CSDL Home IEEE Transactions on Pattern Analysis & Machine Intelligence 2010 vol.32 Issue No.12 - December

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Issue No.12 - December (2010 vol.32)

pp: 2232-2245

Erich Fuchs , University of Passau, Passau

Jiri Nitschke , University of Passau, Passau

Bernhard Sick , Univerisity of Applied Sciences, Deggendorf

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TPAMI.2010.44

ABSTRACT

The paper presents SwiftSeg, a novel technique for online time series segmentation and piecewise polynomial representation. The segmentation approach is based on a least-squares approximation of time series in sliding and/or growing time windows utilizing a basis of orthogonal polynomials. This allows the definition of fast update steps for the approximating polynomial, where the computational effort depends only on the degree of the approximating polynomial and not on the length of the time window. The coefficients of the orthogonal expansion of the approximating polynomial—obtained by means of the update steps—can be interpreted as optimal (in the least-squares sense) estimators for average, slope, curvature, change of curvature, etc., of the signal in the time window considered. These coefficients, as well as the approximation error, may be used in a very intuitive way to define segmentation criteria. The properties of SwiftSeg are evaluated by means of some artificial and real benchmark time series. It is compared to three different offline and online techniques to assess its accuracy and runtime. It is shown that SwiftSeg—which is suitable for many data streaming applications—offers high accuracy at very low computational costs.

INDEX TERMS

Time series, orthogonal polynomials, least-squares approximation, online segmentation, piecewise polynomial representation, SwiftSeg.

CITATION

Erich Fuchs, Jiri Nitschke, Bernhard Sick, "Online Segmentation of Time Series Based on Polynomial Least-Squares Approximations",

*IEEE Transactions on Pattern Analysis & Machine Intelligence*, vol.32, no. 12, pp. 2232-2245, December 2010, doi:10.1109/TPAMI.2010.44REFERENCES

- [1] E. Keogh, S. Chu, D. Hart, and M. Pazzani, "An Online Algorithm for Segmenting Time Series,"
Proc. IEEE Int'l Conf. Data Mining, pp. 289-296, 2001.- [2] E. Keogh, S. Chu, D. Hart, and M. Pazzani, "Segmenting Time Series: A Survey and Novel Approach,"
Data Mining in Time Series Databases, M. Last, A. Kandel, and H. Bunke, eds., vol. 57, ch. 1, pp. 1-22, World Scientific Publishing, 2004.- [3] H.J.L.M. Vullings, M.H.G. Verhaegen, and H.B. Verbruggen, "ECG Segmentation Using Time-Warping,"
Proc. Second Int'l Symp. Advances in Intelligent Data Analysis, Reasoning about Data, pp. 275-285, 1997.- [4] D. Lemire, "A Better Alternative to Piecewise Linear Time Series Segmentation,"
Proc. SIAM Data Mining '07, pp. 545-550, 2007.- [5] X. Liu, Z. Lin, and H. Wang, "Novel Online Methods for Time Series Segmentation,"
IEEE Trans. Knowledge and Data Eng., vol. 20, no. 12, pp. 1616-1626, Dec. 2008.- [6] W. Fitzgerald, D. Lemire, and M. Brooks, "Quasi-Monotonic Segmentation of State Variable Behaviour for Reactive Control,"
Proc. 20th Nat'l Conf. Artificial Intelligence, pp. 1145-1150, 2005.- [7] T. Mori, Y. Nejigane, M. Shimosaka, Y. Segawa, T. Harada, and T. Sato, "Online Recognition and Segmentation for Time-Series Motion with HMM and Conceptual Relation of Actions,"
Proc. IEEE/RSJ Int'l Conf. Intelligent Robots and Systems, pp. 2856-2863, 2005.- [8] J. Kohlmorgen, S. Lemm, K.-R. Muller, S. Liehr, and K. Pawelzik, "Fast Change Point Detection in Switching Dynamics Using a Hidden Markov Model of Prediction Experts,"
Proc. Ninth Int'l Conf. Artificial Neural Networks, vol. 1, pp. 204-209, 1999.- [9] J. Kohlmorgen and S. Lemm, "An On-Line Method for Segmentation and Identification of Non-Stationary Time Series,"
Proc. IEEE Signal Processing Soc. Workshop Neural Networks for Signal Processing XI, pp. 113-122, 2001.- [10] Y. Gorshkov, I. Kokshenev, Y. Bodyanskiy, V. Kolodyazhniy, and O. Shylo, "Robust Recursive Fuzzy Clustering-Based Segmentation of Biological Time Series,"
Proc. Int'l Symp. Evolving Fuzzy Systems, pp. 101-105, 2006.- [11] Y. Rao and J. Principe, "A Fast On-Line Generalized Eigendecomposition Algorithm for Time Series Segmentation,"
Proc. Adaptive Systems for Signal Processing, Comm., and Control Symp., pp. 266-271, 2000.- [12] A. Li, S. He, and Z. Qin, "Real-Time Segmenting Time Series Data,"
Proc. Web Technologies and Applications—Fifth Asian-Pacific Web Conf., X. Zhou, Y. Zhang, and M.E. Orlowska, eds., pp. 178-186, 2003.- [13] R. Bellman, "On the Approximation of Curves by Line Segments Using Dynamic Programming,"
Comm. ACM, vol. 4, no. 6, p. 284, 1961.- [14] L. Tang, B. Cui, H. Li, G. Miao, D. Yang, and X. Zhou, "Effective Variation Management for Pseudo Periodical Streams,"
Proc. ACM SIGMOD '07, pp. 257-268, 2007.- [15] H. Shena, C. Xu, X. Han, and Y. Pan, "Stock Tracking: A New Multi-Dimensional Stock Forecasting Approach,"
Proc. Eighth Int'l Conf. Information Fusion, vol. 2, pp. 1375-1382, 2005.- [16] H. Junker, O. Amft, P. Lukowicz, and G. Tröster, "Gesture Spotting with Body-Worn Inertial Sensors to Detect User Activities,"
Pattern Recognition, vol. 41, pp. 2010-2024, 2008.- [17] G.H. Golub and C.F. van Loan,
Matrix Computations, third ed. Johns Hopkins Univ. Press, 1996.- [18] Å. Björck,
Numerical Methods for Least Squares Problems. SIAM, 1996.- [19] E. Fuchs, C. Gruber, T. Reitmaier, and B. Sick, "Processing Short-Term and Long-Term Information with a Combination of Polynomial Approximation Techniques and Time-Delay Neural Networks,"
IEEE Trans. Neural Networks, vol. 20, no. 9, pp. 1450-1462, Sept. 2009.- [20] H. Liu and H. Motoda,
Feature Selection for Knowledge Discovery and Data Mining. Kluwer Academic Publishers, 1998.- [21]
Feature Extraction, Construction, and Selection: A Data Mining Perspective, H. Liu and H. Motoda, eds. Kluwer Academic Publishers, 1998.- [22] E. Fuchs, "Schnelle Quadratmittelapproximation in gleitenden Zeitfenstern mit diskreten orthogonalen Polynomen," PhD dissertation, Univ. of Passau, 1999.
- [23] E. Fuchs and K. Donner, "Fast Least-Squares Polynomial Approximation in Moving Time Windows,"
Proc. Int'l Conf. Acoustics, Speech, and Signal Processing, vol. 3, pp. 1965-1968, 1997.- [24] E. Fuchs, "On Discrete Polynomial Least-Squares Approximation in Moving Time Windows,"
Applications and Computation of Orthogonal Polynomials, W. Gautschi, G. Golub, and G. Opfer, eds., vol. 131, pp. 93-107, Birkhäuser, 1999.- [25] A. Nikiforov, S. Suslov, and V. Uvarov,
Classical Orthogonal Polynomials of a Discrete Variable. Springer-Verlag, 1991.- [26] S. Elhay, G.H. Golub, and J. Kautsky, "Updating and Downdating of Orthogonal Polynomials with Data Fitting Applications,"
SIAM J. Matrix Analysis and Applications, vol. 12, no. 2, pp. 327-353, 1991.- [27] R.F. Boisvert, J. Hicklin, B. Miller, C. Moler, R. Pozo, K. Remington, and P. Webb, "JAMA: A Java Matrix Package," http://math.nist.gov/javanumericsjama/, 2005.
- [28] A.L. Goldberger, L.A.N. Amaral, L. Glass, J.M. Hausdorff, P.C. Ivanov, R.G. Mark, J.E. Mietus, G.B. Moody, C.-K. Peng, and H.E. Stanley, "PhysioBank, PhysioToolkit, and PhysioNet: Components of a New Research Resource for Complex Physiologic Signals,"
Circulation, vol. 101, no. 23, pp. 215-220, 2000.- [29] R. Klinge,
Das Elektrokardiogramm. Thieme, 2002.- [30] E.N. Lorenz, "Deterministic Nonperiodic Flow,"
J. Atmospheric Sciences, vol. 20, no. 2, pp. 130-141, 1963.- [31] W. Tucker, "A Rigorous ODE Solver and Smale's 14th Problem,"
Foundations of Computational Math., vol. 2, no. 1, pp. 53-117, 2002.- [32] M.R. Henzinger, P. Raghavan, and S. Rajagopalan, "Computing on Data Streams,"
External Memory Algorithms, J.M. Abello and J.S. Vitter, eds., pp. 107-118, Am. Math. Soc., 1999.- [33] H. Ding, G. Trajcevski, P. Scheuermann, X. Wang, and E. Keogh, "Querying and Mining of Time Series Data: Experimental Comparison of Representations and Distance Measures,"
Proc. 34th Int'l Conf. Very Large Data Bases, pp. 1542-1552, 2008.- [34] P.-F. Marteau, "Time Warp Edit Distance with Stiffness Adjustment for Time Series Matching,"
IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 31, no. 2, pp. 306-318, Feb. 2009.- [35] E. Fuchs, T. Hanning, and O. Schwarz, "An Update Algorithm for Fourier Coefficients,"
Proc. 12th European Signal Processing Conf., pp. 1509-1512, 2004.- [36] J.A. Ward, P. Lukowicz, G. Tröster, and T.E. Starner, "Activity Recognition of Assembly Tasks Using Body-Worn Microphones and Accelerometers,"
IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 28, no. 10, pp. 1553-1567, Oct. 2006.- [37] J. Mager, U. Paasche, and B. Sick, "Forecasting Financial Time Series with Support Vector Machines Based on Dynamic Kernels,"
Proc. IEEE Conf. Soft Computing in Industrial Applications, pp. 252-257, 2008.- [38] C. Gruber, T. Gruber, and B. Sick, "Online Signature Verification with New Time Series Kernels for Support Vector Machines,"
Advances in Biometrics, D. Zhang and A.K. Jain, eds., pp. 500-508, Springer-Verlog, 2006.- [39] T. Artieres, S. Marukatat, and P. Gallinari, "Online Handwritten Shape Recognition Using Segmental Hidden Markov Models,"
IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 29, no. 2, pp. 205-217, Feb. 2007. |