# Prognostics of Electromechanical Actuator with Partial Time Scaling Invariant Temporal Alignment
## 3 General Description of Time Series Alignment  
In the previous steps, we have created a database \(\mathfrak{D}_{f}\in\mathfrak{F}_{n,p,q}\) filled with time series. These data originated from accelerating aging scenarios where the monitored system cycled in a harsh environment. Thus, we can hypothesize that this database is filled with probable behaviors for the monitored system. A prognostic method consists of extrapolating the behavior of a system (we consider that the system behavior is characterized by its measurements) and diagnosing these extrapolations to estimate the monitored system's future State of Health (SOH). In other words, two steps are realized: performing a regression and classifying the new measurement. To do this, because we consider the database \(\mathfrak{D}_{f}\) filled with behaviors until the asset's end of life, the extrapolation can be estimated by aligning the new time series (i.e., the new measurement, measurement on a new asset of the same family, the new features) to every one of their counterparts in \(\mathfrak{D}_{f}\). The extrapolated behavior will be the behavior of the best candidate for the alignment method. Furthermore, because the feature database SOH is known for every point, the new measurements can inherit the diagnostic labels of the best-aligned candidate of \(\mathfrak{D}_{f}\). Thus, we develop a prognostic method based on time series alignment, i.e., finding a function \(\tau\) that associates one point from one series with the other. The studied data are then divided into two ensembles:  
* Points of the series to align from, called source points;
* Points of the series to align to, called target points. In practice, let \(X=(x_{i})_{1\leq i\leq n}\) be a time series of length \(n\in\mathbb{N}\) called the source and \(Y=(y_{j})_{1\leq j\leq m}\) a time series of length \(m\in\mathbb{N}\) called the _target_. In the latter section, we consider that \(n\leq m\) and note \(\mathcal{I}\) (resp. \(\mathcal{J}\)) the ensemble of indices of \(X\) (resp. \(\mathcal{Y}\)). The alignment can be framed as an optimization problem in which \(\mathcal{T}\) is found by minimizing an objective function \(\mathcal{L}\). This injective function \(\mathcal{T}:\mathcal{I}\hookrightarrow\mathcal{J}\) associates with every index of each point of \(X\) its associated index in \(Y\). Hence, the association represented by \(\mathcal{T}\) can be incomplete because every point in \(Y\) is not necessarily associated with every point of \(X\). In this PHM framework, we consider the source to have fewer points than the target. Even if this hypothesis is mandatory in our prognostic framework, the time series alignment method is independent of the length of each series. If this hypothesis is unverified, we can increase the sampling of either the source or the target. More precisely, for our study, the vibratory signals captured on \(q=4\) actuators, which are supposed to be independent, are extracted \(p=28\) statistical descriptors of \(m\) points. These signals form a database noted as \(\mathfrak{M}_{m,p,q}(\mathbb{R})\). The subscripts \(1\leq i\leq n\) (or \(m\)), \(1\leq j\leq p\), \(1\leq l\leq q\) and \(1\leq h\leq q-1\) are introduced to realize a consistent notation scheme in the following sections. For this work, an actuator will be selected to simulate a set of data that would have been captured in operational use. Thus, these data will be contained in a matrix \(\mathbb{X}^{l}\in\mathcal{M}_{n,p}(\mathbb{R})\) that will have to be compared to the data contained in a tensor \(\mathfrak{Y}\in\mathfrak{M}_{m,p,(q-1)}(\mathbb{R})\). It is possible to determine the State of Health of the actuator from which the unknown signals come by column indexed by \(j\), and thus a descriptor of the matrix \(\mathbb{X}^{l}\). Hence, \(X^{l}_{j}=\mathbb{X}^{l}[:,j]\in\mathcal{M}_{n,1}(\mathbb{R})\). The target will be noted as \(Y^{h}_{j}=\mathfrak{Y}[:,j,h]\in\mathcal{M}_{m,1}(\mathbb{R})\).