# Prognostics of Electromechanical Actuator with Partial Time Scaling Invariant Temporal Alignment
## 5 Alignment by Successive Displacements
During this first procedure, different points in the source can be assigned to the same point in the target. These repetitions are then counted and stored in the nonInjMatch variable declared on line 5 and updated on line 9 of Algorithm A1. This information allows for the series to be decomposed into intervals and thus reduces the size of the interval considered for alignment. To this effect, the Reduce Range function described by Figure 2 of [11] is used. These new intervals created and representing a bijective assignment function must then go through the application of the Nearest Neighbor Match function in Algorithm A2, at line 17. After this second assignment, the intervals are decomposed into a set of sub-intervals by the Linear Time Decomposition function described by [11], Algorithm 2. This operation represents the third decomposition of a series into distinct intervals. A list of structures containing the start and end of each interval is then stored in the paramListArray variable. On each of these intervals, the nearest neighbor assignment is calculated at line 23, and then the Handle Simple Case function, which replicates the behavior described in Section 3.2 of [11], allows for certain configurations to be solved in linear time. Finally, after one final sequence of procedures, the Simple Solve function described by [11], Algorithm 1, allows for the final assignment to be obtained. Therefore, the function \(\mathcal{T}_{k}\) has been generated. Algorithms A1 and A2 allow us to obtain, for an iteration, the assignment function \(\mathcal{T}_{k}\). Step 2 of Figure 6 is then completed. This function will then be used in the advection model of the final step, step 3. Figure 8: Principle schematics of assignation function \(\mathcal{T}_{k}:\mathcal{I}\hookrightarrow\mathcal{J}\). Step 3--Advection model  
As schematized in Figure 7, the advection model developed differs from the SPOT method in [11], where the source points move towards their neighbors in the target by an amount calculated by solving a one-dimensional transportation problem. Here, a displacement amount is also calculated for each source point, but the points will then move by a given amount in the direction created between them and their respective closest neighbors in the target. Figure 9 illustrates this principle for a source point moving towards two different target points defined as its \(n\)th closest neighbor for each iteration. In Figure 9, the dotted red circles represent the positions of the source point before the current iteration. The function \(\mathcal{T}_{j}^{l}(n)|k=0\) in Figure 9 is the assignment function evaluated at the \(n\)th source point for the first iteration of the alignment of the \(j\)th descriptor of actuator \(l\). At each new iteration of the transportation algorithm, the source cloud will move closer to the target cloud until it ultimately reaches it. In Figure 9, the displacement vectors of the source points are represented by a green arrow. Let \(n(k)\) be the source point considered at iteration \(k\) and \(m(k)\) be the target point corresponding to its closest neighbor at iteration \(k\). Then, for each point, \(n\) and \(m\), the displacement vector has a direction along the line passing through \(n(k)\) and \(m(k)\) is directed towards \(m(k)\) and has a norm, or _movement quantity_, equal to half of the Euclidean distance between \(n(k)\) and \(m(k)\). Thus, the movement quantity (represented by the norm of the green arrow) is calculated using the dichotomy in Figure 9 to simplify the illustration of the concept. The movement quantity of the \(i\)th source point at iteration \(k\) with Euclidean distance \(d_{E}\) is given by Equation (2). \[\zeta_{i}(k)=\frac{d_{E}(n(k),m(k))}{2} \tag{2}\]  
In the case of the method developed in Algorithm A3, the movement quantity is calculated by dividing the remaining distance between \(n(k)\) and \(m(k)\) by a factor \(\alpha\in]1;\infty)\). Let \(d\) be the distance traveled by a source point.