×

Optimal design for adaptive smoothing splines. (English) Zbl 1437.62305

Summary: We consider the design problem of collecting temporal/longitudinal data. The adaptive smoothing spline is used as the analysis model where the prior curvature information can be naturally incorporated as a weighted smoothness penalty. The estimator of the curve is expressed in linear mixed model form, and the information matrix of the parameters is derived. The D-optimality criterion is then used to compute the optimal design points. An extension is considered, for the case where subpopulations exert different prior curvature patterns. We compare properties of the optimal designs with the uniform design using simulated data and apply our method to the Berkeley growth data to estimate the optimal ages to measure heights for males and females. The approach is implemented in an R package called “ODsplines”, which is available from github.com/jialiwang1211/ODsplines.

MSC:

62K05 Optimal statistical designs
62G08 Nonparametric regression and quantile regression
62P10 Applications of statistics to biology and medical sciences; meta analysis
65D07 Numerical computation using splines
62B10 Statistical aspects of information-theoretic topics
62J02 General nonlinear regression
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Atkinson, A.; Donev, A.; Tobias, R., Optimum Experimental Designs, with SAS, Vol. 34 (2007), Oxford University Press · Zbl 1183.62129
[2] Chaloner, K.; Verdinelli, I., Bayesian experimental design: A review, Statist. Sci., 273-304 (1995) · Zbl 0955.62617
[3] Dette, H.; Melas, V. B.; Pepelyshev, A., Optimal designs for free knot least squares splines, Statist. Sinica, 1047-1062 (2008) · Zbl 1149.62064
[4] Dette, H.; Melas, V. B.; Pepelyshev, A., Optimal design for smoothing splines, Ann. Inst. Statist. Math., 63, 5, 981-1003 (2011) · Zbl 1225.62106
[5] DiMatteo, I.; Genovese, C. R.; Kass, R. E., Bayesian curve-fitting with free-knot splines, Biometrika, 88, 4, 1055-1071 (2001) · Zbl 0986.62026
[6] Donev, A. N.; Tobias, R.; Monadjemi, F., Cost-cautious designs for confirmatory bioassay, J. Statist. Plann. Inference, 138, 12, 3805-3812 (2008) · Zbl 1146.62083
[7] Green, P. J.; Silverman, B. W., Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach (1993), CRC Press
[8] Gu, Y.; Jin, Z., Neighborhood preserving d-optimal design for active learning and its application to terrain classification, Neural Comput. Appl., 23, 7-8, 2085-2092 (2013)
[9] He, X., Laplacian regularized d-optimal design for active learning and its application to image retrieval, IEEE Trans. Image Process., 19, 1, 254-263 (2009) · Zbl 1371.94156
[10] Heiligers, B., E-optimal designs for polynomial spline regression, J. Stat. Plan. Inference, 75, 1, 159-172 (1998) · Zbl 0938.62077
[11] Hooks, T.; Marx, D.; Kachman, S.; Pedersen, J., Optimality criteria for models with random effects, Rev. Colombiana Estadíst., 32, 1, 17-31 (2009) · Zbl 07578285
[12] Kaishev, V., Optimal experimental designs for the b-spline regression, Comput. Statist. Data Anal., 8, 1, 39-47 (1989) · Zbl 0702.62068
[13] Kiefer, J., General equivalence theory for optimum designs (approximate theory), Ann. Statist., 849-879 (1974) · Zbl 0291.62093
[14] Li, G., Optimal and efficient designs for gompertz regression models, Ann. Inst. Statist. Math., 64, 5, 945-957 (2012) · Zbl 1254.62086
[15] Li, G.; Majumdar, D., D-optimal designs for logistic models with three and four parameters, J. Statist. Plann. Inference, 138, 7, 1950-1959 (2008) · Zbl 1134.62054
[16] Miyata, S.; Shen, X., Adaptive free-knot splines, J. Comput. Graph. Stat., 12, 1, 197-213 (2003)
[17] Montgomery, D. C., Design and Analysis of Experiments (2017), John wiley & sons
[18] Paine, C. T.; Marthews, T. R.; Vogt, D. R.; Purves, D.; Rees, M.; Hector, A.; Turnbull, L. A., How to fit nonlinear plant growth models and calculate growth rates: an update for ecologists, Methods Ecol. Evol., 3, 2, 245-256 (2012)
[19] Park, S. H., Experimental designs for fitting segmented polynomial regression models, Technometrics, 20, 2, 151-154 (1978) · Zbl 0405.62061
[20] Pintore, A.; Speckman, P.; Holmes, C. C., Spatially adaptive smoothing splines, Biometrika, 93, 1, 113-125 (2006) · Zbl 1152.62331
[21] R Core Team, 2018. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, URL https://www.R-project.org/.
[22] Ramsay, J. O.; Silverman, B. W., Functional Data Analysis (2005), Springer · Zbl 1079.62006
[23] Tsay, J.-Y., On the sequential construction of d-optimal designs, J. Amer. Statist. Assoc., 71, 355, 671-674 (1976) · Zbl 0342.62050
[24] Tuddenham, R. D., Physical growth of california boys and girls from birth to eighteen years, Univ. Calif. Publ. Child Dev., 1, 183-364 (1954)
[25] Varadhan, R., Borchers, H.W., Varadhan, M.R., 2016. Package ‘dfoptim’.
[26] Verbyla, A. P., A note on model selection using information criteria for general linear models estimated using reml, Aust. N. Z. J. Stat., 61, 1, 39-50 (2019) · Zbl 1420.62020
[27] Verbyla, A. P.; Cullis, B. R.; Kenward, M. G.; Welham, S. J., The analysis of designed experiments and longitudinal data by using smoothing splines, J. R. Stat. Soc. Ser. C. Appl. Stat., 48, 3, 269-311 (1999) · Zbl 0956.62062
[28] Wahba, G., Spline Models for Observational Data, Vol. 59 (1990), Siam
[29] Wang, Y., Mixed effects smoothing spline analysis of variance, J. R. Stat. Soc. Ser. B Stat. Methodol., 60, 1, 159-174 (1998) · Zbl 0909.62034
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.