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Given a huge, online stream of time-evolving events with multiple attributes, such as online shopping logs: (item, price, brand, time), and local mobility activities: (pick-up and drop-off locations, time), how can we summarize large, dynamic high-order tensor streams? How can we see any hidden patterns, rules, and anomalies? Our answer is to focus on two types of patterns, i.e., ''regimes'' and ''components'', for which we present CubeScope, an efficient and effective method over high-order tensor streams. Specifically, it identifies any sudden discontinuity and recognizes distinct dynamical patterns, ''regimes'' (e.g., weekday/weekend/holiday patterns). In each regime, it also performs multi-way summarization for all attributes (e.g., item, price, brand, and time) and discovers hidden ''components'' representing latent groups (e.g., item/brand groups) and their relationship. Thanks to its concise but effective summarization, CubeScope can also detect the sudden appearance of anomalies and identify the types of anomalies that occur in practice. Our proposed method has the following properties: (a) Effective: it captures dynamical multi-aspect patterns, i.e., regimes and components, and statistically summarizes all the events; (b) General: it is practical for successful application to data compression, pattern discovery, and anomaly detection on various types of tensor streams; (c) Scalable: our algorithm does not depend on the length of the data stream and its dimensionality. Extensive experiments on real datasets demonstrate that CubeScope finds meaningful patterns and anomalies correctly, and consistently outperforms the state-of-the-art methods as regards accuracy and execution speed., Comment: Accepted by WWW 2023

Large quantifies of online user activity data, such as weekly web search volumes, which co-evolve with the mutual influence of several queries and locations, serve as an important social sensor. It is an important task to accurately forecast the future activity by discovering latent interactions from such data, i.e., the ecosystems between each query and the flow of influences between each area. However, this is a difficult problem in terms of data quantity and complex patterns covering the dynamics. To tackle the problem, we propose FluxCube, which is an effective mining method that forecasts large collections of co-evolving online user activity and provides good interpretability. Our model is the expansion of a combination of two mathematical models: a reaction-diffusion system provides a framework for modeling the flow of influences between local area groups and an ecological system models the latent interactions between each query. Also, by leveraging the concept of physics-informed neural networks, FluxCube achieves high interpretability obtained from the parameters and high forecasting performance, together. Extensive experiments on real datasets showed that FluxCube outperforms comparable models in terms of the forecasting accuracy, and each component in FluxCube contributes to the enhanced performance. We then show some case studies that FluxCube can extract useful latent interactions between queries and area groups., Comment: Accepted by CIKM 2022

Given a large time-evolving event series such as Google web-search logs, which are collected according to various aspects, i.e., timestamps, locations and keywords, how accurately can we forecast their future activities? How can we reveal significant patterns that allow us to long-term forecast from such complex tensor streams? In this paper, we propose a streaming method, namely, CubeCast, that is designed to capture basic trends and seasonality in tensor streams and extract temporal and multi-dimensional relationships between such dynamics. Our proposed method has the following properties: (a) it is effective: it finds both trends and seasonality and summarizes their dynamics into simultaneous non-linear latent space. (b) it is automatic: it automatically recognizes and models such structural patterns without any parameter tuning or prior information. (c) it is scalable: it incrementally and adaptively detects shifting points of patterns for a semi-infinite collection of tensor streams. Extensive experiments that we conducted on real datasets demonstrate that our algorithm can effectively and efficiently find meaningful patterns for generating future values, and outperforms the state-of-the-art algorithms for time series forecasting in terms of forecasting accuracy and computational time.

The recent explosion in the adoption of search engines and new media such as blogs and Twitter have facilitated the faster propagation of news and rumors. How quickly does a piece of news spread over these media? How does its popularity diminish over time? Does the rising and falling pattern follow a simple universal law? In this article, we propose S pike M, a concise yet flexible analytical model of the rise and fall patterns of information diffusion. Our model has the following advantages. First, unification power: it explains earlier empirical observations and generalizes theoretical models including the SI and SIR models. We provide the threshold of the take-off versus die-out conditions for S pike M and discuss the generality of our model by applying it to an arbitrary graph topology. Second, practicality: it matches the observed behavior of diverse sets of real data. Third, parsimony: it requires only a handful of parameters. Fourth, usefulness: it makes it possible to perform analytic tasks such as forecasting, spotting anomalies, and interpretation by reverse engineering the system parameters of interest (quality of news, number of interested bloggers, etc.). We also introduce an efficient and effective algorithm for the real-time monitoring of information diffusion, namely S pike S tream , which identifies multiple diffusion patterns in a large collection of online event streams. Extensive experiments on real datasets demonstrate that S pike M accurately and succinctly describes all patterns of the rise and fall spikes in social networks.

Given a large volume of multi-dimensional data streams, such as that produced by IoT applications, finance and online web-click logs, how can we discover typical patterns and compress them into compact models? In addition, how can we incrementally distinguish multiple patterns while considering the information obtained from a pattern found in a streaming setting? In this paper, we propose a streaming algorithm, namely StreamScope, that is designed to find intuitive patterns efficiently from event streams evolving over time. Our proposed method has the following properties: (a) it is effective: it operates on semi-infinite collections of co-evolving streams and summarizes all the streams into a set of multiple discrete segments grouped by their similarities. (b) it is automatic: it automatically and incrementally recognizes such patterns and generates models for each of them if necessary; (c) it is scalable: the complexity of our method does not depend on the length of the data streams. Our extensive experiments on real data streams demonstrate that StreamScope can find meaningful patterns and achieve great improvements in terms of computational time and memory space over its full batch method competitors.

In recent years, a massive amount of time-stamped sensor data has been generated and collected by many Internet of Things (IoT) applications, such as advanced automobiles and health care devices. Given such a large collection of complex sensor sequences, which consists of multiple attributes (e.g., sensor, user, timestamp), how can we automatically find important dynamic time-series patterns and the points of variation? How can we summarize all the complex sensor sequences, and achieve a meaningful segmentation? Also, can we see any hidden user-specific differences and outliers? In this paper we present CUBEMARKER, an efficient and effective method for capturing such multi-aspect features in sensor sequences. CUBEMARKER performs multi-way summarization for all attributes, namely, sensors, users, and time, and specifically it extracts multi-aspect features, such as important time-series patterns (i.e., time-aspect features) and hidden groups of users (i.e., user-aspect features), in complex sensor sequences. Our proposed method has the following advantages: (a) It is effective: it extracts multi-aspect features from complex sensor sequences and enables the efficient and effective analysis of complicated datasets; (b) It is automatic: it requires no prior training and no parameter tuning; (c) It is scalable: our method is carefully designed to be linear as regards dataset size and applicable to a large number of sensor sequences. Extensive experiments on real datasets show that CUBEMARKER is effective in that it can capture meaningful patterns for various real-world datasets, such as those obtained from smart factories, human activities, and automobiles. CUBEMARKER consistently outperforms the best state-of-the-art methods in terms of both accuracy and execution speed.

Given a large, semi-infinite collection of co-evolving data sequences (e.g., IoT/sensor streams), which contains multiple distinct dynamic time-series patterns, our aim is to incrementally monitor current dynamic patterns and forecast future behavior. We present an intuitive model, namely OrbitMap, which provides a good summary of time-series evolution in streams. We also propose a scalable and effective algorithm for fitting and forecasting time-series data streams. Our method is designed as a dynamic, interactive and flexible system, and is based on latent non-linear differential equations. Our proposed method has the following advantages: (a) It is effective: it captures important time-evolving patterns in data streams and enables real-time, long-range forecasting; (b) It is general: our model is general and practical and can be applied to various types of time-evolving data streams; (c) It is scalable: our algorithm does not depend on data size, and thus is applicable to very large sequences. Extensive experiments on real datasets demonstrate that OrbitMap makes long-range forecasts, and consistently outperforms the best existing state-of-the-art methods as regards accuracy and execution speed.

Given a large collection of co-evolving online activities, such as searches for the keywords "Xbox", "PlayStation" and "Wii", how can we find patterns and rules? Are these keywords related? If so, are they competing against each other? Can we forecast the volume of user activity for the coming month? We conjecture that online activities compete for user attention in the same way that species in an ecosystem compete for food. We present EcoWeb, (i.e., Ecosystem on the Web), which is an intuitive model designed as a non-linear dynamical system for mining large-scale co-evolving online activities. Our second contribution is a novel, parameter-free, and scalable fitting algorithm, EcoWeb-Fit, that estimates the parameters of EcoWeb. Extensive experiments on real data show that EcoWeb is effective, in that it can capture long-range dynamics and meaningful patterns such as seasonalities, and practical, in that it can provide accurate long-range forecasts. EcoWeb consistently outperforms existing methods in terms of both accuracy and execution speed.

Given a large collection of multiple time-evolving sensor sequences, how can we capture the transitions of time series patterns? How can we find individual differences between different sequences? In this paper we present CUBEMARKER, an effective method for capturing multi-aspect features of sensor sequences, which provides a compact and powerful representation of sensor behavior. Our second contribution is a novel, scalable, and parameter-free algorithm. CUBEMARKER performs two-way mining for all attributes. Specifically it discovers multi-aspect time series patterns (human motion, smart factory, etc) and groups of patterns simultaneously. Extensive experiments on real datasets show that CUBEMARKER is effective in that it can capture meaningful patterns for various real sensor datasets.

Given a large collection of time-evolving online user activities, such as Google Search queries for multiple keywords of various categories (celebrities, events, diseases, etc.), which consist of \(d\) keywords/activities, for \(l\) countries/locations of duration \(n\), how can we find patterns and rules? For example, assume that we have the online search volume for “Harry Potter”, “Barack Obama”, and “Amazon”, for 232 countries/territories, from 2004 to 2015, which include external shocks, sudden change of search volume, and more. How do we go about capturing nonlinear evolutions of local activities and forecasting future patterns? In this paper, we present \(\varDelta \)-SPOT, a unifying analytical nonlinear model for analyzing large-scale web search data, which is sensemaking, automatic, scalable, and free of parameters. \(\varDelta \)-SPOT can also forecast long-range future dynamics of the keywords/queries. We use the Google Search, Twitter, and MemeTracker dataset for extensive experiments, which show that our method outperforms other effective methods of nonlinear mining in terms of accuracy and in both fitting and forecasting.

Given a large volume of multiple data streams, such as online web-click logs and sensor data, how can we discover typical patterns and compress them into compact models? How can we incrementally distinguish multiple patterns while considering the information obtained from a pattern found in a streaming setting? In this paper, we propose a streaming algorithm, namely STREAMSCOPE, that can find intuitive patterns efficiently from event streams evolving over time. Our method has the following properties: (a) Effective: It operates on semi-infinite data streams and summarizes all the streams into a set of multiple discrete segments grouped by their similarities. (b) Automatic: It automatically and incrementally recognizes such patterns and generates models for each of them if necessary. (c) Scalable: The complexity of our method does not depend on the length of the input streams. Our experiments on real datasets demonstrate that StreamsCopecan find meaningful patterns and achieve great improvements in terms of computational time and memory space over its full batch method competitors.

Given a large collection of time-evolving online user activities, such as Google Search queries for multiple keywords of various categories (celebrities, events, diseases, etc.), which consist of \(d\) keywords/activities, for \(l\) countries/locations of duration \(n\), how can we find patterns and rules? How do we go about capturing non-linear evolutions of local activities and forecasting future patterns? We also aim to achieve good monitoring of the data sequences statistically, and detection of the patterns immediately. In this paper, we present \({\varDelta \hbox {-}\textsc {SPOT}} \), a unifying analytical non-linear model for analysing large scale web search data, which is sense-making, automatic, scalable and free of parameters. \({\varDelta \hbox {-}\textsc {SPOT}} \) can also forecast long-range future dynamics of the keywords/queries. Besides, we also provide an efficient and effective fitting algorithm, which leads to novel discoveries and sense-making features, and contribute to the need of monitoring multiple co-evolving data sequences.

Given a large, online stream of multiple co-evolving event sequences, such as sensor data and Web-click logs, that contains various types of non-linear dynamic evolving patterns of different durations, how can we efficiently and effectively capture important patterns? How do we go about forecasting long-term future events? In this paper, we present REGIMECAST, an efficient and effective method for forecasting co-evolving data streams. REGIMECAST is designed as an adaptive non-linear dynamical system, which is inspired by the concept of "regime shifts" in natural dynamical systems. Our method has the following properties: (a) Effective: it operates on large data streams, captures important patterns and performs long-term forecasting; (b) Adaptive: it automatically and incrementally recognizes the latent trends and dynamic evolution patterns (i.e., regimes) that are unknown in advance; (c) Scalable: it is fast and the computation cost does not depend on the length of data streams; (d) Any-time: it provides a response at any time and generates long-range future events. Extensive experiments on real datasets demonstrate that REGIMECAST does indeed make long-range forecasts, and it outperforms state-of-the-art competitors as regards accuracy and speed.

Given a large collection of time-evolving activities, such as Google search queries, which consist of d keywords/activities for m locations of duration n, how can we analyze temporal patterns and relationships among all these activities and find location-specific trends? How do we go about capturing non-linear evolutions of local activities and forecasting future patterns? For example, assume that we have the online search volume for multiple keywords, e.g., "Nokia/Nexus/Kindle" or "CNN/BBC" for 236 countries/territories, from 2004 to 2015. Our goal is to analyze a large collection of multi-evolving activities, and specifically, to answer the following questions: (a) Is there any sign of interaction/competition between two different keywords? If so, who competes with whom? (b) In which country is the competition strong? (c) Are there any seasonal/annual activities? (d) How can we automatically detect important world-wide (or local) events? We present COMPCUBE, a unifying non-linear model, which provides a compact and powerful representation of co-evolving activities; and also a novel fitting algorithm, COMPCUBE-FIT, which is parameter-free and scalable. Our method captures the following important patterns: (B)asic trends, i.e., non-linear dynamics of co-evolving activities, signs of (C)ompetition and latent interaction, e.g., Nokia vs. Nexus, (S)easonality, e.g., a Christmas spike for iPod in the U.S. and Europe, and (D)eltas, e.g., unrepeated local events such as the U.S. election in 2008. Thanks to its concise but effective summarization, COMPCUBE can also forecast long-range future activities. Extensive experiments on real datasets demonstrate that COMPCUBE consistently outperforms the best state-of- the-art methods in terms of both accuracy and execution speed.

Online news, blogs, SNS and many other Web-based services has been attracting considerable interest for business and marketing purposes. Given a large collection of time series, such as web-click logs, online search queries, blog and review entries, how can we efficiently and effectively find typical time-series patterns? What are the major tools for mining, forecasting and outlier detection? Time-series data analysis is becoming of increasingly high importance, thanks to the decreasing cost of hardware and the increasing on-line processing capability. The objective of this tutorial is to provide a concise and intuitive overview of the most important tools that can help us find meaningful patterns in large-scale time-series data. Specifically we review the state of the art in three related fields: (1) similarity search, pattern discovery and summarization, (2) non-linear modeling and forecasting, and (3) the extension of time-series mining and tensor analysis. We also introduce case studies that illustrate their practical use for social media and Web-based services.

Given a large collection of time series, such as web-click logs, electric medical records and motion capture sensors, how can we efficiently and effectively find typical patterns? How can we statistically summarize all the sequences, and achieve a meaningful segmentation? What are the major tools for forecasting and outlier detection? Time-series data analysis is becoming of increasingly high importance, thanks to the decreasing cost of hardware and the increasing on-line processing capability. The objective of this tutorial is to provide a concise and intuitive overview of the most important tools that can help us find patterns in large-scale time-series sequences. We review the state of the art in four related fields: (1) similarity search and pattern discovery, (2) linear modeling and summarization, (3) non-linear modeling and forecasting, and (4) the extension of time-series mining and tensor analysis. The emphasis of the tutorial is to provide the intuition behind these powerful tools, which is usually lost in the technical literature, as well as to introduce case studies that illustrate their practical use.

Given a large collection of co-evolving online activities, such as searches for the keywords "Xbox", "PlayStation" and "Wii", how can we find patterns and rules? Are these keywords related? If so, are they competing against each other? Can we forecast the volume of user activity for the coming month? We conjecture that online activities compete for user attention in the same way that species in an ecosystem compete for food. We present ECOWEB, (i.e., Ecosystem on the Web), which is an intuitive model designed as a non-linear dynamical system for mining large-scale co-evolving online activities. Our second contribution is a novel, parameter-free, and scalable fitting algorithm, ECOWEB-FIT, that estimates the parameters of ECOWEB. Extensive experiments on real data show that ECOWEB is effective, in that it can capture long-range dynamics and meaningful patterns such as seasonalities, and practical, in that it can provide accurate long-range forecasts. ECOWEB consistently outperforms existing methods in terms of both accuracy and execution speed.

Given a huge stream of multiple co-evolving sequences, such as motion capture and web-click logs, how can we find meaningful patterns and spot anomalies? Our aim is to monitor data streams statistically, and find sub sequences that have the characteristics of a given hidden Markov model (HMM). For example, consider an online web-click stream, where massive amounts of access logs of millions of users are continuously generated every second. So how can we find meaningful building blocks and typical access patterns such as weekday/weekend patterns, and also, detect anomalies and intrusions? In this paper, we propose Stream Scan, a fast and exact algorithm for monitoring multiple co-evolving data streams. Our method has the following advantages: (a) it is effective, leading to novel discoveries and surprising outliers, (b) it is exact, and we theoretically prove that Stream Scan guarantees the exactness of the output, (c) it is fast, and requires O (1) time and space per time-tick. Our experiments on 67GB of real data illustrate that Stream Scan does indeed detect the qualifying subsequence patterns correctly and that it can offer great improvements in speed (up to 479,000 times) over its competitors.

Given a large collection of epidemiological data consisting of the count of d contagious diseases for l locations of duration n, how can we find patterns, rules and outliers? For example, the Project Tycho provides open access to the count infections for U.S. states from 1888 to 2013, for 56 contagious diseases (e.g., measles, influenza), which include missing values, possible recording errors, sudden spikes (or dives) of infections, etc. So how can we find a combined model, for all these diseases, locations, and time-ticks? In this paper, we present FUNNEL, a unifying analytical model for large scale epidemiological data, as well as a novel fitting algorithm, FUNNELFIT, which solves the above problem. Our method has the following properties: (a) Sense-making: it detects important patterns of epidemics, such as periodicities, the appearance of vaccines, external shock events, and more; (b) Parameter-free: our modeling framework frees the user from providing parameter values; (c) Scalable: FUNNELFIT is carefully designed to be linear on the input size; (d) General: our model is general and practical, which can be applied to various types of epidemics, including computer-virus propagation, as well as human diseases. Extensive experiments on real data demonstrate that FUNNELFIT does indeed discover important properties of epidemics: (P1) disease seasonality, e.g., influenza spikes in January, Lyme disease spikes in July and the absence of yearly periodicity for gonorrhea; (P2) disease reduction effect, e.g., the appearance of vaccines; (P3) local/state-level sensitivity, e.g., many measles cases in NY; (P4) external shock events, e.g., historical flu pandemics; (P5) detect incongruous values, i.e., data reporting errors.

Given a large collection of co-evolving multiple time-series, which contains an unknown number of patterns of different durations, how can we efficiently and effectively find typical patterns and the points of variation? How can we statistically summarize all the sequences, and achieve a meaningful segmentation? In this paper we present AutoPlait, a fully automatic mining algorithm for co-evolving time sequences. Our method has the following properties: (a) effectiveness: it operates on large collections of time-series, and finds similar segment groups that agree with human intuition; (b) scalability: it is linear with the input size, and thus scales up very well; and (c) AutoPlait is parameter-free, and requires no user intervention, no prior training, and no parameter tuning. Extensive experiments on 67GB of real datasets demonstrate that AutoPlait does indeed detect meaningful patterns correctly, and it outperforms state-of-the-art competitors as regards accuracy and speed: AutoPlait achieves near-perfect, over 95% precision and recall, and it is up to 472 times faster than its competitors.

How many listens will an artist receive on a online radio? How about plays on a YouTube video? How many of these visits are new or returning users? Modeling and mining popularity dynamics of social activity has important implications for researchers, content creators and providers. We here investigate the effect of revisits (successive visits from a single user) on content popularity. Using four datasets of social activity, with up to tens of millions media objects (e.g., YouTube videos, Twitter hashtags or LastFM artists), we show the effect of revisits in the popularity evolution of such objects. Secondly, we propose the Phoenix-R model which captures the popularity dynamics of individual objects. Phoenix-R has the desired properties of being: (1) parsimonious, being based on the minimum description length principle, and achieving lower root mean squared error than state-of-the-art baselines; (2) applicable, the model is effective for predicting future popularity values of objects.

Given a large number of taxi trajectories, we would like to find interesting and unexpected patterns from the data. How can we summarize the major trends, and how can we spot anomalies? The analysis of trajectories has been an issue of considerable interest with many applications such as tracking trails of migrating animals and predicting the path of hurricanes. Several recent works propose methods on clustering and indexing trajectories data. However, these approaches are not especially well suited to pattern discovery with respect to the dynamics of social and economic behavior. To further analyze a huge collection of taxi trajectories, we develop a novel method, called F-Trail, which allows us to find meaningful patterns and anomalies. Our approach has the following advantages: (a) it is fast, and scales linearly on the input size, (b) it is effective, leading to novel discoveries, and surprising outliers. We demonstrate the effectiveness of our approach, by performing experiments on real taxi trajectories. In fact, F-Trail does produce concise, informative and interesting patterns.

The recent explosion in the adoption of search engines and new media such as blogs and Twitter have facilitated faster propagation of news and rumors. How quickly does a piece of news spread over these media? How does its popularity diminish over time? Does the rising and falling pattern follow a simple universal law?In this paper, we propose SpikeM, a concise yet flexible analytical model for the rise and fall patterns of influence propagation. Our model has the following advantages: (a) unification power: it generalizes and explains earlier theoretical models and empirical observations; (b) practicality: it matches the observed behavior of diverse sets of real data; (c) parsimony: it requires only a handful of parameters; and (d) usefulness: it enables further analytics tasks such as fore- casting, spotting anomalies, and interpretation by reverse- engineering the system parameters of interest (e.g. quality of news, count of interested bloggers, etc.).Using SpikeM, we analyzed 7.2GB of real data, most of which were collected from the public domain. We have shown that our SpikeM model accurately and succinctly describes all the patterns of the rise-and-fall spikes in these real datasets.

Given huge collections of time-evolving events such as web-click logs, which consist of multiple attributes (e.g., URL, userID, times- tamp), how do we find patterns and trends? How do we go about capturing daily patterns and forecasting future events? We need two properties: (a) effectiveness, that is, the patterns should help us understand the data, discover groups, and enable forecasting, and (b) scalability, that is, the method should be linear with the data size. We introduce TriMine, which performs three-way mining for all three attributes, namely, URLs, users, and time. Specifically TriMine discovers hidden topics, groups of URLs, and groups of users, simultaneously. Thanks to its concise but effective summarization, it makes it possible to accomplish the most challenging and important task, namely, to forecast future events. Extensive experiments on real datasets demonstrate that TriMine discovers meaningful topics and makes long-range forecasts, which are notoriously difficult to achieve. In fact, TriMine consistently outperforms the best state-of-the-art existing methods in terms of accuracy and execution speed (up to 74x faster).

In this paper we deal with a ubiquitous problem in data management: hierarchical model estimation for large distribution sets. This particular problem arises in many applications. Classification, top-k query processing, clustering and outlier detection are just a few possible applications. Our aim is to continuously and incrementally estimate the model parameters of 'typical' distributions that describe the characteristics of a database. Our approach to model estimation can handle arbitrary types of data (e.g., categorical and numerical data) in databases, incrementally, quickly, and with little resource consumption. Moreover, this paper proposes not only incremental algorithms for model fitting, but also a modeling framework in which the learning approach recognizes hierarchical groups, each of whose distributions has similar characteristics, and separately updates the model parameters of each group without scanning all the distributions in the database. Thus, it can provide a response, i.e., the parameters of typical distribution models, with an arbitrary level of granularity, at any time. Just as importantly, we demonstrate the utility of our approach by showing how it can be applied to two specific problems that arise in the context of data management.

Distribution data naturally arise in countless domains, such as meteorology, biology, geology, industry and economics. However, relatively little attention has been paid to data mining for large distribution sets. Given n distributions of multiple categories and a query distribution Q, we want to find similar clouds (i.e., distributions), to discover patterns, rules and outlier clouds. For example, consider the numerical case of sales of items, where, for each item sold, we record the unit price and quantity; then, each customer is represented as a distribution of 2-d points (one for each item he/she bought). We want to find similar users, e.g., for market segmentation, anomaly/fraud detection. We propose to address this problem and present D-Search, which includes fast and effective algorithms for similarity search in large distribution datasets. Our main contributions are (1) approximate KL divergence, which can speed up cloud-similarity computations, (2) multi-step sequential scan, which efficiently prunes a significant number of search candidates and leads to a direct reduction in the search cost. We also introduce an extended version of D-Search: (3) time-series distribution mining, which finds similar subsequences in time-series distribution datasets. Extensive experiments on real multi-dimensional datasets show that our solution achieves up to 2,300 faster wall-clock time over the naive implementation while it does not sacrifice accuracy.