A Novel Method of Blockchain Cryptocurrency Price Prediction Using Fractional Grey Model

Abstract

Cryptocurrency prices have the characteristic of high volatility, which has a specific resistance to cryptocurrency price prediction. Therefore, the appropriate cryptocurrency price predictive method can help reduce the investment risk of investors. In this study, we proposed a novel prediction method using a fractional grey model (FGM (1,1)) to predict the price of blockchain cryptocurrency. Specifically, this study established the FGM (1,1) through the closing price of three representative blockchain cryptocurrencies (Bitcoin (BTC), Ethereum (ETH), and Litecoin (LTC)). It adopted the PSO algorithm to optimize and obtain the optimal order of the model, thereby conducting prediction research on the price of blockchain cryptocurrency. To verify the predictive precision of the FGM (1,1), we mainly took MAPE, MAE, and RMSE as the judging criteria and compared the model’s predictive precision with the GM (1,1) through experiments. The research results indicate that within the data range studied, the predictive accuracy of the FGM (1,1) in the closing price of BTC, ETH, and LTC has reached a “highly accurate” level. Moreover, in contrast to the GM (1,1), the FGM (1,1) outperforms predictive capability in the experiments. This study provides a feasible new method for the price prediction of blockchain cryptocurrency. It has specific references and enlightenment for government departments, investors, and researchers in theory and practice.

1. Introduction

With the transition from printed money to virtual money over the last 20 years, a new kind of money has emerged, namely cryptocurrency [1]. Cryptocurrency involves two elements, with crypto indicating “encryption” and currency indicating “money [2]. “Its attributes are digital assets intended to be used as a medium of exchange, ensuring the security of its transactions through cryptography, limiting the generation of other crypto-currencies, and validating the safe transfer of assets [3]. The success of cryptocurrency mainly comes from blockchain [4]. Cryptocurrency is usually underpinned by blockchain [5]. Blockchain is a distributed database maintained by each user that can store all transaction records of cryptocurrencies, thereby avoiding double expenses and tracking ownership [6]. The technology uses cryptography to encrypt transactions [7]. Therefore, the transactions recorded on the blockchain cannot be modified [8]. Pseudonym Satoshi Nakamoto [9] originally proposed a novel cryptocurrency known as BTC into the blockchain network for peer-to-peer transactions [10]. Since the advent of Bitcoin, many cryptocurrencies have been launched to meet different needs and purposes [4]. In recent years, the cryptocurrency market worldwide has grown significantly, and the market value of cryptocurrency has repeatedly reached record highs [11]. The prosperity of the cryptocurrency market has attracted many investors to invest in it directly or indirectly [12].

Considering that cryptocurrency’s investment risk is higher than other kinds of investments, predicting the price volatility trends of cryptocurrency is significant [13]. Due to its volatility, it is challenging to predict cryptocurrency prices accurately [14]. Correct prediction can provide investors with valuable information to make immediate and informed decision-making, thus avoiding the spreading of false information by speculators and frauds [15]. Given this, the research on predicting cryptocurrency prices has attracted the interest of scholars. At present, most scholars focus on machine learning and deep learning to forecast cryptocurrency prices. The models or algorithms adopted include ARIMA [16], LSTM [14,17], etc. In addition to LSTM neural networks, other types of neural network models are also covered, such as Refs. [11,18,19]. In addition, some scholars used a hybrid model to forecast cryptocurrency prices, such as Refs. [15,20,21,22] Machine learning and deep learning algorithms to forecast cryptocurrency prices are of great significance to enrich the achievements in this research field. However, in general, most machine learning and deep learning methods require large amounts of raw data to train the model to ensure function and prediction accuracy. Therefore, there are specific challenges regarding the amount of data needed for research.

Given the defects of limited data and insufficient information, Deng et al. [23] proposed the grey system theory [24]. Grey system theory can obtain valuable parts under some known information to clarify the internal changing trend of the research object [25]. Grey predicting is one of the most critical aspects of this theory, and it is crucial in addressing the uncertainty problem of small and incomplete samples [26]. Among them, GM (1,1) is the fundamental model of grey predicting, and it is usually adopted to forecast short-term data [27]. GM (1,1) has the benefit of being able to address grey information and insufficient data, but it is primarily suitable for sequences with strong exponential law as well as just depicting monotonous change processes and has significant errors in some particular areas [28]. For the defects existing in the precision of the model, Wu et al. [29] introduced the concept of fractional order into the grey sequence operator and grey forecasting model [30] and developed the FGM (1,1). In this model, every sequence is multiplied by a distinct fractional order [27,31]. This model has been successfully implemented in energy consumption prediction [32], carbon sink capability prediction [33], and other fields.

Presently, there are few research findings in applying grey theory to cryptocurrency prediction. However, using the FGM (1,1) for predicting cryptocurrency is rare. Regarding the findings on applying grey theory to cryptocurrency prediction, for example, Gatabazi et al. took Bitcoin, Litecoin, and Ripple as the research objects. They used GLVM [34], VFGLVM [35], and FGLVM [36] models to predict cryptocurrency. Jalali et al. [37] adopted the GM (1,1) to forecast the price of Bitcoin. Undoubtedly, the above scholars’ research findings provide important enlightenment for the implementation of grey theory in cryptocurrency forecasting. Among them, Refs. [34,35,36] explored the proposed model’s application in cryptocurrency research. However, due to the grey theory being better suited for short-term forecasting, the fact that some types of cryptocurrencies are in the long-term forecast, and their MAPE values are relatively large, the forecasting accuracy is relatively low. Ref. [37] proved the applicability of the GM (1,1) in Bitcoin price prediction. However, compared with the GM (1,1), some research findings have proved that the forecasting performance of the FGM (1,1) is superior to this model, such as Refs. [38,39]. In summary, given the relatively rare research findings of the grey prediction model in cryptocurrency price prediction, the FGM (1,1) has been applied to the research results in this field even less. The above analysis and the fact that the FGM (1,1) has been successfully used in other areas proves that this model has scientific and practical forecasting ability. Therefore, in terms of research methods, this study adopted the FGM (1,1) for forecasting research. Scholars have made relatively numerous research findings on cryptocurrency prediction. However, studies focusing on blockchain cryptocurrency are rare. The first four cryptocurrencies, namely BTC, ETH, Ripple (XRP), and LTC, account for 67%, 8%, 4.5%, and 2% of the cryptocurrency market value worldwide, respectively, totaling about 81.5% [18]. However, XRP does not utilize blockchain due to its technology known as Ripple Protocol Consensus Algorithm [16]. Therefore, this study chose three blockchain cryptocurrencies: BTC, ETH, and LTC, as the research object.

This study adopted the FGM (1,1) to forecast the prices of three representative blockchain cryptocurrencies (BTC, ETH, LTC). This study aims to build the FGM (1,1) based on the closing price data of three representative blockchain cryptocurrencies, verify and test the accuracy of the FGM (1,1) in blockchain cryptocurrency price forecasting, and propose a new and scientific price prediction method of blockchain cryptocurrency. The results will make specific contributions to the model and approach in this field. Moreover, the research findings have reference and learning significance for the management decisions of relevant government departments, the planning of investment strategies, and the advancement of pertinent research. In addition, concerning the verification of the precision of the FGM (1,1), numerous scholars have conducted a comparative analysis between the FGM (1,1) and the GM (1,1)—for example, the Refs. [38,39] mentioned above. Therefore, by utilizing scholars’ verification ideas on the FGM (1,1), this study conducted a comparison of the predictive precision of the FGM (1,1) and the GM (1,1) in blockchain cryptocurrency, thus demonstrating the effectiveness of the FGM (1,1) in blockchain cryptocurrency price forecasting.

The contribution of this study includes the following three aspects: (1) in contrast to the overall perspective of cryptocurrency, the object of this study is based on a more focused and subdivided perspective in cryptocurrency, namely the perspective of blockchain cryptocurrency. (2) This study proposes a new method of blockchain cryptocurrency price prediction using the FGM (1,1), thereby enriching and expanding the research findings in cryptocurrency price prediction. (3) This study takes three representative blockchain cryptocurrencies as experimental objects. It proves that the FGM (1,1) has “highly accurate” prediction performance in blockchain cryptocurrency price prediction through experiments, which provides a new and effective method for blockchain cryptocurrency price prediction.

The remaining sections of this research are structured as follows: Section 2 presents the FGM (1,1) and PSO algorithm principles and processes and the evaluation criteria for model accuracy. Section 3 uses three representative blockchain cryptocurrencies to conduct prediction experiments, thereby assessing and verifying the precision of the two models in the predictive experiment of BTC, ETH, and LTC. Section 4 is the main conclusion of this study.

2. Methods

2.1. FGM (1,1)

The FGM (1,1) modeling procedure is as follows [29,30]:

Step 1: The non-negative time series is established as the original data.

$$X^{(0)}=\left\{x_{}^{(0)}(1),x_{}^{(0)}(2),\cdots ,x_{}^{(0)}(n)\right\}$$

(1)

Through

$$X^{(r)}(k)={\displaystyle \sum _{i=1}^k{C}_{k-i+r-1}^{k-i}}{x}^{(0)}(i){,}^{\hspace{1em}}k=1,2,3,\cdot \cdot \cdot ,n^{}$$

(2)

The r-order accumulation sequence of $X^{(0)}$ is acquired:

$$X^{(r)}=\left\{x_{}^{(r)}(1),x_{}^{(r)}(2),\cdots ,x_{}^{(r)}(n)\right\}$$

(3)

where

$$C_{k-1}^k=0C_{r-1}^0=1C_{k-i+r-1}^{k-i}=\frac{(k-i+r-1)(k-i+r-2)(k-i+r-3)\cdots (r+1)r}{(k-i)!}$$

(4)

Step 2: For the r-order accumulative sequence $X^{(r)}$, the differential equation is:

$$\frac{d{x}^{(r)}(k)}{dt}+a{x}^{(r)}(k)=b$$

(5)

where $a$ and $b$ are constants, $a$ is a coefficient for the development, and $b$ is an endogenous control coefficient and constant input to the system.

Then, solving the equation, the time response function is acquired:

$$x^{(r)}(k+1)=\left[x^{(0)}(1)-\frac{b}{a}\right]e^{-ak}+\frac{b}{a}$$

(6)

Step 3: Using the least square method, the constants $a$ and $b$ are estimated:

$$\left[\begin{array}{c}\widehat{a}\\ \widehat{b}\end{array}\right]={(B^{T}B)}^{-1}{B}^{T}Y$$

(7)

where:

$$B=\left[\begin{array}{cc}-0.5(x^{(r)}(1)+x^{(r)}(2))& 1\\ -0.5(x^{(r)}(2)+x^{(r)}(3))& 1\\ ⋮& ⋮\\ -0.5(x^{(r)}(n-1)+x^{(r)}(n))& 1\end{array}\right]\hspace{1em}Y=\left[\begin{array}{c}{x}^{(r)}(2)-x^{(r)}(1)\\ x^{(r)}(3)-x^{(r)}(2)\\ ⋮\\ x^{(r)}(n)-x^{(r)}(n-1)\end{array}\right]$$

(8)

Step 4: Putting $\widehat{a}$ and $\widehat{b}$ into the Formula (6), the ${\widehat{X}}^{(r)}$ is given.

$${\widehat{x}}^{(r)}(k+1)=\left[x^{(0)}(1)-\frac{\widehat{b}}{\widehat{a}}\right]e^{-\widehat{a}k}+\frac{\widehat{b}}{\widehat{a}}$$

(9)

$${\widehat{X}}^{(r)}=\left\{{\widehat{x}}_{}^{(r)}(1),{\widehat{x}}_{}^{(r)}(2),\cdots ,{\widehat{x}}_{}^{(r)}(n)\right\}$$

(10)

where ${\widehat{x}}^{(r)}(k+1)$ is the fitting value at time $k+1$.

Step 5: Using the reverse accumulation generating operator, the following is acquired:

$${\widehat{X}}^{(1)}=\left\{{\widehat{x}}_{}^{(r)(1-r)}(1),{\widehat{x}}_{}^{(r)(1-r)}(2),\cdots ,{\widehat{x}}_{}^{(r)(1-r)}(n)\right\}$$

(11)

where ${\widehat{x}}^{(0)}(k)={\widehat{x}}^{(1)}(k)-{\widehat{x}}^{(1)}(k-1), k=2,3,4,\cdots ,n,n+1$.

Then, the fitting values for the raw data are acquired: ${\widehat{x}}^{(0)}(1), {\widehat{x}}^{(0)}(2),\cdots ,{\widehat{x}}^{(0)}(n)$.

The predictive values are ${\widehat{x}}^{(0)}(n+1),{\widehat{x}}^{(0)}(n+2),\cdots$.

2.2. PSO Algorithm

Kennedy et al. [40] pioneered the PSO algorithm for the initial time [41]. It belongs to a stochastic method based on population, and it is derived from the social behavior of birds and fish [42]. Every particle in the population indicates an expected solution to an optimization issue [43]. These particles can remember the population’s best position and its own best position and speed [44]. To obtain optimum decision-making, they iterate in a way that tracks local and global optimizations [45]. Meanwhile, the method has the properties of simple design, powerful global search ability, and fast convergence rate. It has particular benefits in handling a variety of complicated nonlinear optimization issues and is well suited for quick optimization in dynamic and multi-target optimizing situations [46].

The steps for utilizing the method to solve global optimum solution are as follows [47,48,49]:

Step 1: In the D-dimensional goal search space with n particles, the $i$th particle’s position vector is $X_i={(x_{i1},x_{i2},\cdot \cdot \cdot ,x_{iD})}^T$, and its velocity vector $i$ is $V_i={(v_{i1},v_{i2},\cdot \cdot \cdot ,v_{iD})}^T$. The optimum position of particle $i$ and whole particle swarm, respectively, for the individual extremum is $P_{Best}={(p_{i1},p_{i2},\cdot \cdot \cdot ,p_{iD})}^T$ and for the global extremum is $G_{Best}={(g_{i1},g_{i2},\cdot \cdot \cdot ,g_{iD})}^T$.

Step 2: Particle $i$ tracks individual and global extremum positions to update its speed and position. The iterative formula is:

$$v_{id}^{k+1}=w{v}_{id}^k+c_1{r}_1(pbes{t}_{id}-x_{id}^k)+c_2{r}_2(gbes{t}_{id}-x_{id}^k)$$

(12)

$$x_{id}^{k+1}=x_{id}^k+v_{id}^{k+1}$$

(13)

where: $k$ symbolizes the number of iterations and $v_{id}^k$ demonstrates the $d$th velocity vector of particle $i$ at the $k$th iteration. Similarly, $x_{id}^k$ denotes the $d$th position vector of particle $i$ at the $k$th iteration.

$c_1$, $c_2$ represent individual and social learning factors, respectively, typically $c_1=c_2=2$.

$r_1$, $r_2$ are two stochastic numbers that follow a uniform distribution between [0, 1].

$w$ is the inertia weight factor and $w{v}_{id}^k$ represents the trend that the particle $i$ maintains its previous motion rate.

2.3. Accuracy Check

This study chose APE, MAPE, MAE, and RMSE as the judging criteria to assess the model’s predictive accuracy. The formulas are as follows:

$$\mathrm{APE}=100\% \times \left|\frac{{\widehat{x}}_{}^{(0)}(i)-x^{(0)}(i)}{x^{(0)}(i)}\right|$$

(14)

$$\mathrm{MAPE}=100\% \times \frac{1}{n}{\displaystyle \sum _{i=1}^n\left|\frac{{\widehat{x}}_{}^{(0)}(i)-x^{(0)}(i)}{x^{(0)}(i)}\right|}$$

(15)

$$\mathrm{MAE}=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|{\widehat{x}}_{}^{(0)}(i)-x^{(0)}(i)\right|}$$

(16)

$$\mathrm{RMSE}=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n({\widehat{x}}^{(0)}(i)-x^{(0)}(i)}{)}^2}$$

(17)

Moreover, the range of MAPE values can reveal the model’s predictive power, as seen in

Table 1. The predictive precision corresponding to MAPE.

MAPE (100%)	Predictive Precision
<10	Highly accurate
10–20	Good
20–50	Reasonable
>50	Inaccurate

[50].

3. Experimental Analysis

3.1. Experiment 1: BTC Price Prediction

Experiment 1 took Bitcoin as the research object to predict its closing price. We selected the closing price data of BTC for 15 days from 11 September 2022 to 25 September 2022. (the data came from https://cn.investing.com, accessed on 15 November 2022). Experiment 1 used the closing price data of 10 days from 11 September 2022 to 20 September 2022 to construct the GM (1,1) and the FGM (1,1) and compare the predictive accuracy. Meanwhile, the experiment adopted the closing price data of 5 days from 21 September 2022 to 25 September 2022 to test the above two models’ forecasting precision. The specific computational procedure of the FGM (1,1) in BTC closing price prediction in experiment 1 is as follows.

The sample set is:

$$X^{(0)}=\left\{21834.9,22395.3,20175.5,20222.5,19701.7,19802.4,20113.5,19418.8,19538.9,18872.4\right\}$$

This study used the PSO to optimize the order of this model to calculate the optimal order r. Among them, the convergence procedure of fractional order r optimized by the PSO is seen in Figure 1.

As seen in Figure 1, according to the optimization of the PSO algorithm, the r presents the law of iterative change. Through the iterative change process, this study finally obtained the optimum order r value as 0.0533.

The following is the 0.0533-order accumulation sequence:

$$\begin{array}{ll}{X}^{(0.0533)}=& \left\{21834.9000,\right.23559.1002,21982.0848,22345.9999,22096.3754,\\ & 22395.7921,22891.2787,22374.9506,22610.9241,\left.22067.2683\right\}\end{array}$$

Using the least squares to calculate $\widehat{a}$ and $\widehat{b}$

$$\left[\begin{array}{c}\widehat{a}\\ \widehat{b}\end{array}\right]={\left(B^{T}B\right)}^{-1}{B}^{T}Y=\left[\begin{array}{c}0.2845\\ 6416.7341\end{array}\right]$$

where

$$B=\left[\begin{array}{cc}-22697.0001& 1\\ -22770.5925& 1\\ -22164.0424& 1\\ -22221.1877& 1\\ -22246.0838& 1\\ -22643.5354& 1\\ -22633.1147& 1\\ -22492.9373& 1\\ -22339.0962& 1\end{array}\right]\hspace{1em}Y=\left[\begin{array}{c}1724.2002\\ -1577.0153\\ 363.9150\\ -249.6245\\ 299.4167\\ 495.4865\\ -516.3280\\ 235.9734\\ -543.6558\end{array}\right]$$

Therefore, the time response function is:

$${\widehat{x}}_{}^{(0.0533)}(k+1)=\left[21834.9-\frac{6416.7341}{0.2845}\right]e^{-0.2845k}+\frac{6416.7341}{0.2845}$$

Then, ${\widehat{X}}^{(0.0533)}=\left\{{\widehat{x}}_{}^{(0.0533)}(1),{\widehat{x}}_{}^{(0.0533)}(2),\cdots ,{\widehat{x}}_{}^{(0.0533)}(15)\right\}$

$$\begin{array}{c}=\left\{21834.9000\right.,22013.9886,22148.7397,22250.1300,22326.4188,\\ 22383.8206,22427.0112,22459.5090,22483.9612,22502.3597,\\ 22516.2032,22526.6195,22534.4569,22540.3540,\left.22544.7912\right\}\end{array}$$

Through the reverse accumulation generating operator, ${\widehat{X}}^{(1)}$ is acquired:

$$\begin{array}{ll}{\widehat{X}}^{(1)}& =\left\{{\widehat{x}}_{}^{(0.0533)(0.9467)}(1),{\widehat{x}}_{}^{(0.0533)(0.9467)}(2),\cdots ,{\widehat{x}}_{}^{(0.0533)(0.9467)}(15)\right\}\\ & =\left\{21834.9000\right.,42685.0884,63109.5977,83266.3276,103224.2720,\\ & 123020.7642,142679.0997,162215.4361,181641.9079,200968.1753,\\ & 220202.2530,239350.9841,258420.3272,277415.5412,\left.296341.3111\right\}\end{array}$$

The predicted values are ${\widehat{X}}^{(0)}=\left\{{\widehat{x}}_{}^{(0)}(1),{\widehat{x}}_{}^{(0)}(2),\cdots ,{\widehat{x}}_{}^{(0)}(15)\right\}$

$$\begin{array}{l}=\left\{21834.9000,\right.20850.1884,20424.5093,20156.7299,19957.9444,\\ 19796.4922,19658.3355,19536.3364,19426.4718,19326.2674,\\ 19234.0777,19148.7311,19069.3431,18995.2140,\left.18925.7699\right\}\end{array}$$

Meanwhile, the values of APE, MAPE, MAE, and RMSE in experiment 1 can be obtained by calculating according to Formulas (14)–(17). The data related to BTC closing price forecasts calculated in this study are shown in

Table 2. The computational data in BTC price prediction (Unit: USD).

Date	Actual Values	GM (1,1)		FGM (1,1)
		Values	APE (%)	Values	APE (%)
2022.9.11	21,834.9000	21,834.9000	0.0000	21,834.9000	0.0000
2022.9.12	22,395.3000	21,205.8070	5.3114	20,850.1884	6.8993
2022.9.13	20,175.5000	20,900.8779	3.5953	20,424.5093	1.2342
2022.9.14	20,222.5000	20,600.3336	1.8684	20,156.7299	0.3252
2022.9.15	19,701.7000	20,304.1110	3.0577	19,957.9444	1.3006
2022.9.16	19,802.4000	20,012.1479	1.0592	19,796.4922	0.0298
2022.9.17	20,113.5000	19,724.3831	1.9346	19,658.3355	2.2630
2022.9.18	19,418.8000	19,440.7562	0.1131	19,536.3364	0.6053
2022.9.19	19,538.9000	19,161.2077	1.9330	19,426.4718	0.5754
2022.9.20	18,872.4000	18,885.6790	0.0704	19,326.2674	2.4049
MAPE (%)		1.8943		1.5638
MAE		390.6908		326.1040
RMSE		527.7727		543.9690
2022.9.21	18,489.0000	186,14.1122	0.6767	19,234.0777	4.0298
2022.9.22	19,404.0000	183,46.4505	5.4502	19,148.7311	1.3155
2022.9.23	19,293.5000	180,82.6375	6.2760	19,069.3431	1.1618
2022.9.24	18,925.2000	178,22.6181	5.8260	18,995.2140	0.3700
2022.9.25	18,803.2000	175,66.3377	6.5779	18,925.7699	0.6519
MAPE (%)		4.9614		1.5058
MAE		946.5937		283.4175
RMSE		1033.9988		371.6113

.

To depict the changing trends of actual and predicted values in the prediction of BTC closing prices and the changes in the APE data for the GM (1,1) and the FGM (1,1), experiment 1 visually presents related data, as seen in Figure 2 and Figure 3.

As seen in Figure 2, the deviation between the values of the two models and actual values is relatively balanced in the fitting stage. In other words, the divergence between the GM (1,1) and the true values is more outstanding on some dates than with the FGM (1,1). On the contrary, the variation is less than the FGM (1,1) on some dates. However, in the forecasting stage, the FGM (1,1) forecasted values are more consistent with the true values of the BTC closing price than the GM (1,1).

This study combines the APE of the two models for analysis to clarify the differences in the prediction ability in the closing price of BTC. As seen from Figure 3, during the fitting phase, the APE data of the GM (1,1) are greater than the FGM (1,1) on five out of nine days, except for the date 11 September. However, there are also four days when the APE values for the GM (1,1) are less than the FGM (1,1) model. Moreover, the magnitude of the difference in APE values shows that the overall difference between the two in the fitting stage is not apparent. This indicates that the difference in APE values between the two in the fitting stage is relatively small, which confirms the above view that the deviation between the two in this stage is relatively balanced. Combined with the values of APE in the forecasting stage, the APE data of the FGM (1,1) are considerably minimal than the GM (1,1) except for the date 21 September. Therefore, comprehensively comparing the APE between the fitting stage and the forecasting stage indicates that the predictive power of the FGM (1,1) in the closing price of BTC is better than the GM (1,1).

To further clarify the difference between the predictive precision of the two, the MAPE data in Table 2 are combined for comparative analysis. As seen from Table 2, according to the evaluation criteria of MAPE prediction accuracy, the MAPE data predicted by the models in the closing price of BTC are both within 5%, and both of them are less than 10%. Therefore, both models have reached the “highly accurate” standard in Bitcoin closing price prediction. However, further comparative analysis shows that the MAPE data of the FGM (1,1) during the fitting stage is 1.5638%. It is lower than the 1.8943% of the GM (1,1). During the predictive stage, the MAPE data of the FGM (1,1) is 1.5058%. It is also lower than the 4.9614% of the GM (1,1).

Combined with the MAE data in Table 2, we can also find that the MAE value of the FGM (1,1) is smaller than the GM (1,1) during the fitting and prediction stage. Meanwhile, the RMSE data in Table 2 indicates that in the fitting phase, the RMSE value of the FGM (1,1) is 543.9690, which is slightly higher than the 527.7727 of the GM (1,1). However, in the prediction phase, the RMSE data of the FGM (1,1) was significantly smaller than the GM (1,1). To further determine the error comparison of the two models in RMSE, through the comprehensive calculation of the overall error of the RMSE of the two models during the fitting and prediction stage, we can conclude that the RMSE value of the FGM (1,1) is 493.2543, which is less than the RMSE value of 686.6783 for the GM (1,1). Therefore, from a comprehensive perspective, the RMSE value of the FGM (1,1) is less than the GM (1,1).

To sum up, combined with the comprehensive comparative analysis regarding trend, APE, MAPE, MAE, and RMSE values indicate that the precision of the FGM (1,1) in the prediction of BTC closing price is better than the GM (1,1). Therefore, it proves the performance of the FGM (1,1) in BTC closing price prediction.

3.2. Experiment 2: ETH Price Prediction

To further validate the effectiveness of the FGM (1,1) in the price prediction of blockchain cryptocurrency, this study took ETH closing price data as the experimental object for prediction research. We also selected the data interval from 11 September 2022 to 25 September 2022 (the data came from https://cn.investing.com, accessed on 15 November 2022). Specifically, this study used the ETH closing price data from 11 September 2022 to 20 September 2022 to construct the GM (1,1) and the FGM (1,1) and test the two accuracies. It used the closing price data from 21 September 2022 to 25 September 2022 to verify the predictive performance.

The calculation process principle of experiment 2 is the same as experiment 1. The optimum order of the FGM (1,1) optimized through the PSO is 0.1545. Meanwhile, APE MAPE, MAE, and RMSE values of ETH closing price forecasts can be obtained based on Formulas (14)–(17). In summary, the relevant data from Experiment 2 calculated in this study are seen in

Table 3. The computational data in ETH price prediction (Unit: USD).

Date	Actual Values	GM (1,1)		FGM (1,1)
		Values	APE (%)	Values	APE (%)
2022.9.11	1766.9300	1766.9300	0.0000	1766.9300	0.0000
2022.9.12	1716.4200	1675.8491	2.3637	1690.0567	1.5359
2022.9.13	1574.4700	1623.7468	3.1297	1622.8950	3.0756
2022.9.14	1637.9200	1573.2643	3.9474	1558.4134	4.8541
2022.9.15	1472.6400	1524.3514	3.5115	1502.6382	2.0370
2022.9.16	1434.0100	1476.9591	2.9950	1456.0791	1.5390
2022.9.17	1468.7900	1431.0403	2.5701	1417.3088	3.5050
2022.9.18	1335.0100	1386.5492	3.8606	1384.6678	3.7197
2022.9.19	1376.0000	1343.4412	2.3662	1356.7640	1.3980
2022.9.20	1323.3300	1301.6735	1.6365	1332.5384	0.6958
MAPE (%)		2.6381		2.2360
MAE		39.2668		33.5946
RMSE		42.8762		40.3971
2022.9.21	1247.7400	1261.2043	1.0791	1311.2092	5.0867
2022.9.22	1326.4400	1221.9934	7.8742	1292.2011	2.5813
2022.9.23	1327.9600	1184.0015	10.8406	1275.0856	3.9816
2022.9.24	1317.0000	1147.1908	12.8936	1259.5389	4.3630
2022.9.25	1294.2600	1111.5245	14.1189	1245.3116	3.7820
MAPE (%)		9.3613		3.9589
MAE		122.8828		51.3984
RMSE		137.1436		52.3337

. Moreover, the actual and forecasted values for the ETH closing price and the dynamic change trend of APE data are seen in Figure 4 and Figure 5, respectively.

Figure 4 illustrates the changing tendency of the two models is relatively synchronous during the fitting and forecasting phase. That is, the values of the two models show a decreasing trend on the whole. Meanwhile, there is some variation between the forecasted values and real values. However, the divergence degree between the forecasted values of the two models and the real values differs during the fitting and forecasting stages. Among them, in the fitting phase, the difference in the deviation between the two is relatively tiny because the GM (1,1) trend is relatively similar to the FGM (1,1). However, the divergence between the models’ predictive and actual values significantly differs during the forecasting stage. Specifically, compared with the GM (1,1), the divergence between the forecasted and real values of the FGM (1,1) is smaller in this phase, and its moving trend is closer to the changing tendency of the actual values.

To further analyze the distinction in predictive effect between the two models, we combine Table 3 and Figure 5 to explore the APE value in the ETH closing price prediction. As seen in Figure 5, the magnitude distinction between the APE data of the two models is relatively minor during the fitting phase. Meanwhile, combined with the APE data in Table 3, we also find that the maximum distinction between the APE data of the above models in the fitting stage is within 2%. However, there is a notable distinction between the models’ APE data in the forecasting stage. Among them, except for date 21 September, the APE values of the GM (1,1) are considerably greater than the FGM (1,1).

Moreover, combined with APE data of the forecasting stage in Table 3, it can be found that, except for the date 21 September, the APE data of the FGM (1,1) range from 2.5813–4.3630%, while the GM (1,1) data range from 7.8752–14.1189%. It follows that the APE values of the FGM (1,1) are markedly less than the GM (1,1) during the forecasting phase. Therefore, comprehensively comparing the APE of the two models during the fitting and forecasting stage demonstrates that the predictive power of the FGM (1,1) in ETH closing price is superior to the GM (1,1).

Furthermore, by combining the MAPE values in Table 3, we find that the models’ MAPE values are both within the range of 10% during the fitting and forecasting stage. Concerning the evaluation criteria of MAPE forecasting accuracy, it indicates that the effects of the models in ETH closing price prediction are both at the level of “highly accurate.” However, the comprehensive comparative analysis of MAPE values of the above two models also found that the MAPE data of the FGM (1,1) during the corresponding fitting stage and forecasting stage is smaller than the GM (1,1). Specifically, the MAPE data for the FGM (1,1) during the fitting and the forecasting stage are 2.2360% and 3.9589%, respectively, while data for the GM (1,1) in the above two phases are 2.6381% and 9.3613%, respectively. In addition, by combining the MAE and RMSE data in Table 3, we also find that in the fitting and prediction stage, the MAE and RMSE data of the FGM (1,1) are smaller than the GM (1,1).

In summary, a comprehensive comparative evaluation of the two models in variation trend, APE, MAPE, MAE, and RMSE values show that the FGM (1,1) outperforms the GM (1,1) in forecasting ETH closing price. Therefore, this confirms the precision of the FGM (1,1) in ETH closing price prediction.

3.3. Experiment 3: LTC Price Prediction

Experiments 1 and 2 separately demonstrate the accuracy of the FGM (1,1) in forecasting the closing price of BTC and ETH. To further confirm the model’s accuracy in blockchain cryptocurrency price prediction, this study used LTC closing price data as the experimental object for prediction research. Among them, the data interval selected for this experiment is consistent with experiments 1 and 2, the period from 11 September 2022 to 25 September 2022. Moreover, the data came from https://cn.investing.com, accessed on 15 November 2022. Specifically, experiment 3 selected the LTC closing price data from 11 September 2022 to 20 September 2022 to build the GM (1,1) and the FGM (1,1) and test their predictive precision. Meanwhile, this experiment used the data from 21 September 2022 to 25 September 2022 to measure the predictive power of the above two models.

The principle of the calculation process of experiment 3 is the same as that of experiments 1 and 2. We used the PSO algorithm to achieve the optimum FGM (1,1) order, which is 0.1384. Meanwhile, this study calculated the APE and MAPE values of LTC closing price prediction using equations (14) and (15), respectively. In summary, the relevant data calculated from this experiment are seen in

Table 4. The computational data in LTC price prediction (Unit: USD).

Date	Actual Values	GM (1,1)		FGM (1,1)
		Values	APE (%)	Values	APE (%)
2022.9.11	62.2100	62.2100	0.0000	62.2100	0.0000
2022.9.12	61.3700	61.1800	0.3096	60.7332	1.0376
2022.9.13	59.1000	59.9561	1.4486	59.9475	1.4340
2022.9.14	60.1800	58.7566	2.3652	58.8233	2.2544
2022.9.15	56.3300	57.5812	2.2212	57.5584	2.1807
2022.9.16	55.9500	56.4293	0.8567	56.3071	0.6382
2022.9.17	57.8200	55.3004	4.3577	55.1421	4.6314
2022.9.18	52.6300	54.1941	2.9719	54.0873	2.7690
2022.9.19	52.8900	53.1099	0.4158	53.1431	0.4785
2022.9.20	52.3000	52.0474	0.4830	52.2998	0.0004
MAPE (%)		1.5429		1.5424
MAE		0.8756		0.8815
RMSE		1.1618		1.1815
2022.9.21	51.1400	51.0062	0.2616	51.5450	0.7919
2022.9.22	53.5500	49.9858	6.6558	50.8659	5.0123
2022.9.23	55.2900	48.9858	11.4021	50.2515	9.1129
2022.9.24	53.4300	48.0059	10.1518	49.6920	6.9961
2022.9.25	52.5200	47.0455	10.4236	49.1795	6.3604
MAPE (%)		7.7790		5.6547
MAE		4.1802		3.0412
RMSE		4.7298		3.4025

. In addition, Figure 6 and Figure 7 present the trends of the actual and forecasted values in LTC closing price prediction and the dynamic comparison of APE data for the two models.

As seen in Figure 6, the trend of predictive values of the two models shows a relatively high synchronization between the fitting and forecasting stages. However, there is also some discrepancy between the forecasted values of the models and the real values. Further analysis reveals a difference in the level of deviation between the models’ predictive and real values. Specifically, during the fitting stage, the movement trends of the two models demonstrate a high degree of synchronization. Therefore, the difference in deviation between the predicted and actual values is minimal in this stage. On the contrary, compared with the FGM (1,1) during the forecasting phase, the forecasted values of the GM (1,1) have a higher degree of deviation from the real values, and its forecasted values have more deviation from the tendency of the real values. Therefore, there exists a comparatively significant distinction in the deviation between the predictive values of the two models and the real values during the forecasting stage.

To further explore the differences between the two models in forecasting performance, we analyze the APE values in LTC closing price forecasting. As seen in Figure 7, during the fitting phase, the APE data from the GM (1,1) and the FGM (1,1) show a high coincidence degree on most dates. Meanwhile, combined with the APE values of the fitting stage in Table 4, we can find that the maximum difference range between the APE of the above models is within 1%, which indicates that the difference in APE between these two models in the fitting stage is minimal. However, in the forecasting stage, the APE of the FGM (1,1) is less than the GM (1,1) on all other dates, except on 21 September when the APE data from the FGM (1,1) is slightly greater than the GM (1,1). Therefore, comprehensively comparing the APE values in the fitting and forecasting stages indicates that the FGM (1,1) is superior to the GM (1,1) in the predictive capability of LTC closing price.

Moreover, the MAPE data from the fitting and forecasting stages indicate that the MAPE of the two models is within 10%. According to the MAPE prediction accuracy criterion, the two models’ performance in predicting LTC closing price is “highly accurate.” Further analysis indicates the MAPE data from the above two models are 1.5429% and 1.5424%, respectively, in the fitting stage, which suggests that the difference in forecasting accuracy between them is minimal. Meanwhile, the predictive precision of the FGM (1,1) slightly exceeds the GM (1,1) during this stage. In the forecasting stage, the two models’ MAPE values are 7.7790% and 5.6547%, respectively, indicating that the predictive precision of the FGM (1,1) outperforms the GM (1,1). Therefore, the comprehensive assessment of the two models’ MAPE data during the fitting and forecasting stages indicates that the predictive performance of the FGM (1,1) is more prominent than the GM (1,1).

In addition, combined with the data from MAE and RMSE in Table 4, we find that in the fitting stage, the values of MAE and RMSE for the FGM (1,1) are 0.8815 and 1.1815, respectively, which are slightly higher than the values of 0.8756 and 1.1618 for the GM (1,1). However, in the prediction stage, the values of MAE and RMSE of the FGM (1,1) were 3.0412 and 3.4025, respectively, which are less than the 4.1802 and 4.7298 values of the GM (1,1). To further judge the error comparison of the two models in MAE and RMSE, through the comprehensive calculation of the overall errors of the fitting and prediction order MAE and RMSE in experiment 3, it is concluded that the MAE and RMSE values of the FGM (1,1) are 1.6014 and 2.1886, respectively, which are less than the 1.9771 and 2.8908 values of the GM (1,1). Therefore, from a comprehensive perspective, the MAE and RMSE data from the FGM (1,1) are smaller than data from the GM (1,1).

After a comprehensive evaluation of the two models’ dynamic trends, APE, MAPE, MAE, and RMSE, the data shows that the FGM (1,1) outperforms the GM (1,1) in LTC closing price forecasting. Therefore, it proves the precision of the FGM (1,1) in LTC closing price prediction.

3.4. Additional Experiments: BTC, ETH, LTC Price Prediction

To further verify the predictive performance of the FGM (1,1), this study again selected two data intervals for additional experiments. Specifically, the data intervals chosen in this study and their reasons are as follows: (1) The overall direction is the downward trend in some fluctuations. Since the market value of cryptocurrencies began to decline significantly in early April 2022, the overall market was extremely unstable and gradually stabilized around mid-to-late June 2022. Therefore, in this study, considering the instability of the overall market of cryptocurrencies, we selected the data within this interval to test the model’s accuracy. The data period chosen in this study is from 13 April to 27 April, 2022 (the data came from https://cn.investing.com, accessed on 4 July 2023). (2) The overall direction is the upward trend in some fluctuations. To further test the model’s predictive performance, taking into account the changes in the data cycle, this study attempted to use the latest data for testing. In the current data range, the market value of the cryptocurrency has shown an apparent upward trend since mid-June 2023. Considering that the data is recent and the direction of market changes is opposite to April 2022, this study selected the data from 16 June to 30 June, 2023, to verify the model’s accuracy. The data are also from https://cn.investing.com, accessed on 4 July 2023.

In this study, the modeling and calculation principles of the additional experiments 1 and 2 on the price prediction of BTC, ETH, and LTC are consistent with experiments 1 to 3, and the related results are shown in

Table 5. Additional Experiment 1: The computational data of price prediction in BTC, ETH, and LTC (Unit: USD).

Date	BTC			ETH			LTC
	X(0)	GM (1,1)	FGM (1,1)	X(0)	GM (1,1)	FGM (1,1)	X(0)	GM (1,1)	FGM (1,1)
2022.4.13	41,133.00	41,133.00	41,133.00	3116.92	3116.92	3116.92	110.50	110.50	110.50
2022.4.14	39,936.00	40,320.16	39,936.12	3021.93	3049.92	3013.08	107.40	111.38	107.40
2022.4.15	40,560.00	40,363.51	40,186.51	3042.01	3045.56	3031.15	110.90	111.05	110.87
2022.4.16	40,382.00	40,406.91	40,493.27	3058.96	3041.21	3051.17	114.30	110.73	112.38
2022.4.17	39,703.00	40,450.35	40,674.82	2989.05	3036.87	3059.27	108.80	110.40	112.66
2022.4.18	40,803.00	40,493.84	40,720.79	3055.64	3032.53	3055.63	111.30	110.07	112.15
2022.4.19	41,503.00	40,537.38	40,655.19	3102.01	3028.19	3042.88	113.80	109.75	111.11
2022.4.20	41,368.00	40,580.96	40,506.15	3076.38	3023.87	3023.68	111.90	109.43	109.69
2022.4.21	40,482.00	40,624.59	40,298.31	2983.95	3019.55	3000.26	106.90	109.11	108.01
2022.4.22	39,709.00	40,668.27	40,051.34	2963.00	3015.23	2974.29	105.40	108.78	106.13
MAPE (%)		1.1145	0.9288		1.1022	0.7801		2.0580	1.2088
MAE		451.6590	377.4600		33.4380	23.7160		2.2640	1.3400
RMSE		577.3478	521.1117		40.1671	34.4310		2.6738	1.8219
2022.4.23	39,418.00	40,712.00	39,780.21	2931.90	3010.92	2947.02	105.00	108.46	104.12
2022.4.24	39,464.00	40,755.77	39,496.04	2922.03	3006.62	2919.32	104.50	108.15	102.02
2022.4.25	40,427.00	40,799.59	39,206.87	3006.04	3002.32	2891.78	104.50	107.83	99.85
2022.4.26	38,113.00	40,843.45	38,918.41	2810.42	2998.03	2864.82	98.60	107.51	97.63
2022.4.27	39,243.00	40,887.37	38,634.58	2888.99	2993.75	2838.70	100.60	107.19	95.40
MAPE (%)		3.7664	1.5364		3.2031	1.6172		5.1124	2.7628
MAE		1466.6360	605.6420		91.9400	47.3560		5.1880	2.8360
RMSE		1651.7408	726.6086		109.1656	61.2858		5.6427	3.3624

Note: Due to format and typesetting issues, the actual and predicted values in Table 5 retain two decimal places.

and

Table 6. Additional Experiment 2: The computational data of price prediction in BTC, ETH, and LTC (Unit: USD).

Date	BTC			ETH			LTC
	X(0)	GM (1,1)	FGM (1,1)	X(0)	GM (1,1)	FGM (1,1)	X(0)	GM (1,1)	FGM (1,1)
2023.6.16	26,341.30	26,341.30	26,341.30	1717.92	1717.92	1717.92	76.12	76.12	76.12
2023.6.17	26,515.00	26,424.51	25,647.88	1727.79	1722.82	1698.68	76.87	76.10	75.11
2023.6.18	26,339.70	26,998.68	26,719.30	1720.98	1747.01	1735.85	77.20	77.87	77.48
2023.6.19	26,845.90	27,585.33	27,809.49	1737.06	1771.53	1776.42	77.51	79.68	80.08
2023.6.20	28,307.70	28,184.73	28,707.35	1791.61	1796.41	1811.83	80.31	81.53	82.41
2023.6.21	29,996.90	28,797.16	29,393.97	1889.87	1821.63	1840.53	85.12	83.42	84.37
2023.6.22	29,890.50	29,422.89	29,890.51	1872.32	1847.21	1862.78	85.97	85.36	85.97
2023.6.23	30,679.40	30,062.22	30,225.90	1891.97	1873.14	1879.34	91.27	87.35	87.24
2023.6.24	30,533.60	30,715.44	30,428.16	1875.00	1899.44	1891.08	89.67	89.37	88.22
2023.6.25	30,465.30	31,382.85	30,522.00	1898.80	1926.11	1898.81	88.22	91.45	88.95
MAPE (%)		1.7215	1.3732		1.2741	1.0589		1.7173	1.6337
MAE		499.5790	382.8540		23.4200	19.1160		1.4580	1.3680
RMSE		625.9578	506.7286		29.8302	24.4338		1.9073	1.8348
2023.6.26	30,267.00	32,064.77	30,528.53	1859.00	1953.15	1903.23	87.19	93.58	89.47
2023.6.27	30,689.10	32,761.50	30,465.38	1889.51	1980.58	1904.99	88.07	95.75	89.81
2023.6.28	30,078.60	33,473.37	30,347.22	1827.95	2008.39	1904.58	82.98	97.98	90.01
2023.6.29	30,445.70	34,200.71	30,186.13	1851.92	2036.58	1902.47	84.71	100.25	90.09
2023.6.30	30,472.90	34,943.86	29,992.06	1933.80	2065.18	1899.00	108.66	102.58	90.06
MAPE (%)		10.1969	0.9833		7.3041	2.3840		11.6132	7.3063
MAE		3098.1820	298.8560		136.3400	44.3380		10.1380	7.0060
RMSE		3260.0879	312.7903		142.1806	48.6506		10.9843	9.3011

Note: Due to format and typesetting issues, the actual and predicted values in Table 6 retain two decimal places.

, respectively.

Table 5 and Table 6 show the relevant data of additional experiments 1 and 2 in the price prediction of BTC, ETH, and LTC in different data intervals, including the original and predicted values, MAPE, MAE, and RMSE in the fitting and prediction stage. As seen from Table 5 and Table 6, the MAPE, MAE, and RMSE data of the FGM (1,1) are smaller than the GM (1,1) in the fitting and prediction stages. In the prediction stage, for the price prediction experiment of BTC and ETH, the MAE and RMSE values of the FGM (1,1) are significantly smaller than the GM (1,1). In the prediction stage, there is no apparent difference between the values of MAE and RMSE for the FGM (1,1) and the GM (1,1) in the price prediction experiment of LTC, which may be mainly related to the small price unit of LTC itself.

Moreover, combined with the MAPE data in Table 5 and Table 6, we also find that the MAPE value of the FGM (1,1) in the fitting and prediction stages is lower than the GM (1,1). Regarding additional experiment 1, in the fitting phase, the MAPE of the FGM (1,1) in the price prediction of BTC, ETH, and LTC are 0.9288%, 0.7801%, and 1.2088%, respectively, which are less than the 1.1145%, 1.1022% and 2.0580% values for the GM (1,1), respectively. In the prediction stage, the MAPE values for the FGM (1,1) in the above three cryptocurrency price prediction experiments are 1.5364%, 1.6172%, and 2.7628%, respectively, and its data are lower than the 3.7664%, 3.2031%, and 5.1124% values for the GM (1,1), respectively. As for additional experiment 2, in the fitting stage, the MAPE values for the FGM (1,1) in the above three cryptocurrency price prediction experiments are 1.3732%, 1.0589%, and 1.6337%, respectively, which are less than the 1.7215%, 1.2741%, and 1.7173% values for the GM (1,1). Similarly, in the prediction stage, the MAPE data of the FGM (1,1) were lower than the GM (1,1).

In addition, in terms of the range of MAPE values, in additional experiments 1 and 2, the range of MAPE values for the FGM (1,1) in the price prediction of BTC, ETH, and LTC is less than 3% in most cases, and their values are within the range of 10%. This indicates that the FGM (1,1) has a “highly accurate” predictive power. However, in the prediction stage of additional experiment 2, the MAPE data for the GM (1,1) in the BTC and LTC price predictions were higher than 10%. Therefore, the assertion that the GM (1,1) is also “highly accurate” is not supported in additional experiment 2.

To sum up, by comparing the data from the MAPE, MAE, and RMSE for the two models in additional experiments 1 and 2 in the price prediction of BTC, ETH, and LTC, we find that the FGM (1,1) also has “highly accurate” prediction performance and the prediction accuracy of this model is better than the GM (1,1).

3.5. Comprehensive Analysis of BTC, ETH, and LTC Experiments

Experiments 1, 2, and 3 and additional experiments 1 and 2 have demonstrated that the FGM (1,1) has a “highly accurate” predictive performance in the price prediction of BTC, ETH, and LTC, and its prediction accuracy is more excellent than the GM (1,1). To further confirm the prediction ability of the FGM (1,1), we comprehensively analyzed the price prediction experiments of BTC, ETH, and LTC under the above three different data periods by comprehensively calculating the overall errors of MAPE, MAE, and RMSE of the two models in the fitting and prediction stages. The calculation results are shown in

Table 7. Overall comparison of predictive performance from 11 September 2022–25 September 2022.

Model	Blockchain Cryptocurrency	MAPE	MAE	RMSE
GM (1,1)	BTC	2.9167	575.9917	686.6783
	ETH	4.8791	67.1388	86.5739
	LTC	3.6216	1.9771	2.8908
FGM (1,1)	BTC	1.5445	311.8751	493.2543
	ETH	2.8103	39.5292	44.4148
	LTC	2.9132	1.6014	2.1886

,

Table 8. Overall comparison of predictive performance from 13 April 2022–27 April 2022.

Model	Blockchain Cryptocurrency	MAPE	MAE	RMSE
GM (1,1)	BTC	1.9985	789.9847	1063.7839
	ETH	1.8025	52.9387	71.0491
	LTC	3.0762	3.2387	3.9217
FGM (1,1)	BTC	1.1313	453.5207	597.5157
	ETH	1.0591	31.5960	45.1919
	LTC	1.7268	1.8387	2.4871

and

Table 9. Overall comparison of predictive performance from 16 June 2023–30 June 2023.

Model	Blockchain Cryptocurrency	MAPE	MAE	RMSE
GM (1,1)	BTC	4.5466	1365.7800	1950.3691
	ETH	3.2841	61.0600	85.6251
	LTC	5.0159	4.3513	6.5302
FGM (1,1)	BTC	1.2433	354.8547	451.4368
	ETH	1.5006	27.5233	34.4524
	LTC	3.5245	3.2473	5.5750

.

Table 7, Table 8 and Table 9 show the overall errors of MAPE, MAE, and RMSE for the two models in the price prediction of BTC, ETH, and LTC under different data cycles. Among them, the above three tables jointly reveal that the MAPE, MAE, and RMSE data for the FGM (1,1) are lower than the GM (1,1) on the whole. Meanwhile, combined with the range of MAPE values, it indicates that from a comprehensive perspective, the MAPE data of the FGM (1,1) are less than 4%, and the MAPE data of the GM (1,1) are less than 6%. According to the judgment standard of prediction accuracy of the MAPE value range, the MAPE data of the above two models are all within the 10% range on the whole. Therefore, from a comprehensive perspective, the prediction performance of the two reached the “highly accurate” level. However, combined with the overall error comparison of MAPE, MAE, and RMSE in Table 7, Table 8 and Table 9, we find that the prediction accuracy of the FGM (1,1) is higher than the GM (1,1).

In summary, based on a comprehensive evaluation of the results of experiments 1 to 3, additional experiments 1 to 2, and a comprehensive analysis of Bitcoin, Ethereum, and Litecoin experiments, this study verified the two models’ predictive effect on the prices of three representative blockchain cryptocurrencies. Among them, the precision of the FGM (1,1) in predicting the closing price for BTC, ETH, and LTC is “highly accurate.” It should be further explained that combined with the analysis in part 3.4, in the prediction stage of additional experiment 2, the MAPE data for the GM (1,1) in price prediction of BTC and LTC is higher than 10%. Therefore, from synthesizing all the above experimental data, this study supports the conclusion that the FGM (1,1) has a “highly accurate” level in the price prediction of blockchain cryptocurrencies. Meanwhile, in contrast to the GM (1,1), the FGM (1,1) has a more outstanding performance in blockchain cryptocurrency price prediction.

Moreover, given the predictive performance of the FGM (1,1), its application to blockchain cryptocurrency price forecasting is scientifically feasible as well as practical significance from an economic perspective. Specifically, for relevant government departments, it may help them make appropriate economic decisions, maintain the stability and healthy operation of the financial market, and thus ensure the sound development of the macroeconomy. For blockchain cryptocurrency investors, applying the model in blockchain cryptocurrency has a specific guiding role. On the one hand, the correct forecasting method will help investors grasp the investment direction better and improve their personal investment ability to make proper and rational investment decisions in the blockchain cryptocurrency market. On the other hand, it helps to ensure the investment return and maximize the return on investment. In addition, relevant researchers in the field can apply the model to blockchain cryptocurrency price prediction.

4. Conclusions

This study used the FGM (1,1) to conduct experimental predicting research on the closing prices of 11 September 2022 to 25 September 2022 for three representative blockchain cryptocurrencies (BTC, ETH, and LTC). To justify the predictive performance of the FGM (1,1) through the utilization of the changing trend of the actual values and predicted values, APE, MAPE, MAE, and RMSE, this study conducted a comparative study between this model and the GM (1,1) in three experiments. By comprehensively comparing the tendency of the forecasted and true values as well as the APE, MAPE, MAE, and RMSE values during the fitting and forecasting stages, we find that the predictive performance of the FGM (1,1) in the three experiments is superior to the GM (1,1). The analysis, in conjunction with the MAPE, reveals that the two models’ MAPE data is within 10%. Therefore, the predictive performance of the models reached the “highly accurate” level in the three experiments. However, the MAPE values for the two stages of the FGM (1,1) were lower than the GM (1,1) in the three experiments. Therefore, the FGM (1,1) has a superior predictive level for blockchain cryptocurrency in the three experiments.

To further verify the prediction ability of the FGM (1,1), this study selected two different time intervals (13 April 2022 to 27 April 2022 and 16 June 2023 to 30 June 2023) and added additional experiments 1 and 2. The results of additional experiments 1 and 2 indicate that the prediction ability of the FGM (1,1) reaches the level of “high accuracy,” and the prediction ability of this model is better than the GM (1,1). In additional experiment 2, it is verified that the prediction ability of the GM (1,1) does not reach the level of “high accuracy.”

Furthermore, to confirm the prediction ability of the FGM (1,1), we comprehensively calculated the overall errors of MAPE, MAE, and RMSE for the two models in the fitting and prediction stages, thereby conducting a comprehensive analysis of the price prediction experiments of the three blockchain cryptocurrencies in the above three different time intervals. The results demonstrate that the prediction accuracy of the FGM (1,1) is better than the GM (1,1). Moreover, on the whole, the two models have reached the level of “high accuracy” in the comprehensive experimental analysis.

In conclusion, by comprehensively analyzing the results of experiments 1 to 3, additional experiments 1 to 2, and the comprehensive research results of the price prediction experiments of BTC, ETH, and LTC, we can conclude that the FGM (1,1) has a “highly accurate” performance in the price prediction of blockchain cryptocurrencies. Moreover, the precision of the FGM (1,1) is superior to the GM (1,1) in blockchain cryptocurrency price prediction. Therefore, this study proves the forecasting performance of the FGM (1,1) in blockchain cryptocurrency price prediction.

This study confirms that the FGM (1,1) has outstanding forecasting performance within the data interval studied, providing a novel and practical forecasting method for blockchain cryptocurrency price forecasting. However, the price of blockchain cryptocurrency is influenced by policy, market sentiment, and other aspects, which may lead to relatively high differences in the changing rules of price data. The regulation and control of monetary policy, the uncertainty of trade policy, the formulation of relevant laws and regulations, the supervision of relevant government departments, and other elements may have an impact on the price of blockchain cryptocurrencies. Therefore, through the perspective of multivariate analysis of blockchain, cryptocurrency prices can be combined in follow-up research, further optimizing and improving the model’s predictive performance.