# American Institute of Mathematical Sciences

doi: 10.3934/bdia.2017018

## Fuzzy temporal meta-clustering of financial trading volatility patterns

 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.

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
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##### References:
Cluster Scatter
DB Index
Centroids of 5 Clusters after Ranking
Average Chronological Daily Patterns
Fuzzy Centroids of 5 Clusters after Ranking
Flowchart of Recursive Meta-clustering
Fuzzy Temporal Meta-clustering Algorithm
Ranks in Final Temporal Cluster
Ranks of day 2012-01-12 and last 10 days of Instrument Z_2
Ranks of day 2011-10-03 and last 10 days of Instrument 3_1
Ranks of day 2011-12-16 and last 10 days of Instrument A_10
Ranks of day 2011-08-16 and last 10 days of Instrument 3_1
Ranks of day 2012-01-04 and last 10 days of Instrument A_10
Ranks of day 2011-11-01 and last 10 days of Instrument A_113
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
 Percentile 10% 25% 50% 75% 90% Percentile of avgp (avgpPerc) 0.9841346 0.9873798 0.9927885 0.9951923 0.9966346
Crisp Cluster Cardinalities
 Cluster number 1 2 3 4 5 Percentile values 14125 8676 3349 817 45 Black Scholes 14182 8990 3061 684 95
 Cluster number 1 2 3 4 5 Percentile values 14125 8676 3349 817 45 Black Scholes 14182 8990 3061 684 95
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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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