# Prognostics of Electromechanical Actuator with Partial Time Scaling Invariant Temporal Alignment
## 8 Uncertainty Measurement  
The preceding sections have expounded upon the alignment procedure, which enables the identification of the most similar actuator within the database (with vibration signals here). For each time series associated with this actuator, we can determine the time index of the point aligned with the current point in the corresponding time series of the actuator under investigation. Consequently, for each parameter, the aforementioned index represents the consumed lifetime as a percentage. The subsequent phase entails amalgamating these distinct values and introducing a measure of uncertainty into the resulting composite. In the ensuing discussion, we will delve into the intricacies of these phases, with a focus on the LLA actuator, serving as an illustrative example. In the case of the LLA, the alignment results yield a vector denoted as \(\mathcal{C}^{\text{LLA}}= \begin{bmatrix}t_{n,1}&t_{n,2}&\dots&t_{n,p}\end{bmatrix}  ^{\top}\in\mathcal{M}_{p,1}(\mathbb{N})\), containing the indices of the last points of the source cloud \(\left(t_{n},x_{n}\right)_{j}^{l}\) aligned for each of the descriptors \(j\). It is important to emphasize that the time indices for both the target and source clouds adhere to the same temporal scale. Consequently, we construct a matrix by concatenating \(p\) Dirac delta functions centered at \(t_{n,j}\). \[\mathbb{I}^{l}=\left(t_{ij}\right)^{l}\text{such as }\mathbb{I}^{l}=\left\{ \begin{aligned} & t_{n,j}=1\text{ si }t_{n,j}=j\\ & 0,\text{ if not}\end{aligned} \right. \tag{11}\]  
We deduce a square matrix from \(\mathbb{I}^{l}\) and the weighting vector \(\mathcal{W}\in\mathcal{M}_{p,1}(\mathbb{R})\):  
\[\mathbb{J}^{l}=\mathbb{I}^{l}\left(\sum_{j=1}^{p}e_{j}^{\top}\mathcal{W}e_{j} e_{j}^{\top}\right)\in\mathcal{M}_{n,p}(\mathbb{R})\]  
Figure 16: Vector \(\mathcal{W}\), normalized values of weights for all selected descriptors. \[\mathbb{J}^{l}= \begin{array}{cccc}\delta_{1}^{\text{LLA}}&\delta_{2}^{\text{ LLA}}&\ldots&\delta_{p}^{\text{LLA}}\\ 0&0&\ldots&0\\ \vdots&\vdots&\vdots&\vdots\\ w_{1}&0&\ldots&0\\ \vdots&\vdots&\vdots&\vdots\\ 0&w_{2}&\ldots&0\\ \vdots&\vdots&\vdots&\vdots\\ 0&0&\ldots&0\end{array}  \tag{12}\]  
The matrix \(\mathbb{J}^{l}\in\mathcal{M}_{n,p}(\mathbb{R})\) is thus created at Equation (12) with as many rows as points in the target series. Here, the set of target series \(\{Y^{\text{LLA}},Y^{\text{URA}},Y^{\text{ULA}}\}\) have the same number of samples. In addition, the matrix \(\mathbb{J}^{l}\) has as many columns as descriptors. Concretely, \(\mathbb{J}^{l}\) is a matrix containing the concatenation of weighted Diracs, each representing the index of the last aligned point of a source descriptor. We have to merge all the information of the columns of \(\mathbb{J}^{l}\) and give an evaluation of the uncertainty. Let \(Z_{j}^{h}\) be the random variable representing the cycle index of the last aligned point at each iteration \(k\). The descriptor of a given actuator, e.g., \(\tilde{X}_{j}^{l}\), is aligned to the descriptor of a given target, \(\tilde{Y}_{j}^{h}\). The alignment algorithm defined in Section 5 is iterative. Thus, as detailed in Algorithm A4, at each iteration, \(k\) is computed for the source cloud with a new assignment of source points to target points by the method A1. An assignment function \(\mathcal{T}_{k}\) being computed at each iteration, during the alignment, is then built with the sequence \(\left(\mathcal{T}_{j}^{l}|_{k}\right)_{1\leq k\leq K}\). The assignment represented by this function is independent of the distance from the points of the source cloud to the points of the target cloud in \(\mathbb{R}^{2}\) because it is computed in the projected space, in \(\mathbb{R}\). The value of interest for the uncertainty measure is, therefore, \(\mathcal{T}_{j}^{l}(n)|_{k}\), which represents the nearest neighbor of the last point of the source cloud \(\tilde{X}_{j}^{l}\in\mathcal{M}_{n,2}(\mathbb{R})\) in the target cloud \(\tilde{Y}_{j}^{h}\in\mathcal{M}_{m,2}(\mathbb{R})\).