Multifractal Characteristics on Multiple Pollution Variables in Malaysia

Abstract

The Gaussian distribution and random-walk behavior are common assumptions used in complex statistical models. However, in some real case, particularly air pollution modeling, the data always indicate the presence of nonlinear characteristics, such as heavy tails, long-term correlation, and volatility clustering. Thus, Gaussian distribution and random walk are often inherently unsuitable for this data description. This study addresses this issue by investigating the nonlinear characteristics of multiple pollution variables using multifractal. Using data from Klang, Malaysia as a case study, this study shows the significant existence of multifractal properties in air pollution series. Thus, several aspects related to multifractal behavior can be derived to effectively understand the mechanisms governing the dynamics of air pollutant series. Overall, the multifractal approach is a good alternative tool for evaluating the characteristics of air pollution data.

1 Introduction

In the context of environmental studies, air pollution systems can be regarded as a complex phenomenon, which is always influenced by various components, including meteorological factors, atmosphere self-purification, and solar radiation [1]. Similarly, air pollution behavior is constantly affected by local interaction factors. For instance, Feng et al. [2] described that air pollutants are influenced by diverse sources, which include vehicular emissions, industrial processes, and energy production from power stations and various complicated physical and chemical processes. All these factors can be correlated among each other in an open and dissipative system for different timescales [3]. These scenarios cannot generally be fully described using only physical methods or linear statistical models [4]. In fact, a widely used statistical assumption in modeling a real-life dataset is whether the data distribution will follow a normal/Gaussian distribution or random-walk hypothesis [5,6,7,8]. Unfortunately, the data commonly found in many environmental processes do not satisfy either normal distribution or random-walk hypothesis [9,10,11,12]. It is attributed to the intrinsic existence of several properties in the data, which include long or heavy tails [13, 14], long-term correlation [15, 16], volatility clustering [17, 18], and nonlinear characteristics [19, 20]. Thus, the application of normal distribution or random-walk hypothesis cannot explain the behavior of environmental events. In particular, the parameters of a complex phenomenon will exhibit random fluctuations on different temporal scales when characterized via multiple interactions among various components and nonlinear behavior. Thus, investigating the structure and temporal variations of air pollutants using common physical methods or linear statistical models will be difficult.

The nonlinear techniques, such as the multifractal approach, could be a good alternative solution to overcome this issue. The multifractal processes generally characterize the fluctuation of the time series by considering the high signal variability on various levels of temporal scales to depict the series as intermittent with sudden bursts of high-frequency fluctuations [21]. Kantelhardt et al. [22] proposed the multifractal detrended fluctuation analysis (MFDFA) method to investigate the characteristics of multifractal processes. The MFDFA is useful in evaluating the fluctuant average of time series considering partition interval as statistical points. Based on the MFDFA analysis, the Generalized Hurst and Rényi exponents can be determined on the basis of these statistical points to represent the power-law property of the fluctuation function. This information is useful in measuring the stationary and nonstationary sequence structures and fluctuation singularity of the dataset [22]. Moreover, the Generalized Hurst exponent can provide information regarding the complexity and persistence of the time series data [23]. Similarly, the MFDFA method has advantages in describing the long-term correlations of nonstationary time series data [24].

This study proposes the application of multifractal analysis related to air pollution scenarios to study the presence of multifractal characteristics in a series of multiple pollutant variables. These series involve the fluctuation of the following five main air pollutant variables in Malaysia: ozone (O₃), sulfur dioxide (SO₂), carbon monoxide (CO), nitrogen dioxide (NO₂), and particulate matter less than 10 microns in size (PM₁₀).

2 Study Area and Data

This study uses the air pollution data of the Klang city, Malaysia, as a case study. Klang, one of the largest cities in Malaysia, is located at latitude 101° 26ʹ 44.023 E and longitude 3° 2ʹ 41.701 N. The city of Klang has a high population density. This city is also a developed economic and industrial center in Malaysia. Klang was once recognized as the center, being the 13th busiest trans-shipment port and the 16th busiest container port in the world [25]. However, Klang is also exposed to the risk of unhealthy air pollution with the rapid progress of development [26]. Therefore, considering the importance of Klang as a major city in Malaysia, monitoring and analyzing their air pollution behaviors are crucial to obtain useful information related to its air pollution behavior. The current study investigates daily data of air pollutants in Klang from January 1, 2002, to August 31, 2020. The data contain the following primary pollutant variables: nitrogen dioxide (NO₂), sulfur dioxide (SO₂), suspended particulate matter less than ten microns in size (PM₁₀), ozone (O₃), and carbon monoxide (CO). The observed data for O₃, CO, NO₂, and SO₂, were measured in terms of parts per million (ppm) units mass of contaminants. Meanwhile, the observed PM₁₀ data were measured in terms of micrograms per cubic meter (μg/m³). Moreover, these pollutant variables were also measured in terms of different averaging time periods (24 h for SO₂ and PM₁₀, 1 h for NO₂ and O₃, while 8 h for CO). Therefore, the original data of these five pollutant variables are not comparable to each other. Thus, each pollutant variable was standardized to obtain an individual index that had unitless measurement [27, 28]. The standardization process makes the value of individual indices for each pollution variable is comparable. Then, based on the individual indices, the API value at a particular time is determined based on the maximum value among these sub-indices. Details regarding the calculation of API data can be referred to Masseran [29]. Figure 1 shows the process of determining the API value in Malaysia [28]. The API provides a basic indicator regarding the status of air quality at a particular time. A high API value generally indicates poor air quality status.

Fig. 1

Process of determining the API value

3 Multifractal Analysis

Multifractal analysis can be conducted via MFDFA. For a series of $N$ lengths, $\left\{ {x\left( t \right)} \right\},\;t = 1,2, \ldots ,N$, a signal profile series can be obtained as followings:

$$ X\left( i \right) = \sum\limits_{t = 1}^{i} {\left( {x\left( t \right) - \overline{x}} \right)} ,\quad i = 1,2, \ldots ,N, $$

(1)

where $\overline{x} = {{\sum\limits_{t = 1}^{N} {x\left( t \right)} } \mathord{\left/ {\vphantom {{\sum\limits_{t = 1}^{N} {x\left( t \right)} } N}} \right. \kern-\nulldelimiterspace} N}$ is an overall mean of the series. The profile of the time series is also known as the cumulative sums of the series. Next, a profile series $X\left( i \right)$ must be partitioned into $N_{s} = \left[ {N/s} \right]$ non-overlapping segments with a length $s$. Based on the literature, a practical range for the value of s for daily time series data is about 10 to 180, for examples, please refer to [30, 31]. However, if the length $N$ is not an integral multiple considering scale $s$, then some portion of data points in the profile will remain unused. A construction procedure will be repeated starting from the opposite end of the dataset to overcome the aforementioned problem [32]. Thus, a total of $2N_{s}$ segments are consequence of starting from both ends.

A profile with $2N_{s}$ segments will then be used in multifractal analysis. In addition, Zhou [33] combined the MFDFA method and trend-related cross-correlation analysis (DCCA) to extract the information regarding the multifractal features of two cross-correlated nonstationary series. This combination is known as multifractal detrended cross-correlation analysis (MFDCCA). Let $\left\{ {y\left( t \right)} \right\},\;t = 1,2, \ldots ,N$ be another series with a corresponding signal profile given as

$$ Y\left( i \right) = \sum\limits_{t = 1}^{i} {\left( {y\left( t \right) - \overline{y}} \right)} ,\quad i = 1,2, \ldots ,N,$$

(2)

where $\overline{y} = {{\sum\limits_{t = 1}^{N} {y\left( t \right)} } \mathord{\left/ {\vphantom {{\sum\limits_{t = 1}^{N} {y\left( t \right)} } N}} \right. \kern-\nulldelimiterspace} N}$ is an overall mean of the series. The same partition procedures are then applied to a profile series $Y\left( i \right)$ to obtain $N_{s} = \left[ {N/s} \right]$ non-overlapping segments with a length s. The local trends on each $2N_{s}$ segment must be determined on the basis of these profile series by fitting the m-th order of the polynomial curve, which is given as follows:

$$ X^{v} \left( i \right) = \hat{a}_{v,0} + \hat{a}_{v,1} i + \hat{a}_{v,2} i^{2} + \cdots + \hat{a}_{v,m} i^{m} ,\quad i = 1,2, \ldots ,s,$$

(3)

$$ Y^{v} \left( i \right) = \hat{a}_{v,0} + \hat{a}_{v,1} i + \hat{a}_{v,2} i^{2} + \cdots + \hat{a}_{v,m} i^{m} ,\quad i = 1,2, \ldots ,s,$$

(4)

where $X^{v} \left( i \right)$ and $Y^{v} \left( i \right)$ represent the m-th polynomial order in segment v, which is also known as the m-order of MFDFA. Meanwhile, the coefficients of $\hat{a}_{v,0} ,\;\hat{a}_{v,1} , \ldots ,\;\hat{a}_{v,m}$ are parameters for the polynomial function which will differ for each segment v. Equations (3) and (4) indicate the local trend function on v-th segment polynomial of the profile series considering constant trend (m = 0), linear trend (m = 1), quadratic trend (m = 2), or any higher-order trends [34]. In parallel with that, the variance for each segment in $v = 1,2, \ldots ,N_{s}$ for both signal profiles can be determined as follows:

$$ F_{X}^{2} \left( {s,v} \right) = \frac{{\sum\nolimits_{i = 1}^{s} {\left\{ {X\left[ {\left( {v - 1} \right)s + i} \right] - X^{v} \left( i \right)} \right\}^{2} } }}{s}, $$

(5)

$$ F_{Y}^{2} \left( {s,v} \right) = \frac{{\sum\nolimits_{i = 1}^{s} {\left\{ {Y\left[ {\left( {v - 1} \right)s + i} \right] - Y^{v} \left( i \right)} \right\}^{2} } }}{s}.$$

(6)

Similarly, the variance for each segment in $v = N_{s} + 1,, \ldots ,2N_{s}$ can be computed as follows:

$$ F_{X}^{2} \left( {s,v} \right) = \frac{{\sum\nolimits_{i = 1}^{s} {\left\{ {X\left[ {N - \left( {v - N_{s} } \right)s + i} \right] - X^{v} \left( i \right)} \right\}^{2} } }}{s}, $$

(7)

$$ F_{Y}^{2} \left( {s,v} \right) = \frac{{\sum\nolimits_{i = 1}^{s} {\left\{ {Y\left[ {N - \left( {v - N_{s} } \right)s + i} \right] - Y^{v} \left( i \right)} \right\}^{2} } }}{s}. $$

(8)

Meanwhile, the cross-covariance for a pair of signal profiles in each segment in $v = 1,2, \ldots ,N_{s}$ can be determined as follows:

$$ F_{X,Y}^{2} \left( {s,v} \right) = \frac{{\sum\nolimits_{i = 1}^{s} {\left\{ {\left| {X\left( {\left( {v - 1} \right)s + i} \right) - X^{v} \left( i \right)} \right| \cdot \left| {Y\left( {\left( {v - 1} \right)s + i} \right) - Y^{v} \left( i \right)} \right|} \right\}^{2} } }}{s}. $$

(9)

Similarly, the cross-covariance of each segment in $v = N_{s} + 1,, \ldots ,2N_{s}$ can be computed as follows:

$$ F_{X,Y}^{2} \left( {s,v} \right) = \frac{{\sum\nolimits_{i = 1}^{s} {\left\{ {\left| {X\left( {N - \left( {v - N_{s} } \right)s + i} \right) - X^{v} \left( i \right)} \right| \cdot \left| {Y\left( {N - \left( {v - N_{s} } \right)s + i} \right) - Y^{v} \left( i \right)} \right|} \right\}^{2} } }}{s}.$$

(10)

Another important piece of information that must be derived is the q-order fluctuation function, which is determined by averaging $F_{X}^{2} \left( {s,v} \right)$ and $F_{Y}^{2} \left( {s,v} \right)$ over all the segments. A q-order fluctuation function for each of the single profile signals $X^{v} \left( i \right)$ and $Y^{v} \left( i \right)$ can then be, respectively, obtained as:

$$ F_{q}^{X} \left( s \right) = \left\{ {\frac{{\sum\nolimits_{v = 1}^{{2N_{s} }} {\left[ {F_{X}^{2} \left( {s,v} \right)} \right]^{{{\raise0.7ex\hbox{$q$} \!\mathord{\left/ {\vphantom {q 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} } }}{{2N_{s} }}} \right\}^{\frac{1}{q}} ,$$

(11)

$$ F_{q}^{Y} \left( s \right) = \left\{ {\frac{{\sum\nolimits_{v = 1}^{{2N_{s} }} {\left[ {F_{Y}^{2} \left( {s,v} \right)} \right]^{{{\raise0.7ex\hbox{$q$} \!\mathord{\left/ {\vphantom {q 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} } }}{{2N_{s} }}} \right\}^{\frac{1}{q}} .$$

(12)

Meanwhile, the q-order fluctuation function for the cross-correlation profile signals $X^{v} \left( i \right)$ and $Y^{v} \left( i \right)$ can be determined as follows:

$$ F_{q} \left( s \right) = \left\{ {\frac{{\sum\nolimits_{v = 1}^{{2N_{s} }} {\left[ {F_{X,Y}^{2} \left( {s,v} \right)} \right]^{{{\raise0.7ex\hbox{$q$} \!\mathord{\left/ {\vphantom {q 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} } }}{{2N_{s} }}} \right\}^{\frac{1}{q}} ,$$

(13)

Different q values have various effects on the fluctuation functions. If q takes the positive value, then the segment v dominates the average of $F_{q} \left( s \right)$ with large variance, which indicates that the series suffers from a large deviation from the corresponding fit. Thus, the scale invariance of the segments with large fluctuation behavior is indicated by the positive q moments. Similarly, the scale invariance of the segments with small fluctuation behavior is denoted by the negative q moments [34].

4 Multifractal Characteristics

Based on Eqs. (11) and (12), if the profile series are found to be power-law correlated with increasing value of s, then the series of $X^{v} \left( i \right)$ and $Y^{v} \left( i \right)$ will follow a power-law principle, which can be, respectively, described as

$$ F_{q}^{X} \left( s \right)\sim s^{{h_{X} \left( q \right)}},$$

(14)

$$ F_{q}^{Y} \left( s \right)\sim s^{{h_{Y} \left( q \right)}},$$

(15)

where $h_{X} \left( q \right)$ and $h_{Y} \left( q \right)$ are generalized Hurst exponent (GHE) considering a series of $X^{v} \left( i \right)$ and $Y^{v} \left( i \right)$, respectively. The solution on GHE can be obtained by implying the $\ln \left( . \right)$ function on Eqs. (14) and (15). The slope of $\ln \left[ {F_{q} \left( s \right)} \right]$ and $\ln \left( s \right)$ is determined as the value for GHE. The value of GHE provides information regarding the scaling behavior of the investigated time series data. If the GHE function is independent of q moments, then the series indicate monofractal characteristics. By contrast, if the GHE function indicates a dependency on its q moments, then the series indicate multifractal characteristics. Moreover, a pair of series $X^{v} \left( i \right)$ and $Y^{v} \left( i \right)$ indicates a long-range cross-correlation, and the scaling behavior of the fluctuations can be represented as power-law behavior for large values of s as follows:

$$ F_{q} \left( s \right)\sim s^{{H_{XY} \left( q \right)}} .$$

(16)

Equation (16) can be simplified as

$$ \log \left[ {F_{q} \left( s \right)} \right] = H_{XY} \left( q \right)\log \left( s \right) + \log \left( A \right), $$

(17)

where the scaling exponent $H_{XY} \left( q \right)$ is a generalized cross-correlation exponent (GCCE). The cross-correlations between a pair of series are monofractal if the value of GCCE is independent of q. On the contrary, if the value of GCCE is dependent on q, then the cross-correlations between a pair of a series indicate multifractal characteristics. Moreover, the GCCE corresponding to a positive value of q provides information regarding the scaling behavior of the segments with large fluctuations. Meanwhile, for a positive value of q, GCCE provides information regarding the scaling behavior of the segments with small fluctuations.

The GHE and GCCE can also be related to the Rényi exponent. The Rényi exponent,$\tau \left( q \right)$ on every single series of $X^{v} \left( i \right)$ and $Y^{v} \left( i \right)$ is, respectively, determined as the following equations:

$$ \tau_{X} \left( q \right) = qh_{X} \left( q \right) - 1,$$

(18)

$$ \tau_{Y} \left( q \right) = qh_{Y} \left( q \right) - 1. $$

(19)

The Rényi exponent $\tau \left( q \right)$ is similarly determined for a cross-correlation series, where its Rényi exponent is given as follows:

$$ \tau_{XY} \left( q \right) = qH_{XY} \left( q \right) - 1.$$

(20)

The Rényi exponent function indicates a nonlinear relationship on q moments for multifractal series, while the emergence of a linear function indicates a monofractal series [35]. The Rényi exponent also provides information regarding the singularity spectrum on $f\left( \alpha \right)$. Using Legendre transformation, the relationship between Rényi exponent and singularity spectrum can be obtained as follows:

$$ \alpha = \frac{d\tau \left( q \right)}{{dq}}, $$

(21)

$$ f\left( \alpha \right) = q\alpha - \tau \left( q \right). $$

(22)

Based on Eqs. (18)–(22), the singularity spectrum on every single series of $X^{v} \left( i \right)$ and $Y^{v} \left( i \right)$ can be determined as

$$ \alpha_{X} = h_{X} \left( q \right) + qh_{X}^{^{\prime}} \left( q \right),$$

(23)

$$ \alpha_{Y} = h_{Y} \left( q \right) + qh_{Y}^{^{\prime}} \left( q \right),$$

(24)

$$ f_{X} \left( \alpha \right) = q\left( {\alpha_{X} - h_{X} \left( q \right)} \right) + 1,$$

(25)

$$ f_{Y} \left( \alpha \right) = q\left( {\alpha_{Y} - h_{Y} \left( q \right)} \right) + 1. $$

(26)

Meanwhile, the singularity spectrum for a cross-correlation series is determined as follows:

$$ \alpha_{XY} = H_{XY} \left( q \right) + qH_{X}^{^{\prime}} \left( q \right),$$

(27)

$$ f_{XY} \left( \alpha \right) = q\left( {\alpha_{XY} - H_{XY} \left( q \right)} \right) + 1. $$

(28)

The parameter $\alpha$ in Eqs. (21)–(28) is known as a Lipschitz–Holder exponent, which represents the singularity strength of the spectrum. As described by Kantelhardt et al. [22], the singularity spectrum for a multifractal series indicates a single-humped function, while the monofractal series is represented by a single point in the $f\left( \alpha \right)$ plane. Thus, complexity of the series quantified by the measurement of singularity spectra commonly can be described by at most fourth-degree of polynomials [36], which is given as

$$ f\left( \alpha \right) = A + B\left( {\alpha - \alpha_{0} } \right) + C\left( {\alpha - \alpha_{0} } \right)^{2} + D\left( {\alpha - \alpha_{0} } \right)^{3} + E\left( {\alpha - \alpha_{0} } \right)^{4} , $$

(29)

where $\alpha_{0}$ is the value corresponding to the maximum of the spectrum, while the coefficients of A, B, C, D, and E are constant parameters. The difference considering singularity spectrum can be determined as the following equation:

$$ \Delta f\left( \alpha \right) = f_{\max } \left( \alpha \right) - f_{\min } \left( \alpha \right), $$

(30)

where $\Delta f\left( \alpha \right)$ in Eq. (30) indicates the differences between maximum and minimum values of the singularity, which denote an estimation of the spread of changes in fractal patterns. $\Delta f\left( \alpha \right)$ denotes the frequency ratio of the largest to the smallest fluctuation. If $\Delta f\left( \alpha \right) > 0$, then the large fluctuations have more influences than small fluctuations. Similarly, if $\Delta f\left( \alpha \right) < 0$, then the small fluctuations have more influences than large fluctuations [34].

Furthermore, the multifractal strength reflected by the width of the spectrum can be determined using the following equation:

$$ \Delta \alpha = \alpha_{\max } - \alpha_{\min } = h\left( { - \infty } \right) - h\left( \infty \right). $$

(31)

The following equation is also considered:

$$ h\left( q \right) = \frac{1}{q} - \frac{{\ln \left( {a^{q} + b^{q} } \right)}}{q\ln \left( 2 \right)},\quad \quad \left( {a > b} \right) $$

(32)

where a and b are the parameters to be estimated. A large value of $\Delta \alpha$ generally implies a stochastic dynamic character, long-term persistence and strong fluctuations of the original series [37]. A large/small multifractal signal depicted by the value of $\Delta \alpha$ implies a high/low heterogeneous signal. As described by Hou et al. [34], a signal is heterogeneous if it is characterized by sudden bursts of high frequency, intermittencies, and/or irregularities. The other indexes that can be used to describe the multifractal characteristics in air pollution series are given as the following equations:

$$ \Delta \alpha_{L} = \alpha_{0} - \alpha_{\min } $$

(33)

$$ \Delta \alpha_{R} = \alpha_{\max } - \alpha_{0}$$

(34)

where $\alpha_{0}$ is the position of maxima in the singularity spectrum, $\alpha_{\max }$ and $\alpha_{\min }$ are the maximum and minimum values of the Holder exponent, respectively. The values determined by $\Delta \alpha_{L}$ and $\Delta \alpha_{R}$ provide knowledge regarding the left- and right-hand branches of the singularity spectrum curve, in which a description regarding the distribution patterns of high and low fluctuations, respectively, is given [38]. The deviations of the singularity spectrum curve can then be determined using the asymmetry index given as

$$ R = \frac{{\Delta \alpha_{L} - \Delta \alpha_{R} }}{{\Delta \alpha_{L} + \Delta \alpha_{R} }} $$

(35)

where the index of R ranges from − 1 to 1 [39]. The value of R = 0 indicates a symmetrical singularity spectrum characteristic. Meanwhile, for R > 0, the series possibly contains a left-hand deviation of the singularity spectrum, which is affected by some degree of local high fluctuations. By contrast, a right-hand deviation with a local low fluctuation is indicated by the value of R < 0 [34].

5 Results and Discussion

Obtaining some insights into the statistical summary on the data is informative before conducting a detailed analysis of the multifractal behavior. Figure 2 shows the time series plots for each pollutant variable in Klang. Results reveal that the series of daily data of API, SO₂, and PM₁₀ demonstrate more volatile fluctuations compared with other pollutant variables. Thus, these series are most suspected to violate the normality or random-walk processes. By contrast, the series of CO₂ and O₃ pollutants indicate moderate volatile fluctuations, while the NO₂ pollution series is found to have the most stable fluctuation. In addition, there is no evidence of deterministic linear trends or of seasonality effect can be found on each series of pollutant variable. Although the SO₂ series appears to be slightly decreasing, however, based on the Cox-Stuart trend test [40, 41], it is not a significant trend. The characteristics demonstrated in the time series plot are supported by the descriptive statistics presented in Table 1.

Fig. 2

Time series plots of daily pollutant variables in Klang for the period of 2002–2020

Table 1 Descriptive statistics for each daily pollutant series in Klang

The skewness and kurtosis for all pollutants series indicate departures from normality. In a similar vein, the Jarque–Bera statistics for normality tests are found to be significant at the 1% significance level. On the other hand, temporal dependency on each pollutant variable has been preliminary checked using autocorrelation function presented in Fig. 3. The autocorrelation function indicates that there is no seasonality behavior on each series of pollutant variable. In fact, an autocorrelation functions slowly decrease as the lags increase. This preliminary result implies that all the pollutant series in Klang may present long-range persistence or long-memory characteristics. Thus, investigating these characteristics using the multifractal approach could be helpful. In this study, R software [42] corresponding to MFDFA package [43] is used to produce the multifractal plots.

Fig. 3

Autocorrelation plots for each pollutant variable

Figure 4 shows linear relationships in the plots of ln(s) versus ln(F_q(s)) for all q moments on different durations of each air pollution variables series, except for the CO variable with moment q =  − 10. These results imply the existence of power-law relationships on each air pollution series in Klang. As reported by Dong et al. [44], if the regression lines for different orders of q tend to converge, then such convergence indicates the multifractality of the original series. Thus, the linear behavior of fluctuation function plots determined on the logarithmic scale for moments q =  − 10, 0, and 10 provides a preliminary indication of the existence of multifractal behavior on the series.

Fig. 4

Fluctuation function plots of air pollutant variable series in Klang

However, the convergences of fluctuation function corresponding to different orders of q for API, CO, and PM₁₀ series are found to be rather slow and vaguely represented. Meanwhile, the fluctuation functions for O₃, SO₂, and NO₂ series do not converge, thus indicating parallel lines for different moments of q =  − 10, 0, and 10, respectively. As described by Weerasinghe et al. [45], if the regression lines are parallel, then the series could be monofractal. Thus, the plots of Rényi and generalized Hurst exponents are, respectively, provided in Figs. 5 and 6 to conduct further investigation.

Fig. 5

Rényi exponent measure for Klang daily air pollution variable series

Fig. 6

GHE measure for Klang daily air pollution variable series

The Rényi exponent provides information regarding the relationships between τ(q) function for different orders of q. The relationship that exhibits nonlinear properties generally implies the existence of multifractal process characteristics. Figure 5 shows that the API, CO, and PM₁₀ indicate nonlinear property relationships between τ(q) function and their q-moment values. Thus, the multifractal properties might exist on these pollution series. Meanwhile, Rényi exponent plots for the O₃, SO₂, and NO₂ series indicate a linear property, which implies that these pollutant series have monofractal processes. The results presented by Rényi exponent plots provide a similar conclusion to the fluctuation function described in Fig. 4. Figure 6 also shows the Hurst exponent plots on air pollutant variable series in Klang.

The GHE in Fig. 6 shows that the h(q) functions are nonlinear on q for all air pollution series. The GHE measure for API, CO, O₃, and PM₁₀ series nonlinearly decreases when the value of q increases. Meanwhile, the GHE measure for the NO₂ and SO₂ series indicates nonlinear irregular patterns over the value of q. As previously described, the decreasing function indicates the presence of multifractal behavior. This result is consistent with that presented by the fluctuation function and Rényi exponent for all pollutant series, except for the O₃ variable. Moreover, the values of GHE for all pollutant variables are larger than 0.5. This finding implies that all air pollution variables in Klang have long-memory properties, wherein their series indicate persistence in large and small fluctuations. In addition, the air series indicate the nonstationary signals with long-range power-law correlations [46]. However, if the h(q) values are < 0.5, then the air pollution series with short-memory properties indicate intermittent or anti-persistent behavior in large and small fluctuations. By contrast, if the value of h(q) is equal to 0.5, then the air pollution series can be described as having complete randomness behavior.

The series for API, CO, O₃, and PM₁₀ indicate the multifractal properties. Thus, Fig. 7a shows the values of $\Delta h\left( q \right)$ on these series. The figure reveals that the $\Delta h\left( q \right)$ value for the daily CO series is the largest, followed by that for API and PM₁₀, and the smallest $\Delta h\left( q \right)$ is determined by the O₃ pollutant series. This result implies that the information provided by the CO series indicates the most heterogeneity and complexity considering the temporal distribution of pollution behavior, while the O₃ series represent the most homogeneous and regular temporal distribution of air pollution behavior in Klang. This result also indicates a weak degree of multifractality on the O₃ pollutant series. Moreover, the changes in $h\left( q \right)$ function provide information regarding the dependency of the small fluctuation variations if q < 0, and the series indicate the substantial influence of variation for large fluctuations if q > 0. Figure 7b indicates that the fluctuation of CO and O₃ pollutant series is influenced by variations due to their small fluctuations. Particularly, the plot of the CO series shows that value of $\Delta h_{q < 0}$ is larger than that of other series. Meanwhile, the fluctuation of the API and PM₁₀ series is mainly influenced by the variation of their large fluctuations. Thus, the API and PM₁₀ series indicate densely clustered time domains that are more heterogeneous and complex than the sparsely populated time axis.

Fig. 7

Plots for (a) $\Delta h\left( q \right)$ value and (b) sectional $\Delta h\left( q \right)$

Another method to be explored in the multifractal features of air pollution series is by evaluating their singularity spectrum. Figure 8 shows the singularity spectrum $f\left( \alpha \right)$ plots on each air pollution variable series that correspond to the values of their Holder exponent $\alpha$. This figure indicates that the shape of $f\left( \alpha \right)$ for API, CO, O₃, and PM₁₀ series indicates a single-humped function. This scenario illustrates that the temporal dynamics for API, CO, O₃, and PM₁₀ series belonging to multifractal processes with a fourth-degree polynomial can represent the singularity spectra that determine the complexity measures. Meanwhile, unhamped function represented by singularity spectra for NO₂ and SO₂ illustrates that these series belong to monofractal processes. The complexity parameters on each pollutant series are presented in Table 2 to elaborate on the singularity spectra. As mentioned previously, the value of $\alpha_{0}$ represents the position of the maxima in a singularity spectrum. Thus, the pollutant series with a value of the exponent $\alpha_{0}$ larger than 0.5 indicate that their fluctuations exhibit persistent long-term correlations. Table 2 shows that all pollutant series have properties of long-term persistent correlations. Next, the value of $\Delta \alpha = \alpha_{\max } - \alpha_{\min }$ represents the width of the spectrum, which determines the strength of the existing multifractality in the series. This value can be estimated by extrapolating the fitted polynomial to $f\left( {\alpha_{\max } } \right) = f\left( {\alpha_{\min } } \right) = 0$. Table 2 shows the hourly CO series with the largest multifractal strength, followed by API, PM₁₀, and O₃ series considering multifractal properties. Meanwhile, considering the pollutant series that indicate monofractal behavior, the SO₂ is found to have higher fractal strength than the NO₂ pollutant series. In addition, the asymmetry parameter index R represents the degree of skewness of the spectrum, where the value R = 0 represents a symmetrical multifractal spectrum, while R > 0 indicates the properties of the left-skewed spectrum, representing the dominance of high fluctuations. This scenario is presented by the series of API and PM₁₀ pollutants. By contrast, the other pollutant series have the value of R < 0, which implies a right-skewed spectrum, indicating the dominance of low fluctuations. These results provide the same argument with the Rényi and GHE analysis.

Fig. 8

Multifractal/Monofractal spectrum on each pollutant variable

Table 2 Multifractal parameters on air pollution series in Klang

The multifractal behavior described above is influenced by two types of sources: (1) such behavior could stem from different long-range temporal correlations for small and large fluctuations and (2) could also be attributed to a fat-tailed probability distribution of variations [47,48,49]. Thus, their original series will be shuffled to destroy any temporal correlations caused by long- or short-range memories in data to investigate the primary multifractal sources on these pollutant variables. However, the shuffled data will retain the same fluctuation distributions but without any memory. In other words, the shuffled series provide close behavior to random-walk processes but retain weak multifractal properties. Figure 8 reveals the range of change in $f\left( \alpha \right)$, particularly for the CO series, which is significantly reduced after the original series is shuffled. This result implies a reduction considering the degree of multifractality. Shuffling the series moves the position of the spectrum to the left for the API, CO, O₃, and PM₁₀ pollutant series. Thus, this scenario concludes that the multifractal behavior on these pollutant series originate from long-term correlation properties, which implies that a nonlinear temporal correlation constitutes the major contributions in the multifractal formation. The change in significantly narrowed widths of spectra also concludes that the multifractal formation is influenced by non-normal distribution. However, the change in spectra for the shuffled and original series of other O₃ series is insignificantly different. Thus, the existence of multifractal behavior in these series is already weak. Nonlinear temporal correlation or fat-tail distribution does not highly influence the O₃ series data. Therefore, exploring the relationship of multifractal characteristics between these pollutant variables will be informative. Figure 9 shows the fluctuation function of the cross-correlation on each pair of pollutant variables.

Fig. 9

Fluctuation function of cross-correlation on each pair of pollutant variables

Figure 9 shows that all fluctuation function curves are linear, which indicates the existence of power-law cross-correlations between all pairs of pollutant variables. Next, Fig. 10 shows the plot of the Hurst exponent (H_xy(q)) corresponding to its q-moment, where q varies from − 10 to 10, to investigate the multifractal strength on cross-correlations for pairs of pollutant variables. The several pairs of pollutant variables have a large range of decreasing function of H_xy(q) over the value of q, and their H_xy(q) values tend to decrease from larger than 1.5 to smaller than 1.0, particularly for pairs, such as CO-PM₁₀, CO-NO₂, and CO-O₃ relationship. These results imply that the cross-relations on these pairs of pollutant variables have a high degree of multifractal properties. In other scenarios, the Hurst exponent for other pairs of pollutant variables only slightly changed over different q moments. This result indicates that the cross-relations for pairs of pollutant variables, such as CO–SO₂, NO₂–O₃, NO₂–SO₂, NO₂–PM₁₀, O₃–SO₃, O₃–PM₁₀, and SO₂–PM₁₀, only have a weak degree of multifractal properties.

Fig. 10

Hurst exponent for cross-correlation on each pair of pollutant variables

Figure 11 shows the singularity spectrum plots on each pair of pollution variables for further investigation. The results indicate that the singularity spectrum for almost all pairs of pollutant series indicates a single-humped function, except for the CO–SO₂ and NO₂–SO₂ pairs. Thus, the majority of cross-relationships among pollutant variables belong to multifractal processes with different magnitudes of multifractal strength. Meanwhile, the unhamped function represented by singularity spectra for CO–SO₂ and NO₂–SO₂ pairs illustrates that these relationships belong to monofractal processes.

Fig. 11

Multispectra of cross-correlation on each pair of pollutant variables

Table 3 determines the complexity parameters on the cross-correlation for each pair of pollutant variables based on the multispectra plot. The results show that exponent $\alpha_{0}$ parameter for all pairs of pollutant variables indicates a value larger than 0.5, which implies that their fluctuations exhibit persistent long-term cross-correlations. Meanwhile, the cross-correlations for a pair of pollutants represented by CO–NO₂, CO–O₃, and CO–PM₁₀ have the largest multifractal strength compared with others based on the parameter of $\Delta \alpha = \alpha_{\max } - \alpha_{\min }$. Similarly, the values of the asymmetry parameter index are R > 0 for all pairs of cross-correlation pollutants. This finding indicates the properties of the left-skewed spectrum, which implies the dominance of high fluctuations among each pair of pollutant variables.

Table 3 Multifractal parameters on cross-correlation for each pair of pollutant variables

6 Conclusion

This study investigates the multifractal properties on a series of multiple air pollution variables in Malaysia. Five main air pollutant variables are considered in this study: ozone (O₃), sulfur dioxide (SO₂), carbon monoxide (CO), nitrogen dioxide (NO₂), and particulate matter less than 10 microns in size (PM₁₀). In addition, the data of the API series are included in the analysis. The MFDFA and MFDCCA methods have been adopted to investigate the characteristics of fractality on these pollution variables. These methods are useful in evaluating the fluctuant of the marginal air pollutant series and each pair of pollutant variables considering partition interval as statistical points. Marginal and cross-correlation multifractal behavior have been evaluated to represent the power-law property of the fluctuation function, measure the stationary and nonstationary sequence structures, assess the fluctuation singularity, and determine the complexity and persistence of behavior of each pollutant variable series based on the derived statistical points. The results showed that the API, CO, O₃, and PM₁₀ for the marginal pollutant series indicate strong multifractal characteristics. Meanwhile, the NO₂ and SO₂ pollutant series are determined to indicate monofractal characteristics. By contrast, the cross-correlations for a pair of pollutants represented by CO–NO₂, CO–O₃, and CO–PM₁₀ are found to have the largest multifractal strength. The other pairs of pollutants indicate a moderate multifractal strength considering their cross-correlations behavior. Meanwhile, the cross-correlations behavior for CO–SO₂ and NO₂–SO₂ belong to the monofractality characteristics. Overall, this study contributes additional knowledge and provides novel insights to the stochastic processes that generate air quality variability in Malaysia over the two previous decades. Particularly, the findings of this study can help to effectively understand the mechanisms governing the dynamics of air pollutant series in Malaysia and perform successful air quality assessment and management.