Alignment-based metrics

The definition of adequate metrics between objects to be compared is at the core of many machine learning methods (e.g., nearest neighbors, kernel machines, etc.). When complex objects (such as time series) are involved, such metrics have to be carefully designed in order to leverage on desired notions of similarity.

Let us illustrate our point with an example.

Fig. 1 \(k\)-means clustering with Euclidean distance. Each subfigure represents series from a given cluster and their centroid (in red).

The figure above is the result of a \(k\)-means clustering that uses Euclidean distance as a base metric. One issue with this metric is that it is not invariant to time shifts, while the dataset at stake clearly holds such invariants.

When using a shift-invariant similarity measure (discussed in our Dynamic Time Warping section) at the core of \(k\)-means, one gets:

Fig. 2 \(k\)-means clustering with Dynamic Time Warping. Each subfigure represents series from a given cluster and their centroid (in red).

This part of the course tackles the definition of adequate similarity measures for time series and their use at the core of machine learning methods.