Modeling Air Pollution Using Partially Varying Coefficient Models with Heavy Tails
Abstract
:1. Introduction
2. The Proposed Model
2.1. The Model
2.2. Distributional Assumption
2.3. Penalized Log-Likelihood Function
3. Parameter Estimation and Inference
3.1. Resolving the Estimation Equations
3.2. Approximate Standard Errors
3.3. On Degrees of Freedom and Smoothing Parameters
3.4. Selecting an Appropriate Model
4. Diagnostics
4.1. Residual Analysis
4.2. Local Influence Method
5. Applications and Results
5.1. Chile Air Pollution
5.2. Lima Air Pollution
6. Conclusions, Limitations, and Future Research
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Penalized Hessian Matrix
References
- MMA. Establishment of Primary Quality Guideline for Inhalable Fine Particulate Matter PM2.5; Technical Report Decree 12; Ministry of Environment of the Chilean Government: Santiago, Chile, 2021.
- Puentes, R.; Marchant, C.; Leiva, V.; Figueroa-Zuñiga, J.; Ruggeri, F. Predicting PM2.5 and PM10 Levels During Critical Episodes Management in Santiago, Chile, with a bivariate Birnbaum-Saunders Log-Linear Model. Mathematics 2021, 9, 645. [Google Scholar] [CrossRef]
- Tapia, V.; Carbajal, L.; Vásquez, V.; Espinoza, R.; Vásquez-Velásquez, C.; Steenland, K.; Gonzales, G. Traffic regulation and environmental pollution by particulate material (2.5 and 10), sulfur dioxide, and nitrogen dioxide in Metropolitan Lima, Peru. Rev. Peru. De Med. Exp. Y Salud Pública 2018, 35, 190. [Google Scholar] [CrossRef]
- Cordova, C.H.; Portocarrero, M.N.L.; Salas, R.; Torres, R.; Rodrigues, P.C.; López-Gonzales, J.L. Air quality assessment and pollution forecasting using artificial neural networks in Metropolitan Lima-Peru. Sci. Rep. 2021, 11, 24232. [Google Scholar] [CrossRef]
- Yáñez, M.; Baettig, R.; Cornejo, J.; Zamudio, F.; Guajardo, J.; Fica, R. Urban airborne matter in central and southern Chile: Effects of meteorological conditions on fine and coarse particulate matter. Atmos. Environ. 2017, 161, 221–234. [Google Scholar] [CrossRef]
- Clements, N.; Hannigan, M.; Miller, S.; Peel, J.; Milford, J. Comparisons of urban and rural PM10-2.5 and PM2.5 mass levels and semi-volatile fractions in northeastern Colorado. Atmos. Chem. Phys. 2016, 16, 7469–7484. [Google Scholar] [CrossRef] [Green Version]
- Carreño, G.; López-Cortés, X.A.; Marchant, C. Machine Learning Models to Predict Critical Episodes of Environmental Pollution for PM2.5 and PM10 in Talca, Chile. Mathematics 2022, 10, 373. [Google Scholar] [CrossRef]
- Hastie, T.; Tibshirani, R. Generalized Additive Models; Chapman and Hall: London, UK, 1990. [Google Scholar]
- Ibacache-Pulgar, G.; Reyes, S. Local influence for elliptical partially varying-coefficient model. Stat. Model. 2018, 18, 149–174. [Google Scholar] [CrossRef]
- Ibacache-Pulgar, G.; Paula, G.; Cysneiros, F. Semiparametric additive models under symmetric distributions. Test 2013, 22, 103–121. [Google Scholar] [CrossRef]
- Ibacache-Pulgar, G.; Paula, G.A.; Galea, M. Influence diagnostics for elliptical semiparametric mixed models. Stat. Model. 2012, 12, 165–193. [Google Scholar] [CrossRef]
- Lira, V.; Ibacache-Pulgar, G.; Villegas, C. Assessing influence in the varying-coefficient generalized linear model. REVSTAT-Stat. J. 2022, in press. [Google Scholar]
- Cook, R.D. Assessment of local influence (with discussion). J. R. Soc. 1986, 48, 133–169. [Google Scholar]
- Zhang, J.; Zhang, X.; Ma, H.; Zhiya, C. Local influence analysis of varying-coefficient linear model. J. Interdiscip. Math. 2015, 3, 293–306. [Google Scholar] [CrossRef]
- Ferreira, C.S.; Paula, G.A. Estimation and diagnostic for skew-normal partially linear models. J. Appl. Stat. 2017, 44, 3033–3053. [Google Scholar] [CrossRef]
- Emami, H. Local influence for Liu estimators in semiparametric linear models. Stat. Pap. 2017, 19, 529–544. [Google Scholar] [CrossRef]
- Ibacache-pulgar, G.; Figueroa-Zúñiga, J.; Marchant, C. Semi-parametric additive beta regression models: Inference and local influence diagnostics. REVSTAT-Stat. J. 2021, 19, 255–274. [Google Scholar]
- Moraga, M.; Ibacache-Pulgar, G.; Nicolis, O. On Elliptical Thin-Plate Spline Partially Varying-Coefficient Model. Chil. J. Stat. 2021, 12, 205–227. [Google Scholar]
- Ibacache-Pulgar, G.; Villegas, C.; López-Gonzales, J.L.; Moraga, M. Influence Measures in Symmetric Nonparametric Regression Model. Stat. Methods Appl. 2022. [Google Scholar] [CrossRef]
- Cárcamo, E.; Marchant, C.; Ibacache-Pulgar, G.; Leiva, V. Birnbaum-Saunders semi-parametric additive modelling: Estimation, smoothing, diagnostics, and application. REVSTAT-Stat. J. 2022, in press. [Google Scholar]
- Cavieres, J.; Ibacache-Pulgar, G.; Contreras-Reyes, J. Smoothing Thin-Plate Spline under Skew Normal setting using Laplace Approximation and Influence Diagnostics Analysis. J. Stat. Comput. Simul. 2022, in press.
- Adams, R.; Fournier, J. Sobolev Spaces; Elsevier: Oxford, UK, 2003. [Google Scholar]
- Green, P.; Silverman, B. Nonparametric Regression and Generalized Linear Models: A Roughness Penalty; Chapman and Hall/CRC: London, UK, 1994. [Google Scholar]
- Berhane, K.; Tibshirani, J. Generalized additive models for longitudinal data. Can. J. Stat. 1998, 26, 517–535. [Google Scholar] [CrossRef]
- Wahba, G. Bayesian confidence intervals for the cross-validated smoothing spline. J. R. Stat. Soc. 1983, 45, 133–150. [Google Scholar] [CrossRef]
- Segal, M.; Bacchetti, P.; Jewell, P. Variances for maximum penalized likelihood estimates obtained via the EM algorithm. J. R. Stat. Soc. 1994, 56, 345–352. [Google Scholar] [CrossRef]
- Hastie, T.; Tibshirani, R. Varying-Coefficient Models. J. R. Stat. Soc. 1993, 55, 757–796. [Google Scholar] [CrossRef]
- Buja, A.; Hastie, T.; Tibshirani, R. Linear smoothers and additive models. Ann. Stat. 1989, 17, 453–555. [Google Scholar] [CrossRef]
- Rigby, R.A.; Stasinopoulos, D.M. Generalized additive models for location, scale and shape. Appl. Stat. 2005, 54, 507–554. [Google Scholar] [CrossRef] [Green Version]
- Opsomer, D. Asymptotic Properties of Backfitting Estimators. J. Multivar. Anal. 2000, 73, 166–179. [Google Scholar] [CrossRef]
- Poon, W.; Poon, Y.S. Conformal normal curvature and assessment of local influence. J. R. Stat. Soc. 1999, 61, 51–61. [Google Scholar] [CrossRef]
- MMA. Establishes a Prevention and Atmospheric Decontamination Plan for the Santiago Metropolitan Region; Technical Report Decree 31; Ministry of Environment of the Chilean Government: Santiago, Chile, 2017.
- Encalada-Malca, A.A.; Cochachi-Bustamante, J.D.; Rodrigues, P.C.; Salas, R.; López-Gonzales, J.L. A Spatio-Temporal Visualization Approach of PM10 Concentration Data in Metropolitan Lima. Atmosphere 2021, 12, 609. [Google Scholar] [CrossRef]
- MINAM. Approval of Environmental Quality Standards for Air and Establish Complementary Provisions; Supreme Decree N 003-2017; MINAM: Lima, Peru, 2017.
- Lange, K.L.; Little, R.J.A.; Taylor, J.M.G. Robust statistical modeling using the t distribution. J. Am. Stat. Assoc. 1989, 84, 881–889. [Google Scholar] [CrossRef]
Variable | n | Min | Max | Range | Mean | Median | SD | CV | CK |
---|---|---|---|---|---|---|---|---|---|
PM | 146 | 13 | 318 | 305 | 107.60 | 102.50 | 57.74 | 0.54 | 1.53 |
Parameter | Estimate | SE | AIC | |
---|---|---|---|---|
17.326 | 0.0168 | −708 | 1437 | |
0.4318 | 0.0101 | |||
559.42 | 0.0000 |
Dropped | |||
---|---|---|---|
20 | |||
33 | |||
58 | |||
70 | |||
75 | |||
I |
Variable | n | Min | Max | Range | Mean | Median | SD | CV | CS | CK |
---|---|---|---|---|---|---|---|---|---|---|
PM | 306 | 22.22 | 294.97 | 272.75 | 108.80 | 93.32 | 59.21 | 0.544 | 0.76 | 2.67 |
Parameter | Estimate | SE | L | AIC |
---|---|---|---|---|
12.16 | 45.800 | −1381 | 2794 | |
0.410 | 0.0104 | |||
281.0 | 30.080 |
Dropped | |||
---|---|---|---|
17 | |||
29 | |||
30 | |||
51 | |||
54 | |||
160 | |||
165 | |||
167 | |||
171 | |||
178 | |||
181 | |||
212 | |||
I |
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Jeldes, N.; Ibacache-Pulgar, G.; Marchant, C.; López-Gonzales, J.L. Modeling Air Pollution Using Partially Varying Coefficient Models with Heavy Tails. Mathematics 2022, 10, 3677. https://doi.org/10.3390/math10193677
Jeldes N, Ibacache-Pulgar G, Marchant C, López-Gonzales JL. Modeling Air Pollution Using Partially Varying Coefficient Models with Heavy Tails. Mathematics. 2022; 10(19):3677. https://doi.org/10.3390/math10193677
Chicago/Turabian StyleJeldes, Nicole, Germán Ibacache-Pulgar, Carolina Marchant, and Javier Linkolk López-Gonzales. 2022. "Modeling Air Pollution Using Partially Varying Coefficient Models with Heavy Tails" Mathematics 10, no. 19: 3677. https://doi.org/10.3390/math10193677
APA StyleJeldes, N., Ibacache-Pulgar, G., Marchant, C., & López-Gonzales, J. L. (2022). Modeling Air Pollution Using Partially Varying Coefficient Models with Heavy Tails. Mathematics, 10(19), 3677. https://doi.org/10.3390/math10193677