Update 2_dtw.md

docs/2_method/2_dtw.md

@@ -10,32 +10,26 @@ Dynamic time warping is a well-known technique for manipulating time series to e
 
 ## DTW Algorithm
 
-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,
 
 $$
-\begin{aligned} 
-x&=(x_1, x_2, ..., x_n)\\
-y&=(y_1, y_2, ..., y_m).
+x=(x_1, x_2, ..., x_n)
+$$
+
+$$
+y=(y_1, y_2, ..., y_m)
 $$
 
 The DTW distance is the sum of the Euclidean distance between each point and its matched point(s) in the other vector. The following constraints must be met: 
 
 1. The first and last elements of each series must be matched.
-2. Only unidirectional forward movement through relative time is allowed, i.e., if $$x_1$$ is mapped to $$y_2$$ then $$x_2$$ may not be mapped to
-    $$y_1$$ (monotonicity). 
+2. Only unidirectional forward movement through relative time is allowed, i.e., if $x_1$ is mapped to $y_2$ then $x_2$ may not be mapped to
+    $y_1$ (monotonicity). 
 3. Each point is mapped to at least one other point, i.e., there are no jumps in time (continuity).
 
-![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).](../../media/Merged_document.png)
-
-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 $$y_j$$:
+![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).](../../media/Merged_document.png)
 
-\[
-c_{i,j} = (x_i-y_j)^2+\min \left\{
-\begin{array}{ccc}
-c_{i-1,j-1} & c_{i-1,j} & c_{i,j-1}
-\end{array}
-\right\}
-\]
+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 $y_j$:
 
 $$
 c_{i,j} = (x_i-y_j)^2+\min \begin{cases}
@@ -45,13 +39,13 @@ c_{i,j} = (x_i-y_j)^2+\min \begin{cases}
     \end{cases}
 $$
 
-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.
+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.
 
-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 $$c_{i-1,j-1}$$, $$c_{i-1,j}$$, $$c_{i,j-1}$$).
+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 $c_{i-1,j-1}$, $c_{i-1,j}$, $c_{i,j-1}$).
 
-The DTW distance $$C_{x,y}$$ is found for each pairwise comparison. 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. 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 ($j$).
 
 ### Warping Window
 
-For longer time series it is possible to speed up the DTW calculation by using a 'wapring window'. This works by restricting which data elements on one seies can be mapped to another based on their proximity. For example, if you have two data series of length 100, and a warping window of 10, only elements with a maximum time shift of 10 between the series can be mapped to each other. So, $$x_{1}$$ can only by mapped to $$y_{1}$$ to $$y_{11}$$. The stricter the wapring window, the greater the increase in speed. However, the data being used must be carefully considered to assertain if this will negatively impact the results. Due to teh fast computation of DTW in Readers are referred [here](https://ieeexplore.ieee.org/abstract/document/1163055) for detailed information on the warping window. 
+For longer time series it is possible to speed up the DTW calculation by using a 'wapring window'. This works by restricting which data elements on one seies can be mapped to another based on their proximity. For example, if you have two data series of length 100, and a warping window of 10, only elements with a maximum time shift of 10 between the series can be mapped to each other. So, $x_{1}$ can only by mapped to $y_{1}$ to $y_{11}$. The stricter the wapring window, the greater the increase in speed. However, the data being used must be carefully considered to assertain if this will negatively impact the results. Due to teh fast computation of DTW in Readers are referred [here](https://ieeexplore.ieee.org/abstract/document/1163055) for detailed information on the warping window.