Precise prediction of urban air pollution is vital not only for people's health but also for environmental management. The present research proposes a hybrid forecasting framework that integrates traditional time-series models (ARIMA, SARIMAX, Prophet), machine-learning techniques (LSTM), and fractal-based indicators to produce forecasts of air pollutant concentrations for Astana, Kazakhstan. The methodology combines fractal analysis with multimodel forecasting, estimating fractal features such as the Hurst exponent, DFA scaling, and fractal dimension to capture both short-term dynamics and long-range dependencies. The results reveal that LSTM networks combined with fractal metrics outperform traditional statistical models, achieving the highest prediction accuracy for PM2.5, PM10, CO, NO2, and SO2 levels. The hybrid ensemble method improved overall accuracy and robustness by capturing persistent nonlinear trends. By combining fractal analysis with machine learning, the approach enables more reliable urban air quality forecasts, supporting timely public health interventions and data-driven pollution mitigation. The results indicate the usefulness of fractal-based models for urban air quality monitoring and, therefore, support the implementation of advanced pollution mitigation strategies.
Urban air pollution is one of the major threats that directly impacts the quality of life in cities,
environmental stability, and regulatory planning [
Forecasting urban air pollution is one of the most challenging tasks in environmental science, largely
due to the complexity of environmental time series data. Nonstationarity, long-range dependence,
and stochastic fluctuations are inherent characteristics that traditional linear models fail to adequately
capture [
Fractal analysis ofers a robust approach to quantifying these structural complexities. Metrics such
as the Hurst exponent and fractal dimension reflect persistence, roughness, and long-memory efects
that frequently appear in pollutant time series [
Machine Learning (ML) and Deep Learning (DL) methods have also proven efective in modeling
nonlinear, high-dimensional time series [
To address these limitations, this study proposes a Hybrid Fractal–Machine Learning (HFML)
approach [
Thus, the aim of this study is to overcome the limitations of existing forecasting methods by developing a framework that combines fractal analysis with machine learning for the analysis of urban air pollution data. To achieve this aim, the following tasks are formulated: 1. To compute the key fractal characteristics of pollutant time series, in particular the Hurst exponent as an indicator of long-term memory, and to assess the degree of nonlinear persistence in the data. 2. To integrate fractal features as a preprocessing stage into predictive models (ARIMA, SARIMAX,
Prophet, and LSTM), thereby forming a hybrid forecasting approach. 3. To evaluate the efectiveness of the proposed hybrid model in comparison with traditional approaches using real-world air pollution data from the city of Astana, testing the hypothesis that hybrid models provide higher forecasting accuracy and robustness.
Unlike previous studies that separately examined fractal properties or applied machine learning techniques to pollution data, this work combines both approaches by directly embedding fractal characteristics into the forecasting process.
Urban air pollution presents a complex and pressing challenge for public health, environmental
management, and regulatory planning. The accurate prediction of the concentrations of pollutants like
PM2.5, PM10, CO, NO2, and SO2 necessitates the application of models that are capable of depicting the
non-linear, non-stationary, and long-range dependent dynamics that are typical for the aforementioned
environmental time series [
Environmental systems are significantly impacted by the interactions of physical, chemical, and
humancaused processes, resulting in irregular and highly dynamic behavior [
The application of fractal theory extensions to urban meteorology, for instance, and anomalous
difusion processes has evidenced that not just the pollutant complexity is characterized by these
parameters but also the interactions with environmental and human factors are disclosed [18]. Besides
air, water, and soil pollution, fractal analysis has been employed in several studies, which showed its
strength in capturing non-linear, persistent and chaotic dynamics that are the basis of environmental
monitoring and forecasting [
Fractal methods are applied as quantitative tools for the analysis of environmental time series data that
are irregular and scale-invariant in nature. The is used for the quantitative expression of a dataset’s
geometric complexity or roughness, while the is associated with the characteristic of long-term
memory and persistence [
Fractal dimension, Hurst exponent, and multifractal spectrum together allow the researchers to spot
the stochastic noise and to distinguish it from the deterministic patterns that lead to the pollution
dynamics. This provides a multi-faceted understanding of the hierarchical and non-linear nature of
pollution dynamics [
Fractal and multifractal techniques have managed to find a wide application in the analysis of urban air
pollution. In the case of Astana, Kazakhstan, Biloshchytskyi and colleagues [
Besides, more and more researchers acknowledged the practical utility of fractal analysis in the management of air quality. Conducted as a typical example, the study by Evagelopoulos et al. [20] in Greece made it possible to distinguish between background and episodic pollution, and the one by Prada et al. [23] in Colombia revealed policy-relevant persistence in PM2.5 and PM10 concentrations. The recent research indicates the incorporation of multi-source data and spatiotemporal analysis for a full-fledged representation of urban pollution dynamics [ 24]. The sum total of these findings is that the use of fractal and multifractal metrics has become indispensable in predictive modeling of urban air quality.
Although fractal methods efectively capture non-linear and persistent dynamics, their power to predict
is still being improved through the combination with ML, DL, and advanced statistical models. The
hybrid frameworks are making use of the transparency of the fractal measures and the power of the
data-driven algorithms to predict urban pollution more accurately [
The integration of fractal and hybrid models is a step forward in urban air pollution monitoring as it is revealing the scale-invariant structures while still making use of the state-of-the-art prediction algorithms. These methods are the best in environmental forecasting and they are also providing a solid base for pollutant dynamics accurately anticipating and informing mitigation strategies [27, 28].
Although previous studies have applied fractal analysis to characterize pollution time series and used modern ML methods for forecasting, only a few works combine these two approaches within a single predictive framework. The literature lacks empirical evidence that fractal indicators can directly improve the predictive performance of machine-learning models for air quality. This article fills this gap.
This study is based on a comprehensive environmental dataset from Astana, Kazakhstan, comprising hourly measurements of key air pollutants and meteorological variables. The dataset provides a detailed representation of urban air quality dynamics, forming the empirical foundation for fractal time series analysis and hybrid machine learning forecasting.
The dataset used in this study includes key air quality and meteorological parameters essential for fractal analysis and hybrid forecasting. The primary air quality indicators comprise the Air Quality Index (AQI), which provides an aggregated measure of overall air pollution, and particulate matter concentrations PM2.5 and PM10, representing particles with aerodynamic diameters less than or equal to 2.5 m and 10 m, respectively.
Gaseous pollutants included in the dataset are carbon monoxide (CO), sulfur dioxide (SO2), and nitrogen dioxide (NO2). In addition, meteorological variables such as temperature, relative humidity, precipitation, wind speed, wind direction, and atmospheric pressure were incorporated to support comprehensive environmental analysis.
Data were collected from multiple publicly accessible and reliable sources to ensure high temporal resolution and broad spatial coverage across the city of Astana: • Kazhydromet National Monitoring Network [29] – providing real-time and historical air quality measurements from automatic and stationary monitoring stations; • AQI India Kazakhstan Dashboard [30] – supplying real-time AQI and pollutant concentration data; • Meteostat Python Library [31] – delivering historical meteorological datasets for preprocessing and analytical integration.
An overview of the dataset structure and representative records is shown in Figure 1. The dataset spans the period from January 1, 2020, to November 9, 2025, enabling the capture of extended environmental variability and supporting robust fractal and predictive modeling. All observations originate from Meteostat monitoring station in Astana “Astana/Prigorodnyy” (ID: UACC0).
The compiled dataset contains 15 columns in tabular form, with each row representing a single hourly measurement. The structure includes: • Air quality indicators: AQI, PM2.5, PM10, CO, SO2, NO2, and O3; • Meteorological variables: temperature, dew point, relative humidity, precipitation, wind speed, wind direction, and atmospheric pressure.
Most columns are numerical types, supporting statistical and fractal analysis. The AQI and pollutant columns reflect hourly concentrations essential for air quality monitoring, while meteorological data (e.g., temperature, humidity, wind) provide key context for interpreting pollution variability. Non-null counts indicate data completeness for each column; minor gaps in meteorological variables (e.g., humidity, precipitation) are addressed during preprocessing. The dataset is structured for straightforward integration into fractal time series analysis pipelines and further modeling.
To ensure the quality and reliability of the dataset for fractal analysis and predictive modeling, a multi-stage data cleaning and preprocessing pipeline was implemented.
Duplicate removal and validation. Duplicate rows, identified via the timestamp column, were removed to prevent redundancy and bias in time series modeling. Pollutant measurements (PM2.5, PM10, CO, SO2, NO2, O3) were checked for invalid values, such as negative or zero readings. These values were replaced with missing entries (NaN) to maintain physical validity.
Temporal regularization. The timestamp column was assessed for uniform hourly intervals. Irregularities in sampling were corrected through interpolation or appropriate handling of missing entries, ensuring a consistent temporal resolution across the dataset.
Handling missing meteorological data. Continuous meteorological variables such as temperature, dew point, relative humidity, atmospheric pressure, and wind speed—were interpolated using timeaware methods based on temporal proximity. Wind direction, being a cyclic variable, was interpolated via trigonometric transformation to avoid discontinuities across the 0°–360° boundary.
Precipitation data treatment. Precipitation (prcp) values required specialized procedures: • short gaps of up to three consecutive hours were interpolated; • single missing entries flanked by zero precipitation were imputed as zero; • longer missing intervals were left unchanged to preserve data integrity.
The resulting dataset is cleaned, regularized, and minimally gapped, providing a reliable foundation for fractal time series characterization and hybrid machine learning forecasting.
Before performing fractal analysis and prediction an exhaustive exploratory study was carried out to describe the statistical and temporal characteristics of the dataset. The case of PM2.5 is taken as an example for its decisive contribution to the evaluation of urban air quality and its very high sensitivity to changes in weather and human activities. The same analysis process was performed on the other pollutant variables.
The temporal evolution of PM2.5 concentrations is illustrated in Figure 2. The hourly time series was visualized to inspect long-term trends, seasonal fluctuations, and short-term irregularities. The plot reveals that there is considerable variability across all hours, days, and seasons, which is characteristic of the emission intensification that is known to occur in wintertime in northern cities such as Astana. The visualization also shows some sharp peaks, which may be the result of pollution episodes or faulty sensors. All these findings point to the need for a more detailed decomposition into the individual components.
The series was broken into parts using Seasonal-Trend decomposition based on Loess (STL) in order to inspect the hidden structure. The trend component shows the changes in the long run driven by heating seasons, industrial cycles, and meteorological variability. The seasonal component shows a very strong intraweek and yearly periodicity, while the residual component takes short-term fluctuations that are not explained by deterministic patterns. This splitting gives a very crucial ground for both fractal analysis—where long-range dependence is sensitive to structural components, and ML models, which are able to separate the signals more clearly as a result. Figure 3 presents the STL decomposition results, consisting of four subplots: the original PM2.5 time series at daily resolution, the extracted long-term trend component, the seasonal component capturing recurring periodic patterns, and the residual component representing short-term irregular fluctuations not explained by the trend or seasonality.
The autocorrelation function (ACF) and partial autocorrelation function (PACF) plots were employed to reveal the characteristics of cyclicity, persistence and lag dependencies. The ACF shows that there is a strong correlation at short as well as long lags thus indicating the presence of slow-decaying memory efects which are typical for fractal and long-range dependent processes. The PACF spots the main short-range dependencies and the corresponding points of the cutof for possible hybrid ARIMA, SARIMA, or SARFIMA components. The periodicity of these structures allows integrating fractal measures in the modeling process, especially the Hurst exponent and the DFA-based scaling exponents. The ACF and PACF plots shown in Figures 4 and 5, respectively, were employed to reveal the characteristics of cyclicity, persistence, and lag dependencies.
For the purpose of forecasting, the time series was assessed through the application of both the Augmented Dickey–Fuller (ADF) and Kwiatkowski–Phillips–Schmidt–Shin (KPSS) tests. The ADF test determines if the series is characterized by a unit root (non-stationarity caused by stochastic trends), while the KPSS test checks for stationarity around a deterministic trend. In accordance with previous similar studies, PM2.5 demonstrated mixed behaviors of stationarity: ADF most often indicates non-stationarity, while KPSS confirms trend-stationary or weakly non-stationary structure most of the time. Detrending, seasonal adjustment, and possibly fractional diferencing in subsequent modeling, are backed by these findings. These results are summarized in Table 1.
Note: Variables in ADF test formula: – observed PM2.5 concentration at time , Δ – first diference, – time index, – intercept, – deterministic trend coeficient, – unit root parameter, – short-term dynamic coeficients, – lag order, – white-noise error. Variables in KPSS test formula – stochastic trend (KPSS), – stationary residual, – random walk innovation, – cumulative sum of residuals, ^2 – long-run variance, – sample size.
Likely stationary
−14.2523
Comparison: −14.2523 < all critical values ⇒ stationary.
Series is stationary because | | is far below critical thresholds.
KPSS tests (random walk): = + , = −1 + (2) KPSS 12 ∑ ︀^2 =1 (3)
=1 2 , = ∑︀ Fail to reject 0: trend-stationary process. Statistic (0.3894) is near but below most thresholds ⇒ stationary.
Series exhibits stationarity under KPSS framework.
Distributional analyses were conducted using histogram estimates, which reveal that PM2.5 exhibits a right-skewed and heavy-tailed distribution. This behavior indicates more often a low concentration of pollutants with rare severe pollution events. Box plots were also used to detect the outliers relying on the interquartile range (IQR), and thus, several very noticeable spikes were pointed out (see Figure 6).
To systematically detect anomalous observations, multiple complementary techniques were applied: • Z-score thresholding, used to identify statistically extreme deviations from the mean; • IQR-based outlier detection, flagging values outside the conventional 1.5 × IQR limits; • Isolation Forest, a robust machine learning algorithm capable of capturing irregular or anomalous patterns in temporal data.
The results of these anomaly detection methods are illustrated in Figure 7.
These combined approaches enable a clear distinction between sensor-induced noise and authentic pollution events. Outliers attributable to measurement errors were corrected or removed during preprocessing, while true high-pollution episodes were retained to preserve the environmental validity of the dataset.
The first step involves characterising the fractal properties of each environmental parameter (AQI, PM 2.5, PM10, CO, SO2, NO2, temperature, humidity, precipitation, atmospheric pressure, soil surface condition). These properties provide foundational insights into the inherent complexity, persistence, and scaling behaviour of the series—features that critically inform subsequent modelling decisions. Identifying long-range dependencies, fluctuation heterogeneity, and multi-scale structure helps determine whether pollution indicators are predictable, mean-reverting, or dominated by stochastic variability.
The Hurst exponent () is the primary metric used to quantify long-term memory in environmental time series.
• > 0.5: persistent behaviour — increases tend to be followed by further increases. • 0.5 ≤ ≤ Th: random behaviour — fluctuations behave similarly to a random walk. • < 0.5: anti-persistent behaviour — increases tend to be followed by decreases.
Given suficiently long and continuous data, the Hurst exponent can be estimated via classical rescaled range analysis.
1. Split the time series into segments of length . 2. For each segment, compute the cumulative deviation:
() = ∑︁( − ¯), = 1, . . . , ,
=1 where is the value of the series at time , ¯ is the mean of the segment, and () is the cumulative deviation. 3. Compute the range:
() = max( ()) − min( ()), where () represents the maximum fluctuation within the segment.
4. Compute the standard deviation: where is a constant of proportionality.
DFA is the primary method used in this study due to its robustness to non-stationarity, missing segments, trends, and structural breaks—conditions frequently observed in pollution monitoring data. 1. Integrate the time series:
where () is the standard deviation of the segment. 5. Estimate the scaling relationship: The exponent is obtained by fitting a log–log regression:
⎯ () = ⎷⎸⎸ 1 ∑=︁1 ( − ¯)2, () () ∼ , log ︂( () )︂ () = log() + ,
() = ∑︁( − ¯),
=1 () = () − ^(), ⎯
() = ⎷⎸⎸ 1 ∑︁ ()2,
=1 where is the original series at time , ¯ is the mean of the series, and () is the cumulative sum (integrated signal). 2. Divide into windows of size and in each window fit a local polynomial trend ^(), where is the window size and ^() is the local trend estimate. 3. Detrend the signal:
where () is the detrended series in window . 4. Compute the fluctuation function: where () measures the root-mean-square fluctuation of the detrended series, and is the total number of points.
(4) (5) (6) (7) (8) (9) (10) (11) 5. Estimate the scaling law: where () and () are the AR and MA polynomials, and is white-noise error.
The autoregressive and moving-average polynomials are:
() ∼ , = .
= 2 − .
= | − 0.5|. ′ = (1 − ) ,
= −1 , ()′ = () , () = 1 − 1 − · · · −
, () = 1 + 1 + · · · + , ^+ℎ = (+ℎ | , −1 , . . . ),
Higher values of indicate stronger deviation from randomness, and thus higher theoretical predictability.
Diferent modeling techniques are utilized to turn the environmental time series characterized by fractal into prediction power. The methods are based on diferent kinds of structural assumptions, diferent degrees of flexibility, and diferent suitability for linear, seasonal or nonlinear dynamics. The chosen ones are ARIMA, SARIMA, LSTM, and Prophet forecasting. Each of the models provides a viewpoint that adds to the understanding of pollution dynamics, thereby making it possible to carry out a solid comparison across the diferent forecasting methods of linear, seasonal, non-linear, and trend-adaptive.
The ARIMA(, , ) model captures linear temporal dependence using autoregression, diferencing, and moving-average components. After diferencing times: where is the original series, ′ is the -times diferenced series, and is the backshift operator.
The ARIMA model satisfies: where is the AR order and are autoregressive coeficients, = 1, . . . , . where is the MA order and are moving-average coeficients, = 1, . . . , .
Forecasts follow the AR–MA recursion:
where is the DFA scaling exponent characterizing long-range correlations.
The exponent is directly related to the Hurst exponent:
From , additional fractal characteristics can be derived: • Fractal dimension: • Predictability index (optional): (12) (13) (14) (15) (16) (17) (18) (19) (20) where ^+ℎ is the ℎ-step ahead forecast and (· | )· denotes conditional expectation given past observations.
ARIMA provides an interpretable baseline, particularly when the diferenced series is stationary and exhibits long-memory behaviour.
subsubsectionSARIMA Model
SARIMA extends ARIMA by incorporating seasonal structure: where , , are non-seasonal AR, diferencing, and MA orders, , , are seasonal AR, diferencing, and MA orders, and is the seasonal period.
Combined diferencing is given by:
SARIMA(, , )(, , ), = (1 − ) (1 − ), (21) (22) (23) (24) (25) (26) (27) (28) (29) where is seasonal AR order and Φ are seasonal AR coeficients, = 1, . . . , .
Φ() = 1 − Φ 1 − · · · − Φ
, Θ() = 1 + Θ1 − · · · − Θ , where is seasonal MA order and Θ are seasonal MA coeficients, = 1, . . . , .
SARIMA is efective for pollutants exhibiting daily or annual periodicity.
LSTM networks model nonlinear and long-range dependencies using gated memory cells. For each time step , with input , previous hidden state ℎ−1 , and cell state −1 :
Forget gate:
= ( + ℎ−1 + ), where is the forget gate vector, is the sigmoid activation, and are input and hidden weights, and is the bias.
Input gate and candidate state: where is the original series, is the diferenced series, is the backshift operator, is non-seasonal diferencing order, is seasonal diferencing order, and is the seasonal period.
The full model is:
Φ()() = Θ()() , where () and () are non-seasonal AR and MA polynomials, Φ() and Θ() are seasonal AR and MA polynomials, and is white-noise error.
Seasonal polynomials:
= ( + ℎ−1 + ), where is the input gate vector, , are weights, and is the bias. where ˜ is the candidate cell state, , are weights, and is the bias.
Cell update: where is the updated cell state and ⊙ denotes element-wise multiplication.
˜ = tanh( + ℎ−1 + ),
= ⊙ −1 + ⊙ ˜ ,
Output gate and hidden state: where is the output gate vector, , are weights, and is the bias.
= ( + ℎ−1 + ),
ℎ = ⊙ tanh( ), where ℎ is the hidden state passed to the next time step or output layer.
LSTMs efectively capture nonlinear patterns, persistence, and extreme pollution events.
Prophet models the time series as an additive decomposition:
() = () + () + ℎ() + , where () is the observed series, () is the trend, () is the seasonal component, ℎ() represents holidays or events, and is the error term.
Trend (piecewise linear): ⎛ ⎞
⎛ () = ⎝ + ∑︁ 1( > )⎠ + ⎝ + ∑︁ 1( > )⎠ , =1 =1 ⎞ where is the base growth rate, is the ofset, is the number of change points at times , and are adjustments to the slope and ofset, and 1(·) is the indicator function.
Seasonality (Fourier series): () = ∑︁ [︂ cos =1 ︂( 2 )︂ + sin ︂( 2 )︂] , where is the number of Fourier components, is the period (e.g., 24 for daily or 365 for yearly), and , are Fourier coeficients.
Events/holidays:
ℎ() = ∑︁ 1( ∈ ),
=1 where is the number of special events, is the set of times corresponding to event , and is the efect size of the event.
Prophet handles multiple seasonalities, trend shifts, missing data, and abrupt environmental changes.
Figure 8 illustrates the overall architecture of the proposed forecasting model. The structure for prediction that is suggested combines classic statistical methods, machine-learning techniques, and fractal indicators into one predictive system. The system is intended to detect not only the short-term but also the long-range dependence of environmental time-series data. The process of forecasting starts with the preprocessing of data where the raw hourly measurements undergo cleaning, alignment, and optional decomposition by STL which separates them into trend, seasonality, and residual components. After this, the fractal analysis is performed on the detrended or residual series to compute the Hurst exponent, fractal dimension, and spectral scaling among other things. These indicators express the amount of power, intricacy, and multi-scale behaviour in the data and thus they are treated as an additional feature set for the forecasting models. Each model is given an input that is specifically curated by merging preprocessed pollutant and meteorological data with fractal-derived features. The (30) (31) (32) (33) (34) (35) architecture incorporates a wide range of forecasting methods, namely ARIMA, SARIMA, ETS, STLbased models, Prophet, tree-based methods, and LSTM networks. Each of the models is responsible for producing its own forecast and uncertainty estimate. A hybrid ensemble layer is then used to combine these outputs through either performance-based weighting or stacked learning methods. This merging step results in higher prediction accuracy, robustness, and adaptability for the air pollution time series that vary in fractal regimes.
In this part, the developed hybrid fractal–machine learning framework’s predictive performance is demonstrated over the four model families: ARIMA, SARIMAX, Prophet, and LSTM. The accuracy of the models was assessed through the use of six complementary metrics—MAE, RMSE, MAPE, sMAPE, MASE, and 2—which allowed an all-encompassing evaluation of both absolute and relative forecasting error. Table 2 summarizes the results.
Let the following notations be defined: • = − −1 • — the observed (actual) value at time ; • ^ — the predicted value at time ; • — the total number of observations in the test set; • ¯ = 1 ∑︀
=1 — the mean of the actual values; • = − ^ — the forecast error;
— the naïve forecast error used in MASE.
MAE = 1 ∑︁ | − ^ | . =1 (36)
⎯ RMSE = ⎷⎸⎸ 1 ∑=︁1 ( − ^ )2.
MAPE = 100 ∑=︁1 ⃒⃒⃒⃒ −^ ⃒⃒⃒⃒ . sMAPE = MASE = . .
The forecasting performance of the diferent models, evaluated using multiple error metrics, is summarized in Table 2.
The results indicate a clear performance hierarchy across model types. The ARIMA model demonstrates the weakest forecasting ability, with negative 2 and the highest error rates, indicating that linear autoregressive patterns alone cannot adequately capture the complexity of urban air pollution dynamics. The ARIMA forecast for AQI, illustrating the model’s limitations in capturing complex temporal patterns, is shown in Figure 9.
The SARIMAX model improves substantially by incorporating exogenous meteorological variables, reducing MAE by approximately 30% compared to ARIMA and achieving a positive 2 of 0.50. Prophet provides further improvements through flexible trend and seasonality modelling, achieving moderate errors and strong generalization performance. The SARIMAX forecast demonstrates improved accuracy over ARIMA, is shown in Figure 10.
The LSTM model augmented with fractal features achieved the highest forecasting accuracy among the evaluated models, with a coeficient of determination of 2 = 0.93 and a mean absolute percentage error (MAPE) of less than 9%. This performance significantly exceeds that of the best-performing traditional Prophet model (2 ≈ 0.76 ) as well as the baseline ARIMA model, which exhibited a negative 2 value, indicating poor approximation quality.
These results confirm the efectiveness of the proposed hybrid approach. The LSTM architecture is capable of capturing complex temporal dependencies, particularly when supplemented with information on long-range dependence through fractal indicators. Notably, high values of the Hurst exponent ( ≈ 0.9 ) observed in particulate matter time series reflect persistent dynamics. Under such conditions, the fractal-enhanced LSTM model demonstrates superior predictive performance, whereas the ARIMA model fails to achieve comparable forecasting accuracy.
Figures 11–13 illustrate the forecasting results obtained from diferent models. Figure 11 presents the LSTM forecast over the full AQI dataset, highlighting the model’s capability to capture complex nonlinear temporal patterns. Figure 12 shows the hybrid LSTM+ARIMA forecast, demonstrating improved performance by combining linear and nonlinear components. Finally, Figure 13 displays the Prophet model forecast, which efectively captures trends and seasonal efects in the AQI series.
In addition to numerical evaluation, forecast performance was assessed visually using: • time-series forecast plots comparing predicted and observed pollutant concentrations, • residual plots highlighting error distribution and autocorrelation, • overlayed multi-model forecast curves illustrating diferences across model families.
The LSTM predictions are always in more detail with both short-term changes and long-term trends than the baseline models as shown by the visualizations. The Prophet and SARIMAX are capturing the medium-term behavior but they are still showing some lag during the rapid regime shifts while ARIMA is unable to model the non-linear variability and strong seasonal components. The hybrid ensemble further reduces the forecasting error by approximately 10–15% compared with the bestperforming individual model, indicating a complementary efect among the constituent predictors. This improvement suggests that each model captures distinct structural characteristics of the data. In particular, linear dynamics and seasonal patterns are efectively modeled by the ARIMA and Prophet approaches, whereas nonlinear long-term dependencies are accounted for by the fractal-enhanced LSTM model. Emphasizing these advantages in the results section makes it possible to clearly demonstrate why the proposed hybrid approach is justified for such a complex urban air pollution forecasting task.
The outcomes demonstrate that incorporating fractal traits and long-range dependence into the prediction models has a remarkable efect on the accuracy of forecasting urban air pollution. Traditional models like ARIMA and SARIMAX are able to capture linear trends and short-term dependencies but are unable to deal with non-linear fluctuations and multi-scale patterns which is expressed in their relatively high errors and low 2 values. Prophet performs better by modelling multiple seasonalities and structural trends, yet its piecewise-linear assumptions limit responsiveness to abrupt regime shifts observed in the dataset.
LSTM achieved the best performance across all metrics, demonstrating its strength in capturing non-linear interactions, long-range dependencies, and complex temporal patterns. This aligns with the persistent behaviour revealed by Hurst exponent and DFA analyses, suggesting that models capable of leveraging fractal features gain a predictive advantage. Incorporating fractal indicators as auxiliary inputs enhances model awareness of scaling behaviour and memory efects, which classical models may overlook.
Overall, these findings confirm that hybrid fractal–machine learning approaches provide a more robust framework for environmental forecasting, particularly in systems with persistence, multi-scale variability, and non-linear dynamics. The study highlights the practical value of combining advanced statistical, deep learning, and fractal-based methods for urban air quality prediction.
Although the hybrid model demonstrates clear advantages, a number of limitations of the study should be noted. The obtained results are based on data from only one city, therefore the proposed approach needs to be tested in other urban conditions to confirm its general applicability. It should also be noted that the proposed framework currently operates in a static mode for the purposes of retrospective forecasting. If the model is to be applied dynamically in real time, it will require adaptation and optimization. This is due to the fact that the implementation of the described models may be computationally demanding for streaming data. In addition, another limitation is that the study applies a univariate approach for each pollutant. The interactions between pollutants were not considered. However, in general, the integration of interdependencies between pollutants is a promising direction for further research. Nevertheless, taking into account the stated limitations, it can be emphasized that the method has high practical potential for real-world air quality management.
One of the priorities for future research is to take into account a broader range of patterns. This can further improve the accuracy of the models, which is important for efective environmental monitoring. An important direction is the development of multidimensional deep learning models that will simultaneously use fractal features, multiple pollutants, and exogenous variables. The integration of this framework with real-time sensor networks is also planned. In addition, an important future step is the quantitative assessment of forecast uncertainty in order to provide decision-makers not only with point forecasts but also with confidence intervals.
In this study, the stated goal of improving the accuracy of urban air pollution forecasting was achieved through the implementation of a new hybrid fractal–machine learning framework. We succeeded in filling the scientific gap by accounting for long-term dependence and multiscale variability using fractal features that traditional models are unable to reproduce. The proposed approach, tested on air quality data from the city of Astana, provided a significant improvement in forecasting accuracy. In particular, the fractal-informed LSTM model reduced MAPE by more than 80% compared with ARIMA and achieved an 2 value of 0.93, which confirmed the proposed hypothesis that fractal metrics enhance the predictive capabilities of the models. By combining this LSTM with classical models in an ensemble, a robust tool was developed that outperforms individual methods while efectively combining interpretability and accuracy in pollution forecasting.
The authors express sincere gratitude to Kazambayev Ilyas for his insightful comments during the development of this work.
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