A Hybrid Approach Combining the Lie Method and Long Short-Term Memory (LSTM) Network for Predicting the Bitcoin Return

Abstract

This paper introduces hybrid models designed to analyze daily and weekly bitcoin return spanning the periods from 18 July 2010 to 28 December 2023 for daily data, and from 18 July 2010 to 24 December 2023 for weekly data. Firstly, the fractal and chaotic structure of the selected variables was explored. Asymmetric Cantor set, Boundary of the Dragon curve, Julia set z² −1, Boundary of the Lévy C curve, von Koch curve, and Brownian function (Wiener process) tests were applied. The R/S and Mandelbrot–Wallis tests confirmed long-term dependence and fractionality. The largest Lyapunov test, the Rosenstein, Collins and DeLuca, and Kantz methods of Lyapunov exponents, and the HCT and Shannon entropy tests tracked by the Kolmogorov–Sinai (KS) complexity test determined the evidence of chaos, entropy, and complexity. The BDS test of independence test approved nonlinearity, and the TeraesvirtaNW and WhiteNW tests, the Tsay test for nonlinearity, the LR test for threshold nonlinearity, and White’s test and Engle test confirmed nonlinearity and heteroskedasticity, in addition to fractionality and chaos. In the second stage, the standard ARFIMA method was applied, and its results were compared to the LieNLS and LieOLS methods. The results showed that, under conditions of chaos, entropy, and complexity, the ARFIMA method did not yield successful results. Both baseline models, LieNLS and LieOLS, are enhanced by integrating them with deep learning methods. The models, LieLSTMOLS and LieLSTMNLS, leverage manifold-based approaches, opting for matrix representations over traditional differential operator representations of Lie algebras were employed. The parameters and coefficients obtained from LieNLS and LieOLS, and the LieLSTMOLS and LieLSTMNLS methods were compared. And the forecasting capabilities of these hybrid models, particularly LieLSTMOLS and LieLSTMNLS, were compared with those of the main models. The in-sample and out-of-sample analyses demonstrated that the LieLSTMOLS and LieLSTMNLS methods outperform the others in terms of MAE and RMSE, thereby offering a more reliable means of assessing the selected data. Our study underscores the importance of employing the LieLSTM method for analyzing the dynamics of bitcoin. Our findings have significant implications for investors, traders, and policymakers.

1. Introduction

Cryptocurrency presents a profitable avenue for speculation owing to its pronounced volatility. While leveraging AI and ML algorithms can enhance the prediction of future prices of cryptocurrencies, the task remains arduous because of the nonlinear and complex nature of price behaviors. Nevertheless, the market value of cryptocurrencies is anticipated to burgeon, with a forecasted annual growth rate of 11.1% [1]. Yet, investors have grappled with challenges stemming from price bubbles that trigger excessive volatility. Addressing these hurdles necessitates the development of a dependable model to aid market participants in discerning trends and making informed predictions. But forecasting bitcoin prices proves daunting given their susceptibility to diverse influences, including governmental policies, technological advancements, public sentiment, and global occurrences.

Forecasting bitcoin’s return and price volatility are challenging due to the influence of numerous factors, including market sentiment, public sentiment, political factors, economic factors, geopolitics risks, and technological advancements. Under the influence of these factors, the return of bitcoin may exhibit a chaotic structure, and it is important to determine its fractal structure. To offer a more realistic description of the cryptocurrency market while avoiding constraints on the statistical and distributional properties of price returns, reference [2] advanced the Fractal Market Hypothesis (FMH). FMH posits that market stability is maintained when market participants make self-similar decisions across various investment time horizons, thereby ensuring market liquidity [3,4].

Reference [5] investigated the cryptocurrency markets using wavelet coherence analysis. The study found evidence of positive correlations between cryptocurrency prices and online factors. These correlations become significantly stronger during bubble-like regimes in the price series. Some papers, such as refs. [4,6], analyzed the fractal dimension of cryptocurrency price series.

Traditional finance research methods are insufficient in accurately predicting bitcoin prices, leading to an increased interest in utilizing machine learning techniques to tackle this issue. Some studies investigating bitcoin used the deep learning method. So, there has been a notable increase in deep learning models tailored for predicting financial time series. Deep learning, which falls under the umbrella of machine learning, entails training Artificial Neural Networks (ANNs) on large datasets. These networks possess the ability to autonomously learn and make informed decisions. Notably, RNNs, for example LSTM and GRU, excel in processing sequential data like time series. The core objective of the DL method is to establish price and return prediction frameworks that cater to the needs of cryptocurrency investors.

In this paper, the core objective of the proposed DL method is to establish price and return prediction frameworks that cater to the needs of cryptocurrency investors and speculators, leveraging historical data for robust insights. Furthermore, our aim is to address the following research inquiry: “How can both AI and Lie algorithms aid investors and speculators in forecasting cryptocurrency prices and returns”? Additionally, we endeavor to determine the optimal Lie algebras with the LSTM method for forecasting the prices of the selected cryptocurrencies.

This paper utilizes the Lie algebra approach to tackle the stochastic differential equation (DE) governing the short-term dynamics of bitcoin. The model posits a stochastic DE framework within a curved state space. Return models for bitcoin are constructed expressing concepts through matrices and operators of differentiation on the $S^2$ manifold. The discoveries of Lie in the latter part of the 19th century revealed special methods for solving DEs as particular instances of a broader integration process reliant on the DE’s invariance under continuous symmetrical groups. Today, Lie groups profoundly influence various fields including mathematics, mechanics, and robotics.

In mathematical finance, several studies have leveraged the Lie method to gain insights into related partial differential equations. Prior research has explored the use of general differentiable manifolds in interest rate models. Reference [7] applied the method of Lie to the BSM equation, while references [8,9] illustrated a model for short rates based on the circle $S^1$. Reference [10] investigated the behavior of interest rates using stochastic differential equations on curved state spaces. Short-term interest rate models were developed for $S^1$ and $S^2$ manifolds employing matrix representations rather than the traditional approach of using differential operator representations of Lie groups. Additionally, reference [11] utilized Lie-LSTMOLS models to analyze oil price volatility.

This paper examines the period between 18 July 2010 and 28 December 2023 for daily data and 18 July 2010 and 24 December 2023 for weekly data, encompassing significant events affecting bitcoin volatility and return such as economic crises, military interventions, and pandemics. These events contribute to the nonlinear nature of bitcoin prices, highlighting the significance of modeling dynamic phenomena and tackling stochastic differential equations (SDEs). Solutions of differential equations are known to yield symmetries corresponding to groups of Lie. In our study, we utilize a model on $S^2$ manifolds utilizing matrix representations instead of representations of Liealgebras through differential operators. Following reference [11], we determine the noise volatility and drift terms of the stochastic state equations to reflect observed phenomena while maintaining simplicity. We explore the fractal and chaos structure of the variables. Asymmetric Cantor set, Boundary of the Dragon curve, Julia set z² −1, Boundary of the Lévy C curve, von Koch curve, and Brownian function (Wiener process) tests are applied. And then the Mandelbrot–Wallis and Lo’s rescaled range(R/S) methods confirm fractionality and long-term dependence. Methods for determining maximum Lyapunov exponents (Rosenstein, Collins and DeLuca, and Kantz), Shannon entropy (SE), and HCT entropy tests followed by the KS complexity measure (KSCM) determine the evidence of chaos, entropy, and complexity. And then parameter estimation is carried out using the OLS method and, in certain cases, nonlinear least square (NLS) methods because of the nonlinear characteristics observed within the designated timeframe. LieOLS and LieNLS methods will determine the parameters of models. The results of the LieOLS and LieNLS methods are compared with the results obtained from the standard econometric model, the ARFIMA model [12]. And the LieOLS and LieNLS methods raise LieLSTMOLS and LieLSTMNLS ones. The obtained parameters are compared with the ones of LieOLS and LieNLS methods. In the last stage, the forecasting performance of our proposed hybrid method is compared with that of the LieOLS and LieNLS standard regression methods.

To our knowledge, this study is the first to combine deep learning with the Lie method. The reason for choosing this hybrid method is to increase the forecast power of deep learning with the Lie algebra method since, as highlighted above, the bitcoin price is very volatile.

The paper is structured as follows. Section 2 is about the literature review. Section 3 discusses Lie groups and algebras of orthogonal matrices, followed by the definition of the bitcoin price model on the Lie groups SO(3). Section 4 presents the data and provides descriptive statistics. Section 5 presents and discusses the model and the results, while Section 6 concludes the paper.

2. Literature Review

Reference [13] used an econophysics model to identify irregular price levels in the bitcoin market.

There are only a few studies on the fractal and multifractal properties of BTC, including DFA analysis [14] and the Hurst exponent [15]. These studies demonstrate strong anti-persistence in BTC returns. Reference [16] analyzed the multifractal properties of the daily returns of BTC by dividing the sample into two periods: the high- and low-price regimes. The researchers found that the price drivers of cryptocurrency markets exhibit medium-term positive correlations with online factors, as discovered in reference [5]. Moreover, these correlations become stronger during bubble periods regime with low/negative returns and high volatility in the price series. The authors propose that these findings elucidate the intermittent nature of these relationships over time.

Similarly, reference [17] established the presence of a persistent “bubble stage” in BTC since surpassing the USD 1000 threshold. The study further observed that certain cryptocurrencies exhibit responsiveness to their internal dynamics, including factors like mining difficulty and hashrate. Reference [18] examined the multifractal properties of bitcoin prices. Utilizing wavelet transform modulus maxima and multifractal detrended fluctuation analysis, their results indicated that bitcoin exhibits significant multifractality across all examined time intervals. This multifractality is primarily attributed to the high kurtosis and fat distributional tails of the series returns. Reference [19] determined the fractal dimension, uncertainty and chaotic structure of bitcoin using Hurst, Lyapunov, and Shannon entropy techniques.

Reference [4] explored fractal dimension for bitcoin and the researchers observed that bitcoin prices display long-term memory, though this tendency has been weakening over time. References [6,20] investigated the fractal dimension of cryptocurrencies. Reference [6] found that bitcoin also exhibits persistent behavior, which is associated with the long memory effect.

Several papers embraced ML methodologies to construct predictive models for financial markets, aiming to capture the nonlinear dynamics inherent in financial time series [19,21,22].

Deep learning (DL) models have gained traction for financial market price forecasting due to their superior performance. Reference [23] integrated autoregressive (AR) features into an LSTM network for daily BTC price prediction, resulting in lower error rates than traditional LSTM models based on metrics like MSE, RMSE, MAPE, and MAE.

Meanwhile, reference [24] utilized a range of features to predict the ETC and BTC prices, choosing dependable predictors using correlation analysis. When employing SVM on these features, linear regression surpassed the other methods. Additionally, LSTM yielded the lowest prediction error for BTC. Reference [25] conducted a comparative analysis of deep learning methods, revealing varied prediction performances across different models. Reference [26] introduced a novel neuro-fuzzy technique combining ANN, demonstrating enhanced prediction accuracy. In another investigation [27], machine learning-based ensemble methods combining a diverse ensemble model, gradient boosted trees, ANN, and KNN were examined for forecasting nine different cryptocurrencies, with the ensemble model displaying the lowest prediction error. Reference [28] employed a hybrid model combining a WNN and an LSTM to predict bitcoin prices. Reference [29] employed machine learning models to predict the volatility of bitcoin. The researchers found that the LSTM model outperformed the other models in terms of accuracy.

Similarly, the researchers in [30] utilized a combined model consisting of Random Forest (RF) and GBM to forecast the prices of ETC, BTC, and XRP, achieving MAPE values ranging from 0.92% to 2.61%.

Reference [31] employed a two-stage methodology to predict BTC prices, using RF and ANN to identify pertinent features for LSTM prediction, surpassing the performance of ARIMA and SVM. Reference [32] utilized a hybrid approach merging LSTM and GRU networks to forecast LTC and XMR prices, demonstrating higher accuracy compared to LSTM alone.

In the study described in reference [33], a hybrid deep learning (DL) model was introduced for forecasting bitcoin volatility. Reference [34] employed the weighted TX Dagum distribution model, known for its heavy-tailed properties, to analyze bitcoin prices.

Reference [1] used three variants of RNNs—LSTM, GRU, and Bi-LST—for predicting the exchange rates of BTC, ETC, and Litecoin. The findings, evaluated through RMSE and MAPE, indicate that Bi-LSTM outperforms LSTM and GRU in prediction accuracy. Bi-LSTM demonstrates superior predictive capability, yielding MAPE values of 0.036 for BTC, 0.041 for LTC, and 0.124 for ETH.

3. Methods and Materials

3.1. Introduction to Orthogonal Lie Group and Its Algebra

This section presents the descriptions of the special orthogonal Lie group SO(3), its algebra, and the relations between stochastic dynamics and this group, as outlined in [11,30,35].

3.2. The Lie Group $SO\left(3\right)$ and Its Algebra

The Lie group $SO\left(3\right)$ and its algebra are defined as follows:

$$SO\left(3\right)=\left\{B\in GL\left(3\right):{BB}^T=IanddetB=1\right\}$$

The Lie group SO(3) is recognized with the unit sphere $S^2=\left\{\left(x_1,x_2,x_3\right):{x_1}^2+{x_2}^2+{x_3}^2=1\right\}$ with parametrization $x_1=cos\theta ,x_2=sin\theta sin\phi$, $x_3=sin\theta cos\phi$.

The Lie algebra of this group is denoted by $so\left(3\right)$ and generated by basis matrix elements.

$$Y_1=\left[\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right],Y_2=\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ -1& 0& 0\end{array}\right],Y_3=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& -1\\ 0& 1& 0\end{array}\right]$$

This algebra is non-commutative, and satisfies the condition $X^T=-X$ for

$$X=\left[\begin{array}{ccc}0& -x_1& x_2\\ x_1& 0& -x_3\\ -x_2& x_3& 0\end{array}\right]\in so\left(3\right)$$

The connection between this algebra and the group is as follows.

$$\mathrm{e}\mathrm{x}\mathrm{p}:so\left(3\right)\to SO\left(3\right)$$

$$\mathrm{e}\mathrm{x}\mathrm{p}\left(X\right)=B\in SO\left(3\right)$$

$$\mathrm{e}\mathrm{x}\mathrm{p}\left[\begin{array}{ccc}0& -x_3& x_2\\ x_3& 0& -x_1\\ -x_2& x_1& 0\end{array}\right]=I_3+sin\theta X\left(x_1,x_2,x_3\right)+\left(1-cos\theta \right)X{\left(x_1,x_2,x_3\right)}^2=BϵSO\left(3\right)$$

(1)

3.3. Stochastic Dynamics on the Lie Group $SO\left(3\right)$ and Its Algebra

On the manifold of SO(3), the bilinear state equation, the bitcoin price, and the dynamics for f are given as follows:

$$dB=BXdt+BdW,$$

(2)

$$r\left(B\right)=\frac{1}{2}Tr\left(MBN{B}^T\right),$$

(3)

$$df=Tr\left[B^{T}MB\left(\left(X-I\right)Ndt+dWN+\frac{1}{2}dWN{dW}^T\right)\right]$$

(4)

where $B\in SO\left(3\right)$, $X,dW\in so\left(3\right)$ and $M,N\in R^{3\times 3}$ are positive definite symmetric matrices.

Consequently, the elements in Equation (4) can be described as follows:

$$X=\left[\begin{array}{ccc}0& -x_3& x_2\\ x_3& 0& -x_1\\ -x_2& x_1& 0\end{array}\right]\mathrm{and}dW=\left[\begin{array}{ccc}0& -{dw}_3& {dw}_2\\ {dw}_3& 0& -{dw}_1\\ -{dw}_2& {dw}_1& 0\end{array}\right]\in so\left(3\right)$$

$$M=\left[\begin{array}{ccc}{m}_{11}& m_{12}& m_{13}\\ m_{12}& m_{22}& m_{23}\\ m_{13}& m_{23}& m_{33}\end{array}\right],N=\left[\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ n_{12}& n_{22}& n_{23}\\ n_{13}& n_{23}& n_{33}\end{array}\right]$$

Hence, we establish the definitions of the bitcoin price, denoted as $r$, and the stochastic dynamic, represented as $dr$ for $f$ on $SO\left(3\right)$, in the following manner:

$$r\left(B\right)=\frac{1}{2}\left[a_{11}{n}_{11}+a_{22}{n}_{22}+a_{33}{n}_{33}+\left(a_{12}+a_{21}\right)n_{12}+\left(a_{13}+a_{31}\right)n_{13}+\left(a_{23}+a_{32}\right)n_{23}\right]$$

where

$$a_{11}=x_{11}{b}_{11}+x_{12}{b}_{21}+x_{13}{b}_{31},a_{22}=x_{22}{b}_{11}+x_{22}{b}_{22}+x_{23}{b}_{32}$$

$$a_{33}=x_{31}{b}_{13}+x_{32}{b}_{23}+x_{33}{b}_{33},a_{12}=x_{11}{b}_{12}+x_{12}{b}_{22}+x_{13}{b}_{32}$$

$$a_{21}=x_{21}{b}_{11}+x_{22}{b}_{21}+x_{23}{b}_{31},a_{13}=x_{11}{b}_{13}+x_{12}{b}_{23}+x_{13}{b}_{33}$$

$$a_{31}=x_{31}{b}_{11}+x_{32}{b}_{21}+x_{33}{b}_{31},a_{23}=x_{21}{b}_{13}+x_{22}{b}_{23}+x_{23}{b}_{33}$$

$$a_{32}=x_{31}{b}_{12}+x_{32}{b}_{22}+x_{33}{b}_{32}$$

$$x_{11}=b_{11}{m}_{11}+b_{21}{m}_{21}+b_{31}{m}_{31},x_{12}=b_{11}{m}_{12}+b_{21}m+a_{31}{q}_{32}$$

$$x_{21}=b_{12}{m}_{11}+b_{22}{m}_{21}+b_{32}{m}_{31},x_{22}=b_{12}{m}_{12}+b_{22}{m}_{22}+b_{23}{m}_{32}$$

$$x_{31}=b_{13}{m}_{11}+b_{23}{m}_{21}+b_{33}{m}_{31},x_{32}=b_{13}{m}_{12}+b_{23}{m}_{22}+b_{33}{m}_{32}$$

$$x_{13}=b_{11}{m}_{13}+b_{21}{m}_{23}+b_{31}{m}_{33},x_{23}=b_{12}{m}_{13}+b_{22}{m}_{23}+b_{32}{m}_{33}$$

$$x_{33}=b_{13}{m}_{13}+b_{23}{m}_{23}+b_{33}{m}_{33}\mathrm{where}{b}_{ij}\in SO\left(3\right)$$

and

$$dr=Tr\left\{\left[\underset{a_{ij}}{\underbrace{B^{T}MB}}\underset{d_{ij}}{\underbrace{\left(X-I\right)N}}\right]dt+\underset{a_{ij}}{\underbrace{B^{T}MB}}\underset{e_{ij}}{\underbrace{\left[dWN\right]}+}\underset{a_{ij}}{\underbrace{B^{T}MB}}\underset{k_{ij}}{\underbrace{\left(dWNd{W}^T\right)}}\right\}$$

$$=\left[a_{11}{d}_{11}+a_{12}{d}_{21}+a_{13}{d}_{31}+a_{12}{d}_{12}+a_{22}{d}_{22}+a_{23}{d}_{32}+a_{13}{d}_{13}+a_{23}{d}_{23}+a_{33}{d}_{33}\right]dt$$

$$+\left[a_{11}{e}_{11}+a_{12}{e}_{21}+a_{13}{e}_{31}+a_{21}{e}_{12}+a_{22}{e}_{22}+a_{23}{e}_{32}+a_{13}{e}_{13}+a_{23}{e}_{23}+a_{33}{e}_{33}\right]$$

$$+\frac{1}{2}\left[a_{11}{k}_{11}+a_{12}{k}_{21}+a_{13}{k}_{31}+a_{21}{k}_{12}+a_{22}{k}_{22}+a_{23}{k}_{32}+a_{31}{k}_{13}+a_{32}{k}_{23}+a_{33}{k}_{33}\right]$$

$$d_{ij}=\left(\begin{array}{ccc}-1& {-b}_3& b_2\\ b_3& -1& {-b}_1\\ {-b}_2& b_1& -1\end{array}\right)\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ n_{21}& n_{22}& n_{23}\\ n_{31}& n_{32}& n_{33}\end{array}\right)$$

$$e_{ij}=\left(\begin{array}{ccc}0& {-dw}_3& {dw}_2\\ {dw}_3& 0& {-dw}_1\\ {dw}_2& {dw}_1& 0\end{array}\right)\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ n_{21}& n_{22}& n_{23}\\ n_{31}& n_{32}& n_{33}\end{array}\right)$$

$$k_{ij}=\left(\begin{array}{ccc}{e}_{11}& e_{12}& e_{13}\\ e_{21}& e_{22}& e_{32}\\ e_{31}& e_{32}& e_{33}\end{array}\right)\left(\begin{array}{ccc}0& {dw}_3& {-dw}_2\\ {-dw}_3& 0& {dw}_1\\ {dw}_2& {-dw}_1& 0\end{array}\right)$$

Thus, the differential expression we give above can be expressed as follows:

$$dr=\left\{\beta \left(t\right)r+{\alpha }_2{r}^2\right\}dt+{\alpha }_3{r}^{3/2}dw$$

(5)

where

$$\beta \left(t\right)={\beta }_1+{\beta }_{2}sin\left(\theta t\right)+{\beta }_{3}cos\left(\theta t\right)+{\beta }_{4}sin\left(2\theta t\right)+{\beta }_{5}cos\left(2\theta t\right)$$

3.4. Hybrid Model

The suggested hybrid model was developed as follows.

$$dr\circ L\left(t\right)$$

(6)

where $L\left(t\right)$ represents LSTM, a type of RNN [36], which leverages the interdependencies among the samples within a segment of the time series on the $SO\left(3\right)$ manifold to achieve precise prediction. The operations of the LSTM can be described by the following equations:

$$m_t=f_t{⨀}_{\ast }{m}_{t-1}+j_t{⨀}_{\ast }{\tilde{m}}_t$$

(7)

$${\tilde{m}}_t=\mathit{tanh}\left(V_m{x}_t+W_m{r}_{t-1}+{\mathcal{b}}_m\right)$$

(8)

$$j_t=\sigma \left(V_j{x}_t+W_j{r}_{t-1}+{\mathcal{b}}_j\right)$$

(9)

$$f_t=\sigma \left(V_f{x}_t+W_f{r}_{t-1}+{\mathcal{b}}_f\right)$$

(10)

$$o_t=\sigma \left(V_o{x}_t+W_o{r}_{t-1}+{\mathcal{b}}_o\right)$$

(11)

$$r_t=o_t{⨀}_{\ast }tanh\left(m_t\right)$$

(12)

where $\mathcal{b}$ is a biasvector, V and W are matrices of weights, σ(·) is the sigmoid function’s symbol, and ${⨀}_{*}$ is the element-wise multiplication’s symbol.

At time step t, the LSTM unit receives the $m_{t-1}$ state vector and the $r_{t-1}$ output vector from the preceding time step, and the input feature vector $x_t$ to produce the state vector $m_t$ and the output vector $r_t$. The LSTM model utilises temporal dependencies by using $x_t$ and $r_{t-1}$ to determine the prior state that needs to be retained through the forget gate $f_t$. The new information is formed in a normalised manner as ${\tilde{m}}_t$, and its intensity is established by applying the input gate activation $j_t$ to it.

4. Data

Bitcoin data (BT) acquired from the OPEN Access Data web site were used. They include daily bitcoin data (Figure 1) and weekly bitcoin data (Figure 2) for the periods 18 July 2010–28 December 2023 and 18 July 2010–24 December 2023, respectively.

The first step was to obtain the descriptive statistics of the bitcoin data and apply a unit root test.

Table 1. Descriptive stats and URT.

Descriptive Stats	Daily	Weekly
Maximum	2.999448	10.99875
Minimum	−0.769551	0.396199
Skewness	−0.240834	−1.042385
Kurtosis	3.075948	2.879419
Jarque–Bera	33.47633	125.0098
Observations	3379	688

shows the obtained statistics. As validated by the JB test, the data’s excess kurtosis prevents them from being modeled by a normal distribution.

Table 2. Heteroskedasticity and unit root tests.

ARCH-LM Test
	BTd	BTw
ARCH-LM (1–5)	389.67	256.03
Results	ARCH effects	ARCH effects
Unit Root Test (URT)
KSS [37]	−16.59016	−9.411
ADF	−9.247	−5.142103

displays the outcomes from the URT tests and heteroskedasticity.

In all cases, the trend specification was found to be insignificant. All results are presented for the first-differenced series.

The second section of Table 2 shows URT for both linear and nonlinear series. The ADF test reveals that all series are I (1) processes. The KSS test, which examines the null hypothesis of a unit root against STAR-type nonlinear processes, also indicates that all series are integrated of order one.

5. Results

The results were derived through a three-stage process. These stages unfolded as follows.

5.1. Determination of Fractal Dimension, Investigation of Long-Term Dependence, Fractionality, and Chaotic Dynamics

Asymmetric Cantor set, Boundary of the Dragon curve, Julia set z² −1, Boundary of the Lévy C curve, von Koch curve, and Brownian function (Wiener process) tests were applied. And then fractionality and long-term dependence were scrutinized using Hurst and Mandelbrot R/S tests along with Lo’s R/S tests.

Chaotic dynamics were detected through the use of Lyapunov (λ) and entropy tests. The Lyapunov analysis revealed the average exponential rate at which nearby orbits in phase space either diverge or converge. A positive λ (λ > 0) indicated chaos, signifying trajectory divergence, while λ ≤ 0 indicated regular motion, and (λ = 0) suggested a bifurcation. A (λ > 1) indicated deterministic chaotic behavior.

Further exploration was halted if (λ ≤ 0) or (λ > 1). However, if λ was between 0 and 1 and positive entropy values were observed, this suggested uncertainty or a random process, prompting further investigation into the variables’ nonlinearities and stochastic processes.

The Hurst exponent measures the irregularity of variables and captures the rate of chaos, while λ reflects how chaos influences future predictions, indicating the exponential divergence of adjacent orbits.

5.1.1. Fractal Dimension, Fractionality, Chaos, and Long-term Dependence

The descriptive tests shown in Table 1 suggested heteroscedasticity in the daily percentage change series. This was confirmed by White and ARCH-LM tests, which tested for nonlinear forms of heteroscedasticity and ARCH-type effects, respectively.

Table 3. Fractal dimensions and R/S test results.

	BTd	BTw
Hurst–Mandelbrot R/S test statistic	0.45	0.51
The actual Hausdorff dimensions
Fractal name	BTd DH	BTw DH
Asymmetric Cantor set	0.7	0.71
Boundary of the Dragon curve	1.54	1.51
Julia set z2 −1	1.3	1.32
Boundary of the Lévy C curve	1.93	1.924
von Koch curve	0.907	0.93
Brownian function (Wiener process)	1.44	1.46

presents the fractal dimensions and Hurst–Mandelbrot R/S test statistics. The table lists fractal names alongside their actual Hausdorff dimensions. The long memory of the series was tested using Lo R/S and Hurst–Mandelbrot R/S tests, with results detailed in Table 3.

The scaled range (R/S) statistic, based on the influential studies of references [38,39,40], is a measure used to assess the dependency structure in non-Gaussian distributed data. In the field of financial research, the Hurst exponent (HE) has been utilized to investigate dependency structures in bitcoin by [20] and in the metaverse by [41].

According to these studies, it has been found that the time series of cryptocurrency prices may exhibit long-range dependence [6,42,43]. Long-range dependence is associated with fractional Brownian motion (fBm), an extension of classical Brownian motion (Bm) that describes a random walk [6]. The properties of Bm and fBm can be quantified using the Hurst exponent (HE), which ranges between 0 and 1. The key characteristics of the HE are as follows: (a) H = 0.5 indicates a Bm process without long-memory behavior; (b) H > 0.5 indicates an fBm process with a long-memory effect; and (c) H < 0.5 indicates an anti-persistent process associated with short-term memory.

The HE and fractal dimension shown in

Table 4. The Hurst exponent and fractal dimension.

Hurst (H)		Fractal Dimension (dA)
BTd	BTw	BTd	BTw
0.53	0.535	1.476	1.469

suggest persistence for BTC.

Reference [4] determined the HE for BT to be 0.61317, while reference [6] found it to be 0.533. Both results are similar to our findings. Additionally, reference [44] observed that bitcoin’s HE has decreased over time. The long-memory HE is locally not Brownian, as evidenced by the decay of the HE over time between regulatory acts, as demonstrated by references [45,46].

5.1.2. Test Results for Entropy and Chaos

Following the examination of non-normality and fractionality, the focus shifted to analyzing the chaotic framework.

Table 5. Results of Largest Lyapunov and the Lyapunov exponent by the Rosenstein, Collins and DeLuca method and Kantz method [47,48].

Largest Lyapunov Exponent (λ)		Lyapunov Exponents by Rosenstein, Collins, DeLuca Method [47]		Lyapunov Exponents by Kantz Method [48]		1/λ
BTd	BTw	BTd	BTw	BTd	BTw	BTd	BTw
0.33	0.35	0.51	0.115	0.53	0.109	3.02	2.857

presents the results of the largest Lyapunov exponents (λ). Since the analyzed period was relatively short, the Largest Lyapunov test results were compared with two separate tests with different structures.

The largest Lyapunov results reveal that all variables exhibit chaotic behavior, as evidenced by λ estimates falling within the range of 0 < λ < 1 across all cases. With λ > 0, the variables display divergence from their initial conditions. Notably, the variables demonstrate clear signs of chaos, with positive λ estimates ranging from 0.33 to 0.35. Based on these findings, we determine that the variables exhibit chaotic dynamics [41].

To test the chaotic structure, two more methods were used after the Lyapunov test. Thus, the existence of a chaotic structure was examined with three different tests. Two methods were employed for estimating the Lyapunov exponents (λ), namely, the Rosenstein et al. [47] and Kantz [48] methods. Moreover, employing two methods serves as a precautionary measure and provides validation for detecting chaos in the series, if present. Both methods have shown good performance in identifying chaotic processes, even when noise is present. The main parameter, embedded dimension, was set with three initial states. Table 5 presents the results obtained from the two methods, focusing solely on the dimension.

Notably, the values determined by the Rosenstein et al. [47] and Kantz [48] methods yielded close and similar results regarding the presence of chaotic dynamics in bitcoin daily data and bitcoin weekly data. The chaos results determined by the three models are positive but less than one.

The value 1/λ can be viewed as an indicator of the predictability of future returns based on past data. For BTC daily data, predictability is limited to 1/λ = 3.020 days.

Therefore, we proceeded to the second stage and obtained the entropy results. We used two different entropy tests: Shanon entropy, and Havrda–Charvât–Tsallis entropy, and additionally the Kolmogorov–Sinai (KS) complexity measure.

Statistical techniques are effectively used to determine complexity in nonlinear systems through entropy [19,49,50]. Entropy measures the rate of information generation within a mechanism, and

Table 6. Entropy test results.

	BTd	BTw
Shannon entropy Shannon, transformed to the [0, 1] range	0.489	0.306
Havrda–Charvât–Tsallis (HCT) measure	51.27	52.36
Kolmogorov–Sinai (KS) complexity measure	7.46	6.93

presents the results of entropy to examine chaotic dynamics.

Positive entropy results suggest uncertainty or a random process. These results are derived from references [51,52,53]. KS complexity is assessed following the method outlined in [54]. These measures provide information on the presence of chaos and quantify the degree of complexity and randomness. Through these stages, we explored the stochastic processes and nonlinearities of the variables.

Our findings indicate that all series display completely random or uncertain behavior. For interpretability, the SE is scaled to the range [0, 1] using minimum and maximum bits. A value of 1 suggests complete randomness and unpredictability, while a value of 0 signifies total predictability. A score approaching 1 shows a greater deviation from long-term equilibrium. Additionally, entropy can be seen as a measure of information distortion, as reflected in BTd and BTw. The significantly positive SE values confirm a high degree of uncertainty in BTd and BTw.

To further verify these findings, we employed the HCT divergence measure, initially developed by Havrda and Charvt [52] and later by Tsalis [53]. The results from the HCT measure corroborate those obtained from SE.

Finally, we applied the Kaspar and Schuster method [54] to evaluate the three series for complexity using the KSCM. Higher values indicate greater unpredictability due to chaotic characteristics. The KSCM relates to the complexity or unpredictability of a system. The KSCM concentrates on particular features of the series in contrast to the Shannon and HCT entropy measures. Although information and uncertainty are the goals of both Shannon entropy and KS measures, their applications differ. While KS entropy studies the behavior of dynamic systems and their sensitivity to initial conditions, SE concentrates on probability distributions and information content. SE quantifies the amount of information present in a source or distribution, whereas KS entropy focuses more on the dynamic characteristics of systems and how sensitive they are to starting conditions.

Overall, chaotic dynamics, randomness, and complexity in the series are confirmed by the KS measure results. Collectively, the results of the Shannon, HCT, and KS measures point to chaos and randomness in the BTd and BTw series under study.

5.2. Nonlinearity Test Results

Since λ was less than 1, we ran the second stage, in which we will assess the stationarity of the variables. Unit Root tests were employed to ascertain stationarity. Linear ADF, and nonlinear KSS, tests were utilized for this purpose. BDS tests were employed to discern the presence of a nonlinear structure. Since BDS tests indicated nonlinearities, TeraesvirtaNW and WhiteNW tests, the Tsay test for nonlinearity, and LR test for threshold nonlinearity were applied.

The results are presented in

Table 7. Statistics for nonlinearity tests.

Tests	X.Squared-Daily	X.Squared-Weekly
TeraesvirtaNW test	11,922.49	58.20973
WhiteNW test	1919.91	49.83447
LR test for threshold nonlinearity	2462.021	99.00979
	F-stats	F-stats
Tsay’s test for nonlinearity	161,606.6	2.96846

and

Table 8. BDS test statistic.

	Daily	Weekly
Dimensions	z-Statistic	z-Statistic
2	149.1646	47.87819
3	159.8742	51.55992
4	173.2034	56.45552
5	192.4045	62.04549
6	218.5781	69.07627

. The BDS test exhibits the highest power in detecting nonlinearity, but it does not explain the type of nonlinearity. For this purpose, some other tests must be used [49]. The TeraesvirtaNW and WhiteNW tests, the Tsay test for nonlinearity, and the LR test for threshold nonlinearity indicate that the linear method is misspecified.

Table 8 shows that all variables are nonlinear for all dimensions according to the BDS test.

5.3. Results of the Hybrid Model

In this subsection, we will proceed with solving the equations in Equations (5) and (6). For the variables, we will use the Lie method, which achieves significant success in solving differential models in the presence of uncertainty and chaotic structure, along with the LSTMLie model, which is an advanced version of the Lie method. A comparison of the LieOLS and LieNLS results with the LieLSTMOLS and LieLSTMNLS results will be performed.

Within the framework of the Forecast evaluation, we will compare the results obtained from LieOLS and LieNLS with those obtained from LieLSTMOLS and LieLSTMNLS. In this subsection, we will give our results in two parts.

5.3.1. Experimental Results

The parameter coefficients for bitcoin daily and bitcoin weekly variables are presented in

Table 9. Lie parameter estimations.

	LieOLS Methods		LieNLS Methods		ARFIMA Methods
	Daily	Weekly	Daily	Weekly
${\mathsf{\alpha }}_{\mathbf{2}}$	0.104384 (5.12)	0.670879 (9.11)	0.277586 (8.33)	0.400311 (3.36)	-	-
${\mathsf{\alpha }}_{\mathbf{3}}$	0.81795 (3.268)	0.507589 (10.47)	0.037505 (7.627)	0.524064 (4.76)	-	-
${\mathsf{\beta }}_{\mathbf{1}}$	0.237793 (4.69)	0.190746 (3.025)	0.037536 (4.931)	0.204789 (5.04)	-	-
${\mathsf{\beta }}_{\mathbf{2}}$	−0.107061 (5.99)	0.565645 (5.068)	−0.100084 (2.556)	0.507935 (5.43)	-	-
${\mathsf{\beta }}_{\mathbf{3}}$	0.056989 (8.79)	−0.158533 (1.92)	0.083018 (1.98)	−0.143251 (0.72)	-	-
${\mathsf{\beta }}_{\mathbf{4}}$	0.056989 (2.17)	0.076463 (1.73)	0.139092 (1.86)	0.066750 (2.012)	-	-
${\mathsf{\beta }}_{\mathbf{5}}$	−0.03626 (1.81)	0.297055 (1.79)	0.064969 (2.096)	−0.109947 (1.96)	-	-
$\mathrm{d}$	-	-	-	-	0.389472 (5.137)	0.483771 (4.45147)
$\mathrm{A}\mathrm{R}\left(1\right)$	-	-	-	-	0.217658 (1.135)	0.59742 (1.403)
$\mathrm{M}\mathrm{A}\left(1\right)$	-	-	-	-	−0.352901 (1.506645)	−0.155307 (4.91951)
SIGMASQ	-	-	-	-	2.208836 (29.87)	3.282842 (33.8377)
AIC	2.68	3.66	2.77	3.49	3.646742	4.04
R2	0.85	0.741	0.87	0.729	0.644671	0.741

. The table gives the results obtained by combining both the OLS and NLS methods with the Lie method. The estimates of the Lie parameters are presented in Table 9. The estimates for the ${\alpha }_2$ coefficient obtained with both methods are close in value for both models, while the estimates for ${\alpha }_3$ exhibits notable differences.

The coefficient of ${\alpha }_3$ for the daily bitcoin value is 0.81795, which is significantly higher than the coefficient value of 0.037505 determined by the LieNLS model. Although both variable values are below 1, there is a substantial difference between the coefficients. For the weekly bitcoin value, the ${\alpha }_3$ coefficient values from the LieOLS and LieNLS methods are close to each other. The coefficient of the ${\beta }_2$ variable for daily data is also close and negative for both the LieOLS and LieNLS methods. However, for the weekly bitcoin variable, the coefficients are close and positive. The R² value is significant for all models, with values of 0.85, 0.741, 0.87, and 0.729 for LieOLSd, LieOLSw, LieNLSd, and LieNLSw, respectively. The models have passed the RESET, BP, and ARCH tests (these values are not shown).

The results of the LieOLS and Lie NLS models were also compared with the results of the ARFIMA model. In the ARFIMA model, the MA(1) values are negative for both daily and weekly values. While the d coefficient is positive in the model for daily, it is not expected that MA(1) values are negative and statistically insignificant. In addition, t value for AR(1) was found as 1.135 in the model for daily. It is statistically insignificant. AR(1) is also statistically insignificant in the model for weekly. The ARFIMA model is insignificant compared to the LieOLS and LieNLS methods. After this stage, the model was not used for further stages.

After obtaining the results of LieOLS and LieNLS models, we obtained the results of the hybrid model in the second stage. The parameters and coefficients obtained by solving the two processes (Lie and deep learning) within the framework of the hybrid method are given in

Table 10. LieLSTM parameter estimations.

	LieLSTMOLS Methods		LieLSTMNLS Methods
	Daily	Weekly	Daily	Weekly
${\mathsf{\alpha }}_{\mathbf{2}}$	0.214 (2.62)	0.18 (2.14)	0.252 (2.07)	0.121 (1.86)
${\mathsf{\alpha }}_{\mathbf{3}}$	0.35 (2.28)	0.259 (2.67)	0.405 (3.17)	0.314 (2.26)
${\mathsf{\beta }}_{\mathbf{1}}$	0.93 (1.89)	0.76 (3.45)	0.86 (2.91)	0.69 (3.27)
${\mathsf{\beta }}_{\mathbf{2}}$	−0.261 (2.76)	0.315 (2.18)	−0.195 (2.44)	0.335 (1.93)
${\mathsf{\beta }}_{\mathbf{3}}$	0.019 (3.28)	0.142 (1.98)	0.028 (1.88)	0.151 (3.22)
${\mathsf{\beta }}_{\mathbf{4}}$	0.10485 (1.88)	0.0663 (1.76)	0.10492 (2.16)	0.0753 (3.48)
${\mathsf{\beta }}_{\mathbf{5}}$	−0.03626 (1.96)	0.297055 (2.063)	0.064969 (1.97)	−0.109947 (2.13)
AIC	1.2	1.95	2.015	2.05
R2	0.85	0.79	0.86	0.791

.

Comparing the results of the hybrid model obtained by enhancing the LieOLS and LieNLS methods with the LSTM method to the results of the LieOLS and LieNLS methods alone revealed several improvements. The most notable improvement was in the coefficients of the variables ${\alpha }_2$ and ${\alpha }_3$, which differed significantly in the results of the LieOLS and LieNLS methods. For ${\alpha }_2$, there was a distinct oscillation between the daily and weekly variables. Although all coefficients were less than 1, there were still noticeable differences. However, in the studies enhanced with the LSTM method, the results showed convergence. Another significant improvement was observed for the ${\alpha }_3$ variable. For instance, while the daily bitcoin data showed different results for the LieOLS and LieNLS methods, the LSTM-enhanced method yielded similar results. This indicates the success of the LSTM method in improving the LieOLS and LieNLS methods.

5.3.2. Forecast Results

According to the results of many studies, the neural network structure is particularly effective in prediction outcomes. Therefore, forecast accuracy is very important in this study. Results were obtained for the in-sample and then for the out-of-sample data.

The forecasting accuracy of the Lie-OLS and LieNLS models was subsequently assessed and enhanced through the utilization of LSTM. To implement the LSTM method, the data were split into two distinct sets: an in-sample training set and an out-sample test set. These sets were created according to time intervals.

The In-Sample Results

Table 11. Forecast results from the in-sample data.

	Lie OLS Method		Lie NLS Method		LieLSTMOLS Method		LieLSTMNLS Method
	Daily	Weekly	Daily	Weekly	Daily	Weekly	Daily	Weekly
RMSE	6.514	5.614	5.43	5.38	1.48	1.489	1.49	1.824
MAE	6.30	4.07	4.97	4.74	1.19	1.199	1.19	1.5071
MAPE	68.86	59.77	51.26	55.84	11.89	12.92	11.76	21.145

displays the results of the LieLSTMOLS and LieLSTMNLS methods. Additionally, it presents the outcomes achieved by employing standard regression techniques with the Lie OLS models for comparison.

The LieLSTMOLS models are more successful than the reference LieOLS models, as shown in Table 10.

The success of models upgraded with LSTM is increasing. This shows us that the success rate will increase in the LSTM out-of-sample model.

The Out-of-Sample Results

The results of MAE and RMSE were calculated for LieOLS, LieNLS, LieLSTMNLS, and LieLSTMOLS to assess their predictive performance for T + 1, T + 10 workdays, and T + 1, and T + 10 weeks, as shown in

Table 12. The performances of the methods compared in out-of-sample situations.

Standard Lie OLS Method					Standard Lie NLS Method
	Daily		Weekly		Daily		Weekly
	T + 1	T + 10	T + 1	T + 10	T + 1	T + 10	T + 1	T + 10
RMSE	5.38	5.46	5.61	5.19	4.21	4.81	3.82	3.91
MAE	4.74	4.91	5.07	5.62	3.38	3.97	2.97	2.65
MAPE	55.84	60.01	61.77	56.51	44.04	51.16	41.58	44.36
LSTM
LieLSTMOLS Method					LieLSTMNLS Method
	Daily		Weekly		Daily		Weekly
	T + 1	T + 10	T + 1	T + 10	T + 1	T + 10	T + 1	T + 10
RMSE	1.517	1.58	1.0487	1.0706	1.67	1.0327	1.8427	1.767
MAE	0.66	0.666	0.75	0.7476	0.65	0.896	0.45850	0.712
MAPE	11.74	11.311	9.185	9.269	12.73	11.9115	12.7872	11.984

. The findings from out-of-sample testing reveal that LieLSTMNLS and LieLSTMOLS yield the highest forecast accuracy.

When comparing the results of LieOLS and LieNLS with those of LieLSTMOLS and LieLSTMNLS, it is evident that the LSTM method provides significant improvement in RMSE, MAE, and MAPE results. For instance, for the daily bitcoin data, the RMSE value is 5.38 for the LieOLS method and 4.81 for the LieNLS method. However, for the LieLSTMOLS method, the RMSE value is 1.517, and for the LieLSTMNLS method, it is 1.67.

In terms of RMSE results, the best performance was obtained with the LieLSTMNLS model for T+10. The second-best result was for the bitcoin weekly data with the LieLSTMNLS model for the T + 1 period, followed by the LieLSTMNLS T + 10 result. The highest value, or in other words, the least successful result, was 1.8427, obtained for the LieLSTMNLS method for the T + 1 period. Generally, the RMSE values for LieLSTMOLS and LieLSTMNLS range from 1.0327 to 1.8427. In contrast, the RMSE values for LieOLS and LieNLS generally range from 3.82 to 5.61. Based on our forecast outcomes, the LSTM models demonstrate superior performance compared to the Lie model, showcasing higher forecasting accuracy.

Assessment of Forecast Accuracy

The equivalence of prediction accuracy (null hypothesis H0) was tested using the WSR and DM tests in

Table 13. The WSR test p-values.

	LieOLS	LieNLS	LieLSTMOLS	LieLSTMNLS
LieOLS	-
LieNLS	0.36	-
LieLSTMOLS	0.008	0.024	-
LieLSTMNLS	0.015	0.023	0.001	-

and

Table 14. The DM test p-values.

	LieOLS	LieNLS	LieLSTMOLS	LieLSTMNLS
LieOLS	-
LieNLS	0.32	-
Lie-LSTMOLS	0.012	0.019	-
Lie-LSTMNLS	0.013	0.027	0.006	-

.

The results show that the DM and WSR statistics p-values are both 0.00, indicating significance at the 1% level in Table 10 and Table 11. The null hypothesis (H0) for these tests assumes that the models possess equivalent levels of accuracy. In most cases, the H0 hypothesis is rejected as the p-value is <0.05. Accordingly, in the RMSE comparison, the p-value is >0.05, indicating that these two models have comparable RMSE performance.

6. Conclusions

We introduced hybrid models for examining the price behavior and return of bitcoin during the period from 18 July 2010 to 28 December 2023 for daily data and 18 July 2010 to 24 December 2023 for weekly data. In our foundational model, the Lie group SO(3) serves as a differential manifold, effectively identified with the unit sphere $S^2$. Furthermore, we integrated this model with LSTM to evaluate its efficacy. In this paper, the focus was on the mean-reverting tendency of bitcoin and a two-factor model for valuing financial and tangible assets based on bitcoin. Our study developed a short-term model and solved it through a hybrid approach combining the Lie method and LSTM network. Upon examination of the bitcoin price time series, we observed excess JB and kurtosis. While alternative methods may be considered for analyzing these processes, our proposed Lie model readily addresses this issue. This is attributed to the modeling of bitcoin prices on the $S^2$.

This study primarily aims to enhance the predictive power of variables exhibiting chaotic structures and confirming the presence of entropy by upgrading the LieOLS and LieNLS methods to the LieLSTMOLS and LieLSTMNLS methods. For this purpose, the existence of chaos was first investigated. The chaotic structure was determined using the Lyapunov, Rosenstein et al. [47], and Kantz [48] methods. The results of entropy were obtained from the Havrda–Charvát–Tsallis (HCT) entropy [52,53] and Shannon entropy (SE) tests [51]. KSCM was applied. The nonlinearity and fractionality tests indicated a preference for fractionality, nonlinearity, and long-term dependence across all analyzed series, particularly in the volatility of each series.

So, the tests confirmed the presence of chaotic structure, complexity, and entropy in Bitcoin, with fractionality being most prominent in Bitcoin data. Firstly, the standard ARFIMA method was applied, and its results were compared with those of the LieNLS and LieOLS methods. The findings indicated that under conditions of chaos, entropy, and complexity, the ARFIMA method did not produce successful outcomes. After, LieOLS and LieNLS models, and the LSTM-enhanced LieLSTMOLS and LieLSTMNLS methods were applied. To evaluate the forecasting success of the models, we calculated RMSE, MAE, and MAPE. When comparing the RMSE, MAE, and MAPE results of the LieOLS and LieNLS models with the LSTM-enhanced LieLSTMOLS and LieLSTMNLS, the success of the LSTM-enhanced models became evident. We advocate for modeling with LieLSTM methods to achieve superior forecasting outcomes. The LieLSTMOLS and LieLSTMNLS methods streamline numerical computations. Both LieLSTMOLS and LieLSTMNLS methods presented improved forecasting performance. Specifically, in forecasting bitcoin prices 1 and 10 days, and 1 and 10 weeks ahead, models incorporating LSTM demonstrated reduced values of RMSE and MAE. According to WSR and DB tests, the Lie-LSTM methods emerge as successful in terms of forecasting performance.

Our study underscores the importance of employing the Lie method for analyzing the dynamics of bitcoin.

Our results have significant implications for traders, investors, and policymakers. Our hybrid LieLSTMOLS and LieLSTMNLS methods showed considerable success. However, our models have certain limitations: they do not incorporate sentiment analysis from the bitcoin market. We propose the following enhancements:

• Expanding the size of the dataset.
• Applying our model to other cryptocurrencies.