zbMATH — the first resource for mathematics

Functional fuzzy clusterwise regression analysis. (English) Zbl 1271.62138
Summary: We propose a functional extension of fuzzy clusterwise regression, which estimates fuzzy memberships of clusters and regression coefficient functions for each cluster simultaneously. The proposed method permits dependent and/or predictor variables to be functional, varying over time, space, and other continua. The fuzzy memberships and clusterwise regression coefficient functions are estimated by minimizing an objective function that adopts a basis function expansion approach to approximating functional data. An alternating least squares algorithm is developed to minimize the objective function. We conduct simulation studies to demonstrate the superior performance of the proposed method compared to its non-functional counterpart and to examine the performance of various cluster validity measures for selecting the optimal number of clusters. We apply the proposed method to real datasets to illustrate the empirical usefulness of the proposed method.

62H30 Classification and discrimination; cluster analysis (statistical aspects)
62J86 Fuzziness, and linear inference and regression
Full Text: DOI
[1] Alva K, Romo J, Ruiz E (2009) Modelling intra-daily volatility by functional data analysis: an empirical application to the Spanish stock market. Statistics and Econometrics Series, vol 9. Universidad Carlos III de Madrid, Madrid
[2] Bezdek JC (1974) Cluster validity with fuzzy sets. J Cybern 3:58–72 · Zbl 0294.68035 · doi:10.1080/01969727308546047
[3] Bezdek JC (1981) Pattern recognition with fuzzy objective function algorithms. Plenum Press, New York · Zbl 0503.68069
[4] Coppi R, Gil MA, Kiers HAL (2006) The fuzzy approach to statistical analysis. Comput Stat Data Anal 51(1):1–14 · Zbl 1157.62308 · doi:10.1016/j.csda.2006.05.012
[5] de Leeuw J, Young FW, Takane Y (1976) Additive structure in qualitative data: An alternating least squares method with optimal scaling features. Psychometrika 41:471–503 · Zbl 0351.92031 · doi:10.1007/BF02296971
[6] DeSarbo WS, Cron WL (1988) A maximum likelihood methodology for clusterwise linear regression. J Classif 5(2):249–282 · Zbl 0692.62052 · doi:10.1007/BF01897167
[7] DeSarbo WD, Jedidi K, Sinha I (2001) Customer value analysis in a heterogeneous market. Strateg Manag J 22:845–857 · doi:10.1002/smj.191
[8] D’Urso P, Massari R, Santoro A (2010) A class of fuzzy clusterwise regression models. Inf Sci 180(24):4737–4762 · Zbl 1204.62112 · doi:10.1016/j.ins.2010.08.018
[9] D’Urso P, Santoro A (2006) Fuzzy clusterwise linear regression analysis with symmetrical fuzzy output variable. Comput Stat Data Anal 51(1):287–313 · Zbl 1157.62461 · doi:10.1016/j.csda.2006.06.001
[10] Fadili MJ, Ruan S, Bloyet D, Mazoyer B (2001) On the number of clusters and the fuzziness index for unsupervised FCA application to BOLD fMRI time series. Med Image Anal 5(1):55–67 · doi:10.1016/S1361-8415(00)00035-9
[11] Gordon AD (1999) Classification. Chapman and Hall/CRC, London
[12] Hastie T, Tibshirani R, Friedman J (2009) The elements of statistical learning: data mining, inference, and prediction, 2nd edn. Springer, New York · Zbl 1273.62005
[13] Hathaway RJ, Bezdek JC (1993) Switching regression models and fuzzy clustering. IEEE Trans Fuzzy Syst 1:195–204 · doi:10.1109/91.236552
[14] Heiser WJ, Groenen PJF (1997) Cluster differences scaling with a within-cluster loss component and a fuzzy successive approximation strategy to avoid local minima. Psychometrika 62:63–83 · Zbl 0889.92037 · doi:10.1007/BF02294781
[15] Hennig C (2000) Identifiability of models for clusterwise linear regression. J Classif 17:273–296 · Zbl 1017.62058 · doi:10.1007/s003570000022
[16] Hoerl AE, Kennard RW (1970) Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12(1):55–67 · Zbl 0202.17205 · doi:10.1080/00401706.1970.10488634
[17] Hosmer DW (1974) Maximum likelihood estimates of the parameters of a mixture of two regression lines. Commun Stat 3:995–1006 · Zbl 0294.62085 · doi:10.1080/03610917408548314
[18] Hruschka H (1986) Market definition and segmentation using fuzzy clustering methods. Int J Res Mark 3:117–134 · doi:10.1016/0167-8116(86)90015-7
[19] Hwang H, Jung K, Takane Y, Woodward TS (2012) Functional multiple-set canonical correlation analysis. Psychometrika 77:48–64 · Zbl 1231.62106 · doi:10.1007/s11336-011-9234-4
[20] Jackson I, Sirois S (2009) Infant cognition: going full factorial with pupil dilation. Dev Sci 12:670–679 · doi:10.1111/j.1467-7687.2008.00805.x
[21] Maharaj EA, D’Urso P (2011) Fuzzy clustering of time series in frequency domain. Inf Sci 181:1187–1211 · Zbl 1215.62061 · doi:10.1016/j.ins.2010.11.031
[22] McBratney AB, Moore AW (1985) Application of fuzzy sets to climatic classification. Agric Forest Meteorol 35:165–185 · doi:10.1016/0168-1923(85)90082-6
[23] Moffitt TE (1993) Adolescent-limited and life-course-persistent antisocial behavior: a developmental taxonomy. Psychol Rev 100:674–701 · doi:10.1037/0033-295X.100.4.674
[24] Ozkan I, Turksen IB (2007) Upper and lower values for the level of fuzziness in FCM. Inf Sci 177(23):5143–5152 · Zbl 1126.68610 · doi:10.1016/j.ins.2007.06.028
[25] Pal NR, Bezdek JC (1995) On cluster validity for the fuzzy c-means model. IEEE Trans Fuzzy Syst 3(3):370–379 · doi:10.1109/91.413225
[26] Preda C, Saporta G (2005) Clusterwise PLS regression on a stochastic process. Comput Stat Data Anal 49:99–108 · Zbl 1429.62299 · doi:10.1016/j.csda.2004.05.002
[27] Quandt RE, Ramsey JB (1978) Estimating mixtures of normal distributions and switching regressions. J Am Stat Assoc 73:730–738 · Zbl 0401.62024 · doi:10.1080/01621459.1978.10480085
[28] Ramsay JO, Ramsey JB (2001) Functional data analysis of the dynamics of the monthly index of non durable goods production. J Econ 107:327–344 · Zbl 1051.62118 · doi:10.1016/S0304-4076(01)00127-0
[29] Ramsay JO, Silverman BW (2005) Functional data analysis. Springer, New York
[30] Roubens M (1982) Fuzzy clustering algorithms and their cluster validity. Eur J Oper Res 10:294–301 · Zbl 0485.62055 · doi:10.1016/0377-2217(82)90228-4
[31] Späth H (1979) Algorithm 39: clusterwise linear regression. Computing 22:367–373 · Zbl 0387.65028 · doi:10.1007/BF02265317
[32] Späth H (1981) Correction to algorithm 39: clusterwise linear regression. Computing 26:275 · Zbl 0444.65020 · doi:10.1007/BF02243486
[33] Späth H (1982) Algorithm 48: a fast algorithm for clusterwise linear regression. Computing 29:175–181 · Zbl 0485.65030 · doi:10.1007/BF02249940
[34] Späth H (1985) Cluster dissection and analysis. Wiley, New York · Zbl 0584.62094
[35] Suk HW, Hwang H (2010) Regularized fuzzy clusterwise ridge regression. Adv Data Anal Classif 4:35–51 · Zbl 1306.62166 · doi:10.1007/s11634-009-0056-5
[36] Tian TS (2010) Functional data analysis in brain imaging studies. Front Quant Psychol Meas 1(Article 35): 1–11
[37] Tokushige S, Yadohisa H, Inada K (2007) Crisp and fuzzy k-means clustering algorithms for multivariate functional data. Comput Stat 22:1–16 · Zbl 1196.62089 · doi:10.1007/s00180-006-0013-0
[38] Wedel M, Steenkamp J-BEM (1989) Fuzzy clusterwise regression approach to benefit segmentation. J Res Mark 6:241–258 · doi:10.1016/0167-8116(89)90052-9
[39] Wedel M, Steenkamp J-BEM (1991) A clusterwise regression method for simultaneous fuzzy market structuring and benefit segmentation. J Mark Res 28:385–396 · doi:10.2307/3172779
[40] Xie XL, Beni G (1991) A validity measures for fuzzy clustering. IEEE Trans Pattern Anal Mach Intell 13(8):841–847 · Zbl 05112080 · doi:10.1109/34.85677
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.