Hierarchical time series clustering on tail dependence with linkage based on a multivariate copula approach. (English) Zbl 1520.62086
Summary: Time series clustering with a dissimilarity matrix based on tail dependence coefficients estimated by copula functions has been proposed in 2011 by the authors, who used a two-step procedure allowing to resort to the \(k\)-means algorithm. The possibility to carry out hierarchical clustering directly on the dissimilarity matrix is still an open issue and the main concerns are relative to the meaning of the most common linkage methods in the context of tail dependence. In this paper, in a multivariate copula approach, we propose a linkage method based on the tail dependence coefficients between the clusters that are agglomerated at each iteration of the hierarchical clustering algorithms.
MSC:
| 62H30 | Classification and discrimination; cluster analysis (statistical aspects) |
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
Keywords:
time series clustering; tail dependence; multivariate copula functions; hierarchical copula functions; hierarchical clustering; linkageReferences:
| [1] | Aghabozorgi, Saeed; Shirkhorshidi, Ali Seyed; Wah, Teh Ying, Time-series clustering-a decade review, Inf. Syst., 53, 16-38 (2015) |
| [2] | Batagelj, V., Note on ultrametric hierarchical clustering algorithms, Psychometrika, 46, 351-352 (1981) |
| [3] | Bauwens, Luc; Laurent, Sébastien, A new class of multivariate skew densities, with application to garch models, J. Bus. Econ. Stat., 23, 346-354 (2005) |
| [4] | Brun, Marcel; Sima, Chao; Hua, Jianping; Lowey, James; Carroll, Brent; Suh, Edward; Dougherty, Edward R., Model-based evaluation of clustering validation measures, Pattern Recognit., 40, 3, 807-824 (2007) · Zbl 1118.68132 |
| [5] | Bruynooghe, Michel, Méthodes nouvelles en classification automatique de données taxinomiques nombreuses, Stat. Anal. Données, 2, 3, 24-42 (1977) |
| [6] | Caliński, Tadeusz; Harabasz, Jerzy, A dendrite method for cluster analysis, Commun. Stat., Theory Methods, 3, 1, 1-27 (1974) · Zbl 0273.62010 |
| [7] | De Luca, Giovanni; Rivieccio, Giorgia, Multivariate tail dependence coefficients for Archimedean copulae, (Advanced Statistical Methods for the Analysis of Large Data-Sets, Studies in Theoretical and Applied Statistics (2012), Springer) · Zbl 1473.62350 |
| [8] | De Luca, Giovanni; Zuccolotto, Paola, A tail dependence-based dissimilarity measure for financial time series clustering, Adv. Data Anal. Classif., 5, 4, 323-340 (2011) |
| [9] | De Luca, Giovanni; Zuccolotto, Paola, Time series clustering on lower tail dependence for portfolio selection, (Mathematical and Statistical Methods for Actuarial Sciences and Finance (2014), Springer), 131-140 · Zbl 1418.91463 |
| [10] | De Luca, Giovanni; Zuccolotto, Paola, A double clustering algorithm for financial time series based on extreme events, Stat. Risk. Model., 34 (2017) · Zbl 1362.60051 |
| [11] | De Luca, Giovanni; Zuccolotto, Paola, Dynamic tail dependence clustering of financial time series, Stat. Pap., 1-17 (2017) · Zbl 1416.62581 |
| [12] | De Luca, Giovanni; Zuccolotto, Paola, Regime dependent interconnectedness among fuzzy clusters of financial time series, Adv. Data Anal. Classif., 1-22 (2020) · Zbl 07363876 |
| [13] | Dhaene, Jan; Denuit, Michel; Goovaerts, Marc J.; Kaas, Rob; Vyncke, David, The concept of comonotonicity in actuarial science and finance: theory, Insur. Math. Econ., 31, 1, 3-33 (2002) · Zbl 1051.62107 |
| [14] | Marta, F.; Di Lascio, L.; Durante, Fabrizio; Pappada, Roberta, Copula-based clustering methods, (Copulas and Dependence Models with Applications (2017), Springer), 49-67 |
| [15] | Marta, F.; Di Lascio, L.; Giannerini, Simone, A copula-based algorithm for discovering patterns of dependent observations, J. Classif., 29, 1, 50-75 (2012) · Zbl 1360.62250 |
| [16] | Marta, F.; Di Lascio, L.; Giannerini, Simone, Clustering dependent observations with copula functions, Stat. Pap., 60, 1, 35-51 (2019) · Zbl 1411.62165 |
| [17] | Disegna, Marta; D’Urso, Pierpaolo; Durante, Fabrizio, Copula-based fuzzy clustering of spatial time series, Spat. Stat., 21, 209-225 (2017) |
| [18] | Dunn, Joseph C., Well-separated clusters and optimal fuzzy partitions, J. Cybern., 4, 1, 95-104 (1974) · Zbl 0304.68093 |
| [19] | Durante, Fabrizio; Fernandez-Sanchez, Juan; Pappada, Roberta, Copulas, diagonals, and tail dependence, Fuzzy Sets Syst., 264, 22-41 (2015) · Zbl 1360.68835 |
| [20] | Durante, Fabrizio; Foscolo, Enrico, An analysis of the dependence among financial markets by spatial contagion, Int. J. Intell. Syst., 28, 4, 319-331 (2013) |
| [21] | Durante, Fabrizio; Foscolo, Enrico; Jaworski, Piotr; Wang, Hao, A spatial contagion measure for financial time series, Expert Syst. Appl., 41, 8, 4023-4034 (2014) |
| [22] | Durante, Fabrizio; Pappadà, Roberta; Torelli, Nicola, Clustering of financial time series in risky scenarios, Adv. Data Anal. Classif., 8, 4, 359-376 (2014) · Zbl 1414.62241 |
| [23] | Durante, Fabrizio; Pappadà, Roberta; Torelli, Nicola, Clustering of time series via non-parametric tail dependence estimation, Stat. Pap., 56, 3, 701-721 (2015) · Zbl 1317.62053 |
| [24] | Embrechts, Paul; Lindskog, Filip; McNeil, Er J., Modeling dependence with copulas and applications to risk management, (Handbook of Heavy Tailed Distributions in Finance (2003), Elsevier), 320-384 |
| [25] | Fisher, Lloyd; Van Ness, John W., Admissible clustering procedures, Biometrika, 58, 1, 91-104 (1971) · Zbl 0224.62030 |
| [26] | Fuchs, Sebastian; Durante, Fabrizio; Marta, F.; Di Lascio, L., Dissimilarity functions for rank-invariant hierarchical clustering of continuous variables, Comput. Stat. Data Anal., Article 107201 pp. (2021) · Zbl 1510.62251 |
| [27] | Glosten, Lawrence R.; Jagannathan, Ravi; Runkle, David, On the relation between the expected value and the volatility of the nominal excess return on stocks, J. Finance, 48, 1779-1801 (1993) |
| [28] | Gordon, A. D., Monographs on Statistics and Applied Probability - Classification (1999), Chapman & Hall/CRC · Zbl 0929.62068 |
| [29] | Halkidi, Maria; Batistakis, Yannis; Vazirgiannis, Michalis, On clustering validation techniques, J. Intell. Inf. Syst., 17, 2, 107-145 (2001) · Zbl 0998.68154 |
| [30] | Hartigan, John A., Clustering Algorithms (1975), John Wiley & Sons, Inc. · Zbl 0372.62040 |
| [31] | Hennig, Christian; Meila, Marina; Murtagh, Fionn; Rocci, Roberto, Handbook of Cluster Analysis (2015), CRC Press · Zbl 1331.68001 |
| [32] | Hofert, Marius, Efficiently sampling nested Archimedean copulas, Comput. Stat. Data Anal., 55, 1, 57-70 (2011) · Zbl 1247.62132 |
| [33] | Hubert, Lawrence J.; Levin, Joel R., A general statistical framework for assessing categorical clustering in free recall, Psychol. Bull., 83, 6, 1072 (1976) |
| [34] | Ji, Hao; Wang, Hao; Liseo, Brunero, Portfolio diversification strategy via tail-dependence clustering and arma-garch vine copula approach, Aust. Econ. Pap., 57, 3, 265-283 (2018) |
| [35] | Jondeau, Eric; Rockinger, Michael, The copula-garch model of conditional dependencies: an international stock market application, J. Int. Money Financ., 25, 827-853 (2006) |
| [36] | Jun, Zhang; Ziping, Du, Distance measure of financial time series based on the coefficients of temporal tail dependence, Int. J. Adv. Manag. Sci., 2, 4, 143-146 (2013) |
| [37] | Kojadinovic, Ivan, Hierarchical clustering of continuous variables based on the empirical copula process and permutation linkages, Comput. Stat. Data Anal., 54, 1, 90-108 (2010) · Zbl 1284.62380 |
| [38] | Krzanowski, Wojtek J.; Lai, Y. T., A criterion for determining the number of groups in a data set using sum-of-squares clustering, Biometrics, 23-34 (1988) · Zbl 0707.62122 |
| [39] | Lafuente-Rego, Borja; D’Urso, Pierpaolo; Vilar, José A., Robust fuzzy clustering based on quantile autocovariances, Stat. Pap. (2018) · Zbl 1397.62233 |
| [40] | Lafuente-Rego, Borja; Vilar, José A., Clustering of time series using quantile autocovariances, Adv. Data Anal. Classif., 10, 3, 391-415 (2016) · Zbl 1414.62372 |
| [41] | Lance, Godfrey N.; Williams, William Thomas, A general theory of classificatory sorting strategies: 1. Hierarchical systems, Comput. J., 9, 4, 373-380 (1967) |
| [42] | Liu, Xin; Wu, Jiang; Yang, Chen; Jiang, Wenjun, A maximal tail dependence-based clustering procedure for financial time series and its applications in portfolio selection, Risks, 6, 4, 115 (2018) |
| [43] | Liu, Yanchi; Li, Zhongmou; Xiong, Hui; Gao, Xuedong; Wu, Junjie; Wu, Sen, Understanding and enhancement of internal clustering validation measures, IEEE Trans. Cybern., 43, 3, 982-994 (2013) |
| [44] | Lohre, Harald; Rother, Carsten; Schäfer, Kilian Axel, Hierarchical risk parity: accounting for tail dependencies in multi-asset multi-factor allocations, (Machine Learning for Asset ManagementNew Developments and Financial Applications (2020)), 329-368 |
| [45] | Meilă, Marina, Comparing clusterings—an information based distance, J. Multivar. Anal., 98, 5, 873-895 (2007) · Zbl 1298.91124 |
| [46] | Morgan, Byron J. T.; Ray, Andrew P. G., Non-uniqueness and inversions in cluster analysis, J. R. Stat. Soc., Ser. C, Appl. Stat., 44, 1, 117-134 (1995) · Zbl 0821.62036 |
| [47] | Murtagh, Fionn; Contreras, Pedro, Algorithms for hierarchical clustering: an overview, ii, Wiley Interdiscip. Rev. Data Min. Knowl. Discov., 7, 6, Article e1219 pp. (2017) |
| [48] | Okhrin, Ostap; Okhrin, Yarema; Schmid, Wolfgang, On the structure and estimation of hierarchical Archimedean copulas, J. Econom., 173, 2, 189-204 (2013) · Zbl 1443.62137 |
| [49] | Okhrin, Ostap; Ristig, Alexander, Hierarchical Archimedean copulae: the hac package, J. Stat. Softw., 58, 4, 1-20 (2014) |
| [50] | Eka Putra, Yogas; Saepudin, Deni; Aditsania, Annisa, Portfolio selection of kompas-100 stocks index using b-spline based clustering, Proc. Comput. Sci., 179, 375-382 (2021) |
| [51] | Rojas-Thomas, J. C.; Santos, Matilde; Mora, M., New internal index for clustering validation based on graphs, Expert Syst. Appl., 86, 334-349 (2017) |
| [52] | Rousseeuw, Peter J., Silhouettes: a graphical aid to the interpretation and validation of cluster analysis, J. Comput. Appl. Math., 20, 53-65 (1987) · Zbl 0636.62059 |
| [53] | Cornella, Savu; Trede, Mark, Hierarchies of Archimedean copulas, Quant. Finance, 10, 295-304 (2010) · Zbl 1270.91086 |
| [54] | Sklar, Abe, Fonctions de répartition án dimensions et leurs marges, Publ. Inst. Stat. Univ. Paris, 8, 229-231 (1959) · Zbl 0100.14202 |
| [55] | Tayalı, Seda Tolun, A novel backtesting methodology for clustering in mean-variance portfolio optimization, Knowl.-Based Syst., 209, Article 106454 pp. (2020) |
| [56] | Vilar, José A.; Lafuente-Rego, Borja; D’Urso, Pierpaolo, Quantile autocovariances: a powerful tool for hard and soft partitional clustering of time series, Fuzzy Sets Syst., 340, 38-72 (2017) · Zbl 1397.62233 |
| [57] | Yang, Chen; Jiang, Wenjun; Wu, Jiang; Liu, Xin; Li, Zhichuan, Clustering of financial instruments using jump tail dependence coefficient, Stat. Methods Appl., 27, 3, 491-513 (2018) · Zbl 1427.62124 |
| [58] | Yang, Han; Wang, Ming-hui; Huang, Nan-jing, The α-tail distance with an application to portfolio optimization under different market conditions, Comput. Econ., 1-30 (2020) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
