Interval Type-2 Fuzzy C-Means (IT2FCM) for Robust Forecasting of Carbon Monoxide Concentrations in Urban Air Quality Monitoring

The increasing severity of urban air pollution has prompted governments and researchers to develop smarter, more proactive monitoring systems [1]. Carbon monoxide (CO), a toxic gas primarily produced from incomplete combustion, is among the most critical pollutants monitored in urban air quality management programs [2]. Short-term exposure to elevated levels of CO can lead to severe cardiovascular and neurological effects, while long-term exposure has been linked to chronic respiratory illnesses and increased mortality risks [3-5]. In countries with dense populations and high vehicle use, such as Indonesia, monitoring and predicting CO concentration is crucial for issuing early warnings and protecting vulnerable groups [6-8].

Despite the availability of environmental sensor networks, the challenge remains in generating reliable, real-time predictions from data that are often noisy, nonlinear, and uncertain [9]. Traditional forecasting methods such as Autoregressive Integrated Moving Average (ARIMA) and Double Exponential Smoothing have been widely used in air quality time series modeling due to their interpretability and mathematical tractability [10-12]. However, these methods typically assume linear relationships and stationary time series, which may not hold in dynamic urban conditions characterized by abrupt changes, sensor drift, and complex meteorological interactions [13].

To overcome these limitations, researchers have increasingly turned to machine learning and soft computing approaches, which offer better adaptability to nonlinearities and uncertainty in the data. Among them, fuzzy clustering, particularly Fuzzy C-Means (FCM), has gained attention due to its capability to partition ambiguous data and represent partial memberships [14, 15]. In environmental contexts, where pollutant levels often fluctuate near decision boundaries and sensor readings may be imprecise, FCM can uncover meaningful data groupings that conventional classifiers might miss [16].

Nevertheless, classical FCM, which relies on Type-1 fuzzy sets, does not fully address the uncertainty in the membership functions themselves. When data are highly overlapped or noisy, the crisp nature of Type-1 memberships may result in unstable or biased clustering outcomes. To address this, Interval Type-2 Fuzzy Sets (IT2FS) have been introduced as an extension that allows membership degrees to be expressed as bounded intervals. This results in a more robust and uncertainty-aware clustering method known as Interval Type-2 Fuzzy C-Means (IT2FCM) [17-19].

IT2FCM provides a more expressive modeling framework by incorporating both lower and upper membership functions, enabling it to better handle overlapping clusters and uncertain boundaries [20, 21]. Several studies have shown that IT2FCM improves classification and clustering stability in applications ranging from medical diagnostics to image processing [22], but its application to environmental time-series prediction—particularly for air pollution forecasting—remains underexplored.

Another challenge lies in the interpretability and computational efficiency of predictive models for deployment in real-time systems, such as IoT-based air quality monitoring stations [23]. Many advanced deep learning models offer high accuracy but at the cost of transparency and processing time, making them less ideal for embedded or edge-computing environments [24, 25]. In contrast, centroid-based prediction models using fuzzy clustering offer a lightweight alternative with strong explanatory value [26]. Rather than fitting complex regression equations, these models predict future values based on cluster centroids [27], ensuring fast [28], interpretable [29], and adaptable forecasting [30].

This study proposes a novel, centroid-based prediction framework combining FCM and IT2FCM for forecasting short-term CO concentration. It introduces a practical approach where the next predicted value is derived from the cluster centroid associated with the latest data point. For IT2FCM, the model aggregates predictions from lower and upper memberships into a single average value, balancing between sensitivity and robustness. The framework is evaluated using real-world CO concentration data from an urban monitoring station in Semarang, Indonesia, and compared with classical time-series models.

This study proposes a comprehensive forecasting framework for air quality time series data, specifically focusing on CO concentration obtained from the AQMS station in Semarang. The methodological flow, as illustrated in Figure 1, comprises five main stages: data preparation, clustering using FCM and IT2FCM, prediction based on cluster centroids, evaluation through MAPE, and comparison with traditional statistical models such as ARIMA and Double Exponential Smoothing.

3.1 Data preparation

In this stage, the collection and preparation of concentration data from the available dataset are performed. Raw data is read and structured into a matrix form suitable for clustering analysis. Subsequently, the optimal number of clusters is determined; in this study, the number of clusters is set as c=3.

The optimal number of clusters was determined not empirically, but through a systematic sensitivity analysis aimed at identifying the parameter that minimizes the Mean Absolute Percentage Error (MAPE) as detailed in Section 3.3.

1.png

Figure 1. Research methodology

3.2 Clustering

This stage involves data clustering using two methods: FCM and IT2FCM.

3.2.1 FCM

FCM is a fuzzy clustering method optimizing the following objective function (1).

$J_m(U, V)=\sum_{i=1}^n \sum_{j=1}^c\left(U_{i j}\right)^m| | x_i-v_j| |^2$     (1)

where, $U_{i j}$ represents the membership degree of data point $x_k$ to cluster $i, v_j$ denotes the centroid of cluster $j, c$ is the number of clusters, and $m$ is the fuzziness parameter, with $\mathrm{m}>1$ in this study.

3.2.2 IT2FCM

IT2FCM extends the FCM concept by incorporating two fuzziness parameters, lower m₁ and upper m₂, to handle high uncertainty in data. While its objective function is conceptually similar to that of FCM, the core of the IT2FCM algorithm lies in its handling of uncertainty through type-reduction. The objective function can be expressed as:

$J_{m 1, m 2}\left(U_L, U_U, V_L, V_U\right)=\sum_{i=1}^n \sum_{j=1}^c\left[\left(U_{L i j}\right)^{m 1}| | x_i-v_{L j}| |^2+\left(U_{U i j}\right)^{m 2}| | x_i-v_{U j}| |^2\right]$     (2)

where, $U_{L i j}$, $U_{U i j}$ are respectively the lower and upper membership degrees for data $x_i$ and $v_{L j}$, $v_{U j}$ are the lower and upper centroids of cluster $j$. It is critical to note that the centroids $v_{L j}$ and $v_{U j}$ are not optimized independently. Instead, they are computed through an iterative type-reduction process, which is a hallmark of type-2 fuzzy systems. This study employs an approach aligned with established literature, where the typereduced centroid for each cluster is first calculated, and then the lower and upper centroids ($v_{L j}$ and $v_{U j}$) are derived from this type-reduced set. Algorithms such as the Karnik-Mendel (KM) algorithm are standard methods for performing this typereduction [48]. This ensures that the centroids properly reflect the footprint of uncertainty modelled by the interval memberships, rather than being simple independent optimizations.

This study set the fuzzifier parameter $\mathrm{m}=2.0$ for standard FCM, aligning with common practice and its robustness to noise in real-world datasets [49, 50]. Following the sensitivity analysis described in Section 3.3, the optimal fuzziness parameters were identified as $\mathrm{m}_1$ and $\mathrm{m}_2$. This interval, in conjunction with c clusters, was proven to provide the lowest prediction error, offering a refined balance between model sensitivity and robustness against data uncertainty.

3.3 Sensitivity analysis for parameter optimization

To ensure the robustness and optimal performance of the proposed model, and in response to reviewer feedback, a sensitivity analysis was conducted. This process involved systematically evaluating the model's performance across a range of key parameters: the number of clusters (c) and the fuzziness interval [m₁, m₂]. The number of clusters c was varied from 2 to 5. The fuzziness interval was tested with three different configurations representing distinct uncertainty footprints: [1.5, 3.0], [1.8, 2.2], and [1.2, 4.0]. The model's performance for each parameter combination was measured using MAPE. This systematic approach allows for the data-driven selection of optimal parameters, moving beyond empirical estimation and enhancing the model's credibility. The results of this analysis are visualized in the heatmap in Figure 2.

2.png

Figure 2. Sensitivity analysis

3.4 Prediction

The prediction process is conducted based on the cluster centroids obtained from the previous clustering results. The predicted value $\widehat{x}_l$ for each data point is computed using the centroid of the cluster with the highest membership degree in function (3).

$\left(\widehat{x_l}\right)=v_j, j=\arg \max _j\left(U_{i j}\right)$     (3)

where, $v_j$ is the centroid of the cluster having the highest membership degree for data point $x_i$. Furthermore, the prediction mechanism is a centroid-based forecasting model that establishes a temporal link between consecutive data points via cluster membership. The process for forecasting the value at time $t+1$ based on the observation at time $t$ is as follows:

1. State Identification: For the current data point $x_t$, its corresponding cluster membership is determined by finding the cluster j with the highest degree of membership.
2. Forecast Generation: The forecast for the next time step, denoted as $\hat{x}_t+1$ then set to be the centroid value of the identified cluster, $v_j$.

This can be formally expressed as: Let $C\left(x_t\right)$ be the cluster assigned to observation $x_t$. Then the forecast is $\hat{x}_t+1=v C_{(x t)}$. This approach effectively treats the cluster centroids as representative states of the system. The forecast is based on the assumption that the system will persist in its current state, represented by its cluster centroid, into the next time step. For the IT2FCM model, the final forecast $\hat{x}_t+1$ is the average of the lower and upper centroids for the assigned cluster, i.e., $\frac{\left(v_{L, C_{(x t)}}+v_{U, C\left(x_t\right)}\right)}{2}$.

3.5 Evaluation

Prediction performance evaluation is conducted using MAPE, computed with the Eq. (4). A lower MAPE indicates better prediction accuracy.

$M A P E=\frac{1}{n} \sum_{i=1}^n \left\lvert\, \frac{x_i-\widehat{x_l}}{x_i} \times 100 \%\right.$     (4)

3.6 Comparison with statistical models

For comparison purposes, two traditional statistical methods, namely ARIMA and Double Exponential Smoothing, are employed.

3.6.1 ARIMA

The ARIMA model is formulated in Eq. (5) with parameters $p$, $d$, $q$ representing autoregressive order, differencing, and moving average order, respectively. This study selected parameters $p, d, q$ for ARIMA via a combination of stationarity tests and inspection of ACF/PACF plots. A grid search over $\mathrm{p}=0-5$, $\mathrm{~d}=0-2$, $\mathrm{q}=0-5$ was conducted to find the model with the lowest AIC; ARIMA (2,1,2) was chosen. Additionally performed residual diagnostics to ensure model adequacy. We also tested auto arima() in pmdarima, confirming similar parameter ranges and AIC values.

$\operatorname{ARIMA}(p, d, q): y_t=c+\sum_{i=1}^p \phi_i y_{t-i}+\sum_{j=1}^q \theta_j \in_{t-j}+\in_t$     (5)

3.6.2 Double Exponential Smoothing

Double Exponential Smoothing is a time series forecasting method that addresses the limitation of simple exponential smoothing by incorporating a trend component. It uses two equations to update both the level and the trend in Eq. (6).

$\begin{aligned} & \hat{y}_t=\alpha y_t+(1-\alpha)\left(\hat{y}_{t-1}+\hat{b}_{t-1}\right) \\ & b_t=\beta\left(\hat{y}_t-\hat{y}_{t-1}\right)+(1-\beta) b_{t-1}\end{aligned}$     (6)

DES parameters $\alpha$ and $\beta$ were tuned via grid search ($\alpha$, $\beta$ $\in$[0.1,0.9]) to minimize MAPE on training data, resulting in $\alpha=0.35$ and $\beta=0.15$. This study further verified that a parameter estimation routine in Statsmodels yielded comparable values, optimizing via log-likelihood, where is the smoothing parameter for the level, and is the smoothing parameter for the trend component [51]. This method improves adaptability to trends but still assumes linearity and may struggle in volatile environments typical of sensorbased pollution monitoring.

The results presented in the previous sections demonstrate that the proposed fuzzy clustering-based forecasting approach—especially the IT2FCM—offers notable advantages over conventional statistical models for short-term prediction of CO concentration levels.

The first key finding is that IT2FCM consistently outperformed both FCM and baseline statistical methods (ARIMA, Exponential Smoothing) in terms of accuracy. This is evident from the MAPE, where the optimized IT2FCM (Average) model achieved a significantly lower MAPE of 3.79%. This result is not just a marginal improvement but a substantial leap in accuracy, with a prediction error less than one-third of that produced by traditional models like ARIMA (13.07%) and Exponential Smoothing (13.75%). These improvements validate the hypothesis that fuzzy clustering methods can more effectively model environmental data, which is often noisy, overlapping, and uncertain [52, 53].

Secondly, the use of interval modeling in IT2FCM introduces flexibility by capturing a range of potential centroid rather than a single point estimate. This design aligns well with real-world air quality monitoring, where variability is common, due to sensor noise, atmospheric conditions, and temporal patterns. By averaging the lower and upper bounds of IT2FCM centroids, the model yields a prediction that is both stable and interpretable, bridging precision and caution—essential in applications involving public health or environmental alerts [54].

A crucial contribution of this revised study is the inclusion of a sensitivity analysis, which addresses the parameter selection limitations of the initial approach. This analysis revealed that increasing the number of clusters from 3 to 5 allows the model to capture more granular patterns in the CO concentration data, likely corresponding to more nuanced states of air quality (e.g., very low, low, medium, high, very high). Similarly, the optimal fuzziness interval of [1.8, 2.2] demonstrates that a more constrained, moderate uncertainty footprint yields better results for this specific dataset compared to wider intervals. This data-driven parameter tuning not only validates the chosen parameters but also enhances the robustness and credibility of the proposed forecasting model.

A particularly insightful contribution is the use of the centroid-based prediction mechanism, which differs from traditional regression-based forecasting [55]. This lightweight mechanism avoids overfitting [56], ensures fast computation [57], and enables deployment in edge-computing scenarios such as IoT-based air quality monitoring stations [58].

A key limitation in the initial version of this study—the empirical selection of parameters—has now been directly addressed. By incorporating a systematic sensitivity analysis to optimize the number of clusters and fuzziness levels, the model's parameters are no longer based on heuristic choices but on a data-driven optimization process. This enhancement significantly strengthens the methodology's rigor. However, a remaining limitation is that the current study focuses solely on univariate prediction. Incorporating multivariate inputs (e.g., temperature, humidity, wind) could further enhance the model’s forecasting capabilities.

In summary, the IT2FCM-based approach introduces a novel direction in environmental time-series prediction by integrating fuzzy uncertainty modeling, interpretability, and computational simplicity. It represents a promising tool for smart environmental monitoring systems and offers potential integration into early warning platforms.

Future enhancements of the proposed IT2FCM model could include the use of multivariate input features, such as incorporating auxiliary environmental parameters (e.g., temperature, wind speed, humidity) to better capture the dynamic dependencies influencing pollutant concentration. Additionally, integrating adaptive clustering mechanisms—such as using validity indices or data-driven cluster evolution—could allow the model to adjust the number of clusters in response to non-stationary changes in environmental patterns. These improvements could significantly increase the model’s predictive robustness and operational flexibility in dynamic IoT deployments.