Functional dimension reduction based on fuzzy partition and transformation. (English) Zbl 1521.62056
Summary: Functional sliced inverse regression (FSIR) is the among most popular methods for the functional dimension reduction. However, FSIR has two evident shortcomings. On the one hand, the number of samples in each slice must not be too small and selecting a suitable \(S\) is difficult, particularly for data with small sample size, where \(S\) indicates the number of slices. On the other hand, FSIR and its related methods are well-known for their poor performance when the link function is an even (or symmetric) dependency. To solve these two problems, we propose three new types of estimation methods. First, we propose the functional fuzzy inverse regression (FFIR) method based on a fuzzy partition. Compared with FSIR that uses a hard partition, the fuzzy partition uses all samples with different weights to estimate the mean in each slice. Therefore, FFIR exhibits good performance even for data with small sample size. Second, we suggest two transformation approaches, namely, FSIRR and FSIRP, avoiding the symmetric dependency between the response and the predictor. FSIRR eliminates the symmetric dependency by transforming the response variable, while FSIRP overcomes the symmetric dependency by transforming the functional predictor. Third, we propose the FFIRR and FFIRP methods by combining the advantages of FFIR and two transformation methods. FFIRR and FFIRP replace the FSIR method on the transformation data via FFIR. Simulation and real data analysis show that three types of proposed methods exhibit better performance than FSIR in terms of the estimation accuracy and stability.
MSC:
| 62G08 | Nonparametric regression and quantile regression |
| 62H25 | Factor analysis and principal components; correspondence analysis |
| 62J86 | Fuzziness, and linear inference and regression |
| 62R10 | Functional data analysis |
Keywords:
FSIR; functional dimension reduction; fuzzy partition; symmetric dependency; transformationReferences:
| [1] | Amato, U., Antoniadis, A. & De Feis, I. (2006). Dimension reduction in functional regression with applications. Computational Statistics & Data Analysis50, 2422-2446. · Zbl 1445.62078 |
| [2] | Baíllo, A. & Grané, A. (2009). Local linear regression for functional predictor and scalar response. Journal of Multivariate Analysis100, 102-111. · Zbl 1151.62028 |
| [3] | Bezdek, J.C. (1981). Pattern Recognition with Fuzzy Objective Function Algorithms. New York: Springer Science & Business Media. · Zbl 0503.68069 |
| [4] | Cai, T.T. & Hall, P. (2006). Prediction in functional linear regression. The Annals of Statistics34, 2159-2179. · Zbl 1106.62036 |
| [5] | Cardot, H., Ferraty, F. & Sarda, P. (2003). Spline estimators for the functional linear model. Statistica Sinica13, 571-591. · Zbl 1050.62041 |
| [6] | Chen, K. & Lei, J. (2015). Localized functional principal component analysis. Journal of the American Statistical Association110, 1266-1275. · Zbl 1373.62293 |
| [7] | Du, J., Cao, R., Kwessi, E. & Zhang, Z. (2019). Estimation for generalized partially functional linear additive regression model. Journal of Applied Statistics46, 914-925. · Zbl 1516.62261 |
| [8] | Du, J., Xu, D. & Cao, R. (2018). Estimation and variable selection for partially functional linear models. Journal of the Korean Statistical Society47, 436-449. · Zbl 1406.62034 |
| [9] | Dunn, J.C. (1974). Some recent investigations of a new fuzzy partitioning algorithm and its application to pattern classification problems. Journal of Cybernetics4, 1-15. · Zbl 0304.68094 |
| [10] | Ferraty, F. & Vieu, P. (2006). Nonparametric Functional Data Analysis: Theory and Practice. New York: Springer Science & Business Media. · Zbl 1119.62046 |
| [11] | Ferré, L. & Yao, A.F. (2003). Functional sliced inverse regression anlysis. Statistics37, 475-488. · Zbl 1032.62052 |
| [12] | Ferré, L. & Yao, A.F. (2005). Smoothed functional inverse regression. Statistica Sinica15, 665-683. · Zbl 1086.62054 |
| [13] | Hu, W., Guo, J., Wang, G. & Zhang, B. (2021). Weight fused functional sliced average variance estimation. Communications in Statistics‐Simulation and Computation, 10.1080/03610918.2020.1752382. · Zbl 07603798 |
| [14] | Horváth, L. & Kokoszka, P. (2012). Inference for Functional Data with Applications. New York: Springer Science & Business Media. · Zbl 1279.62017 |
| [15] | Kaufman, L. & Rousseeuw, P. J. (1990). Finding Groups in Data: An Introduction to Cluster Analysis. New York: John Wiley & Sons. · Zbl 1345.62009 |
| [16] | Kong, D., Xue, K., Yao, F. & Zhang, H.H. (2016). Partially functional linear regression in high dimensions. Biometrika103, 147-159. · Zbl 1452.62500 |
| [17] | Li, K.C. (1991). Sliced inverse regression for dimension reduction. Journal of the American Statistical Association86, 316-327. · Zbl 0742.62044 |
| [18] | Lian, H. & Li, G. (2014). Series expansion for functional sufficient dimension reduction. Journal of Multivariate Analysis124, 150-165. · Zbl 1278.62055 |
| [19] | Prendergast, L.A. & Garnham, A.L. (2016). Response and predictor folding to counter symmetric dependency in dimension reduction. Australian & New Zealand Journal of Statistics58, 515-532. · Zbl 1373.62384 |
| [20] | Ramsay, J.O. & Silverman, B.W. (2002). Applied Functional Data Analysis: Methods and Case Studies. New York: Springer Science & Business Media. · Zbl 1011.62002 |
| [21] | Ramsay, J.O. & Silverman, B.W. (2005). Functional Data Analysis, 2nd edn. New York: Springer Science & Business Media. · Zbl 1079.62006 |
| [22] | Struyf, A., Hubert, M. & Rousseeuw, P.J. (1997). Integrating robust clustering techniques in S‐PLUS. Computational Statistics & Data Analysis26, 17-37. · Zbl 0900.62328 |
| [23] | Thodberg, H.H. (1996). A review of Bayesian neural networks with an application to near infrared spectroscopy. IEEE transactions on Neural Networks7, 56-72. |
| [24] | Wang, G., Feng, X.N. & Chen, M. (2016). Functional partial linear single‐index model. Scandinavian Journal of Statistics43, 261-274. · Zbl 1371.62037 |
| [25] | Wang, G., Lin, N. & Zhang, B. (2013). Dimension reduction in functional regression using mixed data canonical correlation analysis. Statistics and its Interface6, 187-196. · Zbl 1327.62217 |
| [26] | Wang, G. & Song, X. (2018). Functional sufficient dimension reduction for functional data classification. Journal of Classification35, 250-272. · Zbl 1422.62230 |
| [27] | Wang, G., Zhou, Y., Feng, X.N. & Zhang, B. (2015). The hybrid method of FSIR and FSAVE for functional effective dimension reduction. Computational Statistics & Data Analysis91, 64-77. · Zbl 1468.62205 |
| [28] | Wang, T. (2021). Dimension reduction via adaptive slicing. Statistica Sinica,32, 499-516. · Zbl 1524.62194 |
| [29] | Yao, F., Lei, E. & Wu, Y. (2015). Effective dimension reduction for sparse functional data. Biometrika102, 421-437. · Zbl 1452.62996 |
| [30] | Yu, P., Du, J. & Zhang, Z. (2017). Varying‐coefficient partially functional linear quantile regression models. Journal of the Korean Statistical Society46, 462-475. · Zbl 1368.62104 |
| [31] | Yu, P., Du, J. & Zhang, Z. (2020). Single‐index partially functional linear regression model. Statistical Papers61, 1107-1123. · Zbl 1443.62541 |
| [32] | Zipunnikov, V., Caffo, B., Yousem, D.M., Davatzikos, C., Schwartz, B.S. & Crainiceanu, C. (2011). Multilevel functional principal component analysis for high‐dimensional data. Journal of Computational and Graphical Statistics20, 852-873. |
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.
