Managing Partitioning Uncertainty and Transition Ambiguity in Fuzzy Time Series: A Robust IT2FCM-Markov Chain Approach

Time series forecasting (TSF) is a critical tool for decision-making across diverse fields, including energy production [1-4], environmental monitoring [5, 6], and financial analysis [7-9]. While traditional statistical models have been used extensively, they often struggle with the inherent uncertainty and linguistic vagueness of real-world data [10, 11]. The Fuzzy Time Series (FTS) approach, introduced by Song and Chissom [12, 13], offered a paradigm shift by modeling temporal data through fuzzy sets. However, despite its success, the reliability of FTS is persistently hampered by two fundamental challenges: partitioning uncertainty and transition ambiguity.

Partitioning uncertainty arises during the division of the universe of discourse into intervals. Early models relied on rigid, subjective intervals, which were later improved by objective clustering algorithms like Fuzzy C-Means (FCM) [14, 15]. However, FCM remains highly sensitive to noise and the selection of the fuzzifier parameter, leading to unstable partitions [16]. Simultaneously, transition ambiguity occurs when a single fuzzy state leads to multiple potential future states (one-to-many relationships). Standard FTS models often use simple averaging to resolve these Fuzzy Logical Relationship Groups (FLRGs), which can lead to significant loss of information and reduced accuracy [17, 18].

Current literature has addressed these issues primarily in isolation. One group of researchers has focused on improving partitioning robustness by employing advanced fuzzy logic, such as Interval Type-2 Fuzzy Sets (IT2FS), to handle the uncertainty in clustering [18, 19]. For instance, IT2FCM has been recognized for its ability to model the "Footprint of Uncertainty" (FOU) more effectively than traditional type-1 methods [20-23]. Another group has focused on refining transition modeling by integrating Markov Chain (MC) models to calculate probabilities for fuzzy state transitions [20, 21, 24, 25]. While these individual advancements are significant, a critical gap remains: There is a lack of a unified hybrid framework that addresses both partitioning uncertainty and transition ambiguity simultaneously within a single, cohesive architecture.

To fill this gap, this study proposes a novel hybrid model, IT2FCM-MC-FTS. The synergy between IT2FCM’s robust clustering and MC’s probabilistic inference forms the core contribution of this research. Specifically, IT2FCM is utilized to establish noise-resistant intervals, while a first-order MC provides a logical and data-driven mechanism for resolving one-to-many ambiguities. The proposed model is validated using volatile carbon monoxide (CO) concentration data, demonstrating its ability to provide more accurate and stable forecasts than existing hybrid variants.

The remainder of this paper is organized as follows: Section 2 presents the related works. Section 3 details the proposed hybrid model. Section 4 presents the experimental setup and case study. Section 5 discusses the results. Finally, Section 6 concludes the paper.

3.1 The proposed IT2FCM-MC-FTS model

Our proposed model integrates robust partitioning (IT2FCM) with probabilistic forecasting (MC) in a single, cohesive framework. The overall architecture is depicted in Figure 1, and to ensure high transparency and reproducibility, the implementation details are described through the following stages:

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Figure 1. Flowchart of the proposed model

The implementation details, derived from the experimental code, are described below:

3.1.1 Stage 1: Robust partitioning with IT2FCM

The foundation of the model is the quality of fuzzy partitions. To create partitions robust to noise, we employ IT2FCM, which accounts for the uncertainty in the fuzzifier parameter. The algorithm seeks to minimize the objective function:

$J_{m 1, m 2}=\sum_{k=1}^n \sum_{i=1}^c\left(v_{i k}\right)^m\left\|x_k-v_i\right\|^2$         (1)

where, $m \in\left[m_1, m_2\right]$ represents the interval-valued fuzzifier. In our implementation, we utilize the centroid of this interval $(\bar{m}=1.95)$ to achieve a robust representative clustering center. To ensure reproducibility of the fuzzy partitions, a fixed random seed of 7521 is applied. The clustering process is governed by a convergence threshold ($\epsilon=0.005$) and a maximum of 1000 iterations.

The optimal number of clusters $c$ is determined dynamically using a volatility-based heuristic. This is calculated as the ratio between the total range of data and the average volatility (mean absolute difference):

$C_{\text {raw }}=\operatorname{round}\left(\frac{D_{\max }-D_{\min }}{\frac{1}{n-1} \sum_{1=2}^n\left|x_i-x_{i-1}\right|}\right)$          (2)

To ensure model stability across different dataset sizes, a constraint is applied: $\left(c=\max \left(2, \min \left(c_{\text {raw}}, n / 2\right)\right)\right)$. This ensures that the number of fuzzy states is proportional to the dynamic behavior of the series while maintaining enough data points per cluster for statistical validity.

3.1.2 Stage 2: Fuzzification and FLRG establishment

With the cluster centers $v_i^*$ obtained and sorted, the historical time series $Y_t$ is fuzzified. Unlike traditional methods that use equal-length intervals, our model defines the boundaries of fuzzy sets $\left\{A_i, \ldots \ldots, A_c\right\}$ based on the midpoints between adjacent cluster centers:

$B_i=\frac{v_i^*+v_{i+1}^*}{2}, i=1, \ldots, c-1$           (3)

This creates adaptive intervals that are denser where data points are concentrated. Each $Y_t$ is mapped to the corresponding fuzzy set $A_i$ within these boundaries. The sequence of fuzzy sets forms Fuzzy Logical Relationships (FLR), denoted as $A_{t-1} \rightarrow A_t$, which are then grouped by their antecedent to form FLRGs.

3.1.3 Stage 3: Probabilistic forecasting with MC

To resolve the one-to-many ambiguity in FLRGs, w construct a $c \times c$ transition probability matrix $P$. Eacl element $P_{i j}$ represents the probability of moving from state $A_i$ to $A_j$, a first-order MC transition probability matrix $P$ is constructed:

$P_{i j}=\frac{N_{i j}}{\sum_{k=1}^c N_{i k}}$        (4)

where, $N_{i j}$ is the frequency of the transition observed in the training data $A_i \rightarrow A_j$. This ensures that the weights for future states are entirely data-driven.

3.1.4 Forecasting (Defuzzification)

The final forecast $\hat{Y}_{(t+1)}$ is computed as the probabilistic weighted average of the centers of all potential future states:

$\hat{Y}_{(t+1)}=\sum_{j=1}^c v_j^* \cdot P_{i j}$         (5)

This approach avoids the information loss inherent in simple averaging by prioritizing transitions that occur more frequently in the historical data.

3.2 Algorithmic structure of IT2FCM-MC-FTS

The complete logic of the proposed IT2FCM-MC-FTS model is summarized in Algorithm 1.

5.1 Forecasting performance

The forecasting results for all models on the CO concentration dataset are summarized in Table 2. The proposed IT2FCM-MC-FTS model achieved the lowest error on both metrics, demonstrating the highest accuracy among the tested models for this case study, as shown in Figure 2.

Table 2. Forecasting error comparison with FTS models

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Figure 2. Comparison of actual vs. forecasted values for the dataset

5.2 Discussion: The synergistic effect

Visual analysis in Figure 3 provides critical evidence that complements the statistical tests. The Boxplot in Figure 3(a) shows that the proposed model achieves a lower median absolute error and a more compact interquartile range (IQR) compared to FTS-MC, suggesting higher prediction stability.

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(a) Absolute error distribution

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(b) Cumulative absolute error

Figure 3. Statistical significance analysis

The Wilcoxon test comparing the proposed model against IT2FCM-FTS yielded a p-value of 0.0023, indicating a highly significant improvement $(p<0.01)$. This confirms that the MC's probabilistic inference is significantly superior to simple averaging when robust partitions are already established.

For the comparison against FTS-MC, the test resulted in a p-value of 0.1061. Although this exceeds the traditional 0.05 threshold, it represents a marginal significance that suggests a strong trend toward improvement. In the context of TFS with a limited sample size $(n=46)$. A p-value in this range often indicates that the proposed model is consistently outperforming the baseline, but the statistical power is constrained by the number of observations.

5.3 Residual analysis and statistical significance

Statistical test against IT2FCM-FTS yielded $p=0.0023$ (highly significant). Against FTS-MC, $p=0.1061$, showing marginal significance constrained by sample size ($n=46$).

To validate the reliability of the model, we performed a residual analysis. Figure 4(a) shows that the residuals (the difference between actual and predicted values) are randomly scattered around the zero line without obvious heteroscedasticity patterns. This indicates that the model has effectively captured the temporal dynamics of the data. The error distribution in Figure 4(b), the histogram also shows a near-normal shape centered at zero, proving that model bias is very low despite the high volatility of the data.

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(a) Residual analysis

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(b) Error distribution histogram

Figure 4. Residual analysis and statistical significance

5.4 Discussion

The performance metrics in Table 1 not only demonstrate the superiority of the proposed model but also serve as an ablation study to validate the role of its hybrid components. Analysis of RMSE and SMAPE reveals a clear synergistic effect. The FTS-MC model, which utilizes MCs but relies on simple uniform partitioning, yielded the highest error (RMSE 1068.99; SMAPE 20.34%). This confirms that even a sophisticated transition mechanism cannot compensate for non-representative partitioning. Conversely, the IT2FCM-FTS model, using robust partitioning but simple averaging, performed significantly better (RMSE 897.29; SMAPE 19.32%), yet remained less accurate than the proposed IT2FCM-MC-FTS.

Furthermore, the superiority over the IMWPFCM-FTS model (RMSE 914.68 vs. 887.47) suggests that IT2FCM's ability to model the FOU provides a more noise-resistant foundation than other advanced fuzzy clustering variants for this specific volatile dataset.

Beyond these technical benchmarks, the higher accuracy of the IT2FCM-MC-FTS model provides significant practical implications for air quality management in Semarang City:

1. Public Health Early Warning: The reduction of SMAPE to 17.86% enables more accurate detection of CO pollution spikes, providing better response times for health authorities to issue alerts.
2. Traffic Management: Stable and reliable predictions assist the city government in planning adaptive traffic management strategies to reduce vehicle emissions in pollution-prone areas.
3. Environmental Policy: This model provides a more robust data foundation for policymakers to evaluate and refine the effectiveness of vehicle emission regulations and urban sustainability initiatives.