## 2 Theoretical Concept  
Identifying non-linear lead-lag effects of time series data is based on the concept of causal paths. Specifically, we aim at determining the optimal relation structure of two given time series \(x=(x(1),...,x(N))\in\mathbb{R}^{N}\) and \(y=(y(1),...,y(M))\in\mathbb{R}^{M}\). Following [36], a sequence of points\(p=(p_{1},\ldots,p_{l})\) with \(p_{i}=(n_{i},n_{i})\in\{1,\ldots,N\}\times\{1,\ldots,M\}\) for \(i\in\{1,\ldots,I\}\) (\(I\in\{\max(N,M),\ldots,N+M-1\}\)) is called causal path (warping path) if it fulfills the following three characteristics:  
1. \(p_{1}=(1,1)\) and \(p_{I}=(N,M)\) (Boundary condition). 2. \(n_{1}\leq n_{2}\leq\cdots\leq n_{I}\) and \(m_{1}\leq m_{2}\leq\cdots\leq m_{I}\) (Monotonicity condition). 3. \(p_{i+1}-p_{i}\in\{(1,0),(0,1),(1,1)\},\quad\forall i\in\{1,\ldots,I-1\}\) (Step size condition). It is simple to recognize that the step size condition implies the monotonicity condition, which, however, is given for reasons of clarity. We define \(P\) as the set of all possible causal paths between the given time series \(x\) and \(y\). Then, the total cost of a causal path \(p\) (\(p\in P\)) is determined by  
\[c_{p}(x,y)=\sum_{i=1}^{I}c(x(n_{i}),y(m_{i})), \tag{3}\]  
where \(c\) describes the local cost measure. As such, the term \(c(x(n_{i}),y(m_{i}))\) describes the gap between the realizations of \(x\) at time \(n_{i}\) and \(y\) at time \(m_{i}\) (\(i\in\{1,\ldots,I\}\)). Usually, the cost measure is based on the Manhattan distance [9; 35; 37] or the Euclidean distance [38; 39; 40]. The optimal causal path \(p^{*}\) between \(x\) and \(y\) possesses lowest total cost of any possible causal path:  
\[p^{*}=\underset{p\in P}{argmin}\,c_{p}(x,y). \tag{4}\]  
We define the total cost of \(p^{*}\) as \(c_{p^{*}}(x,y)\), which is the sum of all local costs of \(p^{*}\) (see Equation (3)). Figure 1 illustrates the local costs and the optimal causal path \(p^{*}\) of two given time series \(x\) and \(y\). The points of \(p^{*}\) runs along a \"valley\" of low cost (light colors) and avoids \"mountains\" of high cost (dark color). Finding the optimal causal path \(p^{*}\) is a challenging and ambitious research objective. The naive approach would consider the total cost \(c_{p}(x,y)\) for all possible causal paths \(p\in P\)--a complexity of exponential order renders this approach impossible in practice. The most famous way to find the optimal causal path \(p^{*}\) is dynamic programming, i.e., we divide the underlying problem into several sub-problems [41]. To be more specific, we simplify a complicated problem by breaking it down into simpler sub-problems in a recursive manner. Therefore, we first calculate the optimum solutions of the smallest sub-problems and then combine them to form a solution of the next larger sub-problem. This procedure is continued until the original problem is solved. Overall, achieved solutions are stored for future calculations resulting in a time complexity \(\mathcal{O}(NM)\). In addition to the three path conditions described above, academic studies establish global and local constraints with the primary purpose of accelerating computing time. Global constraints  
Figure 1: Local costs of two time series and the identified optimal causal path \(p^{*}\) (solid line). Regions of low cost (high cost) are marked by light colors (dark colors). aim at limiting the deviation of a causal path from the diagonal--important representatives are the Sakoe-Chiba band [42] and the Itakura parallelogram [43] (see Figure 2). Local constraints alter the step size condition by changing the set of potential steps or preferring certain step directions (see [44; 45; 46; 47]). However, we avoid global and local constraints, as both require additional parameter settings and produce inadequate results in most application areas (see [48]). Although dynamic time warping measures a distance-like quantity between two given sequences, it doesn't ensure the triangle inequality to hold [35].