Update paper.md

brackets removed

joss/paper.md

@@ -52,7 +52,7 @@ The current functionality of the software is as follows:
 
 # Mathematical background
 
-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,
+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,
 
 $$
 x=(x_1, x_2, ..., x_n)
@@ -70,7 +70,7 @@ The DTW distance is the sum of the Euclidean distance between each point and its
 
 ![Two time series with DTW pairwise alignment between each element, showing one-to-many mapping properties of DTW (left). Cost matrix $C$ for the two time series, showing the warping path and final DTW cost at $C_{14,13}$ (right). \label{fig:warping_signals}](../media/Merged_document.pdf)
 
-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 $y_j$:
 
 \begin{equation}
     \label{c}
@@ -85,17 +85,17 @@ The final element $c_{n,m}$ is then the total cost, $C_{x,y}$, which provides th
 
 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 ($j$).
+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 $j$.
 
 ![The DTW costs of all the pairwise comparisons between time series in the dataset are combined to make a distance matrix $D$. \label{fig:c_to_d}](../media/distance_matrix_formation_vector.pdf)
 
-Using this matrix, ($D$), the time series can be split into ($k$) separate clusters with integer programming. The problem formulation begins with a binary square matrix $A^{p\times p}$, where $A_{ij}=1$ if time series ($j$) is a member of the $i$th cluster centroid, and 0 otherwise, as shown in \autoref{fig:A_matrix}.
+Using this matrix, $D$, the time series can be split into $k$ separate clusters with integer programming. The problem formulation begins with a binary square matrix $A^{p\times p}$, where $A_{ij}=1$ if time series $j$ is a member of the $i$th cluster centroid, and 0 otherwise, as shown in \autoref{fig:A_matrix}.
 
 ![Example output from the clustering process, where an entry of 1 indicates that time series $j$ belongs to cluster with centroid $i$. \label{fig:A_matrix}](../media/cluster_matrix_formation4.svg){ width=70% }
 
 As each centroid has to be in its own cluster, non-zero diagonal entries in  $A$ represent centroids. In summary, the following constraints apply: 
 
-1. Only ($k$) series can be centroids,
+1. Only $k$ series can be centroids,
 
 $$
 \sum_{i=1}^p A_{ii}=k.
@@ -119,7 +119,7 @@ The optimisation problem to solve, subject to the above constraints, is
     A^\star = \min_{A} \sum_i \sum_j D_{ij} \times A_{ij}.
 \end{equation}
 
-This integer program is solved using Gurobi [@gurobi] or HiGHS [@Huangfu2018]. After solving this integer program, the non-zero diagonal entries of ($A$) represent the centroids, and the non-zero elements in the corresponding columns in ($A$) represent the members of that cluster. In the example in \autoref{fig:A_matrix}, the clusters are time series 1, **2**, 5 and 3, **4** with the bold time series being the centroids.
+This integer program is solved using Gurobi [@gurobi] or HiGHS [@Huangfu2018]. After solving this integer program, the non-zero diagonal entries of $A$ represent the centroids, and the non-zero elements in the corresponding columns in $A$ represent the members of that cluster. In the example in \autoref{fig:A_matrix}, the clusters are time series 1, **2**, 5 and 3, **4** with the bold time series being the centroids.
 
 Finding global optimality can increase the computation time, depending on the number of time series within the dataset and the DTW distances. Therefore, there is also a built-in option to cluster using k-medoids, as used in other packages such as \texttt{DTAIDistance} [@meert2020wannesm]. The k-medoids method is often quicker as it is an iterative approach, however it is subject to getting stuck in local optima. The results in the next section show the timing and memory performance of both MIP clustering and k-medoids clustering using \texttt{DTW-C++} compared to other packages.