Fractal analysis revealed persistent correlations in long-term vegetation fire data in most South and Southeast Asian countries

Vegetation fires are most common in South/Southeast Asian countries (S/SEA). Characterizing the statistical nature of long-term fire datasets can provide valuable information on fire management. Specifically, distinguishing random noise from correlated noise in long-term signals is critical for linking with the underlying processes. Fractal methods can help to assess long-range correlations in long-term timeseries data. This study tested the daily time-series fire data retrieved from the VIIRS satellite (2012–2021) for fractal behavior. Descriptive statistics and popular Detrended Fluctuation Analysis (DFA) were used to assess fire characteristics and persistent versus non-persistent correlations. Results over South Asia (SA) suggested India with the highest mean fire counts (FC) and the least in Bhutan. Fire radiative power (FRP), an indicator of fire intensity, was highest in India and least in Afghanistan. Among Southeast Asia (SEA), Myanmar had the highest mean FC and FRP and least in Timor Leste. The DFA results revealed the fractal nature of FC in different countries. In SA, FC over India, Nepal, Sri Lanka, Afghanistan, and Bhutan showed persistent correlation behavior, whereas anti-persistence in Bangladesh and Pakistan. In addition, FRP showed anti-persistent behavior in Afghanistan, Bangladesh, and Pakistan and a persistent signal for Bhutan, India, Sri Lanka, and Nepal. In contrast to SA, FC and FRP showed persistent behavior in all SEA countries. The persistent or non-persistence nature of the data can help model fire behavior to aid in management and mitigation efforts.


Introduction
The concept of Fractals was first introduced by Mandelbrot (1997) to describe systems that do not have an integer dimension but a fractional one and have the property of self-similarity when observed at different levels of the hierarchy. Since Mandelbrot's pioneering study in mathematics, a new area called fractal geometry developed quickly on the foundation of geometric measure theory, harmonic analysis, dynamical systems, and ergodic theory (Bernaola-Galván et al 1996, Kigami 2001, Gao et al 2007, Feldman 2012. Fractal behavior has been reported earlier for coastlines, clusters of stars, snowflakes, fern leaves, broccoli, and pulmonary vascular trees to the growth and bifurcation of trees and plants, including bacterial colonies and DNA correlations (Mandelbrot 1977, Fujikawa and Matsushita 1989, Matsushita and Fujikawa 1990, Burlando 1993, Voss 1994. Also, the concept of fractals has been applied to a variety of phenomena ranging from electrochemical deposition (Mach et al 1994) to the architecture of physiological systems (Shlesinger 1987, Glenny andRobertson 1991, Carvalho et al 2022). Fractal structures are characterized by the Hausdorff dimension (Balka et al 2015), also known as the 'Fractal Dimension', which is a non-integer dimension that is essentially a measure of 'roughness'. Statistically, self-affine fractals are characterized by the Hurst exponent, which is used as a measure of long-term memory of time series (Schlesinger 1987, Cannon et al 1997. The fractal dimension is evaluated from the slope of the log-log plot of the variance or power spectrum (Pilgrim and Taylor 2019).
Among several methods that can be used to estimate the self-similarity in fractal signals, Detrended Fluctuation Analysis (DFA) is one of the most popular ones (Peng et al 1994, Rigoli et al 2020.
Characterizing the fractal behavior in various datasets is one of the prerequisites to addressing any nonlinearity issues. In this study, fire datasets over South/Southeast Asia (S/SEA) were selected to characterize the fractal behavior. Fires are most common in S/SEA and show significant spatial and temporal variations, including intensity in the region (Justice et al 2015). Figure 1 depicts the spatial variations in fires derived from Visible Infrared Imaging Radiometer Suite (VIIRS) satellite data. Also, figure 2 and3 depict the spatial variation in fires and fire radiative power (FRP, an indicator of intensity) at 0.5degree intervals. The drivers of fires vary in the region (Vadrevu et al 2017). For example, fire is used to clear the forests for agriculture through slash-andburn (Prasad et al 2000, 2002, Biswas et al 2015, Inoue 2018) agricultural residues after crop harvest , Lasko et al 2018a, Vadrevu et al 2017, to clear the land for the next crop, and to clear the forested lands for plantations (Israr et al 2018), promoting the growth of grass in pasture lands for cattle (Stott et al 1990), including reducing weeds before planting of crops (Simorangkir 2007). While most of these fires are anthropogenic, fires can also be natural causes, such as lightning and extreme and prolonged drought conditions. Due to these variations, fires show significant spatial and temporal variations. Especially in the long term, the data sequences can be nonstationary due to the varying nonlinear drivers and mechanisms that control the fire process. Specific to the impacts, fires can threaten human lives and cause severe economic damage, including adverse effects on the ecosystem, such as loss of biodiversity, biomass, nutrients, soil disturbance, etc (Oanh et al 2018. Vegetation fires can also significantly release several greenhouse gas emissions, such as CO, CO2, SO2, hydrocarbons, and other aerosols that can impact local and regional climate (Vadrevu et al 2021a,b). Understanding inherent fire data structure and complexities can provide new insights into fire behavior characteristics useful for fire management and mitigation efforts (Justice et al 2015, Vadrevu et al 2021a. In this study, the daily time-series active fire data retrieved from the VIIRS aboard the joint NASA/NOAA Suomi National Polar-orbiting Partnership (Suomi NPP) from 2012-2021 has been used for the fractal analysis. The first active fires detected with the VIIRS sensor occurred on 19 January 2012, when the instrument was fully commissioned (https://lpdaac.usgs.gov/documents/427/VNP14_User_Guide_V1.pdf). The objective of the study is to characterize fractal behavior (if any) in the daily time-series satellite-derived vegetation fire and fire radiative power (FRP) data in S/SEA countries using the daily long-term datasets (2012)(2013)(2014)(2015)(2016)(2017)(2018)(2019)(2020)(2021). In addition, the implications of the results useful for fire management studies were presented.

Methods
Descriptive statistical metrics of mean, standard deviation, skewness, kurtosis, coefficient of variation, and Mean Absolute Deviation (MAD) were used to understand the statistical nature of the long-term fire datasets (2012-2021). The equations for descriptive statistics are provided in the supplementary material. Then, the fractal behavior in the long-term fire datasets was assessed using the DFA (Peng et al 1994(Peng et al , 1995. The following steps were followed: Consider the correlated signal ( ) u i , where i = 1, K..N max (the total number of points in the series). We integrate the signal ( ) u i and obtain.  1. The integrated signal y(k) is divided into boxes of equal length n; 2. For each n-size box, we fit ( ) y k , using a polynomial function of order l, which represents the trend in the box. The y coordinate of the fitting line in each box is denoted by ( ) y k , n since we use a polynomial fitting of order l, we denote the algorithm by -DFA l 3. The integrated signal ( ) y k is detrended by subtracting the local trend ( ) y k n in each box (of length n); 4. For a given n-size box, the root-mean square fluctuation, F(n), for the integrated and detrended signal is given by, The above computation is repeated for a broad range of scales (n-sizes box) to provide a relationship between ( ) F n and the box size n.
Thus the DFA method provides a relationship between the ( ) F n DFA (Root Mean Square fluctuation) and the time scale n, characterized by a power law: Where µ is the scaling exponent, a self-affinity parameter representing the long-range power-law correlation properties of the signal representing the slope of the line fitting log[F(n)] to log(n). Thus, if µ = 0.5, then the signal is uncorrelated, if µ < 0.5, then the correlation in the signal is anti-persistent, and if µ > 0.5, then the correlation in the signal is persistent.

Results
Vegetation fire counts (FC) and FRP descriptive statistics for different countries based on the daily time series data from 2012-2021 are given in tables 1 and 2 and figures 4, 5(a), (c), (e), (g), (i), (k), (m) and (b), (d), (f), (h), (j), (l), (n), (p), (r). Amongst South Asian countries, India (figure 4(g)) had the highest mean FC, followed by Pakistan, Nepal, etc, and least in Bhutan. On the other hand, the standard deviation and variance in FC were high for India, followed by Nepal, Pakistan, Bangladesh, and others. The horizontal push or pull distortion of a standard distribution curve can be captured by the Skewness, whereas the vertical push and pull distortion is by the Kurtosis measure. Thus, FC in Bhutan and Afghanistan were high in both Skewness and Kurtosis, whereas Bhutan also had the highest coefficient of variation (CV), followed by Bangladesh and Nepal. Mean absolute deviation (MAD) indicates the variability in the average distance between the mean and each observation which was highest for FC in India, followed by Nepal, Bangladesh, and others. Amongst Southeast Asian countries, Myanmar (figure 5(i)) had the highest mean FC, followed by Indonesia, Laos, Cambodia, etc, and least in Timor Leste ( figure 5(o)). The standard deviation and variance in FC for these countries followed a similar trend as the mean. FC in Malaysia, Indonesia, and Timor Leste were high in both Skewness and Kurtosis, whereas Laos, Indonesia, and Timor Leste also had the highest coefficient of variation. MAD was highest for FC in Myanmar, Indonesia, and Cambodia and least for Timor Leste.
In the case of FRP, which is an indicator of fire intensity, among South Asian countries, India (figure 6(g)) had the highest mean FRP, followed by Pakistan, Bangladesh, Nepal, etc, and the least in Afghanistan ( figure 6(a)). In contrast, the standard deviation and variance in FRP were high for India, followed by Bangladesh, Nepal, Pakistan, and others. The FRP in Afghanistan, followed by Bhutan, Bangladesh, and Nepal, were high in both Skewness and Kurtosis, whereas the least for India and Pakistan. Bhutan, Bangladesh, and Afghanistan also had the highest coefficient of variation in FRP. On the other hand, MAD was the highest for FRP in India, followed by Pakistan, Bangladesh, Nepal, and others.
In Southeast Asia, Myanmar had the highest mean FRP (figure 7(i)), followed by Laos, Indonesia, Cambodia, etc, and least in Timor Leste ( figure 7(o)). The standard deviation in FRP was highest for Laos, followed by Myanmar, Indonesia, Cambodia, etc, and least for Timor Leste. Variance in FRP was highest for Laos, Myanmar, Indonesia, Cambodia, etc, and least for Timor Leste. FRP in Timor Leste, Malaysia, and Indonesia was high in Skewness and Timor Leste, followed by Indonesia, Malaysia had the highest Kurtosis. The coefficient of variation was relatively high for Laos, followed by Timor Leste, Indonesia, Myanmar, etc MAD was highest for FRP in Myanmar, Laos, Indonesia, etc, and least for Timor Leste.
The DFA results revealed a fractal nature over several scales for all the fire datasets in different S/SEA countries (figures 4-5(b), (d), (f), (h), (j), (l), (n), (p), (r)). DFA is one of the popular methods for analyzing fractals in time-series datasets, which can help analyze local fluctuations' scaling behavior (Matsoukas et al 2000, Alvarez-Ramirez et al 2008. The DFA approach computes the root mean square (RMS) error of linear fits over progressively larger bins in the time series. A box contains a defined number of samples, and therefore, the scaling properties of the signal can be captured. The relationship between the overall RMS error and the box size is denoted as the fluctuation function. When the fluctuation function follows a linear trend in double-log plots, the slope of this linear trend is a scaling function Alpha which provides the presence of long-range correlations and fractal dynamics (Peng et al 1994). Thus, in the case of long-term fire datasets over South Asia, the alpha (figures 4(a), (c), (e), (g), (i), (k), (m)) was greater than 0.5 for India, Nepal, Sri Lanka, Afghanistan, and Bhutan, suggesting persistent correlation behavior in the long-term fire signal, whereas, for Bangladesh and Pakistan, the correlation in the signal is anti-persistent. The highest persistence was found in fire signals for India (figure 4(g)), and the least persistence was in Bhutan (figure 4(e)). Also, DFA revealed an anti-persistent signal in FRP data for Afghanistan, Bangladesh, and Pakistan (figures 6(a), (c), and (k)). An increasing order in the persistent signal can Table 2. Fire radiative power (FRP in MW) variations for south and southeast asian countries were retrieved using VIIRS active fire data (2012-2021). (Stdev = Standard deviation, CV = coefficient of variation, and MAD is the mean absolute deviation).

Discussion
The persistence versus non-persistence nature of time series datasets is essential in a statistical sense. For example, many statistical signal processing techniques assume the time-series data to be stationary, which may not work in the case of non-stationary data. The fractal or scaling behavior in the data should not be assumed but must be established before further statistical data analysis (Riley et al 2012, Delignières and Marmelat 2013, Pilgrim and Taylor 2019. This is particularly true in the case of long-term fire datasets as they can vary significantly depending on the site conditions, i.e., vegetation type and fuel availability, climate, topography, and  anthropogenic drivers (Telesca and Lasaponara 2006). Also, the vegetation fires in several regions show considerable variations based on seasons and annual cycles with varying amplitudes. Day-to-day variations can also exist based on local management practices (e.g., due to varying agricultural residue-burning practices). In essence, the drivers of the fire are too many, and the data can show both short-term and long-term fluctuations. Thus, it is crucial to understand the scaling behavior in the data, specifically, persistence in the datasets, which indicates non-stationarity. Specifically, persistence behavior in any dataset can be confirmed through the scaling exponent alpha in the DFA. In contrast to persistent processes, which show a positive long-range dependence between the observations, anti-persistent time series depict reverse direction very often and have strong negative autocorrelations (Peng et al 1994, Holl M andKantz H 2015). Thus, persistent signals are relatively easy to predict compared to non-persistent signals because, in the former, small or larger amplitudes are very likely to be followed by even smaller or larger signals (α > 0.5). Further, the higher the α, the stronger the correlations in the signal. In the case of non-persistent signals (α < 0.5), the signal corresponds to long-range anti-correlations, meaning that large values are most likely to be followed by small values and vice versa. The persistence or nonpersistence nature which characterizes the fractal behavior can be helpful in modeling fire behavior. It is essential to document the fractal dimension, as it can change in the forthcoming years based on the driving factors. One way to detect any changes is to plot yearly sequences and compare them with the long-term record of fractal dimension and see if it varied above or below the long-term record. Further, though fires are triggered by a stochastic process and a variety of drivers, past fire activity can influence the evolution of new fires. Since fire regimes (frequency, intensity, timing, duration, and extent) are based on the antecedent conditions, characterizing the fractal characteristics of fires in different landscapes can provide valuable inputs on the better parameterization of the fire behavior models. Users of fire modeling methodologies are mainly fire management agencies, often working under significant time constraints (Pilgrim and Taylor 2019). Thus, incorporating the statistical nature based on the previous history of fires into fire models can increase the robustness of results useful for fire management. For example, (Turco et al 2018) built a non-stationary model to project an increase in future burnt areas (BAs) for the Mediterranean region. Similarly, (Oliveira et al 2021) built a fire spread probability model for the Brazilian Cerrado biome on the non-stationarity relationship between fuel loads and the monthly mean precipitation. More such studies are needed in S/SEA. Specific results from this study on the scaling exponents depicting complex fractal behavior can be used to address fire behavior in these countries. For example, it might be much more challenging to model the fire behavior in Pakistan, which showed antipersistence behavior compared to other countries. Similar is the case with fires in Myanmar and Laos which showed low persistence than in other countries in Southeast Asia. The relative strengths in the scaling behavior characterized in this study can also be used as a metric of fire behavior complexity. Future research focusing on fire behavior modeling will also require a more in-depth multi-fractal assessment of drivers of fires, including biophysical (climate, topography, soils, vegetation, site conditions, etc) and anthropogenic conditions for effective fire management and to determine future fire regimes in the region.  [2012][2013][2014][2015][2016][2017][2018][2019][2020][2021]. India had the highest mean FRP, followed by Pakistan, Bangladesh, Nepal, etc, and least in Afghanistan. An anti-persistent signal in FRP has been found for Afghanistan, Bangladesh, and Pakistan, whereas a persistent signal for Bhutan, India, Sri Lanka, and Nepal.

Conclusions
In this study, the fractal behavior of long-term satellite-derived fire datasets has been tested for long-term correlations. Specifically, daily vegetation fires counts (FC) retrieved from the VIIRS satellite data (2012-2021), including radiative fire radiative power (FRP), an indicator of fire intensity over South/Southeast Asian countries (S/SEA), has been used as a test case. Descriptive statistics and popular Detrended Fluctuation Analysis (DFA) were used to assess fire characteristics and persistent versus non-persistent correlations. It is essential to determine the fractal behavior as typical time-series methods such as autoregressive moving average (ARMA) or ARIMA (I for integrated) can be used for short-range correlations; however, fractal methods are needed to assess long-range correlations. In addition, fractal methods are useful for data that exhibits longmemory processes with self-similar structures. Self-similarity indicates features with the same characteristic value independent of the scale at which the time series is viewed, e.g., independent of the sampling rate of the observations. Using the scaling exponent in DFA, the existence of persistence behavior in any dataset can be confirmed. The DFA results revealed a fractal nature in most countries' fire data. In South Asia, FC over India, Nepal, Sri Lanka, Afghanistan, and Bhutan showed persistent correlation behavior, whereas anti-persistence in Bangladesh and Pakistan. Also, FRP showed anti-persistent behavior in Afghanistan, Bangladesh, and Pakistan, whereas a persistent signal for Bhutan, India, Sri Lanka, and Nepal. In contrast to South Asian countries, FC and FRP in all SEA countries showed persistent behavior. The results suggest caution not to use traditional statistical methods but consider fractal methods to distinguish random noise from structured or correlated noise specific to vegetation fire datasets in S/SEA. The persistence or non-persistence nature of the fires and fire radiative power data highlighted in this study can help build robust fire behavior models to address fire management.