow\mathbb{R}\), and \(f\) can be seen as an aggregation function transforming an input vector to a unique value. Transporting a whole group of raw data into another set is called _feature extraction_. In this study, we concentrate only on univariate time series; hence, the data processed can be extracted from \(\mathfrak{D}^{1}_{a}\in\mathfrak{A}^{1}_{n,1,q,r}\). We then apply feature extraction to a specific signal of size \(r\) taken from \(\mathfrak{A}^{1^{\prime}}_{1,1,1^{\prime}}\). If we calculate only one descriptor for every acquisition for a particular asset, we can reduce the size of the database to \(\mathfrak{A}^{2}_{n,1,q,1}\). Our study calculates \(p\) different descriptors for every time series. Thus, by collapsing the unitary dimension of the tensor and adding a new \(p\) dimension for each feature, we produce the database \(\mathfrak{D}_{f}\in\mathfrak{F}_{n,p,q}\). \(n\) represents the number of points for each time series (now a _feature_), \(p\) is the number of different descriptors calculated, and \(q\) remains the number of assets studied. Note that, in our study, with our subsystem expert knowledge, we select statistical features taken from an acceleration signal in \(\mathrm{m}\,\mathrm{s}^{-2}\). They are presented in Figure 4.  
These descriptors are completed by percentiles of the actuator in deployment mode in Figure 5.  
Finally, we obtain \(p=28\) features for each actuator. In our application, we aged four identical actuators {Lower Right Actuator (LRA), Lower Left Actuator (LLA), Upper Right Actuator (URA), Upper Left Actuator (ULA)} by cycling them acceleratedly in harsh conditions, consisting of using the actuator at high frequencies through high and below-zero temperatures during an extended period of time. Three of them will be the reference actuators, and the fourth will be the actuator with an unknown state. Let us note \(\mathcal{F}_{(1\leq j\leq p,1\leq k<q)}\) with \((i,k)\in\mathbb{N}^{2}\) a _feature_. To avoid notation overloading, let us define \(\mathcal{F}_{t}\), which represents the same preceding feature whose values are indexed by the natural number \(t\in\mathbb{N}\) in ascending order. This indexation could be supersed later by a series of timestamps, creating a new time series.