# 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 |

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**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 |

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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