# Multivariate Multifractal Detrending Moving Average Analysis of Air Pollutants

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## Abstract

**:**

## 1. Introduction

## 2. Multivariate Multifractal Detrending Moving Average Analysis

**Step 1:**- Calculate cumulative sums of each time series $i=1,2,\dots ,p$:$${Y}_{t,i}=\sum _{k=1}^{t}{y}_{k,i}.$$
**Step 2:**- Calculate the moving average function of each time series $i=1,2,\dots ,p$ in a moving window of size n:$${\tilde{Y}}_{t,i}=\frac{1}{n}\sum _{k=-\u230a(n-1)\theta \u230b}^{\u2308(n-1)(1-\theta )\u2309}{Y}_{t-k,i},$$
**Step 3:**- Calculate the series residue by subtracting the moving average function ${\tilde{Y}}_{t,i}$ from ${Y}_{t,i}$:$${e}_{t,i}={Y}_{t,i}-{\tilde{Y}}_{t,i},$$
**Step 4:**- Divide the residue series ${e}_{t,i}$ into ${N}_{n}=\u230aN/n-1\u230b$ non-overlapping segments of equal length n. The segments are denoted by ${e}_{v}$ such that ${e}_{{v}_{t,i}}={e}_{l+t,i}$ for $1\le t\le n$ with $l=(v-1)n$.
**Step 5:**- Calculate the fluctuation variance ${F}^{2}(v,n)$ as a function of n for an arbitrary segment v:$${F}^{2}(v,n)=\frac{1}{n}\sum _{t=1}^{n}\left|\right|{e}_{v}{\left|\right|}^{2},$$
**Step 6:**- Average over all local variances ${F}^{2}(v,n)$ to obtain the ${q}^{th}$ order fluctuation function:$${F}_{q}^{MV-MFDMA}\left(n\right)={\left\{\frac{1}{{N}_{n}}\sum _{v=1}^{{N}_{n}}\left[{F}^{q}(v,n)\right]\right\}}^{\frac{1}{q}}$$$${F}_{0}^{MV-MFDMA}\left(n\right)=\frac{1}{{N}_{n}}\sum _{v=1}^{{N}_{n}}ln\left[F(v,n)\right].$$
**Step 7:**- Vary the values of segment size n to determine the power law relation between the function ${F}_{q}^{MV-MFDMA}\left(n\right)$ and the size scale n. If a time series exhibits multifractal properties, then ${F}_{q}^{MV-MFDMA}\left(n\right)$ for large values of n will follow a power law type of scaling relation, such as:$${F}_{q}^{MV-MFDMA}\left(n\right)\sim {n}^{h\left(q\right)},$$

## 3. Numerical Experiments on Synthetic Data Sets

#### 3.1. Independent Multivariate Monofractal Series

#### 3.2. Correlated Bivariate Monofractal Series

#### 3.3. Multivariate Multifractal Series

## 4. Air Pollutant Time Series via MV-MFDMA

#### 4.1. Data

_{3}levels particularly stand out. Since seasonal trends can sometimes influence the multifractal behavior of time series, it is preferable to perform seasonal decomposition; see for example [15,26,27]. It is interesting to note that a common approach for analyzing trendless fluctuations is absent. Nigmatullin and Vorobev [28] stated that in most cases, authors use traditional methods, such as the Fourier method, the wavelet method, Yulmenteyev’s method, and Timashev’s method or some additional processing algorithms that are also based on conventional methods containing some model assumptions and treatment methods associated with continuous mathematics. Nigmatullin, Lino, and Maione [29] pointed out that these sets of methods solve some specific tasks, but cannot be viewed as universal. To address this gap, a universal “platform” for treating various types of different trendless sequences has been proposed (see [28,29] for more details). For the purposes of this paper, we adjusted the considered time series by performing seasonal and trend decomposition using LOESS (STL) decomposition [30]. Through this algorithm, each time series is additively decomposed into deterministic trends, seasonal components, and stochastic remainder components. Removing seasonal components from the raw time series does not significantly influence the descriptive statistics results. All the descriptive statistics and unit root tests conclusions from the raw data are valid for the seasonally adjusted data as well (see Appendix A). Figure 5 shows that the seasonal components of all time series are characterized by the annual oscillation. Figure 5 also demonstrates, among others, the evolution trend of five air pollutant concentration series. The PM

_{2.5}and SO

_{2}series are more prone to extreme spikes, whereas NO

_{2}, CO, and O

_{3}concentrations have somewhat different trends in winter and summer, indicating a seasonal inclination. On the other hand, the trend components are characterized by a very small range of variability and do not show a large temporal evolution. The range of variability is the largest for series NO

_{2}and PM

_{2.5}. The remainder components of all time series do not display discernible patterns, and there are small fluctuations around zero.

#### 4.2. Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

CO | NO_{2} | O_{3} | PM_{2.5} | SO_{2} | ||
---|---|---|---|---|---|---|

Mean | 0.4381 | 21.4366 | 0.0279 | 8.8198 | 1.9964 | |

Std. Dev. | 0.2422 | 10.7698 | 0.0088 | 7.4037 | 2.6018 | |

Skewness | 1.2054 | 0.3761 | 0.1193 | 6.5486 | 6.0940 | |

Kurtosis | 6.0841 | 3.0578 | 3.7585 | 96.2279 | 102.8836 | |

Jarque–Bera | 2131.856 * | 79.1695 * | 87.9690 * | 1,233,060 * | 1,408,679 * | |

Observations | 3339 | 3339 | 3339 | 3339 | 3339 | |

ADF (AIC) | intercept | −4.3080 * | −4.6241 * | −6.5685 * | −12.3825 * | −7.2438 * |

trend and intercept | −4.4298 * | −4.7864 * | −6.5767 * | −12.3867 * | −8.5387 * | |

ADF (SIC) | intercept | −5.8546 * | −5.7474 * | −7.9224 * | −16.3571 * | −10.2605 * |

trend and intercept | −5.9854 * | −5.9493 * | −7.9653 * | 16.3589 * | −11.4403 * | |

PP | intercept | −26.7990 * | −33.2027 * | −39.6205 * | −19.1607 * | −51.9978 * |

trend and intercept | −26.9846 * | −34.1516 * | −39.8951 * | −19.1657 * | −49.5006 * |

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**Figure 6.**Power law dependence of the fluctuation functions with respect to the scale (log-log plots).

Method | Description | Comments |
---|---|---|

MFDFA [4] | • applies to univariate time series | • requires a larger sample size for accurate estimates |

• detects a long-range correlation and multifractal properties | • non-overlapping segmentation can cause pseudo fluctuation errors | |

• detrends the original series by removing its average | ||

MFDMA [6] | • applies to univariate time series | • requires a larger sample size for accurate estimates |

• detects a long-range correlation and multifractal properties | • the backward MFDMA outperforms the MFDFA | |

• detrends the original series by removing the moving average function | ||

MVDFA [1] | • applies to multivariate time series | • preserves the characteristics of the univariate DFA |

• represents a generalization of the DFA | • characterized by one scaling exponent | |

• describes the autocorrelations’ behavior | • requires an equal length of time series | |

MV-MFDFA [12] | • applies to multivariate time series | • preserves the characteristics of the univariate MFDFA |

• represents a generalization of the MFDFA | • requires an equal length of time series | |

• detects a long-range correlation and the multifractal properties of multichannel data directly | ||

MV-MFDMA Section 2 | • applies to multivariate time series | • preserves the characteristics of the univariate MFDMA |

• represents a generalization of the MFDMA | • requires an equal length of time series | |

• detects a long-range correlation and the multifractal properties of multichannel data directly | • represents an alternative of both the MV-MFDFA and the MFDMA |

Variable | Time-Span | Unit | Frequency | State | Monitor Site |
---|---|---|---|---|---|

Carbon Monoxide (CO) | 2010–2019 | ppm | daily | California | 60010009 |

Nitrogen Dioxide (NO_{2}) | 2010–2019 | ppb | daily | California | 60010009 |

Ozone (O_{3}) | 2010–2019 | ppm | daily | California | 60010009 |

Particulate Matter (PM_{2.5}) | 2010–2019 | g/m^{3} | daily | California | 60010009 |

Sulfur Dioxide (SO_{2}) | 2010–2019 | ppb | daily | California | 60010011 |

CO | NO_{2} | O_{3} | PM_{2.5} | SO_{2} | ||
---|---|---|---|---|---|---|

Mean | 0.4375 | 21.4214 | 0.0280 | 8.8429 | 1.9949 | |

Std. Dev. | 0.2508 | 11.0571 | 0.0091 | 7.7520 | 2.7298 | |

Skewness | 1.4251 | 0.6122 | 0.1371 | 8.5209 | 7.9297 | |

Kurtosis | 7.4013 | 3.4522 | 4.2429 | 131.7860 | 141.7981 | |

Jarque–Bera | 3825.237 * | 237.0335 * | 225.3758 * | 2,347,909 * | 2,715,223 * | |

Observations | 3339 | 3339 | 3339 | 3339 | 3339 | |

ADF (AIC) | intercept | −4.4779 * | −4.8280 * | −6.6122 * | −12.1969 * | −7.4589 * |

trend and intercept | −4.5956 * | −5.0117 * | −6.6436 * | −12.1979 * | −8.6055 * | |

ADF (SIC) | intercept | −5.9964 * | −5.8778 * | −8.0346 * | −16.5856 * | −10.4407 * |

trend and intercept | −6.1243 * | −6.0941 * | −8.0991 * | −16.5850 * | −11.4677 * | |

PP | intercept | −27.8682 * | −34.4109 * | −40.4884 * | −19.5129 * | −52.3557 * |

trend and intercept | −28.07811 * | −35.2493 * | −40.7289 * | −19.5138 * | −50.2368 * |

**Table 4.**The scaling exponent $h\left(2\right)$, multifractality degrees $\mathsf{\Delta}h$, and $\mathsf{\Delta}\alpha $: raw series.

CO | NO_{2} | O_{3} | PM_{2.5} | SO_{2} | Multiv. | ||
---|---|---|---|---|---|---|---|

$\theta =0$ | $h\left(2\right)$ | 0.9980 | 1.0164 | 1.0190 | 0.9823 | 0.9785 | 1.0076 |

$\theta =0.5$ | 0.9295 | 0.8741 | 0.7073 | 0.7071 | 0.6414 | 0.8463 | |

$\theta =1$ | 0.9937 | 1.0126 | 1.0190 | 0.9775 | 0.9837 | 1.0101 | |

$\theta =0$ | $\mathsf{\Delta}h$ | 0.2204 | 0.1544 | 0.0559 | 0.3006 | 0.2016 | 0.1525 |

$\theta =0.5$ | 0.2074 | 0.1658 | 0.0301 | 0.2875 | 0.4126 | 0.1807 | |

$\theta =1$ | 0.2126 | 0.1501 | 0.0559 | 0.2844 | 0.3074 | 0.1387 | |

$\theta =0$ | $\mathsf{\Delta}\alpha $ | 0.3888 | 0.2678 | 0.1078 | 0.6490 | 0.5710 | 0.3099 |

$\theta =0.5$ | 0.3854 | 0.3003 | 0.0650 | 0.6950 | 0.7838 | 0.4080 | |

$\theta =1$ | 0.3667 | 0.2573 | 0.1078 | 0.6045 | 0.6098 | 0.2681 |

**Table 5.**The scaling exponent $h\left(2\right)$, multifractality degrees $\mathsf{\Delta}h$, and $\mathsf{\Delta}\alpha $: seasonally adjusted series.

CO | NO_{2} | O_{3} | PM_{2.5} | SO_{2} | Multiv. | ||
---|---|---|---|---|---|---|---|

$\theta =0$ | $h\left(2\right)$ | 1.1277 | 1.2088 | 1.0583 | 1.0980 | 1.1991 | 1.1140 |

$\theta =0.5$ | 1.1333 | 0.9998 | 0.7086 | 0.8579 | 0.7928 | 0.9192 | |

$\theta =1$ | 1.1399 | 1.1881 | 1.0608 | 1.1425 | 1.2301 | 1.1149 | |

$\theta =0$ | $\mathsf{\Delta}h$ | 0.1758 | 0.2485 | 0.0522 | 0.2778 | 0.2814 | 0.1551 |

$\theta =0.5$ | 0.2988 | 0.1793 | 0.0190 | 0.3510 | 0.2923 | 0.1559 | |

$\theta =1$ | 0.2002 | 0.2162 | 0.0570 | 0.3208 | 0.3142 | 0.1490 | |

$\theta =0$ | $\mathsf{\Delta}\alpha $ | 0.3051 | 0.4907 | 0.1006 | 0.6269 | 0.5462 | 0.3076 |

$\theta =0.5$ | 0.6336 | 0.3357 | 0.0385 | 0.8267 | 0.5706 | 0.3557 | |

$\theta =1$ | 0.3834 | 0.4052 | 0.1150 | 0.7193 | 0.6135 | 0.2891 |

**Table 6.**The scaling exponent $h\left(2\right)$, multifractality degrees $\mathsf{\Delta}h$, and $\mathsf{\Delta}\alpha $: remainder component.

CO | NO_{2} | O_{3} | PM_{2.5} | SO_{2} | Multiv. | ||
---|---|---|---|---|---|---|---|

$\theta =0$ | $h\left(2\right)$ | 0.6209 | 0.6123 | 0.5481 | 0.3940 | 0.4522 | 0.5727 |

$\theta =0.5$ | 1.0275 | 0.9819 | 0.8858 | 0.7224 | 0.6188 | 0.9279 | |

$\theta =1$ | 0.6206 | 0.6065 | 0.5603 | 0.3878 | 0.4435 | 0.5669 | |

$\theta =0$ | $\mathsf{\Delta}h$ | 0.4764 | 0.3253 | 0.3109 | 0.6585 | 0.5222 | 0.3471 |

$\theta =0.5$ | 0.1945 | 0.1448 | 0.1793 | 0.4061 | 0.3391 | 0.1974 | |

$\theta =1$ | 0.4954 | 0.3624 | 0.3160 | 0.6304 | 0.3737 | 0.3632 | |

$\theta =0$ | $\mathsf{\Delta}\alpha $ | 0.8184 | 0.6024 | 0.5106 | 1.1022 | 1.0444 | 0.7440 |

$\theta =0.5$ | 0.3740 | 0.2705 | 0.3500 | 0.7443 | 0.6627 | 0.4341 | |

$\theta =1$ | 0.8240 | 0.6388 | 0.5368 | 1.0571 | 0.7054 | 0.7603 |

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**MDPI and ACS Style**

Kojić, M.; Mitić, P.; Dimovski, M.; Minović, J.
Multivariate Multifractal Detrending Moving Average Analysis of Air Pollutants. *Mathematics* **2021**, *9*, 711.
https://doi.org/10.3390/math9070711

**AMA Style**

Kojić M, Mitić P, Dimovski M, Minović J.
Multivariate Multifractal Detrending Moving Average Analysis of Air Pollutants. *Mathematics*. 2021; 9(7):711.
https://doi.org/10.3390/math9070711

**Chicago/Turabian Style**

Kojić, Milena, Petar Mitić, Marko Dimovski, and Jelena Minović.
2021. "Multivariate Multifractal Detrending Moving Average Analysis of Air Pollutants" *Mathematics* 9, no. 7: 711.
https://doi.org/10.3390/math9070711