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Update 2_dtw.md
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beckyperriment authored Dec 15, 2023
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Expand Up @@ -27,7 +27,7 @@ The DTW distance is the sum of the Euclidean distance between each point and its
$y_1$ (monotonicity).
3. Each point is mapped to at least one other point, i.e., there are no jumps in time (continuity).

Finding the optimal warping arrangement is an optimisation problem that can be solved using dynamic programming, which splits the problem into easier sub-problems and solves them recursively, storing intermediate solutions until the final solution is reached. To understand the memory-efficient method used in ''DTW-C++``, it is useful to first examine the full-cost matrix solution, as follows. For each pairwise comparison, an ($n$) by ($m$) matrix $C^{n\times m}$ is calculated, where each element represents the cumulative cost between series up to the points $x_i$ and $y_j$:
Finding the optimal warping arrangement is an optimisation problem that can be solved using dynamic programming, which splits the problem into easier sub-problems and solves them recursively, storing intermediate solutions until the final solution is reached. To understand the memory-efficient method used in *DTW-C++*, it is useful to first examine the full-cost matrix solution, as follows. For each pairwise comparison, an ($n$) by ($m$) matrix $C^{n\times m}$ is calculated, where each element represents the cumulative cost between series up to the points $x_i$ and $y_j$:

$$
c_{i,j} = (x_i-y_j)^2+\min \begin{cases}
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