Fuzzy Time Series Forecasting Using Semantic Artificial Intelligence Tools

This article is a summary of the scientific work published by Andrii Bielinskyi, Vladimir Soloviev, Victoria Solovieva, Halyna Velykoivanenko

The original source can be accessed via DOI 10.33111/nfmte.2022.157

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In today’s data-driven economy, the ability to anticipate what comes next—market demand, supply-chain disruptions, customer churn, cash-flow needs—often separates industry leaders from the pack. Forecasting is the engine behind that foresight. By mining historical patterns and real-time signals, organizations can project future conditions, allocate resources confidently, and move faster than competitors.

The trouble is that classic statistical tools such as moving averages, linear regression, and exponential smoothing were built for orderly, linear systems. Modern markets rarely behave so neatly. They swing in nonlinear ways, customer sentiment can shift overnight, and external shocks—pandemics, geopolitical turmoil, sudden policy shifts—routinely scramble tidy equations. Traditional models also have a blind spot for the qualitative insights of domain experts, the “gut feel” that seasoned executives rely on but can’t easily encode in a regression line.

Soft-computing techniques—fuzzy logic, neural networks, evolutionary algorithms—were designed for precisely this messy reality. They cope with nonlinear relationships that derail classic models, blend hard numbers with expert judgments expressed in everyday language, and keep learning as fresh data arrives. Because they adapt rather than break when conditions change, these methods give managers forecasts that remain credible even when reality veers off script.

Fuzzy logic, introduced by Lotfi Zadeh in 1965, takes this concept further by translating human vagueness—terms like “high,” “medium,” and “low”—into rigorous mathematics. Marketing teams can grade sentiment as “mostly positive” without forcing a binary score; supply-chain leaders can label risk as “moderate to severe” instead of pinning it to a single percentage; finance chiefs can model revenue scenarios that explicitly include uncertainty instead of wishing it away.

Within the fuzzy-logic family, Fuzzy Time Series (FTS) has emerged as a particularly practical tool for business. It thrives on smaller data sets than deep-learning systems, making it valuable when historical records are thin. It can be built and updated quickly, so forecasts stay relevant in fast-moving markets. And because FTS rules read like plain language—rather than opaque neural-network weights—executives can see the rationale behind every prediction. Companies in finance, retail, logistics, and healthcare are already using FTS to sharpen revenue forecasts, inventory planning, and risk assessments, often matching the accuracy of heavyweight machine-learning models at a fraction of the cost.

Forward-thinking firms no longer treat forecasting as a back-office statistics exercise. By embracing soft-computing tools such as fuzzy logic and FTS, they turn uncertainty into strategic clarity, accelerate decision-making, and build sustainable competitive advantage.

Analysis of recent research and publications

Financial markets generate torrents of data that change by the second. Extracting practical foresight from this information flood now rests heavily on artificial-intelligence techniques. Broadly speaking, two schools of AI dominate the landscape. The first relies on pattern-recognition engines—deep neural networks, gradient-boosting machines, recurrent networks such as LSTM and GRU—that learn directly from raw price series without any preconceived notions about how those series are produced. The second begins with expert knowledge: it embeds well-understood economic relationships or market heuristics into the model’s structure and leaves only the fine-tuning to the algorithm.

Both camps keep evolving. Pattern-recognition models have progressed well beyond the early feed-forward networks that struggled with noisy, non-stationary data. Researchers now deploy self-organizing maps to cluster input vectors by similarity before generating forecasts for each cluster, a strategy that captures high-level regimes such as “growth,” “flat,” or “decline.” Recurrent architectures—Elman, Jordan, and their modern cousins—have become adept at filtering out random spikes and structural breaks. Newer designs like N-BEATSx add explanatory variables to pinpoint how seasonality or macro announcements alter price trajectories. Meanwhile, transformer-based frameworks such as Informer cut the computational burden of traditional attention mechanisms, making long-horizon projections commercially viable. Even more recent entrants like TimeGrad treat forecasting as a denoising-diffusion problem, delivering full probability distributions for thousands of correlated assets in one sweep.

Knowledge-driven models have been just as inventive. Fuzzy-inference systems translate linguistic rules—think “sell when sentiment is high and momentum stalls”—into calculable outputs that adapt to soft boundaries rather than hard thresholds. Some implementations draw on Elliott-wave theory to shape the rule base; others fuse macro indicators into a fuzzy control layer that anticipates foreign-direct-investment flows with surprising accuracy. Multivariate fuzzy-time-series (FTS) frameworks push the concept further by discovering relationships among dozens of variables at once, often after first compressing the data with self-organizing maps or fuzzy C-means clustering. Hybrid designs marry FTS logic with neural networks that handle the final defuzzification step, gaining the clarity of rules without sacrificing predictive muscle.

Our own work has explored a complementary angle: detecting early-warning signals that precede market upheavals. By blending complex-network theory, information-entropy measures, and non-extensive statistics with fuzzy-logic filters, we have built indicators that behaved as credible precursors to crashes in both cryptocurrency and equity indices. We have also shown that changes in time-series irreversibility—captured by metrics such as the Guzik and Porta indices—track the buildup of systemic stress. These complexity gauges answer the question “Is turbulence brewing?” but they do not by themselves project a future price. That task calls for forecasting engines capable of turning today’s signals into tomorrow’s numbers.

The remainder of this article returns to those engines. We begin by revisiting the core building blocks of fuzzy-set theory, from the universe of discourse to membership functions, and then show how FTS algorithms translate those concepts into actionable projections. Using the Bitcoin market as a testbed, we compare pure pattern-recognition models with fuzzy-centric approaches and demonstrate where each excels. Ultimately, we argue that blending interpretable fuzzy logic with data-hungry deep learning produces forecasts robust enough for the trading desk yet transparent enough for the boardroom.

Universe of discourse

Every forecasting model starts with a basic question: “What range of values do we expect to see?” Statisticians call this span the universe of discourse—essentially the lowest and highest numbers the historical data have produced. Even when we assume that a time series is stable, real-world markets still throw the occasional curveball: a rogue trade, a surprise policy announcement, or a one-off data glitch. Those outliers can nudge new observations just outside the historical band.

To keep the model from stalling when that happens, we widen the acceptable range by a small safety margin on both ends. Think of it as building a little extra clearance into a machine part so it doesn’t jam under stress. By defining this expanded “comfort zone” before we begin fuzzification—the step that converts hard numbers into the linguistic categories used by fuzzy logic—we ensure the system stays calm and accurate, even when a fresh data point pushes the edge of what we have seen before.

That extra breathing room makes the entire forecasting process more resilient. Rather than rejecting unexpected values or forcing them into ill-fitting buckets, the model smoothly absorbs the new information, preserving accuracy and avoiding costly misclassifications when markets behave a bit out of character.

Choosing the Right “Lenses” for a Fuzzy Forecast

Once the overall value range—our universe of discourse—is fixed, we still need a way to translate each raw data point into the fuzzy language the model understands. That translation hinges on three design choices: how many fuzzy sets will cover the range, what mathematical shape each set will take, and which partitioning logic will position those sets along the axis.

The membership function is the core of that design. It assigns every crisp input a degree of belonging—from 0 (no membership) to 1 (full membership)—inside each fuzzy set. Different functions offer different trade-offs. Straight-line shapes such as triangles and trapezoids are easy to code and interpret, which explains their popularity in early fuzzy-time-series work. But because they join flat segments at hard corners, they can produce abrupt shifts in membership grades right where the business user expects smooth transitions. For that reason, many analysts now prefer curved alternatives like Gaussian or bell-shaped functions. These rely on only two parameters—the midpoint and the spread—yet deliver a continuous slope that looks and behaves more like a familiar probability distribution.

Although later sections will show that the choice of function rarely moves the final accuracy needle by more than a fraction of a percent, it does influence how readable, and therefore how trustworthy, the model feels to decision-makers. Smooth curves convey gradual change and are often easier for executives to visualize when they ask, “How close are we to tipping from moderate to high risk?” Moreover, any of these functions can be fine-tuned during model training, so the boundaries adapt to the data rather than forcing the data to fit arbitrary boundaries.

Researchers continue to experiment with more exotic shapes—picture fuzzy sets, type-2 and non-stationary fuzzy sets, even hesitant differential variants—each aimed at capturing a specific nuance of uncertainty. Those innovations are useful when your forecasting problem demands extra flexibility, but for mainstream financial time-series work, the well-proven options listed below usually suffice.

With the membership “lenses” selected, the model can now blur crisp market data into overlapping categories without losing critical detail, setting the stage for the rule-generation and prediction steps that follow.

Fuzzy time series idea

A fuzzy-time-series (FTS) model treats every datapoint not as a single hard number but as a member of a descriptive class with soft boundaries—“high,” “medium,” “low,” or, for rate-of-change problems, “rise,” “flat,” “fall.” This simple shift makes the entire forecasting engine behave more like a seasoned analyst than a calculator. The first technical task is to define the span of historical values—the universe of discourse—and then add a small safety buffer above the peak and below the trough so that an unexpected market blip never falls outside the model’s field of vision.

Within that buffered range the universe is sliced into a fixed number of fuzzy sets. Each set is shaped by a membership function, the mathematical lens that says how strongly a particular price belongs to “medium” versus “high,” and so on. Early FTS research relied on triangular and trapezoidal shapes because they are easy to code, but their sharp corners can create awkward discontinuities. Gaussian and bell-shaped curves smooth those edges, require only a midpoint and a spread to set up, and read like the probability distributions executives already know. Although later sections will show that swapping one curve for another rarely moves the accuracy needle by more than a rounding error, the choice has a real impact on how intuitive the model feels to non-technical decision-makers.

Once the fuzzy “lenses” are in place, the historical series is translated—fuzzified—into overlapping linguistic descriptions. From there the training loop looks a lot like teaching a new hire to recognise market regimes. The model watches how today’s description flows into tomorrow’s, storing each transition as a rule. Over thousands of observations those rules coalesce into a knowledge graph that links states such as “moderately bullish” with their most probable successors. Figure 1 sketches the process: the upper panel shows how historical data are partitioned, fuzzified, and distilled into a rule network; the lower panel shows how an incoming price is mapped to its fuzzy state, matched against the learned transitions, and finally converted back—defuzzified—into a concrete forecast.

By allowing crisp numbers to pass through these linguistic layers, the FTS framework captures the way seasoned traders actually think: not in exact decimals but in zones, tendencies, and degrees of confidence. The result is a forecasting system that can digest noisy, non-stationary financial data, express its reasoning in plain language, and still deliver the numeric precision required on the trading floor.

Song and Chissom matrix-based method

The final ingredient in a fuzzy-time-series model is the “playbook” of how one market condition flows into the next. Song and Chissom pioneered a simple way to capture that flow. Each day’s fuzzy description—say, moderately bullish—is paired with the description that follows it the next day. The pairing becomes an if-then rule: If today is moderately bullish, tomorrow is usually strongly bullish.

Stack every such rule in a table and you produce what the researchers call a Fuzzy Relationship Matrix. Think of it as a 360-degree map: every row represents today’s state, every column tomorrow’s, and each cell stores how often that jump has really happened. The more frequently the market made a given move in the past, the stronger the number in the cell.

Using that matrix takes three steps in live trading:

Because the matrix expands automatically as fresh data arrive, it evolves with the market—no manual retuning required. And because every entry is built from an easy-to-read if-then statement, risk managers, traders, and auditors can trace exactly why the system expects the next price point to land where it does. In other words, you get the adaptability of machine learning without losing the hard-won transparency that business leaders demand.

Chen rule-based method

While the Song–Chissom framework laid the theoretical groundwork for fuzzy-time-series forecasting, many practitioners gravitate toward a streamlined variant first popularised by Chen. The appeal is twofold: the rules are still easy for humans to read, yet the underlying computations scale far better when the data set balloons.

A Chen model begins exactly as any fuzzy system does—by fixing a universe of discourse, carving it into fuzzy sets, and converting each historical price into the set where its membership weight is strongest. Once the series has been re-expressed in this linguistic form, consecutive fuzzy states are paired to reveal cause-and-effect patterns. But here is Chen’s twist: instead of storing every one-step transition separately, the method folds all rules with an identical left-hand side into a single “family.” A state like AiA_iAi​ therefore carries one compact list of all the states it has ever led to, ordered by how often each jump occurred, and duplicate links are recorded only once.

At forecast time the engine asks a simple question: What happened most frequently after we last saw state AiA_iAi​? If the historical record shows that AiA_iAi​ has no follow-up, the algorithm plays it safe and assumes inertia, projecting AiA_iAi​ again. If multiple successors exist, it adopts the full family and then defuzzifies, translating that short ranked list into a single price point by averaging the mid-values of the candidate fuzzy sets.

The practical benefits are immediate. Because the entire knowledge base collapses into a handful of rule families, the relationship matrix stays modest even when the universe of discourse is sliced into dozens of fuzzy bands. That restraint keeps memory use in check, shortens training cycles, and slashes response times when new data stream in—advantages that become decisive when analysts must refresh forecasts intraday across hundreds of assets.

In short, Chen’s rule grouping offers an elegant workaround to the curse of dimensionality. It preserves the narrative clarity executives need—“if we sit in a mildly bullish zone, history says we’re likely to edge higher tomorrow”—while delivering the computational economy required for high-frequency or large-portfolio scenarios. For businesses juggling vast data sets but unwilling to sacrifice interpretability, the Chen approach often hits the sweet spot between transparency and speed.

Yu weighted-based rule method

Chen’s streamlined rule‐group model reinvigorated fuzzy-time-series forecasting by shrinking the rule base and speeding computation. Yet its very efficiency hides two blind spots. First, the model forgets how often a particular rule has appeared; a rare, one-off jump is treated with the same importance as a regularly recurring pattern. Second, it ignores when each jump happened, giving something that occurred years ago the same influence as yesterday’s move. In fast-moving markets—where recency and repetition are strong signals—those omissions can dull predictive power.

Yu’s weighted-rule method closes that gap by letting the model remember both how frequently and how recently each fuzzy logical relationship (FLR) has shown up. Instead of discarding duplicate rules, the Yu approach keeps them, tags them with timestamps, and then assigns weights that rise with each recurrence and rise even faster when the recurrence is recent. The weighting can be as simple as numbering the rules in chronological order—one for the oldest, two for the next, three for the most recent—then normalizing so the total weight equals one. Under that scheme a pattern that surfaced last week naturally counts for more than the same pattern seen a year ago, and a pattern that has surfaced five times outweighs one that appeared only once.

When a new price arrives, the procedure still follows Chen’s easy logic—identify today’s fuzzy state, look up the family of next-day states, and compute a forecast—but it now multiplies each candidate successor by its weight before defuzzifying. The arithmetic remains light, yet the model captures the intuitive truth that markets tend to repeat themselves and that fresh evidence carries more credibility than stale anecdotes.

In practice, this simple tweak delivers two business advantages. Forecasts react faster to regime changes because the most recent transitions dominate the calculation. And they become more robust during periods of repetitive volatility, because recurring patterns are no longer watered down by one-time anomalies. The result is a forecasting engine that keeps Chen’s transparency and low computational cost while adding a crucial dose of market memory, providing analysts with predictions that feel both responsive and grounded in real behavioural evidence.

Empirical analysis and forecasting

To move from theory to practical value, we ran the three fuzzy-time-series engines—Song, Chen, and Yu—on Bitcoin’s daily closing prices between 1 January 2016 and 14 September 2022. The data came straight from Yahoo Finance and were split 80/20 into training and test segments so that every model had to prove itself on unseen information.

Smoothing the Noise Before Forecasting

Cryptocurrencies are notoriously volatile. Instead of feeding that raw turbulence into the models, we first converted price levels into daily return on investment (ROI)—the percentage change from one day to the next. The transformation makes the series far more stable, which in turn lets the fuzzy rules lock onto repeatable patterns rather than one-off spikes. After the models generated an ROI forecast, we simply reversed the calculation to obtain a dollar-price projection.

From Prices to Linguistic States

The cleaned ROI series was mapped onto fifteen overlapping Gaussian “states,” each defined by a mean and variance. These states form the linguistic variable BTC-USD fuzzy time series: an intuitive scale that runs from sharp declines through flat periods to strong rallies. Every day’s return joins several states at once but with different membership strengths, mirroring the way a trader might say a move feels “mostly bullish but with a hint of caution.”

Building—and Weighting—the Rule Books

Each engine then learned its own rule book:

• Song stored every one-step transition in a full relationship matrix.
  
• Chen compressed that knowledge by merging all rules with the same starting state, a shortcut that slashed computation time.
  
• Yu kept Chen’s compact format but rewarded rules that appear often and recently, giving the model a “memory” of what matters now.
  
  

Results That Track the Market

Figure 4 plots one-step-ahead forecasts for the 2021–2022 test window. All three traces shadow the true Bitcoin curve closely, capturing the explosive run-up, the protracted slide, and the mid-2022 stabilisation. The Song line drifts slightly higher than reality—a symptom of rigid rule counts—while Chen hugs the actual prices more tightly and Yu stays closest of all thanks to its recency weighting.

Why Diagnostics Matter

Accuracy isn’t just about eye-balling a chart. We ran standard residual tests to check whether each model had squeezed the predictable information out of the series. Song’s residuals failed both the Augmented Dickey–Fuller and Ljung-Box checks, signalling leftover structure the model missed. Chen’s passes were markedly cleaner, and Yu posted the strongest white-noise signature, confirming that its weighted rules had soaked up most of the autocorrelation.

Business Take-Aways

• A simple ROI pre-processing step tames crypto volatility enough for fuzzy models to see clear patterns.
  
• Rule compression (Chen) delivers faster forecasts without sacrificing interpretability.
  
• Recency-weighted rules (Yu) add a crucial adaptive edge, especially in markets that change character overnight.
  
  

In short, if your trading desk or treasury team needs forecasts that stay transparent yet agile, a weighted fuzzy-logic engine like Yu’s offers a compelling blend of speed, clarity, and statistical rigor—capturing the upside of advanced AI while keeping the governance team comfortable with how the numbers are produced.

What the Numbers Say: Which Fuzzy Engine Earns Its Keep?

After six years of Bitcoin data and a head-to-head showdown on the 2021-2022 test window, two names clearly come out on top. Chen’s streamlined rule-group model posts an average forecasting error of roughly seven hundred dollars per day, coupled with a root-mean-squared error just above twelve hundred dollars. That translates into a coefficient of determination of about 0.99 and an overall hit-rate—measured as directional accuracy—north of ninety-eight per cent.

Yu’s recency-weighted upgrade lands close behind. It misses the daily close by just over a thousand dollars on average and records an RMSE near fifteen-hundred dollars. Its R-squared hovers around 0.98, with accuracy a touch above ninety-seven per cent. The marginally larger error is the cost of extra adaptability: because Yu lets recent patterns speak louder than stale ones, it reacts more quickly when the market abruptly changes character.

Song’s original full-matrix approach, while conceptually elegant, falls short when numbers matter. Its mean absolute error tops four thousand dollars, the RMSE approaches forty-seven hundred, and although the R-squared still looks respectable at 0.84, residual diagnostics reveal lingering structure the model failed to capture. In other words, what appears decent on paper would feel shaky on a real trading desk.

Statistical health checks confirm the headline figures. Residuals from Chen and Yu pass both the Augmented Dickey–Fuller and Ljung-Box tests, meaning the left-over errors behave like white noise—stationary, uncorrelated, and thus largely irreducible. Song’s residuals fail both assessments, indicating that significant, predictable information remains untapped.

Year-by-year absolute-percentage-error plots tell the same story. Median errors for Chen and Yu sit at or near zero in 2021 and 2022, whereas the Song model hovers in the ten-to-fifteen-per-cent zone. All three engines experience occasional extreme misses—crypto still surprises even the cleverest algorithms—but Chen and Yu keep those outliers far fewer and farther between.

The practical verdict is straightforward. If you need maximum precision without loading the system down, deploy Chen: it runs light, explains its logic in plain language, and leaves little predictable variance on the table. If your market shifts rapidly and you prize responsiveness slightly more than raw error minimisation, choose Yu: its time-weighted memory costs a few extra dollars of daily error but buys faster adaptation to new regimes. Either way, you gain forecasts that stand up to statistical scrutiny, integrate smoothly into compliance workflows, and arrive soon enough to act upon—benefits the traditional Song framework simply cannot match in a modern, high-frequency setting.

Conclusions

Fuzzy-Time-Series (FTS) forecasting has moved well beyond academic curiosity and is now attracting serious attention from market strategists. Because FTS models learn patterns straight from data rather than relying on a maze of preset parameters, they are quick to build, inexpensive to run and—crucially—easy to explain to stakeholders. In our Bitcoin case study we put three first-order variants through their paces: Song’s original formulation, Chen’s streamlined rule-group version and Yu’s modern, weight-adjusted upgrade. Despite their different rule engines, all three delivered broadly comparable accuracy on daily BTC-USD closes, while retaining the transparency that regulators and auditors increasingly demand.

The deeper dive, however, revealed subtle trade-offs. Chen’s approach offers the leanest run-time footprint, an advantage for desks that update hundreds of symbols intraday. Yu’s weighting scheme reacts faster when fresh market information arrives, making it attractive for highly volatile assets—though at the cost of a small uptick in average error. Song’s full-matrix model, by contrast, lags on both precision and computational efficiency and is harder to justify in a production setting.

Looking ahead, the real opportunity lies in layering these interpretable rule-based engines with multivariate inputs—on-chain metrics, macro signals, even social-media sentiment—and marrying them to early-warning indicators drawn from complexity science. Add a selective dash of deep learning for feature extraction, and businesses could enjoy the rare combination of agility, accuracy and governance-friendly clarity in an industry where black-box models too often leave decision-makers in the dark.

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