×

Identifying local smoothness for spatially inhomogeneous functions. (English) Zbl 1417.62079

Summary: We consider a problem of estimating local smoothness of a spatially inhomogeneous function from noisy data under the framework of smoothing splines. Most existing studies related to this problem deal with estimation induced by a single smoothing parameter or partially local smoothing parameters, which may not be efficient to characterize various degrees of smoothness of the underlying function when it is spatially varying. In this paper, we propose a new nonparametric method to estimate local smoothness of the function based on a moving local risk minimization coupled with spatially adaptive smoothing splines. The proposed method provides full information of the local smoothness at every location on the entire data domain, so that it is able to understand the degrees of spatial inhomogeneity of the function. A successful estimate of the local smoothness is useful for identifying abrupt changes of smoothness of the data, performing functional clustering and improving the uniformity of coverage of the confidence intervals of smoothing splines. We further consider a nontrivial extension of the local smoothness of inhomogeneous two-dimensional functions or spatial fields. Empirical performance of the proposed method is evaluated through numerical examples, which demonstrates promising results of the proposed method.

MSC:

62G08 Nonparametric regression and quantile regression
62G07 Density estimation
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62H30 Classification and discrimination; cluster analysis (statistical aspects)

Software:

SiZer; curvclust
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Antoniadis A, Brossat X, Cugliari J, Poggi JM (2013) Clustering functional data using wavelets. Int J Wavelets Multiresolut Inf Process 11(01):1350003 · Zbl 1271.62131 · doi:10.1142/S0219691313500033
[2] Chaudhuri P, Marron JS (1999) SiZer for exploration of structures in curves. J Am Stat Assoc 94:807-823 · Zbl 1072.62556 · doi:10.1080/01621459.1999.10474186
[3] Cuevas A (2014) A partial overview of the theory of statistics with functional data. J Stat Plan Inference 147:1-23 · Zbl 1278.62012 · doi:10.1016/j.jspi.2013.04.002
[4] Cummins DJ, Filloon TG, Nychka D (2001) Confidence intervals for nonparametric curve estimates: toward more uniform pointwise coverage. J Am Stat Assoc 96:233-246 · Zbl 1015.62049 · doi:10.1198/016214501750332811
[5] Donoho DL, Johnstone IM (1994) Ideal spatial adaptation by wavelet shrinkage. Biometrika 81:425-455 · Zbl 0815.62019 · doi:10.1093/biomet/81.3.425
[6] Donoho DL, Johnstone IM, Kerkyacharian G, Picard D (1995) Wavelet shrinkage: asymptopia? J R Stat Soc Series B 57:301-369 · Zbl 0827.62035
[7] Erästö P, Holmström L (2005) Bayesian multiscale smoothing for making inferences about features in scatter plots. J Comput Graph Stat 14:569-589 · doi:10.1198/106186005X59315
[8] Fan J, Gijbels I (1996) Local polynomial modelling and its applications. Chapman & Hall, London · Zbl 0873.62037
[9] Fryzlewicz P, Oh H-S (2011) Thick-pen transform for time series. J R Stat Soc Series B 73:499-529 · Zbl 1226.62077 · doi:10.1111/j.1467-9868.2011.00773.x
[10] Giacofci M, Lambert-Lacroix S, Marot G, Picard F (2013) Wavelet-based clustering for mixed-effects functional models in high dimension. Biometrics 69:31-40 · Zbl 1274.62774 · doi:10.1111/j.1541-0420.2012.01828.x
[11] Goia A, Vieu P (2016) An introduction to recent advances in high/infinite dimensional statistics. J Multivar Stat 146:1-6 · Zbl 1384.00073 · doi:10.1016/j.jmva.2015.12.001
[12] Green PJ, Silverman BW (1994) Nonparametric regression and generalized linear models. Chapman & Hall, London · Zbl 0832.62032 · doi:10.1007/978-1-4899-4473-3
[13] Hannig J, Lee TCM (2006) Robust SiZer for exploration of regression structures and outlier detection. J Comput Graph Stat 15:101-117 · doi:10.1198/106186006X94676
[14] Hannig J, Lee T, Park C (2013) Metrics for SiZer map comparison. Stat 2:49-60 · doi:10.1002/sta4.17
[15] Holmström L (2010a) BSiZer. Wiley Interdiscip Rev Comput Stat 2:526-534 · doi:10.1002/wics.115
[16] Holmströma L (2010b) Scale space methods. Wiley Interdiscip Rev Comput Stat 2:150-159 · doi:10.1002/wics.79
[17] Holmströma L, Pasanena L (2016) Statistical scale space methods. Int Stat Rev. doi:10.1111/insr.12155 · Zbl 07763504 · doi:10.1111/insr.12155
[18] James GM, Sugar CA (2003) Clustering for sparsely sampled functional data. J Am Stat Assoc 98:397-408 · Zbl 1041.62052 · doi:10.1198/016214503000189
[19] Jang D, Oh H-S (2011) Enhancement of spatially adaptive smoothing splines via parameterization of smoothing parameters. Comput Stat Data Anal 55:1029-1040 · Zbl 1284.62261 · doi:10.1016/j.csda.2010.08.008
[20] Jaques J, Preda C (2013) Functional data clustering: a survey. Adv Data Anal Classif 8:231-255 · Zbl 1414.62018 · doi:10.1007/s11634-013-0158-y
[21] Lee TCM (2004) Improved smoothing spline regression by combining estimates of different smoothness. Stat Probab Lett 67:133-140 · Zbl 1058.62035 · doi:10.1016/j.spl.2004.01.003
[22] Lindeberg T (1994) Scale-space theory in computer vision. Kluwer, Boston · Zbl 0812.68040 · doi:10.1007/978-1-4757-6465-9
[23] Morris JS, Carroll RJ (2006) Wavelet-based functional mixed models. J R Stat Soc Series B 68:179-199 · Zbl 1110.62053 · doi:10.1111/j.1467-9868.2006.00539.x
[24] Pasanena L, Launonena I, Holmströma L (2013) A scale space multiresolution method for extraction of time series features. Stat 2:273-291 · doi:10.1002/sta4.35
[25] Park C, Hannig J, Kang KH (2009) Improved SiZer for time series. Stat Sin 19:1511-1530 · Zbl 1191.62152
[26] Park C, Lee TC, Hannig J (2010) Multiscale exploratory analysis of regression quantiles using quantile SiZer. J Comput Graph Stat 19:497-513 · doi:10.1198/jcgs.2010.09120
[27] Pintore A, Speckman P, Holmes CC (2006) Spatially adaptive smoothing splines. Biometrika 93:113-125 · Zbl 1152.62331 · doi:10.1093/biomet/93.1.113
[28] Ray S, Mallick B (2006) Functional clustering by Bayesian wavelet methods. J R Stat Soc Series B 68:305-332 · Zbl 1100.62058 · doi:10.1111/j.1467-9868.2006.00545.x
[29] Sain SR (2002) Multivariate locally adaptive density estimation. Comput Stat Data Anal 39:165-186 · Zbl 1132.62329 · doi:10.1016/S0167-9473(01)00053-6
[30] Sain SR, Scott DW (1996) On locally adaptive density estimation. J Am Stat Assoc 91:1525-1534 · Zbl 0882.62035 · doi:10.1080/01621459.1996.10476720
[31] Serban N, Wasserman L (2005) CATS: cluster analysis by transformation and smoothing. J Am Stat Assoc 100:990-999 · Zbl 1117.62422 · doi:10.1198/016214504000001574
[32] Smith M, Kohn R (1996) Nonparametric regression using Bayesian variable selection. J Econ 75:317-344 · Zbl 0864.62025 · doi:10.1016/0304-4076(95)01763-1
[33] Wahba G (1983) Bayesian ‘confidence intervals’ for the cross-validated smoothing spline. J R Stat Soc Series B 45:133-150 · Zbl 0538.65006
[34] Wahba G (1990) Spline models for observational data. In: CBMS-NSF, regional conference series in applied mathematics. SIAM, Philadelphia · Zbl 0813.62001
[35] Wahba G (1995) Discussion of a paper by Donoho et al. J R Stat Soc Series B 57:360-361
[36] Wakefield, J.; Zhou, C.; Self, S.; Bernardo, J. (ed.); Bayarri, M. (ed.); Berger, J. (ed.); Dawid, A. (ed.); Heckerman, D. (ed.); Smith, A. (ed.); West, M. (ed.), Modelling gene expression over time: curve clustering with informative prior distributions, No. 7, 721-732 (2003), Oxford
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.