# A time series distance measure for efficient clustering of input output signals by their underlying dynamics

## TL;DR

This paper introduces an extension to the Martin cepstral distance for efficient clustering of input/output time series based on their underlying dynamics, outperforming traditional methods in computational efficiency.

## Contribution

The paper presents a novel extension to the Martin cepstral distance that enables fast and effective clustering of dynamical system-generated time series.

## Key findings

- The new distance measure performs comparably to explicit system modeling.
- It is significantly faster with a complexity of O(N logN).
- Effective for clustering electrical circuit data.

## Abstract

Starting from a dataset with input/output time series generated by multiple deterministic linear dynamical systems, this paper tackles the problem of automatically clustering these time series. We propose an extension to the so-called Martin cepstral distance, that allows to efficiently cluster these time series, and apply it to simulated electrical circuits data. Traditionally, two ways of handling the problem are used. The first class of methods employs a distance measure on time series (e.g. Euclidean, Dynamic Time Warping) and a clustering technique (e.g. k-means, k-medoids, hierarchical clustering) to find natural groups in the dataset. It is, however, often not clear whether these distance measures effectively take into account the specific temporal correlations in these time series. The second class of methods uses the input/output data to identify a dynamic system using an identification scheme, and then applies a model norm-based distance (e.g. H2, H-infinity) to find out which systems are similar. This, however, can be very time consuming for large amounts of long time series data. We show that the new distance measure presented in this paper performs as good as when every input/output pair is modelled explicitly, but remains computationally much less complex. The complexity of calculating this distance between two time series of length N is O(N logN).