Update paper.md

method sub-headings

joss/paper.md

@@ -52,6 +52,8 @@ The current functionality of the software is as follows:
 
 # Mathematical background
 
+## Dynamic time warping
+
 Consider a time series to be a vector of some arbitrary length. Consider that we have $p$ such vectors in total, each possibly differing in length. To find a subset of $k$ clusters within the set of $p$ vectors using MIP formulation, we must first make $\frac{1}{2} {p \choose 2}$ pairwise comparisons between all vectors within the total set and find the `similarity' between each pair. In this case, the similarity is defined as the DTW distance. Consider two time series $x$ and $y$ of differing lengths $n$ and $m$ respectively,
 
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@@ -83,6 +85,8 @@ Finding the optimal warping arrangement is an optimisation problem that can be s
 
 The final element $c_{n,m}$ is then the total cost, $C_{x,y}$, which provides the comparison metric between the two series $x$ and $y$. \autoref{fig:warping_signals} shows an example of this cost matrix $C$ and the warping path through it.
 
+## Clustering 
+
 For the clustering problem, only this final cost for each pairwise comparison is required; the actual warping path (or mapping of each point in one time series to the other) is superfluous for k-medoids clustering. The memory complexity of the cost matrix $C$ is $O(nm)$, so as the length of the time series increases, the memory required increases greatly. Therefore, significant reductions in memory can be made by not storing the entire $C$ matrix. When the warping path is not required, only a vector containing the previous row for the current step of the dynamic programming sub-problem is required (i.e., the previous three values $c_{i-1,j-1}$, $c_{i-1,j}$, $c_{i,j-1}$), as indicated in \autoref{c}.
 
 The DTW distance $C_{x,y}$ is found for each pairwise comparison. As shown in \ref{fig:c_to_d}, pairwise distances are then stored in a separate symmetric matrix, $D^{p\times p}$, where $p$ is the total number of time series in the clustering exercise. In other words, the element $d_{i,j}$ gives the distance between time series $i$ and $j$.