# Prognostics of Electromechanical Actuator with Partial Time Scaling Invariant Temporal Alignment
## 6 Quantification of Time Series Alignment
By normalizing the expression in this way, this effect no longer impacts the value of the cost function. Finally, the sum over the iterations of the alignment algorithm \(\sum_{k=1}^{K}\) aggregates all differences over the total number of iterations. As a reminder, for each iteration \(k\), a function \(\mathcal{T}_{k}\) is calculated that assigns each source point to its neighbor in the target. The value of the function \(\mathcal{L}\) will likely change at each iteration. With the expression of \(\mathcal{L}\) given, evaluating it allows us to answer the different hypotheses made to quantify the value of an alignment. In the case where Hypotheses 1, 2, and 3 are met, the cost function \(\mathcal{L}\) defined in Equation (6) is zero. Indeed, \(\forall(i,k)\in[1;n]\times[1;K],x_{i1}=x_{iK}\), so \((x_{i1}-x_{ik})^{2}=0\). In the case where one of the hypotheses is not met, since \(\mathcal{L}\) is a double sum of squares, we have  
\[\forall\mathbb{X},\mathcal{L}(\mathbb{X})\geq\ 0\]  
A cost function \(\mathcal{L}\) is theoretically defined in Equation (6). Let us now test its relevance with experimental data. To do so, the source to be aligned \(X_{\text{RMS}}^{\text{LLA}}\) will be the RMS descriptor of the LLA actuator and the target will be the same descriptor chosen from the set of actuators \(\{LRA,URA,ULA\}\). Our validations will always follow the same approach. **The LLA descriptors will be taken as the source, and the corresponding descriptors of the other actuators will be taken as potential targets. ** Indeed, we assume a similar behavior exists in the database without being completely identical. The data used in this work come from accelerated aging tests of various electromechanical actuators. The calculated descriptors share the same number of samples and sampling frequency. Let us simulate a source with a uniform compression factor relative to the targets to demonstrate the general nature of the cost function \(\mathcal{L}\). Every other point is then selected in the source \(X_{\text{RMS}}^{\text{LLA}}\). Additionally, this same source is truncated to 900 points. A point cloud is then created using the method described in Section 5 to obtain \(\hat{X}_{\text{RMS}}^{\text{LLA}}[0:900:2]\). The resulting point cloud has 450 samples. This two-dimensional data series represents the behavior of an actuator descriptor evolving from 0 to 900 cycles. Thus, the goal of the alignment method is to align the last point of the source \(\hat{X}_{\text{RMS}}^{\text{LLA}}[0:900:2]\) with, at a minimum, the 900 th point of the uncompressed target \(\hat{Y}_{\text{RMS}}^{\text{LLA}}\). Figure 11 shows the trajectories of the source points as the alignment algorithm converges in two different configurations: an alignment of the compressed source on itself, uncompressed in Figure 10(a), and an alignment of the compressed source on the descriptor of another actuator, uncompressed in Figure 10(b). The two parts of Figure 11 illustrate very different behaviors. Indeed, Figure 10(a) represents the trajectories of source points that are all parallel. The last point of the compressed source reaches the 898 th cycle during its alignment in the target, which differs by only two cycles from reality. On the other hand, Figure 10(b) represents more disparate trajectories of source points. In addition, the 448 th point--in other words, the ante-penultimate aligned point--reaches the 1016 th point in the target. This difference is explained by the fact that the target data differ from the source data, while the source is aligned with its homologous descriptor in the URA actuator. However, this information confirms the idea that, with this method, the quality of a series alignment can be quantified by the behavior of the movement directions of each point. Thus, the behavior of each point could be quantified by the function \(\mathcal{L}\) constructed in Equation (6). The cost values obtained from the two alignments then make it possible to verify the wanted behavior of \(\mathcal{L}\).