# Prognostics of Electromechanical Actuator with Partial Time Scaling Invariant Temporal Alignment
## 5 Alignment by Successive Displacements
In Figure 7, the second and fourth steps of the original method are modified. Instead of choosing a random direction to project each cloud point onto, every point is projected onto an \"optimal\" direction in the sense that it minimizes a specific cost function \(\mathfrak{L}\) in step 1. Moreover, contrary to moving in the same direction, source points now move directly towards each of their neighbors in the target cloud point. During this last step, the displacement quantity is not computed in the one-dimensional projection space but directly in the final two-dimensional space. In doing so, calculating discrete movement amounts in Euclidean space has replaced optimal transport equation solving. Note that, for step 2, we still use the original nearest neighbor algorithm search developed by [11]. Let us look at the detailed method. Step 1--Research for optimal projection space  
This step consists of choosing the projection directions to solve the problem of nearest neighbor assignment in a 1-dimensional space. Let \(\Theta\) be the set of projection directions. This set is a sampling of the unit sphere of directions \(\mathcal{S}^{d-1}\), with \(d\) being the number of dimensions of the problem. For any dimension considered, particularly for our  
Figure 6: Illustration of the principle of SPOT—initial method. 2-dimensional space of interest for time series alignment, \(\Theta\) comprises lines. In the work of [11], \(\Theta\) was a random sampling of \(S^{d-1}\). This randomness did not suit our needs. Indeed, the signals studied have specific dynamics that the projection step sometimes causes to disappear when projecting in a specific direction. The assignment of the nearest neighbors in step 2 was carried out with signals with behaviors too far from the original signals before projection. As a result, the final alignment was not relevant. Therefore, it is necessary to find a set \(\Theta^{*}\subset\Theta\) containing a set of lines providing a good final alignment result. The alignment method being iterative, a direction \(\theta_{k}\) is assigned for each iteration. This is why the set \(\Theta^{*}\) consisting of several directions \(\big{(}\theta_{0}^{*},\theta_{1}^{*},\ldots,\theta_{K}^{*}\big{)}\) must be the solution minimizing a quality criterion. This quality criterion will be the cost function \(\mathfrak{L}\), a variant of the cost function \(\mathcal{L}\) constructed further in Section 6. This optimization problem is formalized in Equation (1). \[ \begin{split}&\Theta_{j}^{h,*}=\min_{(\theta_{1},\theta_{2}, \ldots\theta_{K})_{j}^{l}}\sum_{k=0}^{K}\mathfrak{L}\Big{(}\{\mathbb{X} \mathcal{T}|_{k}\}_{j}^{l}\Big{)}\\ &\text{with }\mathbb{X}_{\mathcal{T}}|_{k}\in\mathcal{M}_{n,k}( \mathbb{R})\text{ such as: }\mathbb{X}_{\mathcal{T}}|_{k}=\big{[}X_{\mathcal{T}}|_{k=0}\quad X_{ \mathcal{T}}|_{k=1}\quad\ldots\quad X_{\mathcal{T}}|_{k=k}\big{]}^{\top}\\ &\text{with }X_{\mathcal{T}}|_{k}\in\mathcal{M}_{n,1}(\mathbb{R}) \text{ such as: }X_{\mathcal{T}}|_{k}=\mathbb{T}\Big{(}\tilde{X}_{j}^{d} \theta_{k,j}^{h}\tilde{Y}_{j}^{h}\theta_{k,j}^{h}\Big{)}\end{split}  \tag{1}\]  
In Equation (1), \(\Theta_{j}^{h}\) represents the optimal set of directions determined for descriptor \(j\) and actuator \(h\). The set is obtained by minimizing the sum of the movement costs, in one dimension, of the different states of the source cloud at each iteration. \(\{\mathbb{X}_{\mathcal{T}}|_{k}\}_{j}^{h}\) gathers these movement costs. For the first iteration, \(k=0\), \(\mathbb{X}_{\mathcal{T}}|_{0}=X_{\mathcal{T}}|_{k=0}\); for the second iteration, \(k=1\), \(\mathbb{X}_{\mathcal{T}}|_{1}=\big{[}X_{\mathcal{T}}|_{k=0}\quad X_{\mathcal{ T}}|_{k=1}\big{]}^{\top}\); and for the last iteration, \(K\)\(\mathbb{X}_{\mathcal{T}}|_{K}=\big{[}X_{\mathcal{T}}|_{k=0}\quad X_{\mathcal{ T}}|_{k=1}\quad\ldots\quad X_{\mathcal{T}}|_{k=K}\big{]}^{\top}\). The set \(\{\mathbb{X}_{\mathcal{T}}|\}_{j}^{l}\) on which the cost function \(\mathfrak{L}\) is calculated is therefore of size \(K\).