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A partial least squares approach for function-on-function interaction regression. (English) Zbl 1505.62064

Summary: A partial least squares regression is proposed for estimating the function-on-function regression model where a functional response and multiple functional predictors consist of random curves with quadratic and interaction effects. The direct estimation of a function-on-function regression model is usually an ill-posed problem. To overcome this difficulty, in practice, the functional data that belong to the infinite-dimensional space are generally projected into a finite-dimensional space of basis functions. The function-on-function regression model is converted to a multivariate regression model of the basis expansion coefficients. In the estimation phase of the proposed method, the functional variables are approximated by a finite-dimensional basis function expansion method. We show that the partial least squares regression constructed via a functional response, multiple functional predictors, and quadratic/interaction terms of the functional predictors is equivalent to the partial least squares regression constructed using basis expansions of functional variables. From the partial least squares regression of the basis expansions of functional variables, we provide an explicit formula for the partial least squares estimate of the coefficient function of the function-on-function regression model. Because the true forms of the models are generally unspecified, we propose a forward procedure for model selection. The finite sample performance of the proposed method is examined using several Monte Carlo experiments and two empirical data analyses, and the results were found to compare favorably with an existing method.

MSC:

62-08 Computational methods for problems pertaining to statistics
62G08 Nonparametric regression and quantile regression
62H25 Factor analysis and principal components; correspondence analysis

Software:

FRegSigCom; fda (R)
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Full Text: DOI arXiv

References:

[1] Aguilera, AM; Ocana, FA; Valderrama, MJ, Forecasting with unequally spaced data by a functional principal component approach, Test, 8, 1, 233-254 (1999) · Zbl 0945.62095
[2] Aguilera, AM; Escabias, M.; Preda, C.; Saporta, G., Using basis expansions for estimating functional PLS regression: applications with chemometric data, Chemom Intell Lab Syst, 104, 2, 289-305 (2010)
[3] Aguilera, AM; Escabias, M.; Preda, C.; Saporta, G., Penalized versions of functional PLS regression, Chemom Intell Lab Syst, 154, 80-92 (2016)
[4] Beyaztas, U.; Shang, HL, On function-on-function regression: partial least squares approach, Environ Ecol Stat, 27, 1, 95-114 (2020)
[5] Chiou, J-M; Müller, HG; Wang, JL, Functional response models, Stat Sin, 14, 675-693 (2004) · Zbl 1073.62098
[6] Chiou, J-M; Yang, Y-F; Chen, Y-T, Multivariate functional linear regression and prediction, J Multivar Anal, 146, 301-312 (2016) · Zbl 1336.62158
[7] Cuevas, A., A partial overview of the theory of statistics with functional data, J Stat Plan Inference, 147, 1-23 (2014) · Zbl 1278.62012
[8] Dayal, BS; MacGregor, JF, Improved PLS algorithms, J Chemom, 11, 1, 73-85 (1997)
[9] de Jong, S., SIMPLS: an alternative approach to partial least squares regression, Chemom Intell Lab Syst, 18, 3, 251-263 (1993)
[10] de Lathuauwer, L.; de Moor, B.; Vandewalle, J., A multilinear singular value decomposition, SIAM J Matrix Anal Appl, 21, 4, 1253-1278 (2000) · Zbl 0962.15005
[11] Delaigle, A.; Hall, P., Achieving near perfect classification for functional data, J R Stat Soc Ser B, 74, 2, 267-286 (2012) · Zbl 1411.62164
[12] Delaigle, A.; Hall, P., Methodology and theory for partial least squares applied to functional data, Ann Stat, 40, 1, 322-352 (2012) · Zbl 1246.62084
[13] Escabias, M.; Aguilera, AM; Valderrama, MJ, Functional PLS logit regression model, Comput Stat Data Anal, 51, 10, 4891-4902 (2007) · Zbl 1162.62392
[14] Escoufier, Y., Echantillonnage dans une population de variables aléatories réelles, Publications de I’Institut de Statistique de I’Université de Paris, 19, 4, 1-47 (1970) · Zbl 0264.62021
[15] Febrero-Bande, M.; Galeano, P.; Gozalez-Manteiga, W., Functional principal component regression and functional partial least-squares regression: an overview and a comparative study, Int Stat Rev, 85, 1, 61-83 (2017) · Zbl 07763512
[16] Ferraty, F.; Vieu, P., Nonparametric functional data analysis (2006), New York: Springer, New York · Zbl 1119.62046
[17] Fuchs, K.; Scheipl, F.; Greven, S., Penalized scalar-on-functions regression with interaction term, Comput Stat Data Anal, 81, 38-51 (2015) · Zbl 1507.62055
[18] He, G.; Müller, HG; Wang, JL; Puri, ML, Extending correlation and regression from multivariate to functional data, Asymptotics in statistics and probability, 197-210 (2000), Leiden: VSP International Science Publishers, Leiden
[19] He, G.; Müller, HG; Wang, JL; Yang, W., Functional linear regression via canonical analysis, Bernoulli, 16, 3, 705-729 (2010) · Zbl 1220.62076
[20] Horvath, L.; Kokoszka, P., Inference for functional data with applications (2012), New York: Springer, New York · Zbl 1279.62017
[21] Hsing, T.; Eubank, R., Theoretical foundations of functional data analysis, with an introduction to linear operators (2015), Chennai: Wiley, Chennai · Zbl 1338.62009
[22] Hyndman, RJ; Shang, HL, Forecasting functional time series, J Korean Stat Soc, 38, 3, 199-211 (2009) · Zbl 1293.62267
[23] Ivanescu, AE; Staicu, A-M; Scheipl, F.; Greven, S., Penalized function-on-function regression, Comput Stat, 30, 2, 539-568 (2015) · Zbl 1317.65037
[24] Kokoszka, P.; Reimherr, M., Introduction to functional data analysis (2017), Boca Raton: CRC Press, Boca Raton · Zbl 1411.62004
[25] Krämer, N.; Boulesteix, A-L; Tutz, G., Penalized partial least squares with applications to B-spline transformations and functional data, Chemom Intell Lab Syst, 94, 1, 60-69 (2008)
[26] Luo R, Qi X (2018) FRegSigCom: functional regression using signal compression approach. R package version 0.3.0. https://CRAN.R-project.org/package=FRegSigCom
[27] Luo, R.; Qi, X., Interaction model and model selection for function-on-function regression, J Comput Graph Stat, 28, 2, 309-322 (2019) · Zbl 07499055
[28] Matsui, H., Quadratic regression for functional response models, Econom Stat, 13, 125-136 (2020)
[29] Matsui, H.; Kawano, S.; Konishi, S., Regularized functional regression modeling for functional response and predictors, J Math Ind, 1, A3, 17-25 (2009) · Zbl 1294.62148
[30] Müller, H-G; Yao, F., Functional additive models, J Am Stat Assoc, 103, 484, 1534-1544 (2008) · Zbl 1286.62040
[31] Pan H (2011) , Bivariate B-splines and its applications in spatial data analysis. PhD thesis, Texas A&M University
[32] Preda, C.; Saporta, G., PLS regression on a stochastic process, Comput Stat Data Anal, 48, 1, 149-158 (2005) · Zbl 1429.62224
[33] Preda, C.; Saporta, G.; Lévéder, C., PLS classification of functional data, Comput Stat, 22, 2, 223-235 (2007) · Zbl 1196.62086
[34] Ramsay, JO; Dalzell, CJ, Some tools for functional data analysis, J R Stat Soc B, 53, 3, 539-572 (1991) · Zbl 0800.62314
[35] Ramsay, JO; Silverman, BW, Applied functional data analysis (2002), New York: Springer, New York · Zbl 1011.62002
[36] Ramsay, JO; Silverman, BW, Functional data analysis (2006), New York: Springer, New York · Zbl 1079.62006
[37] Reiss, PT; Ogden, TR, Functional principal component regression and functional partial least squares, J Am Stat Assoc, 102, 479, 984-996 (2007) · Zbl 1469.62237
[38] Scheipl, F.; Greven, S., Identifiability in penalized function-on-function regression models, Electron J Stat, 10, 495-526 (2016) · Zbl 1332.62249
[39] Sun Y, Wang Q (2020) Function-on-function quadratic regression models. Comput Stat Data Anal 142, 106814 · Zbl 1507.62163
[40] Tucker, RS, The reasons for price rigidity, Am Econ Rev, 28, 1, 41-54 (1938)
[41] Usset, J.; Staicu, A-M; Maity, A., Interaction models for functional regression, Comput Stat Data Anal, 94, 317-329 (2016) · Zbl 1468.62201
[42] Wang, W., Linear mixed function-on-function regression models, Biometrics, 70, 4, 794-801 (2014) · Zbl 1393.62106
[43] Wold, H., Causal flows with latent variables: partings of the ways in the light of NIPALS modelling, Eur Econ Rev, 5, 1, 67-86 (1974)
[44] Wood, SN; Bravington, MV; Hedley, SL, Soap film smoothing, J R Stat Soc B, 70, Part 5, 931-955 (2008) · Zbl 1411.65021
[45] Yao, F.; Müller, H-G, Functional quadratic regression, Biometrika, 97, 1, 49-64 (2010) · Zbl 1183.62113
[46] Yao, F.; Müller, H-G; Wang, J-L, Functional linear regression analysis for longitudinal data, Ann Stat, 33, 6, 2873-2903 (2005) · Zbl 1084.62096
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