# The Fastest Similarity Search Algorithm for Time Series Subsequences under Euclidean Distance and Correlation Coefficient

Similarity search for time series subsequences is THE most important subroutine for time series pattern mining. Subsequence similarity search has been scaled to trillions obsetvations under both DTW (Dynamic Time Warping) and Euclidean distances [a]. The algorithms are ultra fast and efficient. The key technique that makes the algorithms useful is the Early Abandoning technique [b,e] known since 1994. However, the algorithms lack few properties that are useful for many time series data mining algorithms.

1. Early abandoning depends on the dataset. The worst case complexity is still O(nm) where n is the length of the larger time series and m is the length of the short query.
2. The algorithm can produce the most similar subsequence to the query and cannot produce the Distance Profile to all the subssequences given the query.

MASS is an algorithm to create Distance Profile of a query to a long time series. In this page we share a code for The Fastest Similarity Search Algorithm for Time Series Subsequences under Euclidean Distance. Early abandoning can occasionally beat this algorithm on some datasets for some queries. This algorithm is independent of data and query. The underlying concept of the algorithm is known for a long time to the signal processing community. We have used it for the first time on time series subsequence search under z-normalization. The algorithm was used as a subroutine in our papers [c,d] and the code are given below.

1. The algorithm has an overall time complexity of O(n log n) which does not depend on datasets and is the lower bound of similarity search over time series subsequences.
2. The algorithm produces all of the distances from the query to the subsequences of a long time series. In our recent paper, we generalize the usage of the distance profiles calculated using MASS in finding motifs, shapelets and discords. Check out the paper here.

To cite  this code:

Abdullah Mueen, Yan Zhu, Michael Yeh, Kaveh Kamgar, Krishnamurthy Viswanathan, Chetan Kumar Gupta and Eamonn Keogh (2015), The Fastest Similarity Search Algorithm for Time Series Subsequences under Euclidean Distance, URL: http://www.cs.unm.edu/~mueen/FastestSimilaritySearch.html

or use the following bib snippet.

@misc{FastestSimilaritySearch,
title={The Fastest Similarity Search Algorithm for Time Series Subsequences under Euclidean Distance},
author={ Mueen, Abdullah and Zhu, Yan and Yeh, Michael and Kamgar, Kaveh and Viswanathan, Krishnamurthy and Gupta, Chetan and Keogh, Eamonn},
year={2017},
month={August},
note = {\url{http://www.cs.unm.edu/~mueen/FastestSimilaritySearch.html}}
}

Code:

The first version of MASS was developed in 2011 and published in this page in 2015. It was named findNN.

The second version of MASS_V2 was developed in 2017 which is more than 2X faster than the first version. Several people contributed to this code including Yan Zhu, Michael Yeh, Kaveh Kamgar and Eamonn Keogh from UCR.

The third version of MASS_V3 is a piecewise version of MASS that performs better when the size of the pieces are well aligned with the hardware.

MASS_weighted is an extension to create weighted distance profiles.

A C++ version is available upon request.

If you find this code useful for your research, please drop me a line, I would love to know the impact of this code.

Tutorial:

MASS has been featured in several tutorials presented in top data mining venues. A self-contained version is available here.

[a] Thanawin Rakthanmanon, Bilson J. L. Campana, Abdullah Mueen, Gustavo E. A. P. A. Batista, M. Brandon Westover, Qiang Zhu, Jesin Zakaria, Eamonn J. Keogh: Searching and mining trillions of time series subsequences under dynamic time warping. KDD 2012: 262-270
[b] Christos Faloutsos, M. Ranganathan, Yannis Manolopoulos: Fast Subsequence Matching in Time-Series Databases. SIGMOD Conference 1994: 419-429
[c] Abdullah Mueen, Hossein Hamooni, Trilce Estrada: Time Series Join on Subsequence Correlation. ICDM 2014: 450-459
[d] Jesin Zakaria, Abdullah Mueen, Eamonn J. Keogh: Clustering Time Series Using Unsupervised-Shapelets. ICDM 2012: 785-794
[e] Eamonn J. Keogh, Shruti Kasetty: On the need for time series data mining benchmarks: a survey and empirical demonstration. KDD 2002: 102-111

Research and code development funded by NSF IIS - 1161997, NSF SHF 1527127 and an internship at HP Labs.