A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size

Cheng Peng; Zhaohui Tang; Weihua Gui; Qing Chen; Jing He

doi:10.3934/jimo.2019107

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January 2021, 17(1): 205-220. doi: 10.3934/jimo.2019107

A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size

Cheng Peng ^1,2, , Zhaohui Tang ^1,, , Weihua Gui ^1, , Qing Chen ^2, and Jing He ^2,

1.	School of Automation, Central South University, Changsha, Hunan410083, China
2.	School of Computer, Hunan University of Technology, Zhuzhou, Hunan412007, China

^* Corresponding author: Zhaohui Tang

Received October 2018 Revised April 2019 Published January 2021 Early access September 2019

The existing method of determining the size of the time series sliding window by empirical value exists some problems which should be solved urgently, such as when considering a large amount of information and high density of the original measurement data collected from industry equipment, the important information of the data cannot be maximally retained, and the calculation complexity is high. Therefore, by studying the effect of sliding window on time series similarity technology in practical application, an algorithm to determine the initial size of the sliding window is proposed. The upper and lower boundary curves with a higher fitting degree are constructed, and the trend weighting is introduced into the $ LB\_Hust $ distance calculation method to reduce the difficulty of mathematical modeling and improve the efficiency of data similarity computation.

Keywords: Time series, bidirectional boundary distance, sliding window, LBHust distance, similarity computation.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation: Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107

References:

[1]	R. Belohlavek and V. Vychodil, Relational similarity-based model of data part 1: Foundations and query systems, Int. J. General Syst., 46 (2017), 671-751. doi: 10.1080/03081079.2017.1357550. Google Scholar
[2]	W. Bian and D. Tao, Max-Min distance analysis by using sequential sdp relaxation for dimension reduction, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011), 1037-1050. doi: 10.1109/TPAMI.2010.189. Google Scholar
[3]	M. R. Chernick, Wavelet Methods for Time Series Analysis, Technometrics, 43 (2016), 491-508. Google Scholar
[4]	L. Dong, S. Liu and H. Zhang, A method of anomaly detection and fault diagnosis with online adaptive learning under small training samples, Pattern Recognition, 64 (2017), 374-385. doi: 10.1016/j.patcog.2016.11.026. Google Scholar
[5]	G. Hesamian and M. G. Akbari, A semi-parametric model for time series based on fuzzy data, IEEE Transactions on Fuzzy Systems, 99 (2018), 1-10. doi: 10.1109/TFUZZ.2018.2791931. Google Scholar
[6]	K. Hornik, I. Feinerer and M. Kober, Spherical k-means clustering, J. Statistical Software, 50 (2017), 1-22. doi: 10.18637/jss.v050.i10. Google Scholar
[7]	B. Hu, P. C. Dixon and J. V. Jacobs, Machine learning algorithms based on signals from a single wearable inertial sensor can detect surface- and age-related differences in walking, J. Biomechanics, 71 (2018), 36-48. doi: 10.1016/j.jbiomech.2018.01.005. Google Scholar
[8]	I. Güler and M. Meghdadi, A different approach to off-line handwritten signature verification using the optimal dynamic time warping algorithm, Digital Signal Processing, 18 (2008), 940-950. doi: 10.1016/j.dsp.2008.06.005. Google Scholar
[9]	H. Ji, C. Zhou and Z. Liu, An approximate representation method for time series symbol aggregation based on the distance between origin and end, Computer Science, 10 (2018), 135-147. Google Scholar
[10]	R. J. Kate, Using dynamic time warping distances as features for improved time series classification, Data Min. Knowl. Discov., 30 (2016), 283-312. doi: 10.1007/s10618-015-0418-x. Google Scholar
[11]	S. W. Kim, J. Kim and S. Park, Physical database design for efficient time-series similarity search, IEICE Trans Commun., 91 (2008), 1251-1254. doi: 10.1093/ietcom/e91-b.4.1251. Google Scholar
[12]	G. Lee, U. Yun and K. Ryu, Sliding window based weighted maximal frequent pattern mining over data streams, Expert Syst. Appl., 41 (2014), 694-708. doi: 10.1016/j.eswa.2013.07.094. Google Scholar
[13]	R. Li, X. Wu and S. Yang, Dynamic on-state resistance test and evaluation of GaN power devices under hard and soft switching conditions by double and multiple pulses, IEEE Transactions on Power Electronics, 34 (2018), 1-6. doi: 10.1109/TPEL.2018.2844302. Google Scholar
[14]	T. Luo, C. Hou and F. Nie, Dimension reduction for non-Gaussian data by adaptive discriminative analysis, IEEE Transactions on Cybernetics, 49 (2018), 1-14. doi: 10.1109/TCYB.2018.2789524. Google Scholar
[15]	M. D. C. Moura, E. Zio and I. D. Lins, Failure and reliability prediction by support vector machines regression of time series data, Reliability Engineering and Syst. Safety, 96 (2017), 1527-1534. doi: 10.1016/j.ress.2011.06.006. Google Scholar
[16]	S. J. Noh, D. Shim and M. Jeon, Adaptive sliding-window strategy for vehicle detection in highway environments, IEEE Transactions on Intelligent Transportation Syst., 17 (2016), 323-335. doi: 10.1109/TITS.2015.2466652. Google Scholar
[17]	N. M. Parthaláin, Q. Shen and R. Jensen, A distance measure approach to exploring the rough set boundary region for attribute reduction, IEEE Transactions on Knowledge and Data Engineering, 22 (2010), 305-317. doi: 10.1109/TKDE.2009.119. Google Scholar
[18]	F. Petitjean, G. Forestier and G. I. Webb, Faster and more accurate classification of time series by exploiting a novel dynamic time warping averaging algorithm, Knowledge and Information Syst., 47 (2016), 1-26. doi: 10.1007/s10115-015-0878-8. Google Scholar
[19]	H. Ren, M. Liu and Z. Li, A piecewise aggregate pattern representation approach for anomaly detection in time series, Knowledge-Based Syst., 21 (2017), 213-220. doi: 10.1016/j.knosys.2017.07.021. Google Scholar
[20]	J. W. Roh and B. K. Yi, Efficient indexing of interval time sequences, Inform. Process. Lett., 109 (2008), 1-12. doi: 10.1016/j.ipl.2008.08.003. Google Scholar
[21]	H. Ryang and U. Yun, High utility pattern mining over data streams with sliding window technique, Expert Syst. Appl., 57 (2016), 214-231. doi: 10.1016/j.eswa.2016.03.001. Google Scholar
[22]	M. J. Safari, F. A. Davani and H. Afarideh, Discrete fourier transform method for discrimination of digital scintillation pulses in mixed neutron-gamma fields, IEEE Transactions on Nuclear Science, 63 (2016), 325-332. doi: 10.1109/TNS.2016.2514400. Google Scholar
[23]	D. Schultz and B. Jain, Nonsmooth analysis and subgradient methods for averaging in dynamic time warping spaces, Pattern Recognition, 74 (2018), 340-358. doi: 10.1016/j.patcog.2017.08.012. Google Scholar
[24]	Y. Sun, J. Li and J. Liu, An improvement of symbolic aggregate approximation distance measure for time series, Neurocomputing, 102 (2014), 189-198. doi: 10.1016/j.neucom.2014.01.045. Google Scholar
[25]	M. Vafaeipour, O. Rahbari and M. A. Rosen, Application of sliding window technique for prediction of wind velocity time series, Inter. J. Energy and Environmental Engineering, 5 (2014), 105-116. doi: 10.1007/s40095-014-0105-5. Google Scholar
[26]	Y. Xue, X. Mei and Y. Zhi, Method of subway health status recognition based on time series data mining, Information Sciences, 38 (2018), 905-910. Google Scholar
[27]	G. Yuan, P. Sun and J. Zhao, A review of moving object trajectory clustering algorithms, Artificial Intelligence Review, 47 (2017), 1-22. doi: 10.1007/s10462-016-9477-7. Google Scholar
[28]	R. Yao, G. Lin and Q. Shi, Efficient dense labelling of human activity sequences from wearables using fully convolutional networks, Pattern Recognition, 78 (2017), 221-232. doi: 10.1016/j.patcog.2017.12.024. Google Scholar
[29]	S. Yue, Y. Li and Q. Yang, Comparative analysis of core loss calculation methods for magnetic materials under no sinusoidal excitations, IEEE Transactions on Magnetics, 54 (2018), 1-5. doi: 10.1109/TMAG.2018.2842064. Google Scholar
[30]	U. Yun, D. Kim, H. Ryang, G. Lee and K. Lee, Mining recent high average utility patterns based on sliding window from stream data, J. Intelligent and Fuzzy Syst., 30 (2016), 3605-3617. doi: 10.3233/IFS-162106. Google Scholar
[31]	U. Yun, D. Kim, E. Yoon and H. Fujita, Damped window based high average utility pattern mining over data streams, Knowl.-Based Syst., 144 (2018), 188-205. doi: 10.1016/j.knosys.2017.12.029. Google Scholar
[32]	U. Yun and G. Lee, Sliding window based weighted erasable stream pattern mining for stream data applications, Future Generation Comp. Syst., 59 (2016), 1-20. doi: 10.1016/j.future.2015.12.012. Google Scholar
[33]	U. Yun, G. Lee and K. Ryu, Mining maximal frequent patterns by considering weight conditions over data streams, Knowl.-Based Syst., 55 (2014), 49-65. doi: 10.1016/j.knosys.2013.10.011. Google Scholar
[34]	U. Yun, G. Lee and E. Yoon, Efficient high utility pattern mining for establishing manufacturing plans with sliding window control, IEEE Trans. Industrial Electronics, 64 (2017), 7239-7249. doi: 10.1109/TIE.2017.2682782. Google Scholar
[35]	M. Zhu, D. G. M. Mitchell and M. Lentmaier, Braided convolutional codes with sliding window decoding, IEEE Trans. on Communications, 65 (2017), 3645-3658. doi: 10.1109/TCOMM.2017.2707073. Google Scholar

show all references

References:

[1]	R. Belohlavek and V. Vychodil, Relational similarity-based model of data part 1: Foundations and query systems, Int. J. General Syst., 46 (2017), 671-751. doi: 10.1080/03081079.2017.1357550. Google Scholar
[2]	W. Bian and D. Tao, Max-Min distance analysis by using sequential sdp relaxation for dimension reduction, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011), 1037-1050. doi: 10.1109/TPAMI.2010.189. Google Scholar
[3]	M. R. Chernick, Wavelet Methods for Time Series Analysis, Technometrics, 43 (2016), 491-508. Google Scholar
[4]	L. Dong, S. Liu and H. Zhang, A method of anomaly detection and fault diagnosis with online adaptive learning under small training samples, Pattern Recognition, 64 (2017), 374-385. doi: 10.1016/j.patcog.2016.11.026. Google Scholar
[5]	G. Hesamian and M. G. Akbari, A semi-parametric model for time series based on fuzzy data, IEEE Transactions on Fuzzy Systems, 99 (2018), 1-10. doi: 10.1109/TFUZZ.2018.2791931. Google Scholar
[6]	K. Hornik, I. Feinerer and M. Kober, Spherical k-means clustering, J. Statistical Software, 50 (2017), 1-22. doi: 10.18637/jss.v050.i10. Google Scholar
[7]	B. Hu, P. C. Dixon and J. V. Jacobs, Machine learning algorithms based on signals from a single wearable inertial sensor can detect surface- and age-related differences in walking, J. Biomechanics, 71 (2018), 36-48. doi: 10.1016/j.jbiomech.2018.01.005. Google Scholar
[8]	I. Güler and M. Meghdadi, A different approach to off-line handwritten signature verification using the optimal dynamic time warping algorithm, Digital Signal Processing, 18 (2008), 940-950. doi: 10.1016/j.dsp.2008.06.005. Google Scholar
[9]	H. Ji, C. Zhou and Z. Liu, An approximate representation method for time series symbol aggregation based on the distance between origin and end, Computer Science, 10 (2018), 135-147. Google Scholar
[10]	R. J. Kate, Using dynamic time warping distances as features for improved time series classification, Data Min. Knowl. Discov., 30 (2016), 283-312. doi: 10.1007/s10618-015-0418-x. Google Scholar
[11]	S. W. Kim, J. Kim and S. Park, Physical database design for efficient time-series similarity search, IEICE Trans Commun., 91 (2008), 1251-1254. doi: 10.1093/ietcom/e91-b.4.1251. Google Scholar
[12]	G. Lee, U. Yun and K. Ryu, Sliding window based weighted maximal frequent pattern mining over data streams, Expert Syst. Appl., 41 (2014), 694-708. doi: 10.1016/j.eswa.2013.07.094. Google Scholar
[13]	R. Li, X. Wu and S. Yang, Dynamic on-state resistance test and evaluation of GaN power devices under hard and soft switching conditions by double and multiple pulses, IEEE Transactions on Power Electronics, 34 (2018), 1-6. doi: 10.1109/TPEL.2018.2844302. Google Scholar
[14]	T. Luo, C. Hou and F. Nie, Dimension reduction for non-Gaussian data by adaptive discriminative analysis, IEEE Transactions on Cybernetics, 49 (2018), 1-14. doi: 10.1109/TCYB.2018.2789524. Google Scholar
[15]	M. D. C. Moura, E. Zio and I. D. Lins, Failure and reliability prediction by support vector machines regression of time series data, Reliability Engineering and Syst. Safety, 96 (2017), 1527-1534. doi: 10.1016/j.ress.2011.06.006. Google Scholar
[16]	S. J. Noh, D. Shim and M. Jeon, Adaptive sliding-window strategy for vehicle detection in highway environments, IEEE Transactions on Intelligent Transportation Syst., 17 (2016), 323-335. doi: 10.1109/TITS.2015.2466652. Google Scholar
[17]	N. M. Parthaláin, Q. Shen and R. Jensen, A distance measure approach to exploring the rough set boundary region for attribute reduction, IEEE Transactions on Knowledge and Data Engineering, 22 (2010), 305-317. doi: 10.1109/TKDE.2009.119. Google Scholar
[18]	F. Petitjean, G. Forestier and G. I. Webb, Faster and more accurate classification of time series by exploiting a novel dynamic time warping averaging algorithm, Knowledge and Information Syst., 47 (2016), 1-26. doi: 10.1007/s10115-015-0878-8. Google Scholar
[19]	H. Ren, M. Liu and Z. Li, A piecewise aggregate pattern representation approach for anomaly detection in time series, Knowledge-Based Syst., 21 (2017), 213-220. doi: 10.1016/j.knosys.2017.07.021. Google Scholar
[20]	J. W. Roh and B. K. Yi, Efficient indexing of interval time sequences, Inform. Process. Lett., 109 (2008), 1-12. doi: 10.1016/j.ipl.2008.08.003. Google Scholar
[21]	H. Ryang and U. Yun, High utility pattern mining over data streams with sliding window technique, Expert Syst. Appl., 57 (2016), 214-231. doi: 10.1016/j.eswa.2016.03.001. Google Scholar
[22]	M. J. Safari, F. A. Davani and H. Afarideh, Discrete fourier transform method for discrimination of digital scintillation pulses in mixed neutron-gamma fields, IEEE Transactions on Nuclear Science, 63 (2016), 325-332. doi: 10.1109/TNS.2016.2514400. Google Scholar
[23]	D. Schultz and B. Jain, Nonsmooth analysis and subgradient methods for averaging in dynamic time warping spaces, Pattern Recognition, 74 (2018), 340-358. doi: 10.1016/j.patcog.2017.08.012. Google Scholar
[24]	Y. Sun, J. Li and J. Liu, An improvement of symbolic aggregate approximation distance measure for time series, Neurocomputing, 102 (2014), 189-198. doi: 10.1016/j.neucom.2014.01.045. Google Scholar
[25]	M. Vafaeipour, O. Rahbari and M. A. Rosen, Application of sliding window technique for prediction of wind velocity time series, Inter. J. Energy and Environmental Engineering, 5 (2014), 105-116. doi: 10.1007/s40095-014-0105-5. Google Scholar
[26]	Y. Xue, X. Mei and Y. Zhi, Method of subway health status recognition based on time series data mining, Information Sciences, 38 (2018), 905-910. Google Scholar
[27]	G. Yuan, P. Sun and J. Zhao, A review of moving object trajectory clustering algorithms, Artificial Intelligence Review, 47 (2017), 1-22. doi: 10.1007/s10462-016-9477-7. Google Scholar
[28]	R. Yao, G. Lin and Q. Shi, Efficient dense labelling of human activity sequences from wearables using fully convolutional networks, Pattern Recognition, 78 (2017), 221-232. doi: 10.1016/j.patcog.2017.12.024. Google Scholar
[29]	S. Yue, Y. Li and Q. Yang, Comparative analysis of core loss calculation methods for magnetic materials under no sinusoidal excitations, IEEE Transactions on Magnetics, 54 (2018), 1-5. doi: 10.1109/TMAG.2018.2842064. Google Scholar
[30]	U. Yun, D. Kim, H. Ryang, G. Lee and K. Lee, Mining recent high average utility patterns based on sliding window from stream data, J. Intelligent and Fuzzy Syst., 30 (2016), 3605-3617. doi: 10.3233/IFS-162106. Google Scholar
[31]	U. Yun, D. Kim, E. Yoon and H. Fujita, Damped window based high average utility pattern mining over data streams, Knowl.-Based Syst., 144 (2018), 188-205. doi: 10.1016/j.knosys.2017.12.029. Google Scholar
[32]	U. Yun and G. Lee, Sliding window based weighted erasable stream pattern mining for stream data applications, Future Generation Comp. Syst., 59 (2016), 1-20. doi: 10.1016/j.future.2015.12.012. Google Scholar
[33]	U. Yun, G. Lee and K. Ryu, Mining maximal frequent patterns by considering weight conditions over data streams, Knowl.-Based Syst., 55 (2014), 49-65. doi: 10.1016/j.knosys.2013.10.011. Google Scholar
[34]	U. Yun, G. Lee and E. Yoon, Efficient high utility pattern mining for establishing manufacturing plans with sliding window control, IEEE Trans. Industrial Electronics, 64 (2017), 7239-7249. doi: 10.1109/TIE.2017.2682782. Google Scholar
[35]	M. Zhu, D. G. M. Mitchell and M. Lentmaier, Braided convolutional codes with sliding window decoding, IEEE Trans. on Communications, 65 (2017), 3645-3658. doi: 10.1109/TCOMM.2017.2707073. Google Scholar

Figure 1. Distance between three series

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Figure 2. The sliding window principle

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Figure 3. The normalized state of 6 types time serie

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Figure 4. Quadratic distribution of window size

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Figure 5. Steplength range

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Figure 6. Time series of the three generators

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Figure 7. Clustering result comparison

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Figure 8. Clustering error rate with different weight coefficients

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Figure 9. Precision of five methods on5 data sets

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Figure 10. Clustering error rate with different weight coefficients

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Figure 11. Precision of five methods on5 data sets

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Figure 12. Runtime of five methods on 5 data sets

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Table 1. Dataset attribute

Type	Items	Status
1	1-100	Normal
2	101-200	Cyclic
3	201-300	Increasing trend
4	301-400	Decreasing trend
5	401-500	Upward shift
6	501-600	Downward shift

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Type	Items	Status
1	1-100	Normal
2	101-200	Cyclic
3	201-300	Increasing trend
4	301-400	Decreasing trend
5	401-500	Upward shift
6	501-600	Downward shift

Table 2. the combination value of $ {w_s} $ and $ {L_s} $ and the corresponding distance $ {D_T} $

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
2	49.1	-	-	-	-	-	-	-	-	-	-	-	-	-	-
3	55.4	-	-	-	-	-	-	-	-	-	-	-	-	-	-
4	54.8	60.6	-	-	-	-	-	-	-	-	-	-	-	-	-
5	58.7	61.2	-	-	-	-	-	-	-	-	-	-	-	-	-
6	64.3	65.3	60.0	-	-	-	-	-	-	-	-	-	-	-	-
7	62.5	63.9	61.3	-	-	-	-	-	-	-	-	-	-	-	-
8	64.8	65.6	64.0	62.6	-	-	-	-	-	-	-	-	-	-	-
9	62.5	61.2	63.5	62.3	-	-	-	-	-	-	-	-	-	-	-
10	63.4	63.9	59.1	62.9	60.8	-	-	-	-	-	-	-	-	-	-
11	58.0	61.9	62.1	58.5	60.1	-	-	-	-	-	-	-	-	-	-
12	61.3	61.2	58.3	61.1	60.9	59.1	-	-	-	-	-	-	-	-	-
13	60.2	59.9	59.1	49.3	58.2	49.3	-	-	-	-	-	-	-	-	-
14	49.9	59.6	60.2	60.8	59.1	60.1	61.7	-	-	-	-	-	-	-	-
15	55.9	56.2	53.2	48.2	60.2	49.2	58.7	-	-	-	-	-	-	-	-
16	60.9	55.2	58.3	58.0	52.1	50.0	56.1	49.0	-	-	-	-	-	-	-
17	49.2	53.1	54.4	55.2	60.8	59.4	51.7	53.3	-	-	-	-	-	-	-
18	53.7	54.4	52.0	52.3	52.8	60.1	59.2	49.2	49.2	-	-	-	-	-	-
19	58.3	60.8	55.1	50.3	58.7	58.0	58.1	53.0	51.9	-	-	-	-	-	-
20	50.7	53.3	55.8	50.1	42.3	45.7	50.0	51.2	51.2	46.3	-	-	-	-	-
21	49.6	50.2	46.9	46.5	47.2	47.1	47.5	46.0	48.2	43.9	-	-	-	-	-
22	51.3	49.9	48.2	46.7	50.0	45.0	40.0	39.2	39.5	40.7	39.1	-	-	-	-
23	39.9	54.2	50.0	49.8	49.0	36.8	36.5	38.1	42.5	43.9	35.9	-	-	-	-
24	46.8	51.9	48.3	45.0	38.2	42.7	39.4	50.2	41.9	38.4	43.3	51.8	-	-	-
25	51.6	40.0	58.7	43.1	40.0	39.4	35.0	45.3	45.9	41.2	38.1	40.9	-	-	-
26	47.3	48.6	50.3	39.6	42.6	55.2	42.0	36.1	35.0	42.0	43.8	39.5	47.8	-	-
27	47.5	51.3	40.0	41.6	39.5	35.0	50.0	49.2	39.4	38.4	35.6	39.2	49.9	-	-
28	45.0	40.4	38.4	35.0	35.7	46.2	50.6	45.2	39.1	39.6	42.1	48.2	40.0	38.9	-
29	50.1	48.3	40.2	41.6	35.9	36.1	40.3	39.4	50.1	46.3	39.6	35.9	35.9	35.0	-
30	42.2	49.8	45.0	39.2	40.0	38.9	40.4	39.3	37.5	38.6	36.3	36.9	35.0	36.2	35.2

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	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
2	49.1	-	-	-	-	-	-	-	-	-	-	-	-	-	-
3	55.4	-	-	-	-	-	-	-	-	-	-	-	-	-	-
4	54.8	60.6	-	-	-	-	-	-	-	-	-	-	-	-	-
5	58.7	61.2	-	-	-	-	-	-	-	-	-	-	-	-	-
6	64.3	65.3	60.0	-	-	-	-	-	-	-	-	-	-	-	-
7	62.5	63.9	61.3	-	-	-	-	-	-	-	-	-	-	-	-
8	64.8	65.6	64.0	62.6	-	-	-	-	-	-	-	-	-	-	-
9	62.5	61.2	63.5	62.3	-	-	-	-	-	-	-	-	-	-	-
10	63.4	63.9	59.1	62.9	60.8	-	-	-	-	-	-	-	-	-	-
11	58.0	61.9	62.1	58.5	60.1	-	-	-	-	-	-	-	-	-	-
12	61.3	61.2	58.3	61.1	60.9	59.1	-	-	-	-	-	-	-	-	-
13	60.2	59.9	59.1	49.3	58.2	49.3	-	-	-	-	-	-	-	-	-
14	49.9	59.6	60.2	60.8	59.1	60.1	61.7	-	-	-	-	-	-	-	-
15	55.9	56.2	53.2	48.2	60.2	49.2	58.7	-	-	-	-	-	-	-	-
16	60.9	55.2	58.3	58.0	52.1	50.0	56.1	49.0	-	-	-	-	-	-	-
17	49.2	53.1	54.4	55.2	60.8	59.4	51.7	53.3	-	-	-	-	-	-	-
18	53.7	54.4	52.0	52.3	52.8	60.1	59.2	49.2	49.2	-	-	-	-	-	-
19	58.3	60.8	55.1	50.3	58.7	58.0	58.1	53.0	51.9	-	-	-	-	-	-
20	50.7	53.3	55.8	50.1	42.3	45.7	50.0	51.2	51.2	46.3	-	-	-	-	-
21	49.6	50.2	46.9	46.5	47.2	47.1	47.5	46.0	48.2	43.9	-	-	-	-	-
22	51.3	49.9	48.2	46.7	50.0	45.0	40.0	39.2	39.5	40.7	39.1	-	-	-	-
23	39.9	54.2	50.0	49.8	49.0	36.8	36.5	38.1	42.5	43.9	35.9	-	-	-	-
24	46.8	51.9	48.3	45.0	38.2	42.7	39.4	50.2	41.9	38.4	43.3	51.8	-	-	-
25	51.6	40.0	58.7	43.1	40.0	39.4	35.0	45.3	45.9	41.2	38.1	40.9	-	-	-
26	47.3	48.6	50.3	39.6	42.6	55.2	42.0	36.1	35.0	42.0	43.8	39.5	47.8	-	-
27	47.5	51.3	40.0	41.6	39.5	35.0	50.0	49.2	39.4	38.4	35.6	39.2	49.9	-	-
28	45.0	40.4	38.4	35.0	35.7	46.2	50.6	45.2	39.1	39.6	42.1	48.2	40.0	38.9	-
29	50.1	48.3	40.2	41.6	35.9	36.1	40.3	39.4	50.1	46.3	39.6	35.9	35.9	35.0	-
30	42.2	49.8	45.0	39.2	40.0	38.9	40.4	39.3	37.5	38.6	36.3	36.9	35.0	36.2	35.2

Table 3. 5 Groups Dataset

Dataset	Samples	Categories	Attributes
temperature	148	3	2
pressure	169	4	12
position	327	10	17
concentration	112	6	16
flow rate	236	5	7

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Dataset	Samples	Categories	Attributes
temperature	148	3	2
pressure	169	4	12
position	327	10	17
concentration	112	6	16
flow rate	236	5	7

Table 4. Cross-validation results

Dataset	The optimal value			Average precision
Dataset	$ {w_s} $	$ {w_n} $	$ {w_p} $	test set	training set
temperature	8	0.6	0.4	$ {\rm{90\% }} $	$ {\rm{92\% }} $
pressure	9	0.6	0.4	$ {\rm{89\% }} $	$ {\rm{91\% }} $
position	8	0.6	0.4	$ {\rm{91\% }} $	$ {\rm{92\% }} $
concentration	8	0.6	0.4	$ {\rm{90\% }} $	$ {\rm{91\% }} $
flow rate	8	0.6	0.4	$ {\rm{90\% }} $	$ {\rm{92\% }} $

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Dataset	The optimal value			Average precision
Dataset	$ {w_s} $	$ {w_n} $	$ {w_p} $	test set	training set
temperature	8	0.6	0.4	$ {\rm{90\% }} $	$ {\rm{92\% }} $
pressure	9	0.6	0.4	$ {\rm{89\% }} $	$ {\rm{91\% }} $
position	8	0.6	0.4	$ {\rm{91\% }} $	$ {\rm{92\% }} $
concentration	8	0.6	0.4	$ {\rm{90\% }} $	$ {\rm{91\% }} $
flow rate	8	0.6	0.4	$ {\rm{90\% }} $	$ {\rm{92\% }} $

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American Institute of Mathematical Sciences

A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size

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A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size

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