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doi: 10.3934/bdia.2017018

Fuzzy temporal meta-clustering of financial trading volatility patterns

Pawan Lingras , , Farhana Haider and  Matt Triff ,

Department of Mathematics & Computing Science, Saint Mary's University, Halifax, Nova Scotia, B3H3C3, Canada

 

Corresponding author:Pawan Lingras and Matt Triff

Received  July 2017 Revised  March 2018 Published  March 2018

A volatile trading pattern on a given day in a financial market presents an opportunity for traders to maximize the difference between their buying and selling prices. In order to formulate trading strategies it may be advantageous to study typical trading patterns. This paper first describes how clustering can be used to profile typical volatile trading patterns. Fuzzy c-means provides a better description of individual trading patterns, since they can display certain aspects of different trading profiles. While daily volatility profile is a useful indicator for trading a stock, the volatility history is also an important part of the decision making process. This paper further proposes a fuzzy temporal meta-clustering algorithm that not only captures the daily volatility but also puts it in a historical perspective by including the volatility of previous two weeks in the meta-profile.

Keywords: Meta-clustering, fuzzy c-means, financial time series, volatility, trading patterns.
Mathematics Subject Classification: Primary: 91C20, 62-07; Secondary: 91B84.
Citation: Pawan Lingras, Farhana Haider, Matt Triff. Fuzzy temporal meta-clustering of financial trading volatility patterns. Big Data & Information Analytics, doi: 10.3934/bdia.2017018
References:
[1]

J. C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Kluwer Academic Publishers, Norwell, MA, USA, 1981.

[2]

F. Black and M. Scholes, The pricing of options and corporate liabilities, The journal of political economy, 81 (1973), 637-654. doi: 10.1086/260062.

[3]

R. Caruana, M. Elhaway, N. Nguyen and C. Smith, Meta clustering, in Data Mining, 2006. ICDM’06. Sixth International Conference on, IEEE, 2006, 107-118. doi: 10.1109/ICDM.2006.103.

[4]

G. Castellano, A. M. Fanelli and C. Mencar, Generation of interpretable fuzzy granules by a double-clustering technique, Archives of Control Science, 12 (2002), 397-410.

[5]

J. C. Dunn, A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters, Cybernetics, 3 (1973), 32-57. doi: 10.1080/01969727308546046.

[6]

R. El-Yaniv and O. Souroujon, Iterative double clustering for unsupervised and semi-supervised learning, Machine Learning: ECML 2001, Springer, 2001 (2001), 121-132.

[7]

D. Gnatyshak, D. I. Ignatov, A. Semenov and J. Poelmans, Gaining insight in social networks with biclustering and triclustering, Perspectives in Business Informatics Research, Springer, (2012), 162-171.

[8]

D. V. Gnatyshak, D. I. Ignatov and S. O. Kuznetsov, From triadic fca to triclustering: Experimental comparison of some triclustering algorithms, CLA 2013, p249.
[9]

M. Halkidi, Y. Batistakis and M. Vazirgianni, Clustering validity checking methods: Part Ⅱ, ACM SIGMOD Record, 31 (2002), 19-27. doi: 10.1145/601858.601862.

[10]

J. A. Hartigan and M. A. Wong, Algorithm AS136: A K-Means Clustering Algorithm, Applied Statistics, 28 (1979), 100-108.

[11]

D. I. Ignatov, S. O. Kuznetsov and J. Poelmans, Concept-based biclustering for internet advertisement, in Data Mining Workshops (ICDMW), 2012 IEEE 12th International Conference on, IEEE, 2012, 123-130. doi: 10.1109/ICDMW.2012.100.

[12]

D. I. Ignatov, S. O. Kuznetsov, J. Poelmans and L. E. Zhukov, Can triconcepts become triclusters?, International Journal of General Systems, 42 (2013), 572-593. doi: 10.1080/03081079.2013.798899.

[13]

P. Lingras and K. Rathinavel, Recursive Meta-clustering in a Granular Network, in Plenary talk at the Fourth International Conference of Soft Computing and Pattern Recognition, Brunei, 2012. doi: 10.1109/ISDA.2012.6416634.

[14]

P. Lingras and M. Triff, Fuzzy and crisp recursive profiling of online reviewers and businesses, IEEE Transactions on Fuzzy Systems, 23 (2015), 1242-1258. doi: 10.1109/TFUZZ.2014.2349532.

[15]

P. Lingras, A. Elagamy, A. Ammar and Z. Elouedi, Iterative meta-clustering through granular hierarchy of supermarket customers and products, Information Sciences, 257 (2014), 14-31. doi: 10.1016/j.ins.2013.09.018.

[16]

P. Lingras and F. Haider, Recursive temporal meta-clustering, Applied Soft Computing, submitted.
[17]

P. Lingras and K. Rathinavel, Recursive meta-clustering in a granular network, in Intelligent Systems Design and Applications (ISDA), 2012 12th International Conference on, IEEE, 2012, 770-775. doi: 10.1109/ISDA.2012.6416634.

[18]

J. MacQueen, Some methods for classification and analysis of multivariate observations, in Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, 1 (1967), 281-297.

[19]

B. Mirkin, Mathematical Classification and Clustering, Kluwer Academic Publishers, Boston, MA, USA, 1996.

[20]

W. Pedrycz and J. Waletzky, Fuzzy clustering with partial supervision, IEEE Transactions on Systems, Man, and Cybernetics, Part B, 27 (1997), 787-795. doi: 10.1109/3477.623232.

[21]

D. Ramirez-Cano, S. Colton and R. Baumgarten, Player classification using a meta-clustering approach, in Proceedings of the 3rd Annual International Conference Computer Games, Multimedia and Allied Technology, 2010, 297-304.
[22]

K. Rathinavel and P. Lingras, A granular recursive fuzzy meta-clustering algorithm for social networks, in IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), 2013 Joint, IEEE, 2013, 567-572. doi: 10.1109/IFSA-NAFIPS.2013.6608463.

[23]

N. Slonim and N. Tishby, Document clustering using word clusters via the information bottleneck method, in 23rd Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, 2000, 208-215. doi: 10.1145/345508.345578.

[24]

M. Triff and P. Lingras, Recursive profiles of businesses and reviewers on yelp.com, Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, Springer, (2013), 325–336.

show all references

References:
[1]

J. C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Kluwer Academic Publishers, Norwell, MA, USA, 1981.

[2]

F. Black and M. Scholes, The pricing of options and corporate liabilities, The journal of political economy, 81 (1973), 637-654. doi: 10.1086/260062.

[3]

R. Caruana, M. Elhaway, N. Nguyen and C. Smith, Meta clustering, in Data Mining, 2006. ICDM’06. Sixth International Conference on, IEEE, 2006, 107-118. doi: 10.1109/ICDM.2006.103.

[4]

G. Castellano, A. M. Fanelli and C. Mencar, Generation of interpretable fuzzy granules by a double-clustering technique, Archives of Control Science, 12 (2002), 397-410.

[5]

J. C. Dunn, A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters, Cybernetics, 3 (1973), 32-57. doi: 10.1080/01969727308546046.

[6]

R. El-Yaniv and O. Souroujon, Iterative double clustering for unsupervised and semi-supervised learning, Machine Learning: ECML 2001, Springer, 2001 (2001), 121-132.

[7]

D. Gnatyshak, D. I. Ignatov, A. Semenov and J. Poelmans, Gaining insight in social networks with biclustering and triclustering, Perspectives in Business Informatics Research, Springer, (2012), 162-171.

[8]

D. V. Gnatyshak, D. I. Ignatov and S. O. Kuznetsov, From triadic fca to triclustering: Experimental comparison of some triclustering algorithms, CLA 2013, p249.
[9]

M. Halkidi, Y. Batistakis and M. Vazirgianni, Clustering validity checking methods: Part Ⅱ, ACM SIGMOD Record, 31 (2002), 19-27. doi: 10.1145/601858.601862.

[10]

J. A. Hartigan and M. A. Wong, Algorithm AS136: A K-Means Clustering Algorithm, Applied Statistics, 28 (1979), 100-108.

[11]

D. I. Ignatov, S. O. Kuznetsov and J. Poelmans, Concept-based biclustering for internet advertisement, in Data Mining Workshops (ICDMW), 2012 IEEE 12th International Conference on, IEEE, 2012, 123-130. doi: 10.1109/ICDMW.2012.100.

[12]

D. I. Ignatov, S. O. Kuznetsov, J. Poelmans and L. E. Zhukov, Can triconcepts become triclusters?, International Journal of General Systems, 42 (2013), 572-593. doi: 10.1080/03081079.2013.798899.

[13]

P. Lingras and K. Rathinavel, Recursive Meta-clustering in a Granular Network, in Plenary talk at the Fourth International Conference of Soft Computing and Pattern Recognition, Brunei, 2012. doi: 10.1109/ISDA.2012.6416634.

[14]

P. Lingras and M. Triff, Fuzzy and crisp recursive profiling of online reviewers and businesses, IEEE Transactions on Fuzzy Systems, 23 (2015), 1242-1258. doi: 10.1109/TFUZZ.2014.2349532.

[15]

P. Lingras, A. Elagamy, A. Ammar and Z. Elouedi, Iterative meta-clustering through granular hierarchy of supermarket customers and products, Information Sciences, 257 (2014), 14-31. doi: 10.1016/j.ins.2013.09.018.

[16]

P. Lingras and F. Haider, Recursive temporal meta-clustering, Applied Soft Computing, submitted.
[17]

P. Lingras and K. Rathinavel, Recursive meta-clustering in a granular network, in Intelligent Systems Design and Applications (ISDA), 2012 12th International Conference on, IEEE, 2012, 770-775. doi: 10.1109/ISDA.2012.6416634.

[18]

J. MacQueen, Some methods for classification and analysis of multivariate observations, in Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, 1 (1967), 281-297.

[19]

B. Mirkin, Mathematical Classification and Clustering, Kluwer Academic Publishers, Boston, MA, USA, 1996.

[20]

W. Pedrycz and J. Waletzky, Fuzzy clustering with partial supervision, IEEE Transactions on Systems, Man, and Cybernetics, Part B, 27 (1997), 787-795. doi: 10.1109/3477.623232.

[21]

D. Ramirez-Cano, S. Colton and R. Baumgarten, Player classification using a meta-clustering approach, in Proceedings of the 3rd Annual International Conference Computer Games, Multimedia and Allied Technology, 2010, 297-304.
[22]

K. Rathinavel and P. Lingras, A granular recursive fuzzy meta-clustering algorithm for social networks, in IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), 2013 Joint, IEEE, 2013, 567-572. doi: 10.1109/IFSA-NAFIPS.2013.6608463.

[23]

N. Slonim and N. Tishby, Document clustering using word clusters via the information bottleneck method, in 23rd Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, 2000, 208-215. doi: 10.1145/345508.345578.

[24]

M. Triff and P. Lingras, Recursive profiles of businesses and reviewers on yelp.com, Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, Springer, (2013), 325–336.
Figure 1.  Cluster Scatter
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Figure 2.  DB Index
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Figure 3.  Centroids of 5 Clusters after Ranking
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Figure 4.  Average Chronological Daily Patterns
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Figure 5.  Fuzzy Centroids of 5 Clusters after Ranking
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Figure 6.  Flowchart of Recursive Meta-clustering
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Figure 7.  Fuzzy Temporal Meta-clustering Algorithm
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Figure 8.  Ranks in Final Temporal Cluster
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Figure 9.  Ranks of day 2012-01-12 and last 10 days of Instrument Z_2
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Figure 10.  Ranks of day 2011-10-03 and last 10 days of Instrument 3_1
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Figure 11.  Ranks of day 2011-12-16 and last 10 days of Instrument A_10
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Figure 12.  Ranks of day 2011-08-16 and last 10 days of Instrument 3_1
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Figure 13.  Ranks of day 2012-01-04 and last 10 days of Instrument A_10
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Figure 14.  Ranks of day 2011-11-01 and last 10 days of Instrument A_113
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Table 1.  Calculation of Percentiles for a Sample Record
Percentile 10% 25% 50% 75% 90%
Percentile of avgp (avgpPerc) 0.9841346 0.9873798 0.9927885 0.9951923 0.9966346
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Percentile 10% 25% 50% 75% 90%
Percentile of avgp (avgpPerc) 0.9841346 0.9873798 0.9927885 0.9951923 0.9966346
Table 2.  Crisp Cluster Cardinalities
Cluster number 1 2 3 4 5
Percentile values 14125 8676 3349 817 45
Black Scholes 14182 8990 3061 684 95
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Cluster number 1 2 3 4 5
Percentile values 14125 8676 3349 817 45
Black Scholes 14182 8990 3061 684 95
Table 3.  Cluster Intersections
cdvr1 cdvr2 cdvr3 cdvr4 cdvr5
cpr1 10430 3104 519 67 5
cpr2 3411 4047 1089 123 6
cpr3 339 1727 1047 223 13
cpr4 2 112 404 258 41
cpr5 0 0 2 13 30
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cdvr1 cdvr2 cdvr3 cdvr4 cdvr5
cpr1 10430 3104 519 67 5
cpr2 3411 4047 1089 123 6
cpr3 339 1727 1047 223 13
cpr4 2 112 404 258 41
cpr5 0 0 2 13 30
Table 4.  Fuzzy memberships for different stocks
Day:Instrument fcpri fcpr2 fcpr3 fcpr4 fcpr5 Avg Rank
2011-08-16:3_1 0.04 0.06 0.09 0.35 0.46 4.14
2011-08-17:3_1 0.85 0.13 0.03 0 0 1.19
:
2012-01-31:3_1 0.06 0.16 0.65 0.12 0.01 2.86
:
2011-08-16:Z_2 0.97 0.03 0.01 0 0 1.04
:
2012-01-31:Z_2 0.93 0.05 0.01 0 0 1.09
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Day:Instrument fcpri fcpr2 fcpr3 fcpr4 fcpr5 Avg Rank
2011-08-16:3_1 0.04 0.06 0.09 0.35 0.46 4.14
2011-08-17:3_1 0.85 0.13 0.03 0 0 1.19
:
2012-01-31:3_1 0.06 0.16 0.65 0.12 0.01 2.86
:
2011-08-16:Z_2 0.97 0.03 0.01 0 0 1.04
:
2012-01-31:Z_2 0.93 0.05 0.01 0 0 1.09
Table 5.  Static Part of Percentile Data
Day:Instrument p10 p25 p50 p75 p90
2011-08-16:3_1 0 0.28 0.56 0.67 0.78
2011-08-17:3_1 0 0 0.04 0.09 0.11
:
2012-01-31:3_1 0 0 0.15 0.29 0.46
:
2011-08-16:Z_2 0 0.027 0.045 0.05 0.05
:
2012-01-31:Z_2 0 0.01 0.019 0.03 0.11
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Day:Instrument p10 p25 p50 p75 p90
2011-08-16:3_1 0 0.28 0.56 0.67 0.78
2011-08-17:3_1 0 0 0.04 0.09 0.11
:
2012-01-31:3_1 0 0 0.15 0.29 0.46
:
2011-08-16:Z_2 0 0.027 0.045 0.05 0.05
:
2012-01-31:Z_2 0 0.01 0.019 0.03 0.11
Table 6.  Ranked Clusters for Percentile Data after first iteration
Centers
Rank Cluster p10 p25 p50 p75 p90
1 C2 0 0.02 0.03 0.06 0.08
2 C5 0 0.05 0.10 0.16 0.21
3 C4 0 0.09 0.19 0.28 0.35
4 C1 0 0.16 0.35 0.48 0.57
5 C3 0 0.30 0.66 0.88 1.00
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Centers
Rank Cluster p10 p25 p50 p75 p90
1 C2 0 0.02 0.03 0.06 0.08
2 C5 0 0.05 0.10 0.16 0.21
3 C4 0 0.09 0.19 0.28 0.35
4 C1 0 0.16 0.35 0.48 0.57
5 C3 0 0.30 0.66 0.88 1.00
Table 7.  Dynamic Part after first iteration
Daym+1:Instrument dm-9 dm-8 dm-7 dm-6 dm-5 dm-4 dm-3 dm-2 dm-1 dm
2011-08-16:3_1 2.69 2.69 2.70 2.70 2.69 2.70 2.68 2.70 2.70 2.69
2011-08-17:3_1 2.69 2.70 2.70 2.69 2.70 2.68 2.70 2.70 2.69 4.14
:
2012-01-31:3_1 1.07 2.10 3.78 1.25 1.81 3.58 4.06 1.09 1.42 3.56
:
2011-08-16:Z_2 2.69 2.70 2.70 2.70 2.70 2.70 2.70 2.70 2.70 2.69
:
2012-01-31:Z_2 1.09 2.90 2.89 1.15 3.04 1.87 2.00 3.01 2.05 1.71
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Daym+1:Instrument dm-9 dm-8 dm-7 dm-6 dm-5 dm-4 dm-3 dm-2 dm-1 dm
2011-08-16:3_1 2.69 2.69 2.70 2.70 2.69 2.70 2.68 2.70 2.70 2.69
2011-08-17:3_1 2.69 2.70 2.70 2.69 2.70 2.68 2.70 2.70 2.69 4.14
:
2012-01-31:3_1 1.07 2.10 3.78 1.25 1.81 3.58 4.06 1.09 1.42 3.56
:
2011-08-16:Z_2 2.69 2.70 2.70 2.70 2.70 2.70 2.70 2.70 2.70 2.69
:
2012-01-31:Z_2 1.09 2.90 2.89 1.15 3.04 1.87 2.00 3.01 2.05 1.71
Table 8.  Concatenated Static Part(SP) and Dynamic Part(DP) after first iteration
SP DP
Day:Instrument p10 p25 p50 p75 p90 dm-9 dm-8 dm-7 dm-6 dm-5 dm-4 dm-3 dm-2 dm-1 dm
2011-08-16:3_1 0 0.28 0.56 0.67 0.78 2.69 2.69 2.70 2.70 2.69 2.70 2.68 2.70 2.70 2.70
2011-08-17:3_1 0 0 0.04 0.09 0.11 2.69 2.70 2.70 2.69 2.70 2.68 2.70 2.70 2.69 4.14
:
2012-01-31:3_1 0 0 0.15 0.29 0.46 1.07 2.10 3.78 1.25 1.81 3.58 4.06 1.09 1.42 3.56
:
2011-08-16:Z_2 0 0.03 0.045 0.05 0.05 2.69 2.70 2.70 2.70 2.70 2.70 2.70 2.70 2.70 2.69
:
2012-01-31:Z_2 0 0.01 0.02 0.03 0.11 1.09 2.90 2.89 1.15 3.04 1.87 2.00 3.01 2.05 1.71
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SP DP
Day:Instrument p10 p25 p50 p75 p90 dm-9 dm-8 dm-7 dm-6 dm-5 dm-4 dm-3 dm-2 dm-1 dm
2011-08-16:3_1 0 0.28 0.56 0.67 0.78 2.69 2.69 2.70 2.70 2.69 2.70 2.68 2.70 2.70 2.70
2011-08-17:3_1 0 0 0.04 0.09 0.11 2.69 2.70 2.70 2.69 2.70 2.68 2.70 2.70 2.69 4.14
:
2012-01-31:3_1 0 0 0.15 0.29 0.46 1.07 2.10 3.78 1.25 1.81 3.58 4.06 1.09 1.42 3.56
:
2011-08-16:Z_2 0 0.03 0.045 0.05 0.05 2.69 2.70 2.70 2.70 2.70 2.70 2.70 2.70 2.70 2.69
:
2012-01-31:Z_2 0 0.01 0.02 0.03 0.11 1.09 2.90 2.89 1.15 3.04 1.87 2.00 3.01 2.05 1.71
Table 9.  Cluster Centers after clustering with Concatenated Profile
SP DP
Rank Cluster p10 p25 p50 p75 p90 dm-9 dm-8 dm-7 dm-6 dm-5 dm-4 dm-3 dm-2 dm-1 dm
1 C5 0 0.0530 0.1123 0.1720 0.2227 1.9925 1.9849 1.9812 1.9698 1.9645 1.9569 1.9539 1.9481 1.9409 1.9376
2 C2 0 0.0531 0.1124 0.1721 0.2227 1.9933 1.9857 1.9820 1.9706 1.9653 1.9576 1.9546 1.9488 1.9415 1.9382
3 C4 0 0.0531 0.1124 0.1721 0.2228 1.9937 1.9861 1.9824 1.9710 1.9657 1.9581 1.9550 1.9492 1.9419 1.9386
4 C1 0 0.0531 0.1124 0.1722 0.2229 1.9943 1.9867 1.9830 1.9716 1.9663 1.9587 1.9556 1.9498 1.9424 1.9391
5 C3 0 0.0532 0.1124 0.1722 0.2229 1.9946 1.9871 1.9834 1.9720 1.9666 1.9590 1.9501 1.9427 1.9559 1.9393
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SP DP
Rank Cluster p10 p25 p50 p75 p90 dm-9 dm-8 dm-7 dm-6 dm-5 dm-4 dm-3 dm-2 dm-1 dm
1 C5 0 0.0530 0.1123 0.1720 0.2227 1.9925 1.9849 1.9812 1.9698 1.9645 1.9569 1.9539 1.9481 1.9409 1.9376
2 C2 0 0.0531 0.1124 0.1721 0.2227 1.9933 1.9857 1.9820 1.9706 1.9653 1.9576 1.9546 1.9488 1.9415 1.9382
3 C4 0 0.0531 0.1124 0.1721 0.2228 1.9937 1.9861 1.9824 1.9710 1.9657 1.9581 1.9550 1.9492 1.9419 1.9386
4 C1 0 0.0531 0.1124 0.1722 0.2229 1.9943 1.9867 1.9830 1.9716 1.9663 1.9587 1.9556 1.9498 1.9424 1.9391
5 C3 0 0.0532 0.1124 0.1722 0.2229 1.9946 1.9871 1.9834 1.9720 1.9666 1.9590 1.9501 1.9427 1.9559 1.9393
Table 10.  Final Ranked Centers for Percentile Data
Rank Cluster p10 p25 p50 p75 p90 dm-9 dm-8 dm-7 dm-6 dm-5 dm-4 dm-3 dm-2 dm-1 dm
1 C2 0 0.04 0.08 0.12 0.15 1.20 1.17 1.14 1.12 1.11 1.10 1.10 1.11 1.13 1.15
2 C4 0 0.05 0.10 0.15 0.19 2.24 2.20 2.16 2.14 2.11 2.10 2.10 2.11 2.12 2.14
3 C3 0 0.05 0.10 0.16 0.21 3.04 3.03 3.03 3.03 3.02 3.02 3.02 3.02 3.03 3.03
4 C1 0 0.05 0.11 0.17 0.22 3.82 3.86 3.89 3.92 3.94 3.95 3.97 3.98 3.99 3.99
5 C5 0 0.07 0.14 0.21 0.27 4.70 4.75 4.78 4.81 4.83 4.84 4.83 4.82 4.79 4.76
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Rank Cluster p10 p25 p50 p75 p90 dm-9 dm-8 dm-7 dm-6 dm-5 dm-4 dm-3 dm-2 dm-1 dm
1 C2 0 0.04 0.08 0.12 0.15 1.20 1.17 1.14 1.12 1.11 1.10 1.10 1.11 1.13 1.15
2 C4 0 0.05 0.10 0.15 0.19 2.24 2.20 2.16 2.14 2.11 2.10 2.10 2.11 2.12 2.14
3 C3 0 0.05 0.10 0.16 0.21 3.04 3.03 3.03 3.03 3.02 3.02 3.02 3.02 3.03 3.03
4 C1 0 0.05 0.11 0.17 0.22 3.82 3.86 3.89 3.92 3.94 3.95 3.97 3.98 3.99 3.99
5 C5 0 0.07 0.14 0.21 0.27 4.70 4.75 4.78 4.81 4.83 4.84 4.83 4.82 4.79 4.76
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