Matrix profile has been recently proposed as a promising technique to the problem of all-pairs-similarity search on time series. Efficient algorithms have been proposed for computing it, e.g., STAMP, STOMP and SCRIMP++. All these algorithms use the z-normalized Euclidean distance to measure the distance between subsequences. However, as we observed, for some datasets other Euclidean measurements are more useful for knowledge discovery from time series. In this paper, we propose efficient algorithms for computing matrix profile for a general class of Euclidean distances. We first propose a simple but efficient algorithm called AAMP for computing matrix profile with the "pure" (non-normalized) Euclidean distance. Then, we extend our algorithm for the p-norm distance. We also propose an algorithm, called ACAMP, that uses the same principle as AAMP, but for the case of z-normalized Euclidean distance. We implemented our algorithms, and evaluated their performance through experimentation. The experiments show excellent performance results. For example, they show that AAMP is very efficient for computing matrix profile for non-normalized Euclidean distances. The results also show that the ACAMP algorithm is significantly faster than SCRIMP++ (the state of the art matrix profile algorithm) for the case of z-normalized Euclidean distance.
@cite_7: The all-pairs-similarity-search (or similarity join) problem has been extensively studied for text and a handful of other datatypes. However, surprisingly little progress has been made on similarity joins for time series subsequences. The lack of progress probably stems from the daunting nature of the problem. For even modest sized datasets the obvious nested-loop algorithm can take months, and the typical speed-up techniques in this domain (i.e., indexing, lower-bounding, triangular-inequality pruning and early abandoning) at best produce one or two orders of magnitude speedup. In this work we introduce a novel scalable algorithm for time series subsequence all-pairs-similarity-search. For exceptionally large datasets, the algorithm can be trivially cast as an anytime algorithm and produce high-quality approximate solutions in reasonable time. The exact similarity join algorithm computes the answer to the time series motif and time series discord problem as a side-effect, and our algorithm incidentally provides the fastest known algorithm for both these extensively-studied problems. We demonstrate the utility of our ideas for two time series data mining problems, including motif discovery and novelty discovery.
@cite_9: Since their introduction over a decade ago, time series motifs have become a fundamental tool for time series analytics, finding diverse uses in dozens of domains. In this work we introduce Time Series Chains, which are related to, but distinct from, time series motifs. Informally, time series chains are a temporally ordered set of subsequence patterns, such that each pattern is similar to the pattern that preceded it, but the first and last patterns are arbitrarily dissimilar. In the discrete space, this is similar to extracting the text chain "hit, hot, dot, dog" from a paragraph. The first and last words have nothing in common, yet they are connected by a chain of words with a small mutual difference. Time series chains can capture the evolution of systems, and help predict the future. As such, they potentially have implications for prognostics. In this work, we introduce a robust definition of time series chains, and a scalable algorithm that allows us to discover them in massive datasets.
@cite_1: Discovering motifs in sequence databases has been receiving abundant attentions from both database and data mining communities, where the motif is the most correlated pair of subsequences in a sequence object. Motif discovery is expensive for emerging applications which may have very long sequences (e.g., million observations per sequence) or the queries arrive rapidly (e.g., per 10 seconds). Prior works cannot offer fast correlation computations and prune subsequence pairs at the same time, as these two techniques require different orderings on examining subsequence pairs. In this work, we propose a novel framework named Quick-Motif which adopts a two-level approach to enable batch pruning at the outer level and enable fast correlation calculation at the inner level. We further propose two optimization techniques for the outer and the inner level. In our experimental study, our method is up to 3 orders of magnitude faster than the state-of-the-art methods.
@cite_3: Time series motifs are pairs of individual time series, or subsequences of a longer time series, which are very similar to each other. As with their discrete analogues in computational biology, this similarity hints at structure which has been conserved for some reason and may therefore be of interest. Since the formalism of time series motifs in 2002, dozens of researchers have used them for diverse applications in many different domains. Because the obvious algorithm for computing motifs is quadratic in the number of items, more than a dozen approximate algorithms to discover motifs have been proposed in the literature. In this work, for the first time, we show a tractable exact algorithm to find time series motifs. As we shall show through extensive experiments, our algorithm is up to three orders of magnitude faster than brute-force search in large datasets. We further show that our algorithm is fast enough to be used as a subroutine in higher level data mining algorithms for anytime classification, near-duplicate detection and summarization, and we consider detailed case studies in domains as diverse as electroencephalograph interpretation and entomological telemetry data mining.
@cite_5: Mining repeated patterns (often called motifs) in CPU utilization of computers (also called CPU host load) is of fundamental importance. Many recently emerging applications running on high performance computing systems rely on motif discovery for various purposes, including efficient task scheduling, energy saving, etc. In this paper, we propose an efficient motif discovery framework for CPU host load. The framework is elaborately designed to take into account the important properties in host load data. The framework benefits from its ability of on-line discovery and the adaptivity to work with massive data. The experiments are conducted in this paper and the experimental results show that the proposed method is effective and efficient.
In @cite_7 , introduced the theoretical foundations of matrix profile, and proposed a first algorithm, called STAMP, for computing the matrix profile over a time series. The algorithm uses a similarity search algorithm, called MASS, that under z-normalized Euclidean computes the distance of each subsequence to other subsequences by using the Fast Fourier Transform (FFT). Other exact algorithms such as Quick-Motif @cite_1 , IMD @cite_5 , or MK @cite_3 can be fast for cooperative data (those that are relatively smooth data, short motif lengths etc.). But in data (e.g., seismology data) these algorithms are not efficient @cite_9 .