Abstract
This paper considers the estimation and inference of spatially varying coefficient models, while preserving the sign of the coefficient functions. In practice, there are various situations where coefficient functions are assumed to be in a certain subspace. For example, they should be either nonnegative or nonpositive on a domain by their nature. However, optimization on a global space of coefficient functions does not ensure that estimates preserve meaningful features in their signs. In this paper, we propose sign-preserved and efficient estimators of the coefficient functions using novel bivariate spline estimators under their smoothness conditions. Our algorithm, based on the alternating direction method of multipliers, yields estimated coefficient functions that are nonnegative or nonpositive, consistent, and efficient. Simulation studies are conducted to address the advantages of the sign preservation method for a specific situation, where coefficient functions have sign constraints. Furthermore, we propose residual bootstrap-based confidence intervals for sign preserving coefficient functions over the domain of interest after adjusting the inherent bias of penalized smoothing spline techniques. Finally, we evaluate our method in a case study using air temperature, land surface temperature, and elevation in the USA.
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References
Bakar KS, Kokic P, Jin H (2015) A spatiodynamic model for assessing frost risk in South-Eastern Australia. J R Stat Soc Ser C 64:755–778
Bakar KS, Kokic P, Jin H (2016) Hierarchical spatially varying coefficient and temporal dynamic process models using spTDyn. J Stat Comput Simul 86:820–840
Banerjee S, Carlin BP, Gelfand AE (2014) Hierarchical modeling and analysis for spatial data. CRC Press, New York
Bivand R, Yu D, Nakaya T, Garcia-Lopez MA (2020) spgwr: geographically weighted regression. R package version 0.6-34. https://cran.r-project.org/web/packages/spgwr/index.html
Boyd S, Parikh N, Chu E, Peleato B, Eckstein J (2011) Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends® Mach Learn 3:1–122
Brunsdon C, Fotheringham AS, Charlton EM (1996) Geographically weighted regression: a method for exploring spatial nonstationarity. Geogr Anal 28:281–298
Cai Z, Fan J, Yao Q (2000) Functional-coefficient regression models for nonlinear time series. J Am Stat Assoc 95:941–956
Chan E, Ong B (2001) Range restricted scattered data interpolation using convex combination of cubic Bézier triangles. J Comput Appl Math 136:135–147
Dai N (2017) Inference for penalized spline regression: improving confidence intervals by reducing the penalty. arXiv preprint arXiv:1706.00865
Eckstein J (1994) Parallel alternating direction multiplier decomposition of convex programs. J Optim Theory Appl 80:39–62
Eckstein J, Ferris MC (1998) Operator-splitting methods for monotone affine variational inequalities, with a parallel application to optimal control. INFORMS J Comput 10:218–235
Finley AO, Banerjee S (2020) Bayesian spatially varying coefficient models in the spBayes R Package. Environ Model Softw 125:104608
Flores F, Lillo M (2010) Simple air temperature estimation method from MODIS satellite images on a regional scale. Chil J Agric Res 70:436–445
Gelfand AE, Kim H-J, Sirmans C, Banerjee S (2003) Spatial modeling with spatially varying coefficient processes. J Am Stat Assoc 98:387–396
Ghadimi E, Teixeira A, Shames I, Johansson M (2015) Optimal parameter selection for the alternating direction method of multipliers (ADMM): quadratic problems. IEEE Trans Autom Control 60:644–658
Giselsson P, Boyd S (2016) Linear convergence and metric selection for Douglas–Rachford splitting and ADMM. IEEE Trans Autom Control 62:532–544
Hamm N, Finley A, Schaap M, Stein A (2015) A spatially varying coefficient model for mapping PM10 air quality at the European Scale. Atmos Environ 102:393–405
He B, Yang H, Wang S (2000) Alternating direction method with self-adaptive penalty parameters for monotone variational inequalities. J Optim Theory Appl 106:337–356
Lai MJ, Schumaker LL (2007) Spline functions on triangulations. Cambridge University Press, Cambridge
Li X, Zhou Y, Asrar GR, Zhu Z (2018a) Creating a seamless 1 km resolution daily land surface temperature dataset for urban and surrounding areas in the conterminous United States. Remote Sens Environ 206:84–97
Li X, Zhou Y, Asrar GR, Zhu Z (2018b) Developing a 1 km resolution daily air temperature dataset for urban and surrounding areas in the conterminous United States. Remote Sens Environ 215:74–84
Li X, Zhou Y, Yu S, Jia G, Li H, Li W (2019) Urban heat island impacts on building energy consumption: a review of approaches and findings. Energy 174:407–419
Mu J, Wang G, Wang L (2018) Estimation and inference in spatially varying coefficient models. Environmetrics 29:e2485
Mutiibwa D, Strachan S, Albright T (2015) Land surface temperature and surface air temperature in complex terrain. IEEE J Sel Top Appl Earth Obs Remote Sens 8:4762–4774
Nishihara R, Lessard L, Recht B, Packard A, Jordan M (2015) A general analysis of the convergence of ADMM. In: Proceedings of the 32nd international conference on machine learning, pp 343–352
Schumaker LL, Speleers H (2010) Nonnegativity preserving macro-element interpolation of scattered data. Comput Aided Geom Des 27:245–261
Stellato B, Banjac G, Goulart P, Bemporad A, Boyd S (2020a) OSQP: an operator splitting solver for quadratic programs. Math Program Comput 12:637–672
Stellato B, Banjac G, Goulart P, Boyd S, Anderson E (2020b) osqp: quadratic programming solver using the ’OSQP’ Library. R package version 0.6.0.3. https://cran.r-project.org/web/packages/osqp/index.html
Sun Y, Yan H, Zhang W, Lu Z et al (2014) A semiparametric spatial dynamic model. Ann Stat 42:700–727
Wang L, Lai MJ (2019) Triangulation. R package version 1.0. https://github.com/funstatpackages/Triangulation
Wang S, Liao L (2001) Decomposition method with a variable parameter for a class of monotone variational inequality problems. J Optim Theory Appl 109:415–429
Wood NS, Bravington VM, Hedley LS (2008) Soap film smoothing. J R Stat Soc B 70:931–955
Yu S, Wang G, Wang L, Liu C, Yang L (2020) Estimation and inference for generalized geoadditive models. J Am Stat Assoc 115:761–774
ACKNOWLEDGEMENTS
Funding was provided by National Science Foundation awards CCF-1934884, DMS-1916204, Laurence H. Baker Center for Bioinformatics and Biological Statistics at Iowa State University, and College of Liberal Arts and Science’s (LAS) Dean’s Emerging Faculty Leaders award at Iowa State University. The authors would like to thank the editor, the associate editor, and two anonymous referees for their constructive comments.
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Kim, M., Wang, L. & Zhou, Y. Spatially Varying Coefficient Models with Sign Preservation of the Coefficient Functions. JABES 26, 367–386 (2021). https://doi.org/10.1007/s13253-021-00443-5
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DOI: https://doi.org/10.1007/s13253-021-00443-5