# Prognostics of Electromechanical Actuator with Partial Time Scaling Invariant Temporal Alignment
## 6 Quantification of Time Series Alignment  
Each alignment procedure assigns a similarity metric to discern the most appropriate candidates within the database, as expounded upon in Section 3. The establishment of precise criteria to define a meaningful series alignment is paramount in this context. However, the complexity of our particular scenario is compounded by the inherent ambiguity of the data composing the operational series, denoted as \(X_{j}^{l}\). Consequently, when the alignment of \(X_{j}^{l}\) with \(Y_{j}^{h}\) is successfully achieved, no baseline exists to compute the deviation from a known ground truth. In response to this challenge, it becomes imperative to introduce an intrinsic quality metric represented by \(\mathcal{L}\). This metric serves as a quantification that operates independently of prior knowledge concerning the data's inherent characteristics. The incorporation of this methodological approach represents a crucial step in addressing a fundamental challenge inherent in our research endeavors. Let us provide an overview of the method developed in Section 5. The objective is to align a time series, denoted as \(X_{j}^{l}\in\mathcal{M}_{n,1}(\mathbb{R})\), with another time series, referred to as \(Y_{j}^{h}\in\mathcal{M}_{m,1}(\mathbb{R})\). The alignment process commences by transforming both time series into point clouds, resulting in \(\tilde{X}_{j}^{l}\in\mathcal{M}_{n,2}(\mathbb{R})\) and \(\tilde{Y}_{j}^{h}\in\mathcal{M}_{m,2}(\mathbb{R})\). These point clouds are subsequently projected onto a common set of directions, represented by \(\Theta_{j}^{h,*}\in\mathcal{M}_{2,1}(\mathbb{R})\). For a given direction, two new series, \(X\bot,j^{l}\) and \(Y\bot,j^{h}\), are derived, from which an injective assignment function \(\mathcal{T}:\mathcal{I}\hookrightarrow\mathcal{J}\) is constructed. The values of this assignment function play a pivotal role in the advection model, generating point displacements from \(\tilde{X}_{j}^{l}\) to \(\tilde{Y}_{j}^{h}\). Since this displacement transpires within the confines of a two-dimensional space, the advection model calculates a displacement vector for each point within the source cloud, utilizing  
Figure 10: Random alignment of 10 % of selected points of the descriptor RMS of LLA towards the descriptor RMS of LRA. the canonical basis of \(\mathbb{R}^{2}\). Let us denote this orthonormal basis as \(\mathcal{B}_{2}=\big{(}e_{x},e_{y}\big{)}\), with its constituent vectors represented as \(e_{x}=(0,1)\) and \(e_{y}=(1,0)\). The entire process, which includes the projection of the source cloud, construction of the assignment function, and subsequent point movement, is iteratively repeated a total of \(K\) times to achieve convergence between the source and target points. This comprehensive methodology represents a critical step in our research and contributes significantly to the academic discourse in this domain. Within the framework of this method, a crucial step involves extracting pertinent information to assess the quality of an alignment. To shed light on this aspect, we examine the behavior of the assignment function denoted as \(\mathcal{T}\), and its characteristics are visually represented in Figure 8. This function establishes associations between each element in the source and its corresponding counterpart in the target. In the context of an alignment procedure spanning over \(K\) iterations, we introduce the matrix \(\mathbb{T}\in\mathcal{M}_{n,K}(\mathbb{N})\), which aggregates the values of the function \(\mathcal{T}|_{k}\) computed at each iteration. Mathematically, this matrix is expressed as \(\mathbb{T}=\big{[}\mathcal{T}|_{0}\quad\mathcal{T}|_{1}\quad\ldots\quad \mathcal{T}|_{K}\big{]}^{\top}\). This analytical approach is pivotal in our methodology, as it facilitates a deeper understanding of the alignment quality assessment process. Let us further characterize such a function. \(\mathcal{T}|_{k}\) is defined in \(\mathcal{I}_{k}=\{1,2,\ldots,n\}\) with values in \(\mathcal{J}_{k}=\{1,2,\ldots,m\}\).