I'm doing clustering on time-series (each time-series has the information for one day = 24 hours). For the clustering purpose, it's important for me to consider the time period in which the shape of time-series is different from the baseline. So, I'm looking for an appropriate distance metric.
For example, let's consider these 4 time-series, and I want to cluster them into two groups.
Apperaently, for k=2, time-series 1 and 2 are more similar and considered as one cluster and time-series 3 and 4 are inside another cluster. However, when I use the commonly used Euclidean distance metric, the calculated distance between all time-series is the same since it considers one to one mapping between points, and does not account for how far these shapes are. I have tried Dynamic Time Warping (DTW) as well since its application matches what I want to do here, but my clustering results in a larger-scale dataset are even less promising compared to what Euclidean distance obtains (I'm using the default MATLAB command line for DTW, so I'm not sure if I'm utilizing it efficiently).
My question is what distance metric is more appropriate to capture the similarity in this case.