On Clustering Time Series Using Euclidean Distance and Pearson Correlation

TL;DR

This paper reveals that z-score normalized Euclidean distance is mathematically equivalent to Pearson correlation distance for time series, impacting clustering methods like k-Means and providing theoretical insights and experimental validation.

Contribution

It establishes the equivalence between normalized Euclidean distance and Pearson correlation distance, and discusses necessary modifications to k-Means for proper correlation-based clustering.

Findings

Normalized Euclidean distance equals Pearson correlation distance.

Standard k-Means often yields similar clustering results.

Theoretical and experimental validation of the equivalence.

Abstract

For time series comparisons, it has often been observed that z-score normalized Euclidean distances far outperform the unnormalized variant. In this paper we show that a z-score normalized, squared Euclidean Distance is, in fact, equal to a distance based on Pearson Correlation. This has profound impact on many distance-based classification or clustering methods. In addition to this theoretically sound result we also show that the often used k-Means algorithm formally needs a mod ification to keep the interpretation as Pearson correlation strictly valid. Experimental results demonstrate that in many cases the standard k-Means algorithm generally produces the same results.