key: cord-1047517-39plzp61
authors: Li, Xing
title: On the multifractal analysis of air quality index time series before and during COVID-19 partial lockdown: A case study of Shanghai, China
date: 2020-11-23
journal: Physica A
DOI: 10.1016/j.physa.2020.125551
sha: 4fa87fc5e53a2112be3ced48588b439e278c752b
doc_id: 1047517
cord_uid: 39plzp61

Due to the COVID-19 pandemic, human activities are largely restricted in Shanghai, China and it is a valuable experiment to testify the correlation of air quality and human activities. In consideration of the complexity of air pollution, this study aims to compare the multifractal characteristics of air quality index (AQI) time series before and during COVID-19 partial lockdown, and analyze the correlations between multifractal parameters of AQI time series and human activities in Shanghai, China. The hourly AQI series in Shanghai from November 27, 2019 to March 23, 2020 is used for this study. Firstly, using the MF-DFA method, the multifractal characteristics of the AQI series are explored. Secondly, the causes of the multifractality of the AQI series are determined. Finally, the correlations between multifractal parameters of AQI time series and human activities are investigated. The multifractal analysis results reveal that the AQI series during COVID-19 partial lockdown also has multifractal characteristics, and the slightly weaker multifractal characteristics and marginally smaller multifractal degree are obtained in comparison with the pre-lockdown phase. However, the contribution of the effective or intrinsic multifractality before and during COVID-19 partial lockdown are very close. The results via the sliding window procedure indicate that the multifractal parameters ([Formula: see text]) show the similar fluctuations along with the fluctuations of passenger volume in Shanghai Metro. Furthermore, it is found that [Formula: see text] and [Formula: see text] and adjusted passenger volume in Shanghai Metro are positively correlated. The possible trend is that the higher adjusted passenger volume is, the larger the value of [Formula: see text] , [Formula: see text] becomes, which means the stronger multifractal characteristics and larger multifractal degree of air quality system.

Here within enclosed is my revised paper for consideration to be published on "Physica A: Statistical Mechanics and Its Applications". The further information about the paper is in the following:

The Title: On the multifractal analysis of air quality index time series before and during COVID-19 partial lockdown: a case study of Shanghai, China The Author: Xing Li Due to the COVID-19 pandemic, human activities are largely restricted in Shanghai, China and it is a valuable experiment to testify the correlation of air quality and human activities. In consideration of the complexity of air pollution, this study aims to compare the multifractal characteristics of air quality index(AQI) time series before and during COVID-19 partial lockdown, and analyze the correlations between multifractal parameters of AQI time series and human activities in Shanghai, China.

The revised main works and conclusions are as follows: 1. the AQI series during COVID-19 partial lockdown also has multifractal characteristics, and the slightly weaker multifractal characteristics and marginally smaller multifractal degree are obtained in comparison with the pre-lockdown phase. 2. The contribution of the effective or intrinsic multifractality before and during COVID-19 partial lockdown are very close. 3 . The three multifractal parameters show the similar fluctuations along with the fluctuations of passenger volume in Shanghai Metro. 4 . The three multifractal parameters and adjusted passenger volume in Shanghai Metro are positively correlated. The possible trend is that the higher adjusted passenger volume is, the stronger multifractal characteristics and larger multifractal degree of air quality system.

The multifractal analysis of AQI series in Shanghai before and after COVID-19 partial lockdown and correlation analysis between multifractal characteristic and human activities provide new insights into the evolution of urban air quality, which can obtain a better understanding of the complex structure of environmental condition. This study is helpful in providing objective guidance and credible decision making support in environmental condition forecasting and regulation.

The author claims that none of the material in the paper has been published or is under consideration for publication elsewhere. I am the corresponding author and my address and other information is as follows, ity index(AQI) time series before and during COVID-19 partial lockdown, and analyze the correlations between multifractal parameters of AQI time series and human activities in Shanghai, China. The hourly AQI series in Shanghai from November 27, 2019 to March 23, 2020 is used for this study. Firstly, using the MF-DFA method, the multifractal characteristics of the AQI series are explored. Secondly, the causes of the multifractality of the AQI series are determined. Finally, the correlations between multifractal parameters of AQI time series and human activities are investigated. The multifractal analysis results reveal that the AQI series during COVID-19 partial lockdown also has multifractal characteristics, and the slightly weaker multifractal characteristics and marginally smaller multifractal degree are obtained in comparison with the pre-lockdown phase. However, the contribution of the effective or intrinsic multifractality before and during COVID-19 partial lockdown are very close. The results via the sliding window procedure indicate that the multifractal parameters(∆H, ∆α, ∆f ) show the similar fluctuations along with the

Since the first infection was diagnosed in Wuhan in December 2019, the novel coronavirus(COVID -19) has been spreading at a speed even beyond the expectation of medical experts. In response, the Chinese government mandated a quarantine of the Wuhan city on Jan 23, 2020. 31 provinces and municipalities, including Beijing and Shanghai, have also taken measures to prevent the spread of the outbreak. Shanghai imposed its highest-level response mechanism for major public health emergencies on January 24, 2020, announcing the most strict prevention and control measures to contain the spread of infection, including health screenings conducted at all road entrances to Shanghai, temporarily suspension of all large public activities, the operation of companies, elementary, middle and high schools, universities and other educational institutions . The public transportation started working with reduced hours. Based on the COVID-19 situation gradually being brought under control, Shanghai decided to lower the provincial public health emergency level from Level I to Level II since 0 am February 24, 2020. This period is a valuable experiment to testify the correlation of air quality and human activities. In consideration of the complexity of air pollution, nonlinear methods to study the evolution of air pollutant have been adopted in the researches. Of these methods, the multifractal method provides a powerful tool for people to deal with complex objects. Detrended fluctuation analysis (DFA) [3] proposed by PENG et al. is suitable for the long-range power law correlation analysis of non-stationary time series. Based on DFA, Kantelhardt et al. [4] gave an improved version, The multifractal detrended fluctuation analysis(MF-DFA), which is capable of studying the multipoint correlation of the non-stationary series. This method can accurately quantify the long-range correlation of nonstationary time series, which is based on random walk theory and can avoid artificially induced time series instability. The MF-DFA has been successfully used in various fields, such as stock market [5] , traffic flow [6] , wind speed [7] etc.

In recent years, more and more scholars have adopted this method to study environmental problems. C.-K. Lee et al. [8] find that there exist multifractal characteristics in the ozone concentration time series in Taipei. A. M. Diosdado et al. [9] provide evidence that the concentration time series of atmospheric pollutants have multifractal characteristics. Z. Liu et al. [10] prove the air pollution index (API), SO 2 , N O 2 and P M 10 time series in Shanghai, China have multifractal characteristics. Q. Dong et al. [11] show P M 2.5 and P M 10 time series in Shanghai, China have multifractal characteristics. C. Shen et al. [12] illustrate the air pollution index (API) time series in Nanjing, China has multifractal characteristics. Q. Wang [13] analyze the multifractal characteristics of polluted time series in Beijing, Zhengzhou, and Jinan.

The plan of this study is to investigate and compare the multifractal characteristics of AQI time series before and during COVID-19 partial lockdown in 

Considering the time series x = {x 1 , x 2 , · · · , x N }, MF-DFA method can be described as following five steps.

(1) Calculate the profile:

The profile is subdivided into N s = int(N/s) non-overlapping windows of equal length s . Since the length N of the series may not be an integer multiple of the window size s, and a short part of the profile y i at the end may be disregarded by the procedure. In order not to disregard this part of the series, the same procedure is repeated starting from the opposite end, obtaining a total of 2N s segments.

(3) Calculate the local trend for each of the 2N s segments by a least-square fit of the series. Then the variance can be determined as

where y v i is the fitting polynomial in segment v.

J o u r n a l P r e -p r o o f (4) Averaging over all segments to get qth order fluctuation function:

Repeat step (2) to (4) with different time scales s, to see how different s affects the dependency of F q (s) on q.

(5) Analyze the log-log plots F q (s) versus s for different q. If the series are long-range power-law correlated, for large values of s, the generalized Hurst

The generalized Hurst exponent defines the fractal structure of the time series is more likely followed by another increase (decrease). When H(2) < 0.5, the time series has an long range anti-correlation structure, an increase (decrease) is more likely followed by another decrease (increase). The range of H(q),

indicates the extent to which the series is multifractal. Higher ∆H means stronger multifractal characteristics. In addition, scaling exponent τ (q) is defined by:

The singularity strength α and the multifractal spectrum f (α) can be calculated via Legendre transform, or be related with H(q) with the following equations:

Multifractal spectrum f (α) describes the fractal dimension of the ensemble formed by all the points that share the same singularity exponent α. Fractal dimension f (α) ∼ α is shaped like a single-peaked bell. The difference between maximum and minimum singularity exponent, ∆α = α max − α min , is called the multifractal spectrum width, that represents the interval between the maximum probability and the minimum probability and measures the degree of the multifractality property. The type of the extreme fluctuation rate occurring with a higher probability can be qualified using ∆f = f (α min ) − f (α max ) which is the difference between the fractal dimension of the maximum probability subset and that of the minimum probability subset. When ∆f > 0, the maximal fluctuation rate occurs with a higher possibility than that of the minimal fluctuation rate and vice versa.

It is usually argued that the sources of multifractality in time series are the fat tails and/or the long-range temporal correlations [4] . More recent studies think the multifractality in time series one observes may originate from long-range nonlinear autocorrelations, the presence of fat tails in probability distributions of data, or linear autocorrelations present in shorter (finite) time series [14] . However, many studies shows that empirical multifractal analysis of time series generated from monofractal models and mathematical models can produce spurious multifractality [15, 16] . Possessing linear correlations or long memory in time series is not sufficient for the emergence of multifractality and a nonlinear process is required to have intrinsic multifractality [17] . The nonlinear correlations are the genuine source of the multifractality [18] [19] . An even more critical question is to ask whether the empirical extracted multifractality is intrinsic or apparent. Understanding the origin of the measured multifractality in time series is an important problem which has attracted many researchers' attention and interests. [14, 18, 19, 20, 21] 

the contribution of the broad probability distribution and probably the finitesize effect can be qualified by [4] ∆α P DF = ∆α shuf .

The Fourier transform surrogate time series can contain the same linear correlations as the original data, while any nonlinear correlations are eliminated and the amplitude distribution becomes Gaussian [22] . The most commonly used methods for generating surrogates include the Fourier transform algorithm [23] , the amplitude adjusted Fourier transform (AAFT) algorithm [23] , and the iterative amplitude adjusted Fourier transform (IAAFT) algorithm [24] .

The shuffling procedure and the Fourier transform surrogate procedure can be combined to understand the sources of the empirically estimated apparent multifractality.

The generalized Hurst exponents can be utilized to measure the multifractality of the time series to analyze the sources of the multifractality. On one hand, the multifractality of broad probability distribution, will not be affected That is, if only the correlation multifractality exists in the time series, H corr (q)

is not zero and depends on q, and H shuf (q) = 0.5 and H(q) = 0.5 + h corr (q)

hold.

The multifractality of the time series can also be characterized by the singularity spectrum, i.e. the width of the multifractal spectrum, with which the quantitative analysis of the origin of multifractality can be done. The width of the original series' multifractal spectrum ∆α can be decomposed to the three main sources of multifractality, namely the nonlinear correlation ∆α N L , the linear correlation ∆α LM and fat-tailed probability distribution ∆α P DF and can be expressed as the following equation [21, 25] :

Because the AAFT algorithm can produce the same power spectrum only for time series with infinite length in the limit N → ∞ and does not usually result in the same sample power spectra [24] . To overcome this shortcoming, the IAAFT algorithm is introduced, which improves the AAFT algorithm [24] . It should be noted that in this study, a more practical and convenient way, which is directly introducing linear correlations into random time series generated from original time series, has been applied to construct the surrogate time series.

By construction, the surrogate series have the same distribution and the same linear correlations as the raw time series, but do not possess any nonlinear correlations [24] . The width of multifractal spectrum of surrogate series can reflect the multifractality degree of the linear correlation part and the PDF part and can be expressed as follows [26] ∆α surr = ∆α LM + ∆α P DF . 

The intrinsic multifractal nature is characterized by the effective multifractality ∆α EF F composing of the nonlinearity component ∆α N L and the PDF component ∆α P DF [21, 25] ∆α EF F = ∆α − ∆α LM = ∆α N L + ∆α P DF .

According to Equations (10)(11)(12), the effective multifractality can also be expressed as:

Based on the previous researches, the simulation procedure and multifractal causes analysis performed in this study are as follows:

Step 1: The origial series is shuffled to remove any potential correlations. The multifractal analysis is conducted on the shuffled series and the multifractal characteristics H shuf (q), ∆α shuf is determined;

Step 2: The surrogate series are obtained by phase-randomizing the orininal series using the IAAFT algorithm. The multifractal analysis is carried out on the surrogate series and H surr (q), ∆α surr are calculated;

Step 3: The steps 1-2 are repeated until 20000 sets of {H shuf (q), H surr (q), ∆α shuf , ∆α surr } of the the original series before and during COVID-19 partial lockdown are accumulated .

Step 4: Then the difference between H shuf (q), H surr (q), ∆α shuf , ∆α surr are checked to determine the components of the multifractality and intrinsic multifractality of the original series before and during COVID-19 partial lockdown respectively.

The original hourly air quality index(AQI) data applied to the multifrac- and 1416/57(before COVID-19 partial lockdown). Compared with the daily data in Air Quality Publishing Platform of China, 2 outlier data is found in the hourly data during COVID-19 partial lockdown. In this paper, arithmetic mean is adopted to handle the missing and outlier data.

The hourly air quality index(AQI) series before and during COVID-19 partial lockdown are illustrated in Fig.1 and their probability density function curve(B and C). 

The generalized Hurst exponent of order q and the multifractal spectrum of the hourly AQI series before and during COVID-19 partial lockdown are shown in Fig. 2 . In addition, in Fig.2 -B, the multifractal spectrum is of a bell shape. Therefore, it can be concluded that the hourly AQI series before COVID-19 partial lockdown has multifractal characteristics. Accordingly, it is not appropriate to describe the hourly AQI series before COVID-19 partial lockdown with a simple single fractal model.

The width of the multifractal spectrum of the hourly AQI series before COVID-19 partial lockdown ∆α bl = 1.09275 and the difference of extreme fractal dimensions ∆f bl = 0.4710. The chance of maximization of the hourly AQI series before COVID-19 partial lockdown is greater than the chance of being at a minimum in that ∆f bl > 0.

For the hourly AQI series during COVID-19 partial lockdown, the similar conclusions from Fig.2 -C and D can be drawn.

In summary, the hourly AQI series in Shanghai both before and during COVID-19 partial lockdown have multifractal characteristics and both are longrange correlated. Moreover, H dl (2) < H bl (2) and ∆α dl < ∆α bl illustrate that the slightly weaker multifractal characteristics and marginally smaller multifractal degree are obtained during COVID-19 partial lockdown.

After shuffling and phase-randomizing the original hourly AQI series before and during COVID-19 partial lockdown 10000 times respectively, the multifractal analysis is conducted on these series and the multifractal characteristics, The multifractal spectra of the shuffled and the surrogate hourly AQI series before and during COVID-19 partial lockdown are shown in Fig.4-A and 4 -B and the numerical results are shown in Table 2 . Table 2 shows that the total multifractality of hourly AQI series before COVID-19 partial lockdown is ∆α bl = 1.09275, and the multifractality con- Table 3 . Table 3 shows that the effective multifractality contributes to 65.38% ± 19.82% of the total multifractality before COVID-19 partial lockdown, and accounts for 65.64% ± 19.33% of the multifractality during COVID-19 partial lockdown. It is obvious that the effective multifractality, composed of the nonlinear correlation part and the PDF part, occupies a larger proportion of the total multifractality. Therefore, it can be concluded that the impact of multifractality generated from the nonlinear correlation part and the PDF part is greater than the linear correlation multifractality. partial lockdown. However, the difference of ∆α EF F are very small in the two periods. Therefore, different from the slightly weaker apparent multifractality in hourly AQI series, the contribution of the effective or intrinsic multifractality before and during COVID-19 partial lockdown are very close. J o u r n a l P r e -p r o o f 

In this section, a sliding (moving) window procedure is utilized to investi- Furthermore, the correlations between multifractal parameters of AQI series and human activities are investigated and the resulting are shown in Fig. 6 .

It is found that ∆H, ∆α, ∆f of AQI series and adjusted passenger volume series in Shanghai Metro are positively correlated. The correlation coefficients, standard errors and t-test on the correlations between ∆H, ∆α, ∆f and adjusted passenger volume are illustrated in Fig.4 . The null hypothesis of t-test is H 0 :

ρ ≤ 0 and the alternative hypothesis is H 1 : ρ > 0.

From Fig.4 , it can be shown that at the significant level of 1%, the null hy-

J o u r n a l P r e -p r o o f three multifractal parameters ∆H, ∆α, ∆f and the adjusted passenger volume are positively correlated. The possible trend is that the higher adjusted passenger volume is, the larger the value of ∆H, ∆α, ∆f becomes, which means the stronger multifractal characteristics and larger multifractal degree of air quality system. It is observed that the correlation coefficient between ∆f and adjusted passenger volume is the largest, which illustrates the ∆f is also a useful indicator of multifractality of AQI series.

Due to the COVID-19 pandemic, human activities are largely restricted in follows:

(1) The hourly AQI series in Shanghai both before and during COVID-19 partial lockdown have multifractal characteristics and both are long-range correlated. Moreover, the slightly weaker multifractal characteristics and marginally smaller multifractal degree are obtained during COVID-19 partial lockdown.

(2) The impact of multifractality generated from the nonlinear correlation part and the PDF part is greater than the linear correlation multifractality. Different from the slightly weaker apparent multifractality in hourly AQI series, the contribution of the effective or intrinsic multifractality before and during COVID-19 partial lockdown are very close. (4) Furthermore, it is found that ∆H, ∆α and ∆f and adjusted passenger volume in Shanghai Metro are positively correlated. The possible trend is that the higher adjusted passenger volume is, the larger the value of ∆H, ∆α, ∆f becomes, which means the stronger multifractal characteristics and larger multifractal degree of air quality system. Therefore, the multifractal analysis of AQI series in Shanghai before and during COVID-19 partial lockdown and correlation analysis between multifractal characteristic and human activities provide new insights into the evolution of urban air quality, which can obtain a better understanding of the complex structure of environmental condition. This study is helpful in providing objective guidance and credible decision making support in environmental condition forecasting and regulation.

Severe air pollution events not avoided by reduced anthropogenic activities during covid-19 outbreak, Resources, Conservation and Recycling 158 (2020) 104814

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