Multifractal Analysis and Machine Learning-Based Predictive Modeling of the Tanker Freight Market: Unraveling Complexity Across Four Breaks

1. Methods

1.1. The Multifractal Detrended Fluctuation Analysis Method

The following introduction of MF-DFA method is based on the work from Kantelhardt, et.al. (2002)[34].

Here are the general steps of MF-DFA method on the series$\mathrm{x}\left(\mathrm{i}\right)$, where $i=\mathrm{1,2},...,N$and $N$ is the length of the series. $\bar{x}$ stands for the average value of series$x\left(i\right)$.

Assuming that $\mathrm{x}\left(\mathrm{i}\right)$ are increments of a random walk process around the mean $\bar{x}$, then by the signal integration, the "trajectory" or "profile" could be expressed as

(1)

Next, we divide the integrated series into $N_s=\mathrm{i}\mathrm{n}\mathrm{t}\left(N/s\right)$, non-overlapping segments of equal length $s$. Generally, the length $N$ of the series is not a multiple of the considered time scale $s$, a short part may remain at the end of the profile $y\left(i\right)$. Not to disregard this remaining part, this procedure is repeated oppositely starting from the end. So $2N_s$ segments are obtained. Next, the local trend for each of the $2N_s$ segments could be calculated by a least-square fit of the series. Then the variance is determined by

(2)

for each segment $v$, $v=1,...,Ns$ and

(3)

For $\mathrm{v}=\mathrm{N}\mathrm{s}+1,...,2\mathrm{N}\mathrm{s}$. Here, $y_v\left(i\right)$ is the fitting line in segment $v$. Next, over all segments are averaged to obtain the $q$-th order fluctuation function by

(4)

where, the index variable $q$ can generally take any real value except zero. Repeating the above steps for several time scales$\mathrm{s}$, $F_q\left(s\right)$ will increase as $s$ increases. The scaling behavior of the fluctuation functions could be analyzed by log-log plots $F_q\left(s\right)$ versus $s$ for each value of $q$. A power-law between $F_q\left(s\right)$and $s$ exists as the Eq. (5) when the series $x\left(i\right)$ is long-range power-law correlated.

(5)

However, because of the diverging exponent, the averaging procedure of Eq. (4) could not be applied directly to calculate the value $h_0$ corresponds to the limit $h_q$ as $q\to 0$. Instead, we must employ a logarithmic averaging procedure by Eq. (6).

(6)

The exponent $h_q$ generally depends on $q$. For stationary series, $h_2$ is the well-defined Hurst exponent $H$. So $h_q$ is called the generalized Hurst exponent. In a special case, when $h_q$ is independent from $q$, it is defined as monofractal series. The distinct scaling patterns exhibited by small and large fluctuations have a substantial impact on the relationship between the $q$-th order Hurst exponent $h_q$ and the scaling parameter $q$. In the case of positive $q$, segments $v$ characterized by a significant deviation from the expected trend, i.e., those with large variances, will exert a dominant influence on the average $q$-order Hurst exponent $Fq\left(s\right)$. Consequently, a positive $q$ captures the scaling behavior of these segments $v$ with notable fluctuations, which typically correspond to smaller scaling exponents in multifractal time series. Conversely, for negative $q$ values, segments $v$ with smaller variances take precedence in determining the average $q$-order Hurst exponent $Fq\left(s\right)$. Hence, a negative $q$ describes the scaling behavior of segments $v$ with minor fluctuations, which generally exhibit larger scaling exponents in multifractal time series. This intricate interplay between $q$, the scaling behavior of different segments $v$, and the corresponding fluctuations provides valuable insights into the multifractal nature of the time series, shedding light on how various levels of variance impact the overall scaling exponents.

The multifractal spectrum $f\left(\alpha \right)$ is another tool to characterize multifractality in a series. $f\left(\alpha \right)$ can be obtained by the Eq. (7)

(7)

and then the Legendre transform

(8)

(9)

where $\alpha$ is the Holder exponent value which indicates the strength of singularity. When the $f\left(\alpha \right)$ is broader, it indicates a stronger multifractality or complexity.

The width of the spectrum could be

(10)

where ${\alpha }_{max}$and ${\alpha }_{min}$ indicate the maximum and minimum values respectively.

We name MF-DFA1, MFDFA2 and MFDFA3 separately with polynomial order $m=\mathrm{1,2},3$. Here we apply MF-DFA1 and MF-DFA2 to investigate the BCTI, BDTI and specific routes of TC2 and TD7.

1.2. The Multifractal Detrending Moving Average Method

Following brief introduction of MF-DMA method is based on works of Gu and Zhou (2010) [41].

Assuming time series $x\left(t\right)$ , $t=\mathrm{1,2},\cdots ,N.$ and $N$ is the length of the series. We construct a new series

(11)

Next step, $\tilde{y}\left(t\right)$ indicates the moving average function. To calculate the sequence of cumulative totals, we slide a window of fixed size across the sequence.

(12)

where $n$ is the size of window, $⌊x⌋$ is the largest integer but not greater than $x$, $⌈x⌉$ is the smallest integer but not smaller than $x$, and $\theta$ is the position parameter, varying from 0 to 1. Here $\tilde{y}\left(t\right)$ is calculated over $⌈\left(n-1\right)\left(1-\theta \right)⌉$ data points from the preceding period while $⌊\left(n-1\right)\theta ⌋$ data points from the subsequent period. We have to notice three special cases with different $\theta$ values. The backward moving average, where $\theta =0$ and $\tilde{y}\left(t\right)$ is calculated by all the past data points. $\theta =0.5$refers to the centered moving average, where $\tilde{y}\left(t\right)$ is calculated over half past and half future data points. $\theta =1$ means the forward moving average, where $\tilde{y}\left(t\right)$ is based on the trend of future data points. In this context, we utilize the selected case $\theta =0$, as it has demonstrated superior performance compared to the other two alternatives, based on evidence presented in references [37,41,43].

Subsequently, we eliminate the moving average component $\tilde{y}\left(i\right)$ from the series $y\left(i\right)$ to eliminate any underlying trend, resulting in a residual sequence $\epsilon \left(i\right)$.

(13)

where $\mathrm{n}-⌊\left(\mathrm{n}-1\right)\mathsf{\theta }⌋\le \mathrm{i}\le \mathrm{N}-⌊(\mathrm{n}-1)\mathsf{\theta }⌋$.

Then, the residual series $\epsilon \left(i\right)$ is divided into $N_n$ ($N_n=⌊N/n-1⌋$) non-overlapping segments, each of equal length $n$. These segments can be represented as ${\epsilon }_v\left(i\right)=\epsilon (l+i)$ for$1\le i\le n$, where $l=(v-1)n$. We can get the root-mean-square function $F_v\left(n\right)$ by Eq. (14).

(14)

Additionally, the $q$-th order overall fluctuation function $F_q\left(n\right)$ is expressed as

(15)

(16)

Next step, when the values of $n$ varies, we can get the power-law relation between $F_q\left(n\right)$ and $n$ in Eq. (17)

(17)

Finally, the multifractal scaling exponent $\tau \left(q\right)$ and multifractal spectrum $f\left(q\right)$ could be defined similarly with that of above MF-DFA.

1.3. The Effective Multifractality

According to the references [58,61], the total multifractal spectrum could be intricately divided into three parts: the non-linear and linear correlation, and the PDF. This decomposition is captured by the Eq. (18).

(18)

It is important to emphasize that both the linear correlation component $\Delta {\alpha }_{LM}$ and the nonlinear correlation component $\Delta {\alpha }_{NL}$represent temporal correlations [2,5]. Specifically, the linear correlation component is attributed to finite-size effects [54,63]. Furthermore, it is noteworthy that $\Delta {\alpha }_{LM}$ indicating the linear correlation component, can be computed by semi-analytical formulas of an explicit form, offering a comprehensive quantitative characterization of this phenomenon [39]. A type of computational deviation stemming from the sample number constraints is defined as the finite-size effect in reference [26]. In essence, smaller time series sizes lead to greater computation deviations. To mitigate the impact of sample size limitations, especially for small sample sizes (<10000), it is necessary to calculate and exclude the linear correlation component from the true multifractality. Consequently, the true multifractality, denoted as $\Delta {\alpha }_{eff}$, which encompasses the nonlinearity component $\Delta {\alpha }_{NL}$, and the PDF component $\Delta {\alpha }_{PDF}$, is determined[40,59,61].

To depict the spectrum of multifractality, it is important to conduct an analysis that involves both the elimination of the linear correlation component stemming from the sample size limitations (sample size < 10000 points) and the decomposition of the remaining two effective parts [59,61,62]. This quantitative analysis can be achieved through the creation of two new series: the shuffled and the surrogated time series. The shuffled time series is generated through the shuffled original series. During the process, the temporal correlations are disrupted, while the probability distribution remains unaltered [43,61].

The creation of surrogate data is accomplished through a two-stage procedure. Initially, the process ensures that the surrogate data matches the original volatility time series in terms of probability distribution, which is executed through a transformation technique as described in reference [49]. Subsequently, the surrogate time series is manipulated to include linear correlations by applying an improved version of the amplitude-adjusted Fourier transform (IAAFT), as detailed in reference [59]. To gain a thorough grasp of the surrogate time series construction process, it is recommended that readers refer to the comprehensive explanation in the reference [26].

1.4. Machine Learning-Three Learners

Overall, we selected and combined three learners(XGBoost, LightBGM and CatBoost) from machine learning to form our stacking regression model. Therefore, the introduction of machine learning is based on three researches which are respectively done by Chen T , Guestrin C[54]; Ke G, Meng Q, Finley T[64]; Prokhorenkova L, Gusev G, Vorobev A[65].

XGBoost is a scalable tree boosting system, which firstly use tree boosting in a nutsheel to regularize learning objective:

(19)

where $\Omega \left(f\right)={\rm Y}{\rm T}+{\displaystyle \frac{1}{2}}\lambda {‖\omega ‖}^2$, $l(\widehat{y_i},y_i)$ is the loss function, $y^i$ is the predicted value, $y_i$ is the target value, ${\rm Y}$ and $\lambda$ are regularization parameters, ${\rm T}$ is the number of trees, and ${‖\omega ‖}^2$ represents the square of the output score on each tree's leaf nodes (equivalent to L2 regularization) .

Then, the system add ft to minimize objective and use second-order approximation to quickly optimize the objective in the general setting, the corresponding optimal is:

(20)

In the final stage, it is necessary to scale the newly added weights and perform column sampling to prevent overfitting (similar to random forests). XGBoost also includes a split finding algorithm, where the basic greedy algorithm enumerates all possible splits, calculates the gain for each split, and then selects the split with the maximum gain. The approximate algorithm, on the other hand, proposes candidate split points by mapping continuous features into bins and then aggregating statistics to find the optimal solution. In summary, XGBoost introduces a new sparse-aware algorithm and weighted quantile sketch, where caching access patterns, data compression, and partitioning are key, thus enabling the solution of real-world scale problems with minimal resources.

LightGBM is an efficient Gradient Boosting Decision Tree (GBDT) algorithm proposed by Ke et al. at the NIPS conference in 2017. It addresses the efficiency and scalability issues associated with high-dimensional features and large datasets by introducing two innovative techniques: Gradient-based One-Side Sampling (GOSS) and Exclusive Feature Bundling (EFB). The GOSS technique excludes data instances with small gradients, using only the remaining instances to estimate the information gain, thereby significantly reducing the amount of data processed while maintaining the accuracy of information gain estimation. The EFB technique reduces the number of features by bundling mutually exclusive features that rarely take non-zero values simultaneously, such as one-hot encoded features in text mining. LightGBM safely bundles these exclusive features together, constructing the same feature histograms from feature bundles as from individual features through a carefully designed feature scanning algorithm, thus reducing the complexity of histogram construction from O(#data×#feature) to O(#data×#bundle), where #bundle is much smaller than #feature, significantly accelerating the training of GBDT. Experimental results show that LightGBM is over 20 times faster in training on multiple public datasets while achieving nearly the same accuracy as traditional GBDT. These achievements not only demonstrate the superior performance of LightGBM in handling large-scale datasets but also provide new directions for the optimization of GBDT algorithms.

CatBoost introduced two algorithmic improvements: ordered boosting and an innovative algorithm for handling categorical features, corresponding to the Ordered mode and Plain mode (built-in ordered TS standard GBDT algorithm). For the Plain mode, multiple random permutations are first used to calculate gradients and TS, evaluate candidate splits, update the support model to construct decision trees, and then a complexity comparison and analysis with the standard GBDT algorithm is performed, culminating in the greedy construction of high-order feature combinations. CatBoost identified and analyzed the problem of prediction shift, proposing ordered boosting and ordered TS as solutions, and demonstrated superior performance in multiple benchmark tests.

1.5. Predictive Methodology Overview

To investigate how freight rates can help predict Brent oil prices in different periods, we employed the following methodology across four distinct periods (Period I-IV). In Table 1, each of these periods corresponds to a significant global event that profoundly affected both freight rates and oil prices.

For each of these periods, we employed the same predictive approach to understand the effect of multifractal features on forecasting oil prices. Specifically, the freight rates (BDTI) were utilized alongside different feature sets to predict Brent oil prices, with the following steps:

Direct Prediction Using BDTI: As a baseline, we predicted Brent prices using only BDTI data.

Addition of Crisis Period Indicator: We added an indicator for the crisis period to account for the impact of significant events (e.g., the financial crisis or the pandemic) on market dynamics.

Addition of Multifractal Features: Multifractal features extracted using the Multifractal Detrended Fluctuation Analysis (MFDFA) method were included to capture complex market behaviors.

Combination of Crisis Indicators and Multifractal Features: Both crisis indicators and multifractal features were included to examine their combined effect on prediction accuracy.

To provide a clear visual representation of the methodology, the following flowchart outlines the framework for multifractal analysis and predictive modeling using the Baltic Dirty Tanker Index (BDTI) and its application to Brent crude oil price prediction. This framework includes three key components: data processing, multifractal analysis, and predictive modeling and validation.

Figure 1. Framework for Multifractal Analysis and Predictive Brent Crude Oil Prices Using BDTI. This visual representation complements the detailed methodology described above, offering readers an intuitive understanding of the step-by-step process adopted in this study, particularly how multifractal indicators and machine learning models are integrated for predictive modeling across the four structural break periods.

Figure 1. Framework for Multifractal Analysis and Predictive Brent Crude Oil Prices Using BDTI. This visual representation complements the detailed methodology described above, offering readers an intuitive understanding of the step-by-step process adopted in this study, particularly how multifractal indicators and machine learning models are integrated for predictive modeling across the four structural break periods.

2. Data Description

The Baltic Clean Tanker Index (BCTI) is a widely tracked benchmark that measures the cost of shipping clean petroleum products, such as refined oil, on specific routes within the Baltic region [2,7,10,14]. It serves as a vital indicator for gauging freight rates and understanding the supply and demand dynamics within the clean tanker market. The BCTI's fluctuations influence various economic sectors, making it an essential tool for industry stakeholders, analysts, and investors seeking insights into energy market trends and shipping conditions. Therefore, we pay great attention to the BCTI and specific route of TC2 from Continent to USAC with clean tanker size of 37,000mt.

Similarly, The Baltic Dirty Tanker Index (BDTI) is a vital benchmark that assesses the cost of shipping dirty petroleum products, including crude oil, on selected routes within the Baltic region. It serves as a key indicator for understanding freight rates and evaluating the supply and demand dynamics within the dirty tanker market. A specific route of TD7 from North Sea to Continent with dirty tanker size of 80,000mt is selected to analyze.

The sample for daily BCTI and BDTI covers the period from Jan 28, 1998 to Jan 12, 2024. The sample size includes 6413 and 6358 points separately, which is enough for multifractal models [36,37,38,39,40,41]. However, the data for BCTI TC2 is 5014 points as the data begins from March 4, 2004. The data of BDTI TD7 includes 6234 points as we could not get the records for year 2023. The statistics results of sample time series are listed in Table 2. Figure 2 describes the BCTI and BDTI of the sample observations. There are big volatilities in the year 2008 under global financial crisis, in the year 2020 when the COVID-19 break out and in the year 2022 when geographic conflict happened. Figure 3 records the BCTI and BDTI logarithmic changes (that is $lnP\left(t\right)/lnP\left(t-1\right)$), which is widely applied to calculate daily volatilities and the returns often help to decrease non-stationarities though it is not required in multifractal methods [12,39].

3. Analysis of Results

3.1. Multifractality in Dirty and Clean Tanker Freight Rate Returns

The Hurst exponent is a measure that characterizes the rate at which the root-mean-square (RMS) deviation of a time series expands as the size of the observational window (i.e., the scale) grows, revealing the monofractal nature of the data [11,28,33,34,35]. In the context of a multifractal time series, the localized variations, or the RMS deviations, exhibit significantly large values for segments coinciding with periods of high fluctuation, and similarly, they show significantly small values for segments during periods of low fluctuation [32,33,34]. The q-order Hurst exponent is calculated by examining the slopes of the regression lines that relate the q-order RMS to the scale of the observation [36,37,38,39,40,41]. Figure 4(a) to (d) on the left illustrate that for multifractal time series, these slopes $Hq$ vary depending on the value of q. The distinction between the q-order RMS for positive and negative fluctuations is more pronounced at smaller segment widths than at larger ones. This is because smaller segments are more sensitive to the local variability within a specific period, whereas larger segments encompass multiple periods and tend to average out the differences in fluctuation magnitude. As q increases, the q-order RMS for a multifractal time series generally decreases, as shown on the right side of Figure 4(a) to (d). Both BCTI and BDTI returns exhibit multifractal characteristics, indicating that their fluctuation patterns are not uniform across different scales and require a more nuanced analysis to understand their underlying dynamics [12,43].

3.2. Multifractal Characteristics of Tanker Freight Fluctuation Under Structural Breaks

The logarithmic return of BCTI and BDTI and the specific routes in Figure 2 suggests that there exist big fluctuations in year 2008, year 2020 and year 2022 with events of financial crisis, COVID-19 and geographic conflict, which is in accordance with the references [7,9,12,46]. According to references, big events such as global financial crisis can lead to structural breaks in time series [45,47,48], we divide the sample data BDTI into two parts: period I from Jan 28, 1998 to Dec 01, 2010 and period II from Dec 02, 2010 to Jan 12, 2024 to delve into the changing external factors’ effects on the multifractalities.

The study of multifractal time series often involves various scaling exponents, among which the q-order Hurst exponent $Hq$ is prominent. However, the local Hurst exponent $Ht$ has proven advantageous in detecting specific time points of structural change within a time series [45,48,49,50], [58,59]. This local perspective aligns with q-order Hurst values $Hq$ for extreme fluctuations, correlating positively or negatively depending on the q value's sign. The utility of the local Hurst exponent $Ht$ is particularly evident when financial time series experience sudden disturbances. It pinpoints how these shocks modify the series' inherent scale-invariant features on a localized level [23,24,25,26,27]. Visualized through histograms, the temporal variations of the local Hurst exponent $Ht$ offer a probability distribution $Ph$ of these changes (illustrated in Figure 5 (e) for the first period and (f) for the second period). Complementing this, the multifractal spectrum – delineated by the parameters $f\left(\alpha \right)$and $\alpha$ captures the breadth of multifractality within the series (depicted in Figure 5 (e) for the first period and (f) for the second period). An increasing spectrum breadth denotes growing structural disparities between periods marked by minor and major fluctuations [25,26,27,28]. The research employs multifractal spectrum width $\Delta \alpha$ as a measure of multifractality level, which, according to results presented in Figure 4 (e) and (f), confirms strong multifractality in both examined periods of the Baltic Dry Index (BDTI) returns. These findings align with previous research outlined in references [17,18], solidifying the observed characteristics of the market across different analytic methodologies. From Figure 5 (a) and (c), both the BCTI and BDTI markets are with multifractal characteristics though they are different from the specific routes of TC2 and TD7 in Figure 5 (b) and (d).

3.3. Temporal Dynamics of Tanker Freight Market Complexity with MF-DMA Method

In Figure 6 (a), the multifractality from PDF is $\Delta {\alpha }_{PDF}=0.48$, while the total multifractality from the three parts is $\Delta \alpha =0.90$. The multifractality from linear correlation and PDF are $\Delta {\alpha }_{LM}+\Delta {\alpha }_{PDF}=0.61$, so the multifractality from the non-linear correlation $\Delta {\alpha }_{NL}=0.29$. The true multifractality$\Delta {\alpha }_{eff}=\Delta {\alpha }_{PDF}+\Delta {\alpha }_{NL}=0.77$. The results in Figure 6 (a)-(f) are listed in Table 3.

The multifractality analysis of the Baltic Clean Tankers market in Figure 6 (a) reveals significant insights into its dynamics. A multifractality value of 0.90 for the original data points to a highly complex market with pronounced multifractal behavior due to strong correlations at various time scales. A lesser, but still substantial, multifractality value of 0.61 in the surrogated data indicates that multifractality persists without temporal correlations, suggesting that price change distributions inherently contribute to market complexity. The shuffled data's multifractality value at 0.48, the lowest observed, illustrates the crucial role of temporal ordering in market behavior. These values collectively attest to the market's intricate and nonlinear interaction across scales [43].

The multifractality analysis for the Baltic Clean Tankers in Figure 6 (a) and (b), inclusive of the specific 37000 tonnage route, elucidates distinct market dynamics. The original data for the entire market and the targeted route yield high multifractality values of 0.9 and 0.86 respectively, underscoring considerable multifractal behavior. However, subsequent surrogated and shuffled data yield diminished multifractality values of 0.61 and 0.48 for the broader market, and 0.39 and 0.23 for the specific route, respectively. These reductions upon data modification suggest inherent temporal organization as a key contributor to multifractality [26,27,28,43]. The analysis underscores the influence of data structure on the assessment of market complexity and multifractal characteristics.

The multifractal analysis of the Baltic Dirty Tankers market data in Figure 6 (c) reveals varying degrees of market complexity. An original multifractality value of 0.58 indicates a moderate complex market structure with self-similarity across time scales. The surrogated data's 0.52 multifractality value suggests that nontrivial scaling behavior persists even after the removal of some structural correlations. This denotes inherent complexity within the price change distribution itself [14,26,27,28]. A notably lower multifractality value of 0.28 in the shuffled data emphasizes the importance of chronological order, indicating that temporal organization significantly contributes to the market’s multifractal nature [43].

The multifractal analysis for the Baltic Dirty Tankers market in a specific route TD7 at the 80000-tonnage level in Figure6 (d) exhibits a high degree of market complexity, with original data yielding a multifractality value of 1.03. This denotes a rich multifractal structure and extensive self-similarity across temporal scales. Upon surrogate treatment, multifractality is markedly reduced to 0.5, indicating a diminished multifractal behavior upon the exclusion of certain structural and temporal correlations. Further declines to a multifractality value of 0.28 in the shuffled data underscore the pivotal role of temporal sequencing in fostering multifractal properties.

3.4. Predictive Applications: Using BDTI to Predict Brent Oil Prices

3.4.1. Understanding a Complexity for the Specific Periods and Motivation for Predictive Modeling

The comparative assessment of the Baltic Dirty Tankers from the specified periods of 1998 to 2010 and 2010 to 2023 in Figure 6 (e) and (f) may provide valuable insights into the temporal changes in multifractal nature and complexity within the market dynamics. The original multifractality value of 0.72 in the first period decreases to 0.60, suggesting a potential reduction in complexity and multifractal behavior. For the period of 1998 to 2010, the surrogated data yields a multifractality value of 0.62, indicating a decrease in complexity compared to the original data; similar for the period from 2010 to 2023. The shuffled data provides additional insights into the temporal changes in multifractality [26,43]. For the period of 1998 to 2010, the shuffled data yields a multifractality value of 0.35, reflecting a notable reduction in complexity compared to the original and surrogated data. Likewise, for the period from 2010 to 2023, the value further decreases to 0.29, signaling a continued decrease in complexity during the later period. Overall, the period from 2010 to 2023 exhibits higher multifractality values across all data types, indicating stronger multifractal nature and greater complexity [16,26,27,28].

These findings prompted our interest in understanding the underlying driving factors behind this marked increase in market complexity post-2010. To gain deeper insights, we apply a novel predictive approach using freight rates (BDTI) to predict Brent oil prices during distinct periods, aiming to explore the predictive power of freight rates in different market phases.

Traditionally, most studies have focused on forecasting freight rates based on oil price movements, which aligns with the conventional economic logic that oil prices drive downstream costs, including shipping. However, freight rates can be highly responsive to immediate changes in supply-demand dynamics, vessel utilization rates, and macroeconomic shifts, making them potentially valuable indicators for predicting oil prices as well. Thus, in this study, we take an innovative stance by attempting to predict oil prices from freight rates, with the intention of providing additional insights and decision-making tools for governments, businesses, and individual investors, especially during periods of significant market volatility and complexity.

3.4.2. Case Study: Prediction for Period III (2019-01-01 - 2021-01-01)

The choice of Period III (2019-01-01 to 2021-01-01) as the initial focus for analysis does not carry any special significance. In this study, we consider all four structural break periods to be equally important. Thus, selecting any of them as the starting point for analysis would have been equally valid. Period III was simply chosen at random as the first period to examine.

The predictive process was performed using Baltic Dirty Tanker Index (BDTI) as the main feature, and the following additional features were systematically introduced to improve prediction performance:

Crisis Period Indicator: An indicator variable was introduced to capture the effect of the 2020 COVID-19 pandemic, particularly between December 2019 and June 2020. This indicator took the value of 1 during the crisis and 0 otherwise, to help the model account for the dramatic market shifts.

Multifractal Features: Multifractal characteristics of the BDTI time series were extracted using the MFDFA method. We computed the Hurst exponent for various q-values (ranging from -5 to 5) to quantify the complexity and self-similarity within the time series.

To combine these features for predicting Brent oil prices, we employed a stacking regression model, consisting of the following base learners:

XGBoost: Capable of handling high-dimensional feature spaces and mitigating overfitting, XGBoost was used with 300 estimators, a learning rate of 0.01, and a maximum depth of 3.

LightGBM: Known for its efficient handling of large datasets, LightGBM was configured similarly to XGBoost, to provide complementary strengths in feature learning.

CatBoost: Particularly effective in dealing with categorical features and reducing preprocessing requirements, CatBoost was employed with 300 iterations and a learning rate of 0.01.

The predictions from these base learners were then integrated using a Ridge Regression model as the meta-learner. This stacking approach allows the model to capture a diverse range of data patterns and interactions, improving the robustness of predictions.

The resulting predictions for Period III are illustrated in Figure 7, where multiple models are compared against the actual Brent oil prices. The models include:

Predicted_Direct: Predictions made using only the BDTI data.

Predicted_Time_Indicator: Incorporating the crisis time indicator to assess the effect of the COVID-19 pandemic.

Predicted_Multifractal: Including multifractal features derived from the BDTI to understand the impact of market complexity.

Predicted_Multifractal_Indicator: Utilizing both the crisis indicator and multifractal features to determine their combined influence.

While the Predicted_Direct model shows some deviations, particularly during major disruptions, the inclusion of multifractal features (Predicted_Multifractal) allows the model to better capture the intricate market dynamics, resulting in predictions that align more closely with the actual price trends. The Predicted_Multifractal_Indicator model demonstrates the best performance, especially during the high-volatility period in early 2020 when the COVID-19 pandemic caused a sharp drop in oil prices.

3.4.3. Predictive Results and Analysis for Period I - IV

Following the same methodology as applied in Period III, we conducted predictive modeling for Periods I, II, and IV, with the results summarized as follows:

Period I (2006-01-01 - 2010-12-31): The 2008 Global Financial Crisis had a profound effect on both freight rates and oil prices. Figure 8 shows the predictive results, indicating that the inclusion of multifractal features and crisis indicators significantly improved prediction accuracy compared to the direct approach.

Period II (2013-06-30 - 2016-06-30): During the 2014 Shale Oil Revolution, the predictive model that included multifractal features outperformed the direct prediction, as shown in Figure 9. The Predicted_Multifractal_Indicator model captured the complex fluctuations caused by shifts in the energy supply landscape more effectively.

Period IV (2021-01-01 - 2024-01-12): The 2022 Russia-Ukraine conflict introduced significant geopolitical uncertainty. Figure 10 illustrates that the Predicted_Multifractal_Indicator model again provided the best fit, with reduced error margins and greater alignment with actual price movements, highlighting the value of multifractal features during periods of high volatility.

3.4.4. Robustness Analysis of Prediction Methods

Above, we presented several intuitive figures that provide a visual comparison of the predictive performance of different models across the four periods. Now, we proceed to a more detailed quantitative analysis to corroborate these visual observations with empirical metrics. Specifically, we focus on the predictive accuracy as measured by the Mean Squared Error (MSE) and R² values, which offer critical insights into the effectiveness of incorporating multifractal features and crisis indicators into the forecasting models.

Figure 11 presents the Mean Squared Error (MSE) values for both the training and test sets across all periods, demonstrating the improvement in predictive accuracy after incorporating multifractal features. As shown in the figure, the inclusion of these features significantly reduces the MSE values across all four periods, with the greatest improvements observed in Periods II, III, and IV. This suggests that the predictive models benefited substantially from the added complexity of multifractal characteristics, especially in the post-2010 periods where market dynamics became increasingly intricate.

Figure 12 displays the R² scores for both the training and test sets, providing insights into the proportion of variance explained by the models. The predictive models with multifractal features exhibit substantially higher R² values compared to the direct prediction models. Specifically, for Periods II, III, and IV, the Predicted_Multifractal_Indicator model achieves R² values close to or exceeding 0.9, indicating a strong fit to the actual data. Interestingly, the enhancement in predictive performance is less pronounced for Period I, suggesting that the multifractal characteristics during this earlier period were less influential compared to the more recent periods. This aligns with our previous observations that post-2010 periods exhibited greater multifractality and complexity, thereby benefiting more from the incorporation of multifractal features.

To provide a more comprehensive understanding, Figure 13 illustrates the substantial percentage improvements in MSE and R² when multifractal features are incorporated into the predictive models for each period. For instance, during Period II, the R² for training improves by approximately 182.54%, and the R² for testing by 167.62%, highlighting a dramatic enhancement in model performance when multifractal features are added. Similarly, Period III shows an 88.60% improvement in training R² and a 117.14% increase in testing R², emphasizing the importance of capturing complex dynamics during high volatility phases, such as those triggered by the COVID-19 pandemic. Notably, the improvements in Periods II, III, and IV are significantly greater compared to Period I, which only shows modest gains, reinforcing the notion that market dynamics after 2010 became more intricate and multifaceted. This distinct difference in performance can be attributed to increased globalization, advancements in technology, and heightened geopolitical sensitivity after 2010, all contributing to more complex and unpredictable market interactions. These findings underscore that the multifractal characteristics are especially relevant for capturing the nuances of post-2010 market behavior, making the predictive models significantly more effective for later periods.

Such findings are critical, as they highlight the evolving nature of the oil market and underscore the importance of adapting predictive methodologies to account for changes in market structure and dynamics. The superior performance of models incorporating multifractal features during periods of heightened market complexity suggests that traditional linear models may be insufficient for capturing the nuanced behaviors of modern financial markets. By utilizing multifractal analysis, stakeholders can achieve a deeper understanding of market dynamics and enhance their decision-making processes, particularly during times of instability.

4. Discussion of Results

From the discussion, one may conclude that this study successfully unravels the multifaceted complexity of the Baltic Tanker Freight market using multifractal analysis techniques and advanced machine learning models. By integrating the Baltic Dirty Tanker Index (BDTI) as a leading indicator, we demonstrate that freight rates can effectively predict Brent oil prices, particularly during heightened market volatility caused by global crises such as the 2008 Financial Crisis, the 2014 Shale Oil Revolution, the COVID-19 pandemic, and the Russia-Ukraine conflict. The findings reveal that multifractal characteristics, such as the generalized Hurst exponent and multifractal spectrum, significantly enhance the predictive accuracy of the models, outperforming traditional approaches that rely solely on linear or unidirectional relationships [7,9]. Moreover, the stacking regression framework combining XGBoost, LightGBM, CatBoost, and Ridge Regression validates the robustness of the proposed methodology, aligning closely with contemporary machine learning advancements [26,62,63,64]. These results provide actionable insights for policymakers, energy companies, and investors, emphasizing the utility of multifractal analysis in managing systemic risks and navigating energy market volatility [4,6].

Comparison of the findings with those of other studies confirms the progressive advantage of integrating multifractal analysis with predictive modeling. Previous research has largely focused on forecasting freight rates based on oil prices, reflecting a conventional perspective of oil price-driven costs in shipping [17,18]. However, this study advances the discussion by demonstrating that BDTI provides valuable information for predicting Brent oil prices. This finding aligns with recent studies that highlight the sensitivity of freight rates to immediate supply-demand imbalances and macroeconomic shocks [74]. Unlike earlier studies that overlooked nonlinear dynamics, the inclusion of multifractal features offers a substantial improvement in capturing complex market behaviors. Thus, the predictive performance achieved in this study reflects methodological progress compared to traditional econometric models or linear approaches, which often fail under volatile conditions [12,18].

Nevertheless, there are limitations to the current study that warrant consideration. First, while the study leverages BDTI and multifractal features, it does not account for other exogenous factors such as regional economic policies, environmental regulations, or vessel fleet dynamics, which may further influence oil and freight markets [8]. Second, the analysis is constrained to the Baltic indices (BDTI and BCTI), which, although significant, may not fully represent global tanker market dynamics. Third, the use of historical data assumes that past patterns remain valid predictors for future trends, which may not hold during unprecedented disruptions or structural market changes [19,33]. Finally, the reliance on machine learning models, though effective, introduces the risk of overfitting, particularly when applied to smaller or less volatile datasets [51,52].

To address these limitations, several potential solutions can be proposed. Expanding the dataset to include additional variables such as global trade volumes, bunker fuel prices, or geopolitical risk indices may enhance the robustness of the models [19,21]. Incorporating real-time data streams or satellite-based tracking of vessel movements could also improve predictive accuracy [52,53]. Furthermore, the application of hybrid methods, combining deep learning approaches like Long Short-Term Memory (LSTM) networks with multifractal analysis, could address the limitations of traditional machine learning models.

Research questions that could be asked include whether integrating additional market indicators, such as energy derivatives or macroeconomic indicators, could further refine predictive outcomes [14]. One important future direction of this research is to explore the interplay between multifractality and emerging market phenomena, such as carbon emissions trading and the adoption of alternative fuels in shipping. Moreover, investigating the applicability of multifractal analysis in other energy markets, such as natural gas or LNG freight rates, may provide deeper insights into the broader dynamics of energy transportation [16,19]. These experimental research results will hopefully serve as useful feedback information for future iterations of predictive frameworks.

In conclusion, this study demonstrates the significant potential of multifractal analysis and machine learning in accurately forecasting energy prices during periods of high volatility. The findings not only advance the theoretical understanding of tanker freight markets but also provide practical tools for stakeholders to manage uncertainty and enhance decision-making. By successfully integrating multifractal features and advanced predictive models, this study offers a robust framework that can serve as a foundation for future research. Moving forward, further refinements and broader applications of the proposed methodology may uncover additional insights into the intricate relationships shaping global energy markets, paving the way for more resilient and adaptive forecasting strategies.

5. Conclusions

This study utilizes MF-DMA methodology to unravel the multifaceted nature of the Baltic Tanker freight market. Initial findings suggest an overarching multifractal nature within the market, with total multifractality reaching up to 0.90 in the Clean Tankers market. This complexity arises partly from a fat-tailed probability distribution (0.48) and non-linear correlations (0.29), indicating sophisticated temporal organization and inherent volatility as core components of market behavior. A closer examination of a specific route (TC2, 37,000-tonnage) shows that the market retains its multifractal attributes, albeit with reduced magnitudes upon data manipulation. This consistent reduction in multifractality values—evident upon shuffling and surrogating—reinforces the significant contribution of temporal arrangement to the market's complex structure. The world clean tanker market reflects a tight supply-demand balance, with freight rates fluctuating significantly in response to external changes.

In the case of the Dirty Tankers market, the study highlights moderate complexity, with an original multifractality value of 0.58, diminishing under surrogate and shuffled scenarios. This suggests that chronological sequencing is crucial for preserving multifractal properties in this segment. This assertion is further substantiated by focusing on specific routes like TD7 (80,000-tonnage), where multifractality peaks at 1.03 but declines markedly when temporal and structural correlations are disrupted.

The transition in multifractal dynamics between 1998–2010 and 2010–2024 reflects a shift from crisis-induced market behaviors to diversified and complex dynamics influenced by technological advancements, regulatory changes, and environmental policies.

Building on these findings, the study proposes a novel predictive framework that integrates the Baltic Dirty Tanker Index (BDTI), multifractal features, and crisis period indicators to forecast Brent oil prices during periods of heightened volatility. An analysis of four major global events—the 2008 Financial Crisis, the 2014 Shale Oil Revolution, the COVID-19 pandemic, and the Russia-Ukraine conflict—illustrates how external factors influence market dynamics and energy prices. The advanced machine learning models employed, including stacking regression with XGBoost, LightGBM, CatBoost, and Ridge Regression, enhance the robustness and reliability of predictions. These methods highlight the potential of freight rates as leading indicators for energy markets, providing actionable insights for policymakers and market participants.

In conclusion, this study combines multifractal analysis and predictive modeling to provide a comprehensive framework for understanding and navigating the complexities of the Baltic Tanker freight market. By revealing the evolving multifractal dynamics and demonstrating the predictive power of freight rates, the research underscores the importance of integrating multifractal characteristics into forecasting models. The findings offer practical implications for strategic decision-making, operational resilience, and risk management in the shipping and energy industries. Future studies should build on this framework by incorporating additional datasets, refining predictive algorithms, and exploring the interplay between multifractality and other market indicators to further enhance prediction accuracy and application scope.