## Locally Adaptive Dimensionality Reduction for Indexing Large Time Series Databases (2002)

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Venue: | In proceedings of ACM SIGMOD Conference on Management of Data |

Citations: | 237 - 28 self |

### BibTeX

@INPROCEEDINGS{Chakrabarti02locallyadaptive,

author = {Kaushik Chakrabarti and Eamonn Keogh and Sharad Mehrotra and Michael Pazzani},

title = {Locally Adaptive Dimensionality Reduction for Indexing Large Time Series Databases},

booktitle = {In proceedings of ACM SIGMOD Conference on Management of Data},

year = {2002},

pages = {151--162}

}

### Years of Citing Articles

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### Abstract

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 searchin& 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.

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Citation Context ...ltidimensional index structure is shown in Table 6 3 . The above algorithm is an optimization on the GEMINI K-NN algorithm described in Table 3 and was proposed in [41]. Like the basic K-NN algorithm =-=[19,40]-=-, the algorithm uses a priority queue queue to navigate nodes/objects in the index in the increasing order of their distances from Q in the indexed (i.e. APCA) space. The distance of an object (i.e. A... |

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Citation Context ...rier Transform (DFT) to perform the dimensionality reduction, but other techniques have been suggested, including Singular Value Decomposition (SVD) [28, 24, 23], the Discrete Wavelet Transform (DWT) =-=[9, 49, 22]-=- and Piecewise Aggregate Approximation (PAA) [24, 52]. For a given index structure, the efficiency of indexing depends only on the fidelity of the approximation in the reduced dimensionality space. Ho... |

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Citation Context ...ngular inequality. ■ 2 + (-1-1) 2 + (-12) 2 10sThe failure of DAE to obey the triangular inequality means that it may not lower bound the Euclidean distance and thus cannot be used for exact indexin=-=g [51]-=-. However, we will demonstrate later that it is very useful for approximate search. 3.3.2 An lower-bounding measure DLB To define DLB(Q,C) we must first introduce a special version of the APCA. Normal... |

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Citation Context ... time axis L= { <l 1 , l 2 >, <l 3 , l 4 >, <l 5 , l 6 >} APCA segments Figure 10: The M Regions associated with a 2M-dimensional MBR. The boundary of a region G is denoted by G = {G[1], G[2], G[3], G=-=[4]-=-} a 2-d rectangle, is defined by 4 numbers: the low bounds G[1] and G[2] and the high bounds G[3] and G[4] along the value and time axes respectively. By definition, R G i [ 1] = minC under U ( cmini ... |

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Citation Context ...ses. Similarity search is useful in its own right as a tool for exploratory data analysis, and it is also an important element of many data mining applications such as clustering [13], classification =-=[26, 33]-=- and mining of association rules [12]. The similarity between two time series is typically measured with Euclidean distance, which can be calculated very efficiently. However the volume of data typica... |

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Citation Context ...space and (more complex) indexing structures”. Singular value decomposition is also a data adaptive technique used for time series [28, 24, 23], but it is globally, not locally, adaptive. Recent wor=-=k [7]-=- has suggested first clustering a multi-dimensional space and then doing SVD on local clusters, making it a semi-local approach. It is not clear however that this approach can be made work for time se... |

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Citation Context ...Since realistic queries typically contain 20 to 1,000 datapoints (i.e. n varies from 20 to 1000) and most multidimensional index structures have poor performance at dimensionalities greater than 8-12 =-=[6]-=-, we need to first perform dimensionality reduction in order to exploit multidimensional index structures to index time series data. In [16] the authors introduced GEneric Multimedia INdexIng method (... |

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Citation Context ...l work by Agrawal et. al. utilizes the Discrete Fourier Transform (DFT) to perform the dimensionality reduction, but other techniques have been suggested, including Singular Value Decomposition (SVD) =-=[28, 24, 23]-=-, the Discrete Wavelet Transform (DWT) [9, 49, 22] and Piecewise Aggregate Approximation (PAA) [24, 52]. For a given index structure, the efficiency of indexing depends only on the fidelity of the app... |

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Citation Context ... returns approximately the same results. Typically this involves transforming the data with a lossy compression scheme, then doing a sequential search on the compressed data. Typical examples include =-=[42, 27, 30, 46]-=-, who all utilize a piecewise linear approximation. Others have suggested transforming the data into a discrete alphabet and using string-matching algorithms [2, 20, 34, 29, 21, 38]. All these approac... |

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Citation Context ... returns approximately the same results. Typically this involves transforming the data with a lossy compression scheme, then doing a sequential search on the compressed data. Typical examples include =-=[42, 27, 30, 46]-=-, who all utilize a piecewise linear approximation. Others have suggested transforming the data into a discrete alphabet and using string-matching algorithms [2, 20, 34, 29, 21, 38]. All these approac... |

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Citation Context ... data and is completely independent of any implementation choices, including spatial access method, page size, computer language or hardware. A similar idea for evaluating indexing schemes appears in =-=[18]-=-. Figure 13 shows the value of P over a range of query lengths and dimensionalities for the experiments that were conducted the Mixed Bag dataset. FastMap 0.5 0.4 0.3 0.2 0.1 0 1024 512 256 Figure 13:... |

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Citation Context ...e to scale to large databases. Further, the systems are evaluated without considering precision and recall, thus we can say little or nothing about the quality of the returned answer set. The work of =-=[3, 36, 45, 25, 26] d-=-iffers from the above in that they focus in providing a more flexible query language and not on performance issues. 2.2 Exact techniques for similarity searching. A time series C = {c1,…,cn} with n ... |

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Citation Context ...rier Transform (DFT) to perform the dimensionality reduction, but other techniques have been suggested, including Singular Value Decomposition (SVD) [28, 24, 23], the Discrete Wavelet Transform (DWT) =-=[9, 49, 22]-=- and Piecewise Aggregate Approximation (PAA) [24, 52]. For a given index structure, the efficiency of indexing depends only on the fidelity of the approximation in the reduced dimensionality space. Ho... |

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Citation Context ...e note that its time complexity is O(SnN) while that of APCA is O(Snlog(n)). 25spartitioning index structure (“dimensionality-independent” fanout) and has been shown to scale to high dimensionalit=-=ies [6, 37, 24]-=-. Since we had access to the source code of the index structure (http://www-db.ics.uci.edu) we implemented the optimization discussed in Section 4 (i.e. to increase leaf node fanout) for our experimen... |

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Citation Context ...rier Transform (DFT) to perform the dimensionality reduction, but other techniques have been suggested, including Singular Value Decomposition (SVD) [28, 24, 23], the Discrete Wavelet Transform (DWT) =-=[9, 49, 22]-=- and Piecewise Aggregate Approximation (PAA) [24, 52]. For a given index structure, the efficiency of indexing depends only on the fidelity of the approximation in the reduced dimensionality space. Ho... |

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Citation Context ... returns approximately the same results. Typically this involves transforming the data with a lossy compression scheme, then doing a sequential search on the compressed data. Typical examples include =-=[42, 27, 30, 46]-=-, who all utilize a piecewise linear approximation. Others have suggested transforming the data into a discrete alphabet and using string-matching algorithms [2, 20, 34, 29, 21, 38]. All these approac... |

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Citation Context ... Typical examples include [42, 27, 30, 46], who all utilize a piecewise linear approximation. Others have suggested transforming the data into a discrete alphabet and using string-matching algorithms =-=[2, 20, 34, 29, 21, 38]-=-. All these approaches suffer from some limitations. They are all evaluated on small datasets residing in main memory, and it is unclear if they can be made to scale to large databases. Further, the s... |

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Citation Context ...g all possible Msegment APCA representations of C. Finding the optimal piecewise polynomial representation of cr1 cr2 cr3 cv3 cr4 cv4 7sa time series requires a O(Mn 2 ) dynamic programming algorithm =-=[15, 35]-=-. This is too slow for high dimensional data. In this paper, we propose a new algorithm to produce almost optimal APCA representations in O(nlog(n)) time. The algorithm works by first converting the p... |

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Citation Context ...e indexing time series indexing. A locally adaptive representation for 2-dimensional shapes was suggested in [8] but no indexing technique was proposed. Also in the context of images, it was noted by =-=[50]-=- that the use of the first N Fourier coefficients does not guarantee the optimal pruning power. They introduced a technique where they adaptively choose which coefficients to keep after looking at the... |

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Citation Context ... Typical examples include [42, 27, 30, 46], who all utilize a piecewise linear approximation. Others have suggested transforming the data into a discrete alphabet and using string-matching algorithms =-=[2, 20, 34, 29, 21, 38]-=-. All these approaches suffer from some limitations. They are all evaluated on small datasets residing in main memory, and it is unclear if they can be made to scale to large databases. Further, the s... |

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Citation Context ...e to scale to large databases. Further, the systems are evaluated without considering precision and recall, thus we can say little or nothing about the quality of the returned answer set. The work of =-=[3, 36, 45, 25, 26] d-=-iffers from the above in that they focus in providing a more flexible query language and not on performance issues. 2.2 Exact techniques for similarity searching. A time series C = {c1,…,cn} with n ... |

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Citation Context ...rocessing performed was to insure that each time series had a mean of zero and a standard deviation of one (otherwise many queries become pathologically easy). The 7 datasets are, Space Shuttle STS57 =-=[27, 25]-=-, Arrhythmia [32], Random Walk [46, 34, 52, 24], INTERBALL Plasma processes (figure 4) [43], Astrophysical data (figure 1) [47], Pseudo Periodic Synthetic Time Series [4]. Exchange rate (figure 4) [47... |

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Citation Context ...in time series databases. Similarity search is useful in its own right as a tool for exploratory data analysis, and it is also an important element of many data mining applications such as clustering =-=[13]-=-, classification [26, 33] and mining of association rules [12]. The similarity between two time series is typically measured with Euclidean distance, which can be calculated very efficiently. However ... |

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Citation Context ...g all possible Msegment APCA representations of C. Finding the optimal piecewise polynomial representation of cr1 cr2 cr3 cv3 cr4 cv4 7sa time series requires a O(Mn 2 ) dynamic programming algorithm =-=[15, 35]-=-. This is too slow for high dimensional data. In this paper, we propose a new algorithm to produce almost optimal APCA representations in O(nlog(n)) time. The algorithm works by first converting the p... |

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Citation Context ...chniques that perform dimensionality reduction on the data, then use a multidimensional index structure to index the data in the transformed space. The technique was introduced in [1] and extended in =-=[39, 31,11]-=-. The original work by Agrawal et. al. utilizes the Discrete Fourier Transform (DFT) to perform the dimensionality reduction, but other techniques have been suggested, including Singular Value Decompo... |

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Citation Context ...ved from the physically stored information. For example, space partitioning index structures like the hB-tree and the Hybrid Tree store the partitioning information inside the index nodes as kd-trees =-=[14, 6]-=-. Since the MBRs can be derived from the kd-trees, the techniques discussed here are applicable to such index structures [6]. 14spopping out the item from the top of queue at each step (Line 9). If th... |

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Citation Context ...ses. Similarity search is useful in its own right as a tool for exploratory data analysis, and it is also an important element of many data mining applications such as clustering [13], classification =-=[26, 33]-=- and mining of association rules [12]. The similarity between two time series is typically measured with Euclidean distance, which can be calculated very efficiently. However the volume of data typica... |

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Citation Context ...l work by Agrawal et. al. utilizes the Discrete Fourier Transform (DFT) to perform the dimensionality reduction, but other techniques have been suggested, including Singular Value Decomposition (SVD) =-=[28, 24, 23]-=-, the Discrete Wavelet Transform (DWT) [9, 49, 22] and Piecewise Aggregate Approximation (PAA) [24, 52]. For a given index structure, the efficiency of indexing depends only on the fidelity of the app... |

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Citation Context ...chniques that perform dimensionality reduction on the data, then use a multidimensional index structure to index the data in the transformed space. The technique was introduced in [1] and extended in =-=[39, 31,11]-=-. The original work by Agrawal et. al. utilizes the Discrete Fourier Transform (DFT) to perform the dimensionality reduction, but other techniques have been suggested, including Singular Value Decompo... |

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Citation Context ...ly revisit related work. We believe that this paper is the first to suggest locally adaptive indexing time series indexing. A locally adaptive representation for 2-dimensional shapes was suggested in =-=[8]-=- but no indexing technique was proposed. Also in the context of images, it was noted by [50] that the use of the first N Fourier coefficients does not guarantee the optimal pruning power. They introdu... |

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Citation Context ...nsertion is the Achilles heel of SVD, a single insertion requires recomputing the entire index. Faster methods do exist for incremental updates, but they introduce the possibility of false dismissals =-=[10]-=-. 7. Conclusions and directions for future work The main contribution of this paper is to show that a simple, novel dimensionality reduction technique, namely APCA, can outperform more sophisticated t... |

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Citation Context ...l work by Agrawal et. al. utilizes the Discrete Fourier Transform (DFT) to perform the dimensionality reduction, but other techniques have been suggested, including Singular Value Decomposition (SVD) =-=[28, 24, 23]-=-, the Discrete Wavelet Transform (DWT) [9, 49, 22] and Piecewise Aggregate Approximation (PAA) [24, 52]. For a given index structure, the efficiency of indexing depends only on the fidelity of the app... |

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Citation Context ...tion of one (otherwise many queries become pathologically easy). The 7 datasets are, Space Shuttle STS57 [27, 25], Arrhythmia [32], Random Walk [46, 34, 52, 24], INTERBALL Plasma processes (figure 4) =-=[43]-=-, Astrophysical data (figure 1) [47], Pseudo Periodic Synthetic Time Series [4]. Exchange rate (figure 4) [47]. Once again, we generated data of 3 different dimensionalities: n=1024, n=512 and n=256 a... |

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Citation Context ...nal for any level of compression (i.e., #retained coefficients/n) can be obtained by sorting the coefficients in order of decreasing normalized magnitude, then truncating off the smaller coefficients =-=[44]-=-. The segments in the reconstructed signal may have approximate mean values (due to truncation); we replace them by the exact mean values to get a valid APCA representation as defined in Equation 2. T... |