We propose an approximation algorithm for efficient correlation search in time series data. In our method, we use Fourier transform and neural network to embed time series into a low-dimensional Euclidean space. The given space is learned such that time series correlation can be effectively approximated from Euclidean distance between corresponding embedded vectors. Therefore, search for correlated time series can be done using an index in the embedding space for efficient nearest neighbor search. Our theoretical analysis illustrates that our method's accuracy can be guaranteed under certain regularity conditions. We further conduct experiments on real-world datasets and the results show that our method indeed outperforms the baseline solution. In particular, for approximation of correlation, our method reduces the approximation loss by a half in most test cases compared to the baseline solution. For top- @math highest correlation search, our method improves the precision from 5 to 20 while the query time is similar to the baseline approach query time.
We also notice that since Agrawal @cite_29 researchers suggested many different dimensionality reduction techniques to provide tighter approximation bound for pruning. To name a few, there are Adaptive Piecewise Constant Approximation (APCA) @cite_40 , Piecewise Aggregate Approximation (PAA) @cite_5 , multi-scale segment mean (MSM) @cite_7 , Chebyshev Polynomials (CP) @cite_19 , and Piecewise Linear Approximation (PLA) @cite_31 . However, according to the comparison work by Ding @cite_4 , none of these significantly outperform DFT and DWT.
@cite_4 @cite_7 @cite_29 @cite_19 @cite_40 @cite_5 @cite_31
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The last decade has witnessed a tremendous growths of interests in applications that deal with querying and mining of time series data. Numerous representation methods for dimensionality reduction and similarity measures geared towards time series have been introduced. Each individual work introducing a particular method has made specific claims and, aside from the occasional theoretical justifications, provided quantitative experimental observations. However, for the most part, the comparative aspects of these experiments were too narrowly focused on demonstrating the benefits of the proposed methods over some of the previously introduced ones. In order to provide a comprehensive validation, we conducted an extensive set of time series experiments re-implementing 8 different representation methods and 9 similarity measures and their variants, and testing their effectiveness on 38 time series data sets from a wide variety of application domains. In this paper, we give an overview of these different techniques and present our comparative experimental findings regarding their effectiveness. Our experiments have provided both a unified validation of some of the existing achievements, and in some cases, suggested that certain claims in the literature may be unduly optimistic. Similarity-based time-series retrieval has been a subject of long-term study due to its wide usage in many applications, such as financial data analysis, weather data forecasting, and multimedia data retrieval. Its original task was to find those time series similar to a pattern (query) time-series data, where both the pattern and data time series are static. Recently, with an increasing demand on stream data management, similarity-based stream time-series retrieval has raised new research issues due to its unique requirements during the stream processing, such as one-pass search and fast response. In this paper, we address the problem of matching both static and dynamic patterns over stream time-series data. We will develop a novel multiscale representation, called multiscale segment mean, for stream time-series data, which can be incrementally computed and thus perfectly adapted to the stream characteristics. Most importantly, we propose a novel multistep filtering mechanism, step by step, over the multiscale representation. Analysis indicates that the mechanism can greatly prune the search space and thus offer fast response. Furthermore, batch processing optimization and the dynamic case where patterns are also from stream time series are discussed. Extensive experiments show the multiscale representation together with the multistep filtering scheme can efficiently filter out false candidates and detect patterns, compared to the multiscale wavelet. We propose an indexing method for time sequences for processing similarity queries. We use the Discrete Fourier Transform (DFT) to map time sequences to the frequency domain, the crucial observation being that, for most sequences of practical interest, only the first few frequencies are strong. Another important observation is Parseval's theorem, which specifies that the Fourier transform preserves the Euclidean distance in the time or frequency domain. Having thus mapped sequences to a lower-dimensionality space by using only the first few Fourier coefficients, we use R * -trees to index the sequences and efficiently answer similarity queries. We provide experimental results which show that our method is superior to search based on sequential scanning. Our experiments show that a few coefficients (1–3) are adequate to provide good performance. The performance gain of our method increases with the number and length of sequences. In this paper, we attempt to approximate and index a d- dimensional (d ≥ 1) spatio-temporal trajectory with a low order continuous polynomial. There are many possible ways to choose the polynomial, including (continuous)Fourier transforms, splines, non-linear regressino, etc. Some of these possiblities have indeed been studied beofre. We hypothesize that one of the best possibilities is the polynomial that minimizes the maximum deviation from the true value, which is called the minimax polynomial. Minimax approximation is particularly meaningful for indexing because in a branch-and-bound search (i.e., for finding nearest neighbours), the smaller the maximum deviation, the more pruning opportunities there exist. However, in general, among all the polynomials of the same degree, the optimal minimax polynomial is very hard to compute. However, it has been shown thta the Chebyshev approximation is almost identical to the optimal minimax polynomial, and is easy to compute [16]. Thus, in this paper, we explore how to use the Chebyshev polynomials as a basis for approximating and indexing d-dimenstional trajectories.The key analytic result of this paper is the Lower Bounding Lemma. that is, we show that the Euclidean distance between two d-dimensional trajectories is lower bounded by the weighted Euclidean distance between the two vectors of Chebyshev coefficients. this lemma is not trivial to show, and it ensures that indexing with Chebyshev cofficients aedmits no false negatives. To complement that analystic result, we conducted comprehensive experimental evaluation with real and generated 1-dimensional to 4-dimensional data sets. We compared the proposed schem with the Adaptive Piecewise Constant Approximation (APCA) scheme. Our preliminary results indicate that in all situations we tested, Chebyshev indexing dominates APCA in pruning power, I O and CPU costs. Similarity search in large time series databases has attracted much research interest recently. It is a difficult problem because of the typically high dimensionality of the data.. The most promising solutions involve performing dimensionality reduction on the data, then indexing the reduced data with a multidimensional index structure. Many dimensionality reduction techniques have been proposed, including Singular Value Decomposition (SVD), the Discrete Fourier transform (DFT), and the Discrete Wavelet Transform (DWT). In this work we introduce a new dimensionality reduction technique which we call Adaptive Piecewise Constant Approximation (APCA). While previous techniques (e.g., SVD, DFT and DWT) choose a common representation for all the items in the database that minimizes the global reconstruction error, APCA approximates each time series by a set of constant value segments of varying lengths such that their individual reconstruction errors are minimal. We show how APCA can be indexed using a multidimensional index structure. We propose two distance measures in the indexed space that exploit the high fidelity of APCA for fast searching: a lower bounding Euclidean distance approximation, and a non-lower bounding, but very tight Euclidean distance approximation and show how they can support fast exact searching, and even faster approximate searching on the same index structure. We theoretically and empirically compare APCA to all the other techniques and demonstrate its superiority. The problem of similarity search in large time series databases has attracted much attention recently. It is a non-trivial problem because of the inherent high dimensionality of the data. The most promising solutions involve first performing dimensionality reduction on the data, and then indexing the reduced data with a spatial access method. Three major dimensionality reduction techniques have been proposed: Singular Value Decomposition (SVD), the Discrete Fourier transform (DFT), and more recently the Discrete Wavelet Transform (DWT). In this work we introduce a new dimensionality reduction technique which we call Piecewise Aggregate Approximation (PAA). We theoretically and empirically compare it to the other techniques and demonstrate its superiority. In addition to being competitive with or faster than the other methods, our approach has numerous other advantages. It is simple to understand and to implement, it allows more flexible distance measures, including weighted Euclidean queries, and the index can be built in linear time. Similarity-based search over time-series databases has been a hot research topic for a long history, which is widely used in many applications, including multimedia retrieval, data mining, web search and retrieval, and so on. However, due to high dimensionality (i.e. length) of the time series, the similarity search over directly indexed time series usually encounters a serious problem, known as the "dimensionality curse". Thus, many dimensionality reduction techniques are proposed to break such curse by reducing the dimensionality of time series. Among all the proposed methods, only Piecewise Linear Approximation (PLA) does not have indexing mechanisms to support similarity queries, which prevents it from efficiently searching over very large time-series databases. Our initial studies on the effectiveness of different reduction methods, however, show that PLA performs no worse than others. Motivated by this, in this paper, we re-investigate PLA for approximating and indexing time series. Specifically, we propose a novel distance function in the reduced PLA-space, and prove that this function indeed results in a lower bound of the Euclidean distance between the original time series, which can lead to no false dismissals during the similarity search. As a second step, we develop an effective approach to index these lower bounds to improve the search efficiency. Our extensive experiments over a wide spectrum of real and synthetic data sets have demonstrated the efficiency and effectiveness of PLA together with the newly proposed lower bound distance, in terms of both pruning power and wall clock time, compared with two state-of-the-art reduction methods, Adaptive Piecewise Constant Approximation (APCA) and Chebyshev Polynomials (CP).