ERK'2020, Portorož, 367-370 367 A review and comparison of time series similarity measures Maˇ sa Kljun 1 , Matija Terˇ sek 1 , Erik ˇ Strumbelj 1 1 FacultyofComputerandInformationscience,UniversityofLjubljana E-mail: mk2700@student.uni-lj.si,mt2421@student.uni-lj.si,erik.strumbelj@fri.uni-lj.si Abstract We review 12 time series similarity measures and inves- tigate their time complexity, normalization, invariance with respect to warping and scaling, support of time se- riesofdifferentlengths,andotherproperties. Weshowon simulated data that several similarity measures perform wellonaverage,butnoneperformwellinallcasesandin some cases measures that typically perform poorly, such ascompression-basedsimilarity,areabetteralternative. 1 Introduction Measuring similarity between time series is an important component in time series data analysis, especially unsu- pervised learning. Many different measures exist and it is often not clear which measure is the best choice for the test at hand or how different measures compare with respect to relevant properties such as invariance to scal- ing/warping and time complexity. The few related works are Wang et al. [13] who com- pare 9 measures but omit those based on correlation coef- ficients or compression. Serra and Arcos [10] and G´ orecki and Piasecki [6] compare 7 and 30 measures, respectively, but focus on 1-NN classification performance and not clustering performance and other properties as we do. Es- ling and Agon [4] do focus on other properties, but not on clustering performance. We aim to provide a compact review and classifica- tion of the most commonly used similarity measures and relevant properties which are often excluded in related work. Furthermore, we use several simulated data sets to empirically evaluate how different measures compare to each other and how well they perform in clustering. 2 Distance measure features In this paper we will view time series similarity measures in terms of these properties, which are relevant to choos- ing the best similarity measure for the task at hand: Time complexity. Can compare time series of different lengths. Normalization. Does increasing the length or sam- pling frequency of the time series, without chang- ing any other properties, change the value? If so, we provide a factor that normalizes the measure and facilitates comparison across time series of dif- ferent lengths. Invariance/robustness with respect to warping and scaling. Warping is a change of the time se- ries’ times that preserves the ordering. Scaling is multiplication of the time series’ values with a con- stant. Related work is inconsistent about warp- ing, so we additionally defineweakinvariance (the same change is applied to both compared time se- ries) and strong invariance (the change is applied to only one of the time series). Strong invariance to warping implies weak invariance to warping. A summary of similarity measures is in Table 1. 3 Distance measures LetX = x 1 ;:::;x n andY = y 1 ;:::;y m be the two time series whose similarity we are interested in. We also use X n and X 1 to represent X without the last and first point, respectively. 3.1 L p norms/distances Depending on the value ofp, we have: Manhattan (p = 1): P n i=1 jx i y i j. Minkowski (1