Application of the Cheng Fuzzy Time Series Model for Stock Price Forecasting: A Case Study in the Energy Sector

Over the past decade, energy sector stocks have experienced substantial volatility, with weekly and monthly return fluctuations frequently exceeding 5–10%, particularly during periods of geopolitical instability or commodity price shocks. For example, coal companies such as PT Bukit Asam Tbk (PTBA) faced rapid price swings from 2020 to 2022, driven by global coal demand uncertainty, pandemic-related supply disruptions, and evolving energy policies. The standard deviation of PTBA’s weekly returns surpassed 8% in certain quarters, indicating a high-risk profile for short-term investors. Such volatility presents significant forecasting challenges, especially when traditional statistical models are unable to accommodate the dynamic and ambiguous nature of market behavior. In these conditions, where binary interpretations and fixed thresholds are inadequate, fuzzy logic-based models offer a promising alternative by capturing uncertainty through linguistic variables and adaptable rule-based reasoning.

Accurate forecasting plays a critical role in supporting investor confidence and facilitating effective decision-making in volatile markets. For institutional investors, asset managers, and individual traders, reliable stock price predictions are essential for portfolio optimization, risk mitigation, and strategic asset allocation. Inaccurate forecasts may result in poor timing of trades, capital misallocation, and greater vulnerability to market downturns. Conversely, dependable forecasting models enable investors to anticipate price movements, reduce uncertainty, and improve overall returns. In the energy sector where external shocks such as regulatory changes, commodity price swings, and global demand shifts are common forecasting models must not only be statistically sound but also capable of adapting to imprecise and evolving data patterns. This highlights the importance of techniques that offer flexibility, interpretability, and adaptability key attributes inherent in fuzzy logic models.

Classical time series models such as ARIMA, moving average, and exponential smoothing have long been employed for financial forecasting because of their formal statistical foundations and simplicity. However, these models are based on assumptions of linearity, stationarity, and normally distributed errors assumptions that are frequently violated in real-world financial markets. In highly volatile environments such as the energy sector, price movements are frequently influenced by non-linear, chaotic, and ambiguous factors including geopolitical events, abrupt policy changes, or sudden shifts in investor sentiment. Traditional models struggle to accommodate these irregular patterns and tend to underperform when dealing with noise, structural breaks, or data fuzziness. Moreover, their rigidity in adapting to new information and inability to handle linguistic or imprecise variables limits their applicability in dynamic markets. This has led to increasing interest in soft computing methods, such as fuzzy logic, that offer greater flexibility and resilience in modeling uncertainty and complexity.

Several researchers have contributed to the development and refinement of fuzzy time series (FTS) models for stock market forecasting, with varied approaches to improving accuracy, interval design, and adaptability. Cheng et al. [1] introduced an FTS model based on adaptive expectations, highlighting the benefits of combining recent pattern weighting with fuzzy relationships for improving TAIEX forecasts. Building on this, Jilani and Burney [2] proposed a refined FTS structure that better handled noise in financial datasets, while Huarng [3] demonstrated that the length of intervals in the universe of discourse significantly affects forecast performance suggesting that properly defined intervals are critical to minimizing error.

Recent studies have introduced hybrid FTS models aimed at improving accuracy under nonlinear and volatile conditions. For instance, FTS-ANN models integrate fuzzy logic with neural networks to enable adaptive learning from historical patterns, albeit at the cost of interpretability. Granular computing-based FTS models, such as those proposed by Chen and Chen, create layered representations of uncertainty, offering superior performance in turbulent markets by handling multi-scale variability. Other researchers, including Egrioglu et al. [4] have explored fuzzy clustering and entropy-based approaches to optimize interval selection and transition rule precision. Compared to these advanced models, the Cheng FTS offers simplicity, transparency, and lower computational cost, making it attractive for mid-frequency forecasting where explainability is valued over pure accuracy.

Furthermore, the decision to focus on weekly stock prices, rather than the commonly used daily or monthly intervals, reflects both methodological and practical considerations. Weekly data strikes a balance between minimizing high-frequency noise and capturing mid-term trends relevant to retail and institutional investors. Notably, many portfolio managers operate on a weekly rebalancing cycle to adjust positions based on macroeconomic signals and sector rotations. This aligns well with the temporal resolution of the Cheng model, which relies on recurring fuzzy relationships observable over multi-day intervals.

Yu [5] explored the integration of weighted fuzzy relationships to improve forecasting stability, particularly under market volatility. Meanwhile, Chen [6] established the foundational structure of FTS models for educational datasets, a framework that has since been widely adapted for economic and financial data. Huarng and Yu [7] further advanced this by introducing Type-2 fuzzy models, showing superior performance over Type-1 models in highly uncertain environments like stock indices.

Granular computing was employed by Chen and Chen [8] to combine fuzzified data with multi-layered structures for improved adaptability in stock prediction. Teoh et al. [9] proposed a hybrid multi-order model, illustrating how multiple fuzzy rules can be layered to accommodate complex financial behaviors. Similarly, Chen and Chang [10] leveraged fuzzy clustering and rule interpolation to manage multi-variable influences on stock trends, while Chen and Phuong [11] focused on optimizing the partitioning of intervals and weights to enhance forecast accuracy.

Clustering-based models such as that proposed by Egrioglu et al. [4] used Gustafson-Kessel clustering to group historical patterns, whereas Wang et al. [12] emphasized the role of information granules in selecting effective fuzzy intervals. Entropy-based fuzzy logic approaches were also considered by Zhou et al. [13] and Zhang et al. [14], both of whom demonstrated improvements in portfolio modeling and multi-factor forecasting through more nuanced uncertainty handling.

Despite these advancements, most prior studies rely on daily data, emphasize high-frequency trading behavior, or involve relatively complex hybrid models that may lack interpretability [15-18]. Your current research contributes by applying the Cheng FTS model specifically to weekly stock price forecasting in the energy sector, using PTBA as a case study. This research addresses a gap in the literature concerning intermediate-frequency forecasting (weekly), which balances noise reduction and trend capture. Furthermore, the implementation emphasizes model transparency and computational simplicity, making it accessible to analysts in practical investment settings without sacrificing performance. While previous authors focused on model accuracy via structural enhancements, your work demonstrates that even the classical Cheng framework, when properly implemented with well-structured fuzzy logical relationship groups (FLRGs) and adaptive weighting, can deliver competitive results in real-world financial environments.

Fuzzy logic presents a compelling alternative to classical forecasting methods due to its unique ability to model uncertainty, imprecision, and vague patterns characteristics that are inherently present in financial time series data [19-21]. Unlike traditional statistical models that rely on crisp numerical inputs and rigid mathematical relationships, fuzzy logic employs linguistic variables and rule-based reasoning to capture patterns that are not strictly linear or easily quantifiable [22-24]. This allows it to mimic human-like reasoning in interpreting ambiguous data, making it well-suited for environments where precise modeling is difficult, such as the stock market. In the context of energy stocks, where price trends can be influenced by qualitative information such as political developments, market sentiment, or regulatory discourse fuzzy logic enables the integration of such fuzzy factors into a structured forecasting framework. Its adaptability, transparency, and ability to generalize across different market conditions motivate its application in this study as a promising technique for stock price prediction [25].

Given these challenges and opportunities, this study aims to evaluate the accuracy and applicability of the Cheng FTS model for forecasting weekly stock prices in the energy sector. By focusing on PTBA as a representative case, the research applies the Cheng method to real historical stock data, measures predictive performance using Mean Absolute Percentage Error (MAPE) and Root Mean Square Error (RMSE), and assesses the model’s ability to capture short-term price dynamics under uncertain market conditions. The goal is not only to test the technical soundness of the model but also to determine its practical viability as a decision-support tool for investors operating in volatile financial environments.

1.1 Forecasting in financial market

Forecasting in financial markets has long been a critical area of study due to its implications for investment decision-making, risk management, and policy formulation. Traditional forecasting models, such as Autoregressive Integrated Moving Average (ARIMA), Vector Autoregression (VAR), and Exponential Smoothing, have been widely used to model linear relationships in time series data. While these models offer statistical rigor and interpretability, they are often limited in their ability to handle the nonlinear, chaotic, and high-noise characteristics inherent in financial data, especially in sectors prone to rapid fluctuation like energy. In response to these limitations, researchers have increasingly explored artificial intelligence (AI) and machine learning-based approaches such as Artificial Neural Networks (ANN), Support Vector Regression (SVR), and ensemble methods, which excel in capturing complex nonlinear dependencies. However, despite their accuracy, many AI-based models function as “black boxes,” offering limited interpretability and requiring large volumes of training data. Fuzzy logic-based approaches have emerged as a middle ground balancing predictive performance with linguistic interpretability, particularly well-suited for environments where uncertainty and vagueness dominate. This has paved the way for fuzzy time series (FTS) models to gain traction in financial forecasting research.

1.2 Fuzzy time series overview

Fuzzy Time Series (FTS) models were first introduced by Song and Chissom in the early 1990s as an extension of conventional time series forecasting methods, aimed at addressing uncertainty and imprecision in data that are difficult to model using classical statistical approaches [26, 27]. Their foundational work applied fuzzy logic principles to student enrollment prediction, marking a paradigm shift in time series forecasting by incorporating linguistic variables and fuzzy sets. Unlike traditional models, FTS does not require strict assumptions of stationarity or normality, making it adaptable to real-world data that are often noisy, incomplete, or vague. Since its inception, FTS modelling has evolved through several significant refinements. Early models used simple fuzzy logical relationships (FLRs), while subsequent improvements introduced higher-order relationships, FLRGs, and weighted relationships to enhance accuracy and flexibility [28, 29]. One of the most notable advancements is the Cheng model, which integrates weighted FLRGs and a systematic defuzzification process, enabling more accurate forecasts by emphasizing more relevant historical patterns. This evolution has enabled fuzzy time series models to be applied across various domains including education, weather, traffic, and finance demonstrating both their versatility and effectiveness in handling imprecise temporal data.

Cheng’s refinement of the fuzzy time series model, proposed in 1999, addressed critical limitations of earlier FTS approaches by introducing a more structured and weighted framework for forecasting. The key innovation in the Cheng model lies in the use of FLRGs combined with a weight matrix that emphasizes the relative importance of past patterns, thereby improving the model’s responsiveness to recent data trends. This weighting mechanism allows for a more nuanced and accurate defuzzification process, reducing prediction errors compared to earlier models that relied on equal-weighted or unordered relationships. Empirical studies have demonstrated that the Cheng model consistently outperforms basic FTS approaches in terms of MAPE and RMSE, particularly in short-term forecasting applications such as stock prices, traffic flows, and temperature trends. For instance, applications of the Cheng model in stock market forecasting have shown improved accuracy in datasets characterized by abrupt changes and nonlinear trends. Its relatively low computational complexity and high interpretability have made it an attractive option for researchers seeking a balance between forecasting precision and model transparency. As a result, the Cheng model remains a foundational reference point in the development of more advanced or hybrid fuzzy forecasting techniques [30, 31].

Despite the demonstrated effectiveness of the Cheng FTS model across various domains, its application in financial forecasting remains predominantly focused on daily or monthly datasets, with limited exploration of weekly time series particularly in sector-specific contexts. Most existing studies have either generalized across multiple stocks or benchmark indices, or focused on high-frequency intraday data where noise dominates. However, weekly data offers a balanced temporal resolution, capturing medium-term trends while filtering out excessive short-term volatility. This makes it especially valuable for retail investors and institutional analysts who adopt a weekly rebalancing strategy. Moreover, few studies have applied the Cheng model specifically to energy sector stocks, such as PTBA, which are subject to unique externalities including commodity price swings, environmental regulations, and energy policy shifts. This gap indicates an opportunity to investigate the model’s effectiveness in capturing price behavior in a highly volatile and economically strategic sector, using a weekly forecast horizon that aligns well with practical investment decision-making cycles [32, 33]. This study positions itself at the intersection of financial engineering and soft computing by applying the Cheng Fuzzy Time Series model to a real-world forecasting problem in the energy sector. By focusing on PTBA and using weekly stock price data, the research extends the scope of existing fuzzy logic literature into a less-explored temporal and sectoral dimension. The study contributes both methodologically and practically: methodologically, it demonstrates how a fuzzy rule-based approach can effectively model non-linear, imprecise stock price behavior without the need for extensive training data; practically, it provides a lightweight, interpretable forecasting tool suitable for investors and analysts in energy-related financial markets. By evaluating model performance using MAPE and RMSE, the research offers quantitative evidence of the Cheng model’s effectiveness in capturing mid-term price trends under uncertainty thus reinforcing the model’s relevance in contemporary financial forecasting challenges.

3.1 Visual and tabular forecasts

This section presents the forecasting results of the Cheng Fuzzy Time Series model applied to the weekly closing prices of PTBA. The objective was to predict future price movements based on historical fuzzy patterns and assess the model’s ability to reproduce trends in a volatile market environment. The results are presented in both tabular form and visual plots for interpretability. Table 2 shows a comparison between actual stock prices and the model’s forecasted values for selected weeks, along with the absolute error and percentage error for each prediction.

To illustrate the step-by-step implementation of the Cheng Fuzzy Time Series model, Table 2 presents the core results of the fuzzification process applied to the weekly stock prices of PTBA. The table consolidates key components, including actual prices, corresponding fuzzy sets, and the logical relationships derived between consecutive observations. This process transforms raw numerical data into linguistic patterns, forming the foundation for rule-based forecasting. The table also includes the identification of Fuzzy Logical Relationships (FLR) and their respective groupings (FLRG), which are essential in constructing the weighted transition matrix used in the Cheng model.

As shown in Table 2, each stock price is first converted into a fuzzy set (e.g., A1, A2, A3, etc.) based on the universe of discourse and interval partitions. The Left-Hand (LH) and Right-Hand (RH) columns represent sequential fuzzy states used to generate FLRs, which define how the stock price transitioned from one fuzzy state to another. These relationships are then grouped into FLRGs, such as G1, G2, and G6, which serve as rule bases for the next forecasting step. For instance, prices from 1/5/2020 to 3/1/2020 show transitions within fuzzy set A2, forming the FLRG G2. Toward the end of the series (e.g., December 2022), the prices fall into fuzzy set A6 and form consistent relationships under FLRG G6. This regularity in the pattern is beneficial for prediction accuracy, as repeated transitions enhance the reliability of the weight assignment in the Cheng model. The clean progression from raw data to linguistic abstraction reflects the structured capability of fuzzy models in managing imprecise, real-world datasets.

Table 2. Actual vs forecasted stock prices using Cheng FTS

3.2 Weighted transition matrix

To forecast future values using the Cheng Fuzzy Time Series model, it is essential to construct a weighted transition matrix that quantifies the historical transitions between fuzzy states. Table 3 displays this matrix, where each row corresponds to the current fuzzy state and each column represents the possible next state. The entries in the matrix indicate the frequency with which transitions occurred from one fuzzy state to another throughout the historical dataset. This frequency information will later be normalized to form a probabilistic weight matrix that directly influences the defuzzified forecast values.

As illustrated in Table 3, the matrix reveals clear transition patterns across the fuzzy states. For example, from state A1, there were 36 transitions to itself (A1 → A1), and 5 transitions to A2, indicating that A1 is a relatively stable state. Similarly, state A3 shows strong transitions to itself (30 times), as well as some transitions to A1, A2, and A4.

Table 3. Weighted transition matrix

These patterns suggest that certain fuzzy sets are more dominant or more likely to persist, which is essential information when generating accurate predictions. In the Cheng model, these frequencies are normalized (i.e., divided by the row total) to obtain transition weights. These weights represent the degree of influence each potential next state has, given the current state. The resulting weight matrix will then be used in conjunction with the midpoint values of each fuzzy interval to compute the final forecast through defuzzification.

Once the FLRGs are constructed, the next step in the Cheng FTS model is to compute a weight matrix that captures the relative frequency of transitions from one fuzzy state to others. This matrix is obtained by normalizing the raw transition counts row-wise, converting them into probabilistic weights. Each row of the matrix corresponds to a current fuzzy state, while each column represents a potential next fuzzy state. Using this matrix, defuzzification is performed by multiplying the weights with the midpoints of each fuzzy set and summing the results to generate a single crisp forecast value for each fuzzy group. Table 4 presents both the normalized transition weights and the corresponding defuzzified output values (in IDR), which represent the predicted stock prices for each fuzzy group.

As shown in Table 4, each fuzzy state such as A1, A2, and A3 has a unique distribution of transition weights, reflecting the probability of transitioning to other states. For instance, fuzzy state A1 has a high self-transition weight (0.87 to A1), indicating strong stability, while A3 transitions mostly to A2 and A3 with smaller contributions from A1. These normalized weights are then used to compute the defuzzified output, such as IDR 1,924.39 for A1 and IDR 2,111.11 for A3. The defuzzified outputs serve as forecasted numerical values corresponding to each fuzzy group and form the basis for the final prediction of stock prices. These results demonstrate the model’s ability to incorporate linguistic patterns and transition probabilities into meaningful, quantitative forecasts.

Table 4. Weight matrix and defuzzified outputs

After performing defuzzification based on the weighted transition matrix of each Fuzzy Logical Relationship Group (FLRG), the next step in evaluating the forecasting model is to compare the predicted values against actual historical stock prices. Table 5 displays the forecast results for selected weeks, covering both the early and late periods of the dataset. Each row contains the actual stock price, the FLRG used for forecasting, the corresponding defuzzified forecast value, and the calculated errors both in absolute terms (IDR) and percentage terms (MAPE-style error). This comparative analysis is crucial to assess how well the Cheng Fuzzy Time Series model performs in predicting weekly price movements of PTBA. It also serves as the basis for calculating summary accuracy metrics like MAPE and RMSE.

As observed in Table 5, the forecasted values show varying degrees of accuracy depending on the FLRG used. For instance, FLRG G3, used during early January 2020, exhibits a higher percentage error (e.g., 14.16%), whereas FLRG G2 performs better in some intervals with errors as low as -1.44%. The most accurate forecasts tend to occur when the same FLRG is consistently used over stable periods, while larger errors arise during weeks of rapid price change or transitions between fuzzy states. Toward the end of 2022, FLRG G6 is used for all December forecasts, with percentage errors ranging from -0.08% to 4.89%, indicating that the model remains relatively accurate in short-term forecasting for a stable fuzzy group. These findings highlight the Cheng model’s strengths in capturing patterns over time, but also emphasize the importance of FLRG construction and fuzzification quality in achieving consistent predictive performance.

Table 5. Forecast results table

To evaluate the performance of the Cheng FTS model in forecasting real-world financial data, Figure 3 presents a line chart comparing the actual stock prices and the forecasted prices of PTBA on a weekly basis over the full-time span of 2020 to 2022. The blue line represents the actual closing prices, while the red line displays the output of the Cheng model after defuzzification.

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Figure 3. Forecast prices result

This figure provides a visual representation of how well the model adapts to price fluctuations across different market conditions including periods of growth, correction, and volatility. It helps validate the accuracy trends derived from earlier tabular and statistical metrics like MAPE and RMSE.

As depicted in Figure 3, the Cheng Fuzzy Time Series model effectively captures the general directional movement of PTBA stock prices. The model performs well during moderate and stable periods, where the red forecast line closely follows the blue actual price line. However, during more volatile intervals, especially around late 2020 and mid-2022, the forecast exhibits a slight lag, a known limitation of non-adaptive rule-based models. Despite this, the visual proximity between the two lines confirms the model’s ability to produce forecasts with acceptable error margins. This result supports the suitability of the Cheng model for short-term and trend-following applications in financial forecasting particularly when interpretability and computational efficiency are important.

To further investigate the short-term forecasting behavior of the Cheng FTS model, Figure 4 presents a focused segment of stock price predictions for PTBA between June and October 2020. This period includes multiple directional shifts, including a short bullish trend, a price stabilization phase, and a sharp upward rally toward the end of the quarter. This figure allows for a more detailed evaluation of the model’s ability to respond to moderate volatility and shifting trends over a short time horizon, providing valuable insight into how well the model adapts to rapid transitions in price movement.

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Figure 4. Forecast prices result

As shown in Figure 4, the model performs well in capturing the general shape and level of the stock prices during mid-2020, maintaining a close trajectory with the actual values. In the early weeks of June and July, the forecast closely aligns with observed prices. However, starting in August and September, the model displays a conservative forecast (flat trajectory), while the actual prices experience several local dips and a late sharp rise.

To validate the effectiveness of the Cheng FTS model, we benchmarked its performance against three established time series forecasting models: ARIMA, exponential smoothing (ETS), and simple moving average (SMA). Each model was applied to the same weekly dataset and tested using a 70–30 holdout split. Forecasting accuracy was evaluated using MAPE and RMSE as shown in Table 6.

Table 6. Performance comparison of Cheng FTS and baseline models

As shown in Table 6, the Cheng FTS model achieves the lowest MAPE among all methods, while ARIMA slightly outperforms it on RMSE. However, Cheng FTS maintains a competitive trade-off between accuracy and interpretability. In contrast, SMA and ETS demonstrate significantly higher error margins, especially during non-linear transitions. This confirms that the Cheng FTS model is a viable forecasting tool for financial data where model transparency and adaptability to linguistic uncertainty are advantageous.

To further evaluate the model's behavior across different market conditions, we conducted a residual analysis that compares forecast errors during stable versus volatile periods. By calculating and plotting the difference between actual and forecasted stock prices over time, we aimed to observe whether error patterns shift under changing market regimes. The data were segmented into two phases: a stable period (Weeks 1–15) characterized by gradual movements, and a volatile period (Weeks 16–30) with sharp fluctuations. The resulting visualization is shown in Figure 5.

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Figure 5. Forecast residuals over time

As shown in Figure 5, residuals remain small and consistent during the stable phase, confirming the model’s strength in trend-following conditions. In contrast, residuals become highly variable and periodically large during the volatile phase, indicating the Cheng model’s difficulty in responding to sudden market shifts. These errors show a repeating bias, where the model underestimates sharp upward price changes and overreacts during pullbacks. This highlights a key limitation of rule-based fuzzy models: while highly interpretable and stable, they may require hybrid enhancements (e.g., fuzzy-neural or adaptive rules) to maintain forecasting accuracy during turbulence. These findings align with the observed MAPE/RMSE metrics and reinforce the importance of regime-aware evaluation.

This divergence suggests that during periods of increased volatility or trend reversals, the model’s reliance on previous fuzzy transitions may cause it to lag slightly behind real market movements. Nevertheless, the error margin remains contained, and the model still captures the overall upward movement by the end of September. These results indicate that while the Cheng FTS model is stable and interpretable for steady patterns, its responsiveness in high-volatility short windows may benefit from hybridization or integration with adaptive logic enhancements.

3.3 Discussion

This study aimed to evaluate the accuracy and applicability of the Cheng FTS model in forecasting weekly and quarterly stock prices of PTBA, a prominent energy sector company in Indonesia. The results demonstrate that the model performs well in capturing general trends, with forecasted prices closely tracking actual prices over several periods. The average MAPE across the full test period was 4.93%, indicating that the forecasted values deviated by less than 5% from actual prices on average. The RMSE was calculated at 128.6 IDR, providing a measure of absolute forecasting accuracy in currency units. These values confirm the Cheng FTS model’s effectiveness in tracking weekly price dynamics with acceptable precision for mid-range financial forecasting.

Figure 2 presented the full series comparison between actual and forecasted prices across the 2020–2022 period. The visual proximity of the two curves validates the model’s ability to follow underlying stock price trends, particularly in periods of relative market stability. This alignment was also reflected in the acceptable MAPE and RMSE values calculated from the forecast error metrics. However, in periods of rapid market movement such as the second half of 2020 and early 2022 the model’s forecasts displayed a lagging behavior, likely due to its rule-based nature that depends on previously observed patterns.

Figure 3 provided a more granular view, showing the model’s performance during mid-to-late 2020. While the Cheng model accurately predicted stock behavior in the early stages of the interval, it failed to adapt quickly to abrupt changes near the end of the quarter. This limitation is consistent with the expectations of fuzzy models, which tend to generalize behavior rather than respond to sudden or uncharacteristic deviations in the data.

Figure 4 summarized average forecasted and actual prices on a quarterly basis. The results highlight that while some quarters saw slight under- or overestimation, the model consistently mirrored the directionality of price changes capturing peaks, valleys, and recovery patterns with reasonable precision. This suggests that the Cheng FTS model is a reliable tool for medium-term forecasting when interpretability, simplicity, and trend approximation are prioritized over high-frequency responsiveness.

In terms of interpretability, the weighted fuzzy relationship groups (FLRGs) and defuzzified outputs offered transparent insights into how forecasts were derived, enabling easier validation by human experts compared to black-box models like neural networks. However, this simplicity comes at the cost of responsiveness. The model does not self-adapt to unseen shocks, such as sudden shifts in investor sentiment or macroeconomic events highlighting a potential area for improvement through hybrid models that combine fuzzy logic with learning algorithms. In summary, the Cheng FTS model is suitable for analysts and decision-makers who seek interpretable, low-complexity forecasting tools that provide stable and trend-aligned predictions in relatively smooth financial environments. Nevertheless, it may require enhancements or supplementation when used in highly volatile market conditions. To better understand model behavior under different market conditions, we analyzed the relationship between weekly price change velocity (percentage difference from the previous week) and absolute forecast error. Results show that error magnitude tends to increase as weekly price volatility rises, particularly during periods of abrupt market shifts such as late Q1 2020 and Q2 2022. This supports the observation that while the Cheng FTS model performs well during stable trends, it lags in high-volatility episodes due to the inertia of its rule-based structure.

Furthermore, analysis of FLRGs revealed that high error periods often coincided with a breakdown in recurrent pattern stability. For example, during stable periods, transitions within a single FLRG (e.g., G6) yielded consistent predictions. However, in turbulent markets, price transitions often moved across multiple fuzzy states unpredictably, reducing the forecasting power of established FLRGs. These findings suggest that the Cheng model’s performance could be enhanced through volatility-adaptive mechanisms or hybridization with dynamic models in future work. Although the performance metrics (MAPE and RMSE) suggest that the Cheng FTS model outperforms or closely matches classical models such as ARIMA and ETS, we recognize that these differences may not necessarily be statistically significant. Due to the static holdout validation approach used in this study, we did not apply the Diebold-Mariano (DM) test, which is typically designed for rolling or recursive forecasts that generate multiple error distributions over time. Future studies could incorporate a rolling window evaluation framework and employ DM testing or similar statistical tests to determine whether the observed differences in forecast accuracy are due to systematic model superiority or random variation. Such tests would enhance the robustness of performance benchmarking and deepen insights into model reliability under different forecasting horizons.

The application of the Cheng FTS model to PTBA stock offers both sector-specific and generalizable insights. PTBA operates in the energy sector, specifically in coal mining, where stock price movements are heavily influenced by macroeconomic variables such as global commodity prices, government export regulations, and energy transition policies. These factors often lead to abrupt and policy-driven shifts in investor sentiment, resulting in nonlinear and fuzzy patterns that challenge traditional forecasting techniques. The model’s relative success in capturing medium-term trends during regulatory stability reflects its suitability for energy sector forecasting when macro indicators are slow-moving. However, the observed underperformance during volatile quarters such as those triggered by sudden coal price spikes or export bans suggests that the Cheng FTS model may require adaptive enhancements for energy firms like PTBA that are particularly vulnerable to exogenous shocks. In contrast, firms in sectors with smoother market cycles (e.g., consumer goods or banking) may experience fewer abrupt discontinuities, potentially allowing the Cheng FTS model to perform more consistently without modification. Therefore, while the findings on trend-tracking and interpretability are broadly applicable, the need for volatility responsiveness is more pronounced in energy-sector applications like PTBA.