# Multifractal Characteristics on Temporal Maximum of Air Pollution Series

## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data

_{10}), nitrogen dioxide (NO

_{2}), ozone (O

_{3}), sulfur dioxide (SO

_{2}), and carbon monoxide (CO). Figure 2 shows the details. The Department of Environment Malaysia uses API as an indicator of the status of air quality at a particular time. In general, a high API indicates poor air quality [51]. Details regarding the calculation of API data can be referred to in Masseran and Safari [52] and Masseran [53].

## 3. Multifractal Spectrum Analysis

## 4. Multifractality Characteristics

## 5. Results and Discussion

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Nomenclature | |

$R$ | Asymmetry index |

${\widehat{a}}_{i}$ | Estimated polynomial coefficient |

$h\left(q\right)$ | Generalized Hurst exponent |

$\overline{y}$ | Mean of the series |

${Y}^{v}\left(i\right)$ | m-th polynomial order in segment v |

${N}_{s}$ | Non-overlapping segments with a length s |

$y\left(t\right)$ | Observed data |

${F}_{q}\left(s\right)$ | q-order fluctuation function |

$\tau \left(q\right)$ | Rényi exponent |

$Y\left(i\right)$ | Signal profile series |

$f\left(\alpha \right)$ | Singularity spectrum |

${F}^{2}\left(s,v\right)$ | Variance for segment v |

Greek symbols | |

$\alpha $ | Lipschitz–Hölder exponent |

${\alpha}_{0}$ | Maxima position in the singularity spectrum |

${\alpha}_{\mathrm{max}}$ | Maximum value of the Hölder exponent |

${\alpha}_{\mathrm{min}}$ | Minimum value of the Hölder exponent |

$\mathsf{\Delta}{\alpha}_{L}$ | Left-hand branch of the singularity spectrum curve |

$\mathsf{\Delta}{\alpha}_{R}$ | Right-hand branch of the singularity spectrum curve |

$\mathsf{\Delta}\alpha $ | Spectrum width |

Acronyms | |

API | Air pollution index |

CO | Carbon monoxide |

MFDFA | Multifractal detrended fluctuation analysis |

NO_{2} | Nitrogen dioxide |

O_{3} | Ozone |

SO_{2} | Sulfur dioxide |

PM_{10} | Suspended particulate matter with size less than 10 microns |

Subscripts | |

s | Length of segment |

q | Fluctuation order |

Superscript | |

v | Segment of the series |

m | Polynomial order |

## References

- Bhat, T.H.; Jiawen, G.; Farzaneh, H. Air Pollution Health Risk Assessment (AP-HRA), Principles and Applications. Int. J. Environ. Res. Public Health
**2021**, 18, 1935. [Google Scholar] [CrossRef] [PubMed] - Chen, F.; Chen, Z. Cost of economic growth: Air pollution and health expenditure. Sci. Total Environ.
**2021**, 755, 142543. [Google Scholar] [CrossRef] [PubMed] - Chen, Y.; Xu, Y.; Wang, F. Air pollution effects of industrial transformation in the Yangtze River Delta from the perspective of spatial spillover. J. Geogr. Sci.
**2022**, 32, 156–176. [Google Scholar] [CrossRef] - Zeng, J.; Wen, Y.; Bi, C.; Feiock, R. Effect of tourism development on urban air pollution in China: The moderating role of tourism infrastructure. J. Clean. Prod.
**2021**, 280, 124397. [Google Scholar] [CrossRef] - Lin, Y.; Huang, R.; Yao, X. Air pollution and environmental information disclosure: An empirical study based on heavy polluting industries. J. Clean. Prod.
**2021**, 278, 124313. [Google Scholar] [CrossRef] - Cariolet, J.-M.; Colombert, M.; Vuillet, M.; Diab, Y. Assessing the resilience of urban areas to traffic-related air pollution: Application in Greater Paris. Sci. Total Environ.
**2018**, 615, 588–596. [Google Scholar] [CrossRef] - Yang, J.; Shi, B.; Shi, Y.; Marvin, S.; Zheng, Y.; Xia, G. Air pollution dispersal in high density urban areas: Research on the triadic relation of wind, air pollution, and urban form. Sustain. Cities Soc.
**2020**, 54, 101941. [Google Scholar] [CrossRef] - Afroz, R.; Hassan, M.N.; Ibrahim, N.A. Review of air pollution and health impacts in Malaysia. Environ. Res.
**2003**, 92, 71–77. [Google Scholar] [CrossRef] - Gocheva-Ilieva, S.G.; Ivanov, A.V.; Voynikova, D.S.; Boyadzhiev, D.T. Time series analysis and forecasting for air pollution in small urban area: An SARIMA and factor analysis approach. Stoch. Environ. Res. Risk Assess.
**2014**, 28, 1045–1060. [Google Scholar] [CrossRef] - Masseran, N. Modeling fluctuation of PM10 data with existence of volatility effect. Environ. Eng. Sci.
**2017**, 34, 816–827. [Google Scholar] [CrossRef] - Masseran, N.; Hussain, S.I. Copula modelling on the dynamic dependence structure of multiple air pollutant variables. Mathematics
**2020**, 8, 1910. [Google Scholar] [CrossRef] - Liu, H.; Yan, G.; Duan, Z.; Chen, C. Intelligent modeling strategies for forecasting air quality time series: A review. Appl. Soft Comput.
**2021**, 102, 106957. [Google Scholar] [CrossRef] - Ravindra, K.; Rattan, P.; Mor, S.; Aggarwal, A.N. Generalized additive models: Building evidence of air pollution, climate change and human health. Environ. Int.
**2019**, 132, 104987. [Google Scholar] [CrossRef] [PubMed] - Roca-Pardiñas, J.; Ordóñez, C. Predicting pollution incidents through semiparametric quantile regression models. Stoch. Environ. Res. Risk Assess.
**2019**, 33, 673–685. [Google Scholar] [CrossRef] - Masseran, N. Modeling the characteristics of unhealthy air pollution events: A copula approach. Int. J. Environ. Res. Public Health
**2021**, 18, 8751. [Google Scholar] [CrossRef] [PubMed] - Álvarez-Liébana, J.; Ruiz-Medina, M.D. Prediction of air pollutants PM
_{10}by ARBX(1) processes. Stoch. Environ. Res. Risk Assess.**2019**, 33, 1721–1736. [Google Scholar] [CrossRef] - Huang, C.; Zhao, X.; Cheng, W.; Ji, Q.; Duan, Q.; Han, Y. Statistical Inference of dynamic conditional Generalized Pareto Distribution with weather and air quality factors. Mathematics
**2022**, 10, 1433. [Google Scholar] [CrossRef] - Jiang, P.; Li, C.; Li, R.; Yang, H. An innovative hybrid air pollution early-warning system based on pollutants forecasting and Extenics evaluation. Knowl. Based. Syst.
**2019**, 164, 174–192. [Google Scholar] [CrossRef] - Masseran, N. Power-law behaviors of the duration size of unhealthy air pollution events. Stoch. Environ. Res. Risk Assess.
**2021**, 35, 1499–1508. [Google Scholar] [CrossRef] - Zhou, Y.; Chang, F.-J.; Chang, L.-C.; Kao, I.-F.; Wang, Y.-S. Explore a deep learning multi-output neural network for regional multi-step-ahead air quality forecasts. J. Clean. Prod.
**2019**, 209, 134–145. [Google Scholar] [CrossRef] - Al-Janabi, S.; Mohammad, M.; Al-Sultan, A. A new method for prediction of air pollution based on intelligent computation. Soft Comput.
**2020**, 24, 661–680. [Google Scholar] [CrossRef] - Sayeed, A.; Choi, Y.; Eslami, E.; Lops, Y.; Roy, A.; Jung, J. Using a deep convolutional neural network to predict 2017 ozone concentrations, 24 hours in advance. Neural Netw.
**2020**, 121, 396–408. [Google Scholar] [CrossRef] - Saez, M.; Barceló, M.A. Spatial prediction of air pollution levels using a hierarchical Bayesian spatiotemporal model in Catalonia, Spain. Environ. Model. Softw.
**2002**, 151, 105369. [Google Scholar] [CrossRef] - Ding, W.; Leung, Y.; Zhang, J.; Fung, T. A hierarchical Bayesian model for the analysis of space-time air pollutant concentrations and an application to air pollution analysis in Northern China. Stoch. Environ. Res. Risk Assess.
**2021**, 35, 2237–2271. [Google Scholar] [CrossRef] - Liu, C.-C.; Lin, T.-C.; Yuan, K.-Y.; Chiueh, P.-T. Spatio-temporal prediction and factor identification of urban air quality using support vector machine. Urban Clim.
**2022**, 41, 101055. [Google Scholar] [CrossRef] - Gyarmati-Szabó, J.; Bogachev, L.V.; Chen, H. Nonstationary POT modelling of air pollution concentrations: Statistical analysis of the traffic and meteorological impact. Environmetrics
**2017**, 28, e2449. [Google Scholar] [CrossRef] [Green Version] - Masseran, N.; Mohd Safari, M.A. Intensity–duration–frequency approach for risk assessment of air pollution events. J. Environ. Manag.
**2020**, 264, 110429. [Google Scholar] [CrossRef] - Vettori, S.; Huser, R.; Genton, M.G. Bayesian modeling of air pollution extremes using nested multivariate max-stable processes. Biometrics
**2019**, 75, 831–841. [Google Scholar] [CrossRef] - Yadav, M.; Singh, N.K.; Sahu, S.P.; Padhiyar, H. Investigations on air quality of a critically polluted industrial city using multivariate statistical methods: Way forward for future sustainability. Chemosphere
**2022**, 291, 133024. [Google Scholar] [CrossRef] - Wang, Q. Multifractal characterization of air polluted time series in China. Phys. A Stat. Mech. Appl.
**2019**, 514, 167–180. [Google Scholar] [CrossRef] - Li, X. On the multifractal analysis of air quality index time series before and during COVID-19 partial lockdown: A case study of Shanghai, China. Phys. A Stat. Mech. Appl.
**2021**, 565, 125551. [Google Scholar] [CrossRef] [PubMed] - Frenzel, S.; Pompe, B. Partial mutual information for coupling analysis of multivariate time series. Phys. Rev. Lett.
**2007**, 99, 204101. [Google Scholar] [CrossRef] [PubMed] - Jauregui, M.; Zunino, L.; Lenzi, E.K.; Mendes, R.S.; Ribeiro, H.V. Characterization of time series via Rényi complexity–entropy curves. Phys. A Stat. Mech. Appl.
**2018**, 498, 74–85. [Google Scholar] [CrossRef] [Green Version] - Kantelhardt, J.W.; Zschiegner, S.A.; Koscielny-Bunde, E.; Havlin, S.; Bunde, A.; Stanley, H.E. Multifractal detrended fluctuation analysis of nonstationary time series. Phys. A Stat. Mech. Appl.
**2002**, 316, 87–114. [Google Scholar] [CrossRef] [Green Version] - Carrizales-Velazquez, C.; Donner, R.V.; Guzmán-Vargas, L. Generalization of Higuchi’s fractal dimension for multifractal analysis of time series with limited length. Nonlinear Dyn.
**2022**, 108, 417–431. [Google Scholar] [CrossRef] - Jiang, P.; Wang, B.; Li, H.; Lu, H. Modeling for chaotic time series based on linear and nonlinear framework: Application to wind speed forecasting. Energy
**2019**, 173, 468–482. [Google Scholar] [CrossRef] - Zou, Y.; Donner, R.V.; Marwan, N.; Donges, J.F.; Kurths, J. Complex network approaches to nonlinear time series analysis. Phys. Rep.
**2019**, 787, 1–97. [Google Scholar] [CrossRef] - Chen-hua, S.; Yi, H.; Ya-ni, Y. An analysis of multifractal characteristics of API time series in Nanjing, China. Phys. A Stat. Mech. Appl.
**2016**, 451, 171–179. [Google Scholar] - Liu, Z.; Wang, L.; Zhu, H. A time–scaling property of air pollution indices: A case study of Shanghai, China. Atmos. Pollut. Res.
**2015**, 6, 886–892. [Google Scholar] [CrossRef] - Xu, W.; Liu, C.; Shi, K.; Liu, Y. Multifractal detrended cross-correlation analysis on NO, NO
_{2}and O_{3}concentrations at traffic sites. Phys. A Stat. Mech. Appl.**2018**, 502, 605–612. [Google Scholar] [CrossRef] - Manimaran, P.; Narayana, A.C. Multifractal detrended cross-correlation analysis on air pollutants of University of Hyderabad Campus, India. Phys. A Stat. Mech. Appl.
**2018**, 502, 228–235. [Google Scholar] [CrossRef] - Cárdenas-Moreno, P.R.; Moreno-Torres, L.R.; Lovallo, M.; Telesca, L.; Ramírez-Rojas, A. Spectral, multifractal and informational analysis of PM
_{10}time series measured in Mexico City Metropolitan Area. Phys. A Stat. Mech. Appl.**2021**, 565, 125545. [Google Scholar] [CrossRef] - Masseran, N. Multifractal characteristics on multiple pollution variables in Malaysia. Bull. Malays. Math. Sci. Soc.
**2022**, 45, 325–344. [Google Scholar] [CrossRef] - Plocoste, T.; Pavón-Domínguez, P. Temporal scaling study of particulate matter (PM
_{10}) and solar radiation influences on air temperature in the Caribbean basin using a 3D joint multifractal analysis. Atmos. Environ.**2020**, 222, 117115. [Google Scholar] [CrossRef] - Wang, J.; Shao, W.; Kim, J. Multifractal detrended cross-correlation analysis between respiratory diseases and haze in South Korea. Chaos Solitons Fractals
**2020**, 135, 109781. [Google Scholar] [CrossRef] - Wang, J.; Kim, J.; Shao, W. Investigation of the implications of “Haze Special Law” on air quality in South Korea. Complexity
**2020**, 2022, 6193016. [Google Scholar] [CrossRef] [Green Version] - Zhang, C.; Ni, Z.; Ni, L. Multifractal detrended cross-correlation analysis between PM2.5 and meteorological factors. Phys. A Stat. Mech. Appl.
**2015**, 438, 114–123. [Google Scholar] [CrossRef] - Gin, O.K. Historical Dictionary of Malaysia; Scarecrow Press: Lanham, MD, USA, 2009; pp. 157–158. [Google Scholar]
- Masseran, N.; Safari, M.A.M. Risk assessment of extreme air pollution based on partial duration series: IDF approach. Stoch. Environ. Res. Risk Assess.
**2020**, 34, 545–559. [Google Scholar] [CrossRef] - Google. 2019. Available online: https://maps.googleapis.com/maps/api/geocode/json?address=Klang%2CSelangor&key=xxx (accessed on 13 April 2022).
- Department of Environment. A Guide to Air Pollutant Index in Malaysia (API); Ministry of Science, Technology and the Environment: Kuala Lumpur, Malaysia, 1997. Available online: https://aqicn.org/images/aqi-scales/malaysia-api-guide.pdf (accessed on 13 February 2022).
- Masseran, N.; Safari, M.A.M. Mixed POT-BM approach for modeling unhealthy air pollution events. Int. J. Environ. Res. Public Health
**2021**, 18, 6754. [Google Scholar] [CrossRef] - Masseran, N. Power-law behaviors of the severity levels of unhealthy air pollution events. Nat. Hazards
**2022**, 112, 1749–1766. [Google Scholar] [CrossRef] - Cao, G.; He, L.-Y.; Cao, J. Multifractal Detrended Analysis Method and Its Application in Financial Markets; Springer: Singapore, 2018. [Google Scholar]
- Hou, W.; Feng, G.; Yan, P.; Li, S. Multifractal analysis of the drought area in seven large regions of China from 1961 to 2012. Meteorol. Atmos. Phys.
**2018**, 130, 459–471. [Google Scholar] [CrossRef] - da Silva, H.S.; Silva, J.R.S.; Stosic, T. Multifractal analysis of air temperature in Brazil. Phys. A Stat. Mech. Appl.
**2020**, 549, 124333. [Google Scholar] [CrossRef] - Chattopadhyay, A.; Khondekar, M.H.; Bhattacharjee, A.R. Fractality and singularity in CME linear speed signal: Cycle 23. Chaos Solit. Fractals
**2018**, 114, 542–550. [Google Scholar] [CrossRef] - Mali, P.; Manna, S.K.; Mukhopadhyay, A.; Haldar, P.K.; Singh, G. Multifractal analysis of multiparticle emission data in the framework of visibility graph and sandbox algorithm. Phys. A Stat. Mech. Appl.
**2018**, 493, 253–266. [Google Scholar] [CrossRef] - Sun, Y.; Yuan, X. Nonlinear relationship between money market rate and stock market liquidity in China: A multifractal analysis. PLoS ONE
**2021**, 16, e0249852. [Google Scholar] [CrossRef] [PubMed] - Wu, W.; Yuan, N.; Xie, F.; Qi, Y. Understanding long-term persistence and multifractal behaviors in river runoff: A detailed study over eastern China. Phys. A Stat. Mech. Appl.
**2019**, 533, 122042. [Google Scholar] [CrossRef] - Adarsh, S.; Nourani, V.; Archana, D.S.; Dharan, D.S. Multifractal description of daily rainfall fields over India. J. Hydrol.
**2020**, 586, 124913. [Google Scholar] [CrossRef] - Xie, S.; Bao, Z. Fractal and multifractal properties of geochemical fields. Math. Geol.
**2004**, 36, 847–864. [Google Scholar] [CrossRef] - Dong, Q.; Wang, Y.; Li, P. Multifractal behavior of an air pollutant time series and the relevance to the predictability. Environ. Pollut.
**2017**, 222, 444–457. [Google Scholar] [CrossRef] - Weerasinghe, R.M.; Pannila, A.S.; Jayananda, M.K.; Sonnadara, D.U.J. Multifractal behavior of wind speed and wind direction. Fractals
**2016**, 24, 1650003. [Google Scholar] [CrossRef] - Miloş, L.R.; Haţiegan, C.; Miloş, M.C.; Barna, F.M.; Boțoc, C. Multifractal detrended fluctuation analysis (MF-DFA) of stock market indexes. Empirical evidence from seven central and Eastern European markets. Sustainability
**2020**, 12, 535. [Google Scholar] [CrossRef] - Shi, K. Multifractal processes and self-organized criticality of PM
_{2.5}during a typical haze period in chengdu, China. Aerosol Air Qual. Res.**2015**, 15, 926–934. [Google Scholar] [CrossRef] [Green Version] - Xue, Y.; Pan, W.; Lu, W.-Z.; He, H.-D. Multifractal nature of particulate matters (PMs) in Hong Kong urban air. Sci. Total Environ.
**2015**, 532, 744–751. [Google Scholar] [CrossRef] [PubMed] - Kwapien, J.; Oswiecimka, P.; Drozdz, S. Components of multifractality in high-frequency stock returns. Phys. A Stat. Mech. Appl.
**2005**, 350, 466–474. [Google Scholar] [CrossRef]

**Figure 2.**Process of determining the API value [53].

**Figure 7.**(

**a**) $\mathsf{\Delta}h\left(q\right)$ values for API series with different durations in Klang, (

**b**) sectional $\mathsf{\Delta}h\left(q\right)$.

Variable | Mean | Variance | Min. | Max. | Median | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

Hourly API | 55.735 | 434.448 | 0 | 543 | 54 | 4.738 | 68.370 |

Max. Daily API | 65.530 | 548.758 | 21 | 543 | 61 | 5.561 | 74.658 |

Max. Weekly API | 83.382 | 1129.12 | 40 | 543 | 76 | 5.608 | 55.403 |

Max. Monthly API | 105.861 | 2444.431 | 60 | 543 | 93 | 4.779 | 33.222 |

Duration | ${\mathit{\alpha}}_{\mathbf{min}}$ | ${\mathit{\alpha}}_{\mathbf{max}}$ | ${\mathit{\alpha}}_{0}$ | $\mathsf{\Delta}{\mathit{\alpha}}_{\mathit{L}}$ | $\mathsf{\Delta}{\mathit{\alpha}}_{\mathit{R}}$ | $\mathsf{\Delta}{\mathit{\alpha}}_{}$ | ${\mathit{R}}_{}$ | $\mathsf{\Delta}\mathit{f}\left(\mathit{\alpha}\right)$ |
---|---|---|---|---|---|---|---|---|

Hourly | 1.322 | 12.746 | 1.511 | 0.189 | 11.235 | 11.424 | −0.967 | −3.779 |

Daily | 0.237 | 1.180 | 0.952 | 0.715 | 0.228 | 0.943 | 0.516 | −1.490 |

Weekly | 0.157 | 1.144 | 0.843 | 0.686 | 0.301 | 0.988 | 0.390 | −1.254 |

Monthly | 0.190 | 0.259 | 0.844 | 0.654 | 0.585 | 1.239 | 0.056 | −1.107 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. 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

**MDPI and ACS Style**

Masseran, N.
Multifractal Characteristics on Temporal Maximum of Air Pollution Series. *Mathematics* **2022**, *10*, 3910.
https://doi.org/10.3390/math10203910

**AMA Style**

Masseran N.
Multifractal Characteristics on Temporal Maximum of Air Pollution Series. *Mathematics*. 2022; 10(20):3910.
https://doi.org/10.3390/math10203910

**Chicago/Turabian Style**

Masseran, Nurulkamal.
2022. "Multifractal Characteristics on Temporal Maximum of Air Pollution Series" *Mathematics* 10, no. 20: 3910.
https://doi.org/10.3390/math10203910