A New Dual Normalization for Enhancing the Bitcoin Pricing Capability of an Optimized Low Complexity Neural Net with TOPSIS Evaluation

Abstract

Bitcoin, the largest cryptocurrency, is extremely volatile and hence needs a better model for its pricing. In the literature, many researchers have studied the effect of data normalization on regression analysis for stock price prediction. How has data normalization affected Bitcoin price prediction? To answer this question, this study analyzed the prediction accuracy of a Legendre polynomial-based neural network optimized by the mutated climb monkey algorithm using nine existing data normalization techniques. A new dual normalization technique was proposed to improve the efficiency of this model. The 10 normalization techniques were evaluated using 15 error metrics using a multi-criteria decision-making (MCDM) approach called technique for order performance by similarity to ideal solution (TOPSIS). The effect of the top three normalization techniques along with the min–max normalization was further studied for Chebyshev, Laguerre, and trigonometric polynomial-based neural networks in three different datasets. The prediction accuracy of the 16 models (each of the four polynomial-based neural networks with four different normalization techniques) was calculated using 15 error metrics. A 16 × 15 TOPSIS analysis was conducted to rank the models. The convergence plot and the ranking of the models indicated that data normalization plays a significant role in the prediction capability of a Bitcoin price predictor. This paper can significantly contribute to the research with a new normalization technique for utilization in varied fields of research. It can also contribute to international finance as a decision-making tool for different investors as well as stakeholders for Bitcoin pricing.

1. Introduction

Digitization of the global economy is steadily progressing into a new future. We are shifting from traditional ways of transacting and investing to digitized ones. This signifies the importance of digital currency in the current economic scenario. Nowadays, every business wants a more digitized form of implementation using digital technology in some forms. To enhance their efficiency, many business enterprises are using digital information systems and software such as Systems Applications and Products (SAP) [1]. To explore the benefits of digital currency, a review of centralization in a decentralized ledger is extensively presented [2]. Such currency can be assumed to be goods as well as money at a time and is known as cryptocurrency [3].

However, the growth of the Bitcoin market is heavily affected by its volatility. It is eight times more volatile than the stock market. Many investors and stakeholders are still very skeptical about the acceptance of Bitcoin as a reliable asset. A better model for predicting Bitcoin price can be a great help in the expansion of the crypto market. Many researchers have used various artificial neural networks (ANNs)-based models for Bitcoin prediction. A few machine learning and deep learning methods have been used to predict the Bitcoin prices as well as other cryptocurrencies [4,5,6]. We proposed a low complexity polynomial-based neural network optimized by a new mutated climb monkey algorithm for an efficient prediction of daily Bitcoin closing price [7]. Baser and Sadorsky [8] studied the effect of technical indicators, macroeconomic variables and multi-step forecast horizon (from 1 day to 20 days) on Bitcoin price direction prediction using random forests. Erfanian et al. [9] analyzed and compared the forecasting efficiency of machine learning algorithms with statistical analysis by using technical indicators, microeconomic variables, and macroeconomic variables for short-term and long-term Bitcoin price prediction. Rathore et al. [10] demonstrated the qualitative prediction of Bitcoin price by using seasonal inputs for training.

Although many recent studies have addressed the volatility of Bitcoin price by using different input variables, machine learning, and deep learning algorithms, there is very little literature available that predicts the Bitcoin price by analyzing the data preprocessing techniques.

Data preprocessing is a process of cleaning, reducing, extracting, scaling, and handling missing values in a raw dataset. Many researchers have used feature extraction methods to improve the prediction accuracy of Bitcoin [11,12]. Rajabi et al. [13] proposed the use of deep learning in selecting an optimal window size for prediction and then using the optimal size for actual Bitcoin price forecasting.

Data normalization is a vital preprocessing step for the majority of classification and regression problems. This method utilizes various ways to map the highly volatile data to a limited range to improve the prediction capability of various models. In the literature, Shanker et al. [14] studied the data standardization on the training of a neural network and concluded that with the increase in network size, the self-scalability of the network increases, and hence, there is no effect of scalability on network performance.

In contrast, many researchers have demonstrated the significant role played by various data-scaling methods in classification problems [15,16]. It has been observed that the data normalization type may be used to improve classification accuracy. Some researchers have studied the impact of various normalization techniques on stock price prediction and had suggested that the data normalization type plays a very significant role in prediction and so it should be appropriately selected [17,18].

However, the effect of data normalization on Bitcoin pricing is yet to be analyzed. As the Bitcoin price is highly volatile, to obtain an efficient model for prediction, the normalization of the Bitcoin dataset needs to be explored. The main goal of this investigation is to analyze the influence of data normalization techniques on the prediction capability of a Bitcoin price predictor. Namely, this study investigated the effect of nine different data normalization techniques on the Legendre-based neural network model for daily Bitcoin price prediction.

The novelty of this approach is that it proposes a new dual normalization technique for efficient Bitcoin price prediction. Hybrid normalizations have been extensively used in the medical domain, but in cryptocurrency prediction, researchers have mostly used min–max normalization or z-score normalization. This is the first paper to suggest the analysis of 10 normalization methods using 15 performance metrics for efficient Bitcoin price prediction.

The main contributions of this paper are enlisted as follows:

• Nine data normalization types were used to process three different datasets.
• Each normalized dataset was used as input for the Legendre-based polynomial neural network model trained by the mutated climb monkey algorithm (LMCMA) to predict the closing Bitcoin price.
• A new dual normalization technique was proposed for an improved prediction.
• The proposed normalization technique was tested in three different functional link neural networks, an econometric model and a Support Vector Regression model
• A TOPSIS-based approach was applied for ranking the normalization type using 15 performance criteria.
• The LMCMA model with the proposed dual normalization was also tested for predicting daily Bitcoin log returns for the three datasets under study.

This empirical research analyzed the following research hypotheses:

Hypothesis 1. 

There is a significant relationship between data normalization and prediction accuracy of Bitcoin Price Predictor models.

Hypothesis 2. 

A dual normalization improves the prediction capability of Bitcoin price predictor models.

The research methodology is varied in that the Bitcoin predictor models in this study were evaluated using a TOPSIS-based ranking. In addition, the models used datasets scaled using a new normalization technique.

The importance of this research lies in the need for a highly efficient predictor for Bitcoin prices to handle the high volatility feature of Bitcoin prices. The research proposes a dual normalization that significantly improves the accuracy of prediction of Bitcoin closing price as well as daily log returns. The use of the new normalization technique indicated a root mean square error gain of 81.37%, 98.13%, and 98.73% over min–max normalization in the Bitfinex, Binance, and Coinbase Pro datasets, respectively.

Section 2 gives a review of the literature on Bitcoin pricing and normalization techniques used in this paper. Section 3 introduces the basic model used in this study. Section 4 explains the workflow of this study along with explanations of related techniques. It then introduces the proposed normalization and the experimental setup. Section 5 presents the experimental results followed by a thorough discussion in Section 6. The last section concludes this study and suggests various open research issues for future work.

2. Literature Review

Bitcoin is highly volatile and dynamic in nature. Many forecasting models have been presented using machine learning for Bitcoin price prediction. The volatility has been proved by a study of a Support Vector Machine (SVM)-based predictor [19]. Aggarwal et al. [20] demonstrated Bitcoin pricing using an SVM and ensemble approach. Many researchers have used LSTM and Bayesian networks for Bitcoin prediction [6,9]. Deep learning has also been used to predict Bitcoin prices efficiently by using autoencoders and LSTM [5,21]. Rathore et al. [10] have presented a more real-world model by using a FbProphet model for better prediction in comparison to LSTM.

Recently, we proposed a functional artificial link neural network (FLANN) based on Chebyshev and Legendre polynomial functions as a simple yet efficient model for Bitcoin pricing [7]. An optimal FLANN has been proposed to predict Bitcoin price movement by using a genetic algorithm-based optimization of the network [6].

Nayak et al. [22] presented the hybridization of high-order neural network (HONN) models with the evolutionary algorithms for efficient stock price prediction. They have explained the reliable and simple architecture of various HONNs such as pi-sigma neural network (PSNN), sigma-pi neural network (SPNN), and FLANNs based on Chebyshev, Laguerre, and Legendre polynomials. Ye et al. [23] presented a stable and more accurate wind power forecasting using a Laguerre polynomials-based neural network. Many other researchers have utilized the flat architecture, fast learning capabilities, and reliable features of FLANN models for stock price, gold price, and mutual fund net asset value prediction [24,25,26,27,28,29]. This motivated us to use the LMCMA model for Bitcoin prediction in this study [7].

In the literature, many researchers have studied the effect of data normalization on various classifications, regression, and decision-making models. Jain et al. [30] explored min–max and z-score normalization with 12 data complexity measures in 14 classification models to find the best normalization selected dynamically. They used Friedman’s test for evaluation [31]. Alshdaifat et al. [32] analyzed three normalization methods with three missing value handling methods on nine different ANN and SVM models for 18 benchmark datasets. They concluded that the z-score performed the best and decimal scaling was the worst normalization under study. The evaluation was based on Friedman’s test and Nemenyi’s posthoc test [33]. Singh and Singh [16] investigated 14 normalization methods to study their effect on classification accuracy by integrating normalization with weighted features. In [34], they proposed a feature-wise normalization technique. They explained that entire data normalization can be replaced with feature-wise normalization for efficient training.

Sola and Sevilla [35] studied the impact of data normalization on two neural networks trained by using backpropagation in nuclear power plant applications and concluded that normalization reduces errors and error computation time as well as enhances the network performance. The effect of normalization has been studied for many other applications such as linear ordering and 2Dcoordinate transformation [36,37].

The selection of normalization also plays a vital role in enhancing the performance of multi-criteria decision-making models (MCDM) [38,39,40]. The ranking process of MCDM models is also enhanced by selecting the appropriate normalization technique [41,42].

Very few hybrid normalization techniques have been proposed by combining different available normalization methods [41,43]. Recently, Izonin et al. [44] proposed two-step normalization by combining max-abs scaler and vector scaler, which improved the classification accuracy for medical applications. This motivated us to study the effect of normalization on Bitcoin pricing and propose a new dual normalization technique for enhanced prediction capability.

In the literature, econometric models have been extensively used in the finance domain. Batrancea [45] demonstrated the effectiveness of two econometric models in predicting the influence of many economic indicators on liquidity and solvency ratios in the healthcare industries. In [46], the influence of the financial statement of a bank on its assets, liabilities, and performance is analyzed using an econometric model. For Bitcoin price prediction, many studies have been used to study the impact of technical indicators and macroeconomic indicators. In [9], the effect of technical, macroeconomic, microstructure, and blockchain indicators on Bitcoin pricing has been analyzed using SVR (Support Vector Regression), MLP, OLS and an ensemble model. The SVR model was efficient in comparison to other models.

In light of the above review, it can be stated that this research will be very useful in the designing of new models by researchers for efficient prediction using the proposed dual normalization. In addition, it can help investors make decisions on Bitcoin investment with a more accurate prediction of future prices.

3. Materials and Methods

In [7], we proposed the use of a new mutated climb monkey algorithm-based Legendre neural network (LMCMA) model for efficient Bitcoin pricing. This model uses the benefits of a FLANN with efficient training capability of the MCMA algorithm. The basic architecture of the LMCMA model is shown in Figure 1.

The LMCMA model contains an input expansion block (IEB) which expands each input x to expanded polynomials L_i(x) where i = 1, 2, … r, for expansion order r. This model utilizes polynomial expansion of order 2. As shown in Figure 1, the inputs x₁ and x₂ are expanded into L₁(x₁), L₂(x₁) and L₁(x₂), L₂(x₂), respectively, for r = 2. A random weight matrix is applied to the expanded inputs. The weighted sum of the expanded input is passed to the tanh activation function. It derives the predicted output, which is then subtracted from the actual output to generate an error. This error is used by the MCMA algorithm for weight updates.

In this model, the expanded Legendre polynomials are generated by using the recursive formula given in Equation (1):

$${\mathrm{L}}_{\mathrm{i}+1}\left(\mathrm{x}\right)=\frac{1}{\mathrm{i}+1}\times \left[\left(2\mathrm{i}+1\right)\times \mathrm{x}\times {\mathrm{L}}_{\mathrm{i}}\left(\mathrm{x}\right)-\mathrm{i}\times {\mathrm{L}}_{\mathrm{i}-1}\left(\mathrm{x}\right)\right]$$

(1)

where x is the input data and L_i represents the ith Legendre polynomial for i = 0, 1, 2, … and L₀(x) = 1 and L₁(x) = x.

After training the model, the optimized weight giving minimum root mean square error (RMSE) is frozen for testing the network for the test dataset.

The MCMA algorithm used in this model uses a mutated climb operation in the original monkey algorithm. This reduces the time spent on the climb operation and also gives optimal weights for prediction [7].

4. Proposed Work

This section describes the steps involved in this study. It then provides a dataset analysis of the used datasets. The succeeding subsection describes the normalization types used in this study and their analysis. It is followed by the new proposed normalization and the performance evaluation methods involved in this study.

4.1. Procedural Analysis

This study analyzes the normalization types for the prediction of Bitcoin prices and then proposes a new normalization type. The steps involved in this study are presented in Figure 2.

The proposed study starts with normalizing the datasets using nine normalization methods. The normalized dataset is then divided into input and output data using the sliding window technique. The data are then divided into training and testing datasets.

The LMCMA model is then trained using each of the normalized training datasets. The minimum RMSE for each normalized training dataset is calculated, and the best two normalization types are used to design a dual normalization technique. The LMCMA model is then trained on the new normalized dataset. All 10 normalization techniques are applied on testing datasets and then evaluated based on 15 different error metrics. The top-ranked normalization along with the min–max normalization technique, which was originally proposed in the LMCMA model [7], is then tested with three other FLANNs.

The recursive Chebyshev polynomials are generated using Equation (2):

$${\mathrm{C}}_{\mathrm{i}+1}\left(\mathrm{x}\right)=2\times \mathrm{x}\times {\mathrm{C}}_{\mathrm{i}}\left(\mathrm{x}\right)-{\mathrm{C}}_{\mathrm{i}-1}\left(\mathrm{x}\right)$$

(2)

where x is the input data and C_i represents the ith Chebyshev polynomial for i = 0, 1, 2, … and C₀(x) = 1 and C₁(x) = x.

The recursive Laguerre polynomials are generated using Equation (3):

$${\mathrm{La}}_{\mathrm{i}+1}\left(\mathrm{x}\right)=\frac{1}{\mathrm{i}+1}\times \left[\left(2\mathrm{i}+1\right)\times {\mathrm{La}}_{\mathrm{i}}\left(\mathrm{x}\right)-\mathrm{i}\times {\mathrm{La}}_{\mathrm{i}-1}\left(\mathrm{x}\right)\right]$$

(3)

where x is the input data and La_i represents the ith Laguerre polynomial for i = 0, 1, 2, … and La₀(x) = 1 and La₁(x) = 1 − x.

The trigonometric functions used for input expansion are described in Equation (4):

T₁(x) = x, T₂(x) = sine(x), T₃(x) = cosine(x), T₄(x) = sine(πx), T₅(x) = cosine(πx)

(4)

After testing the four networks with min–max normalization, the proposed dual normalization, and the top two ranked normalization ranked in the training of the LMCMA model were evaluated using 15 error measures. A 16 × 15 TOPSIS analysis was conducted to select the best normalization technique in a FLANN model.

4.2. Dataset Analysis

For the simulation of this study, historical daily closing Bitcoin prices in US dollars (USD) were collected from http://www.investing.com/crypto/bitcoin/historical-data (accessed on 15 November 2022) for three different crypto-markets. They are Bitfinex, Binance and, Coinbase Pro datasets. The most recent data were collected for this study within the time range of 15 November 2018 to 15 November 2022. The datasets cover prices from pre-COVID to post-COVID durations and hence map the high volatility. The dataset descriptions are shown in

Table 1. Dataset Descriptions.

Datasets	Total Samples	Training Dataset Size	Testing Dataset Size
Bitfinex	1462	971	486
Binance	1462	971	486
Coinbase Pro	1462	971	486

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The datasets were analyzed statistically to obtain an insight into the data represented in each of the datasets. The statistical analysis of the three datasets is presented in

Table 2. Statistical Analysis of Datasets.

Datasets	MIN 1	MAX 2	MEAN	MEDIAN	STD 3	MODE	SKEWNESS	KURTOSIS
Bitfinex	3282.8	67,526	23,115.3	16,176.2	17,837.9	3929.8	0.7109	2.13189
Binance	3212.7	67,520	23,091.2	16,178.6	17,844.3	3631.9	0.7096	2.13184
Coinbase Pro	4838.5	67,557	26,684.6	21,392.1	16,769.5	8697.5	0.4445	1.94083

¹ Minimum value ² Maximum value ³ Standard deviation value.

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4.3. Normalization Methods

Data normalization is a vital process in any regression analysis. It transforms data into a fixed interval to decrease the training time and errors. This study analyzes the impact of nine data normalization techniques in the LMCMA model for the prediction of Bitcoin closing prices.

4.3.1. Min–Max Normalization

Most of the regression analysis uses the min–max normalization technique. It is a process of converting data by using the minimum and maximum values in a dataset [7,47]. The normalized data are calculated by using Equation (5):

$${\mathrm{x}}^{\prime }=\frac{\mathrm{x}-\min }{\max -\min }$$

(5)

where x is the original data, x′ is the normalized data, and min and max represent the minimum and maximum values in the dataset, respectively.

4.3.2. Decimal Scaling

In decimal scaling normalization, each data is normalized using a maximum value of a dataset and the radix of the dataset used [32,48]. The normalized data are calculated by using Equation (6):

$${\mathrm{x}}^{\prime }=\frac{\mathrm{x}}{{10}^{\mathrm{d}}}$$

(6)

where x is the original data, x′ is the normalized data and d is the number of digits present in the maximum value of the dataset.

4.3.3. Vector Normalization

Vector normalization is a sum-based normalization. Here, the data are transformed by using the absolute sum of the squares of each data [44,49]. The normalized data are calculated by using Equation (7):

$${\mathrm{x}}^{\prime }=\frac{\mathrm{x}}{\sqrt{{\sum }_1^{\mathrm{n}}{\left|\mathrm{X}\right|}^2}}$$

(7)

where x is the original data, x′ is the normalized data and n is the total number of data in dataset X.

4.3.4. Maximum Linear Normalization

The maximum linear normalization is also known as MaxAbs Scaler. It uses absolute values for mapping [44,50]. This normalization is based on the maximum value in the dataset. The normalized data are calculated by using Equation (8).

$${\mathrm{x}}^{\prime }=\frac{\mathrm{x}}{\max \left(\left|\mathrm{X}\right|\right)}$$

(8)

where x is the original data, x′ is the normalized data and max(X) is a function to find themaximum value in dataset X.

4.3.5. Juttler–Korth Normalization

In this normalization, the data are normalized using the maximum value in the dataset. It also uses the absolute values of the data for normalization [49,51]. The normalized data are calculated by using Equation (9):

$${\mathrm{x}}^{\prime }=1-\left|\frac{\max \left(\left|\mathrm{X}\right|\right)-\mathrm{x}}{\max \left(\mathrm{X}\right)}\right|$$

(9)

where x is the original data, x′ is the normalized data and max(X) is a function to find the maximum value in dataset X.

4.3.6. Peldschus Normalization

This is a square-based normalization technique that also uses the maximum value in a dataset for scaling [49,51]. The normalized data are calculated by using Equation (10):

$${\mathrm{x}}^{\prime }={\left(\frac{\mathrm{x}}{\max \left(\mathrm{X}\right)}\right)}^2$$

(10)

where x is the original data, x′ is the normalized data and max(X) is a function to find the maximum value in dataset X.

4.3.7. Tanh Estimator

A tanh estimator is an efficient normalization technique based on the mean and standard deviation of the dataset [52]. This normalization has been used in many stock price predictions. The normalized data are calculated by using Equation (11):

$${\mathrm{x}}^{\prime }=0.5\times \left[\tanh \left[\frac{0.01\times \left(\mathrm{x}-\mathsf{\mu }\right)}{\mathsf{\sigma }}+1\right]\right]$$

(11)

where x is the original data, x′ is the normalized data, and μ and σ are the mean and standard deviation of dataset X, respectively.

4.3.8. Logistic Sigmoidal Normalization

This normalization is based on the mean and standard deviation of the dataset. It uses a sigmoidal-based approach for normalization [53]. The normalized data are calculated by using Equation (12):

$${\mathrm{x}}^{\prime }=\frac{1}{1+{\mathrm{e}}^{\left(\frac{\mathsf{\mu }-\mathrm{x}}{\mathsf{\sigma }}\right)}}$$

(12)

where x is the original data, x′ is the normalized data, and μ and σ are the mean and standard deviation of dataset X, respectively.

4.3.9. Hyperbolic Tangent Function-Based Normalization

This normalization is a variant of logistic sigmoidal normalization and is based on the mean standard deviation of the dataset [54]. The normalized data are calculated by using Equation (12).

$${\mathrm{x}}^{\prime }=\frac{1-{\mathrm{e}}^{\left(\frac{\mathsf{\mu }-\mathrm{x}}{\mathsf{\sigma }}\right)}}{1+{\mathrm{e}}^{\left(\frac{\mathsf{\mu }-\mathrm{x}}{\mathsf{\sigma }}\right)}}$$

(13)

where x is the original data, x′ is the normalized data, and μ and σ are the mean and standard deviation of dataset X, respectively.

4.4. Proposed Dual Normalization

The LMCMA model was trained by using the nine different normalized datasets using Equations (5)–(13). After training, the minimum root mean squared error (RMSE) was calculated for each model based on different normalized data. The minimum RMSE after training is shown in

Table 3. Minimum root mean square error for each normalization based Legendre neural network for 3 datasets.

Normalization Type	Minimum RMSE
	Bitfinex	Binance	Coinbase Pro
N1 1	0.0227	0.0199	0.0352
N2 2	0.0110	0.0124	0.0230
N3 3	0.0011	0.0013	0.0027
N4 4	0.0208	0.0210	0.0326
N5 5	0.0194	0.0233	0.0329
N6 6	0.0270	0.0205	0.0369
N7 7	0.0008	0.0005	0.0012
N8 8	0.0150	0.0141	0.0261
N9 9	0.0328	0.0326	0.0512

¹ Min–max normalization, ² Decimal scaling, ³ Vector normalization, ⁴ Maxabs normalization, ⁵ Juttler–Korth normalization, ⁶ Peldschus normalization, ⁷ Tanh estimator, ⁸ Sigmoidal normalization, ⁹ Hyperbolic tangent-based normalization.

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Table 3 indicates that vector normalization and the tanh estimator give the minimum RMSE for all the three datasets under study. This table also validates the first hypothesis as different normalizations affect the RMSE value in the LMCMA model. As the tanh estimator has the disadvantage of giving unexpected results for negative inputs as well as for very higher values, it is not always reliable for predicting highly volatile Bitcoin prices. However, vector normalization can convert larger values to smaller ones as it is a sum-based model. As both these normalizations are the best ones in this training, the concept of dual normalization was formulated to utilize the benefits of both techniques.

The proposed dual normalization converts the input into an intermediate form using vector normalization using Equation (14):

$$\mathrm{t}=\frac{\mathrm{x}}{\sqrt{{\sum }_1^{\mathrm{n}}{\left|\mathrm{X}\right|}^2}}$$

(14)

where x is the original data, t is the intermediate data and n is the total number of data in the original dataset X.

This intermediate value is then re-normalized using the tanh estimator. The formula for converting intermediate data into the final normalized form using dual normalization is given in Equation (15):

$${\mathrm{x}}^{\prime }=0.5\times \left[\tanh \left[\frac{0.01\times \left(\mathrm{t}-\mathsf{{\rm Y}}\right)}{\mathsf{\beta }}+1\right]\right]$$

(15)

where t is the intermediate data, x′ is the final normalized data, and Υ and β are the mean and standard deviation of the intermediate dataset, respectively.

A good normalization technique must provide a perfect denormalization method. This helps with predicting the actual data at the end of the study. The denormalization method must be able to map the actual dataset distribution. The denormalization method of this dual normalization technique is completed in three steps.

• Find the vector data from the original data using Equation (16):

  $$\mathrm{A}=\frac{\mathrm{x}}{\sqrt{\sum {\mathrm{x}}^2}}$$

  (16)

  where x is the original data and A is the vector data.
• Find the inverse tanh value for the normalized data using the mean and standard deviation of vector data as given in Equation (17):

  $$\mathrm{z}=\left(100\times \mathsf{\phi }\times {\tanh }^{-1}\left(2{\mathrm{x}}^{\prime }-1\right)\right)+\overline{\mathrm{A}}$$

  (17)

  where x′ is the normalized data, and $\overline{\mathrm{A}}$ and φ are the mean and standard deviation of the vectored data A.
• Find the final denormalized data dx using Equation (18):

  $$\mathrm{dx}=\mathrm{z}\times \sqrt{\sum {\mathrm{x}}^2}$$

  (18)

The symbolic notations used for each normalization method in this study have been enlisted in

Table 4. Symbolic Notation for Normalization Types.

Normalization Type	Symbolic Notation
Min–max normalization	N1
Decimal scaling	N2
Vector normalization	N3
Maxabs normalization	N4
Juttler–Korth normalization	N5
Peldschus normalization	N6
Tanh estimator	N7
Sigmoidal normalization	N8
Hyperbolic tangent-based normalization	N9
Proposed dual normalization	N10

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The stability of the proposed dual normalization technique was also tested by plotting the normalized dataset using dual normalization in all three datasets. The normalized plot along with the corresponding original dataset plot is presented in Figure 3, Figure 4 and Figure 5.

The proposed normalization was also analyzed by comparing the minimum and maximum values of each of the 10 normalizations under study. The normalized dataset analysis is presented in

Table 5. Minimum and maximum value in the normalized dataset.

Normalization Type	Bitfinex		Binance		Coinbase Pro
	MIN	MAX	MIN	MAX	MIN	MAX
N1	0	1	0	1	0	1
N2	0.0328	0.6753	0.0321	0.6752	0.0485	0.6756
N3	0.0029	0.0605	0.0029	0.0605	0.0040	0.0561
N4	0.0486	1	0.0476	1	0.0716	1
N5	0.0486	1	0.0476	1	0.0716	1
N6	0.0024	1	0.0023	1	0.0051	1
N7	0.4944	0.5124	0.4944	0.5124	0.4935	0.5122
N8	0.2475	0.9234	0.2471	0.9234	0.2137	0.9196
N9	0.5049	0.8468	0.5057	0.8468	0.5726	0.8393
N10	0.4944	0.5124	0.4944	0.5124	0.4935	0.5122

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Table 5 indicates that the proposed normalization maps the value within the same range as other normalizations under study. All the normalization scales the values between 0 and 1.

4.5. Performance Evaluation

The new dual normalization method was used to generate a dataset for training and testing of the LMCMA model. The performance of the normalization methods was calculated by using 15 different error metrics, as shown in

Table 6. FifteenError Metrics with Formulas.

Error #	Name of the Error	Formula
E1	Root Mean Square Error (RMSE)	$\mathrm{RMSE}=\sqrt{\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left(\mathrm{F}\left(\mathrm{i}\right)-\mathrm{A}\left(\mathrm{i}\right)\right)}^2}{\mathrm{n}}}$
E2	Mean Square Error (MSE)	$\mathrm{MSE}=\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left(\mathrm{F}\left(\mathrm{i}\right)-\mathrm{A}\left(\mathrm{i}\right)\right)}^2}{\mathrm{n}}$
E3	Mean Absolute Error (MAE)	$\mathrm{MAE}=\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left|\mathrm{F}\left(\mathrm{i}\right)-\mathrm{A}\left(\mathrm{i}\right)\right|}{\mathrm{n}}$
E4	Theil’sU Error(TU)	$\mathrm{TU}=\frac{\sqrt{\frac{1}{\mathrm{n}}\times {\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left(\mathrm{F}\left(\mathrm{i}\right)-\mathrm{A}\left(\mathrm{i}\right)\right)}^2}}{\sqrt{\frac{1}{\mathrm{n}}\times {\sum }_{\mathrm{i}=1}^{\mathrm{n}}\mathrm{A}{\left(\mathrm{i}\right)}^2}+\sqrt{\frac{1}{\mathrm{n}}\times {\sum }_{\mathrm{i}=1}^{\mathrm{n}}\mathrm{F}{\left(\mathrm{i}\right)}^2}}$
E5	R-Square Error (R2)	${\mathrm{R}}^2=1-\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left(\mathrm{A}\left(\mathrm{i}\right)-\mathrm{F}\left(\mathrm{i}\right)\right)}^2}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left(\mathrm{A}\left(\mathrm{i}\right)-\overline{\mathrm{A}}\right)}^2}$
E6	Mean Percentage Error (MPE)	$\mathrm{MPE}=\frac{100}{\mathrm{n}}\times \sum _{\mathrm{i}=1}^{\mathrm{n}}\frac{\mathrm{A}\left(\mathrm{i}\right)-\mathrm{F}\left(\mathrm{i}\right)}{\mathrm{A}\left(\mathrm{i}\right)}$
E7	Mean Absolute Percentage Error (MAPE)	$\mathrm{MAPE}=\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left|\mathrm{F}\left(\mathrm{i}\right)-\mathrm{A}\left(\mathrm{i}\right)\right|}{\mathrm{n}}\times 100$
E8	Symmetric Mean Absolute Percentage Error (SMAPE)	$\mathrm{SMAPE}=\frac{100}{\mathrm{n}}\times \sum _{\mathrm{i}=1}^{\mathrm{n}}\frac{\left|\mathrm{F}\left(\mathrm{i}\right)-\mathrm{A}\left(\mathrm{i}\right)\right|}{\frac{\left|\mathrm{A}\left(\mathrm{i}\right)\right|+\left|\mathrm{F}\left(\mathrm{i}\right)\right|}{2}}$
E9	Mean Absolute Scaled Error (MASE)	$\mathrm{MASE}=\frac{\frac{1}{\mathrm{n}}\times {\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left|\mathrm{A}\left(\mathrm{i}\right)-\mathrm{F}\left(\mathrm{i}\right)\right|}{\frac{1}{\mathrm{n}-1}\times {\sum }_{\mathrm{i}=2}^{\mathrm{n}}\left|\mathrm{A}\left(\mathrm{i}\right)-\mathrm{A}\left(\mathrm{i}-1\right)\right|}$
E10	Sum Squares Error (SSE)	$\mathrm{SSE}=\sum _{\mathrm{i}=1}^{\mathrm{n}}{\left(\mathrm{A}\left(\mathrm{i}\right)-\mathrm{F}\left(\mathrm{i}\right)\right)}^2$
E11	Root Squared Sum Error (RSSE)	$\mathrm{RSSE}=\sqrt{\sum _{\mathrm{i}=1}^{\mathrm{n}}{\left(\mathrm{A}\left(\mathrm{i}\right)-\mathrm{F}\left(\mathrm{i}\right)\right)}^2}$
E12	Mean Relative Absolute Error (MRAE)	$\mathrm{MRAE}=\frac{1}{\mathrm{n}}\times \sum _{\mathrm{i}=2}^{\mathrm{n}}\left|\frac{\mathrm{A}\left(\mathrm{i}\right)-\mathrm{F}\left(\mathrm{i}\right)}{\mathrm{A}\left(\mathrm{i}\right)-\mathrm{A}\left(\mathrm{i}-1\right)}\right|$
E13	Mean Signed Deviation (MSD)	$\mathrm{MSD}=\frac{1}{\mathrm{n}}\times \sum _{\mathrm{i}=1}^{\mathrm{n}}\left(\mathrm{A}\left(\mathrm{i}\right)-\mathrm{F}\left(\mathrm{i}\right)\right)$
E14	Average Relative Variance (ARV)	$\mathrm{ARV}=\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left(\mathrm{F}\left(\mathrm{i}\right)-\mathrm{A}\left(\mathrm{i}\right)\right)}^2}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left(\mathrm{F}\left(\mathrm{i}\right)-\overline{\mathrm{A}}\right)}$
E15	Root Relative Square Error (RRSE)	$\mathrm{RRSE}=\sqrt{\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left(\mathrm{F}\left(\mathrm{i}\right)-\mathrm{A}\left(\mathrm{i}\right)\right)}^2}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left(\mathrm{F}\left(\mathrm{i}\right)-\overline{\mathrm{A}}\right)}^2}}$

A(i) is the actual output, F(i) is the predicted output and n is the dataset size.

.

Out of the 15 error metrics, E5 is negatively oriented, and so a higher value is better. In contrast, the remaining 14 error measures are positively oriented.

After training, the performance of any single normalization method was not optimal. Some normalization gave better error values for certain error metrics but the worst values for some other metrics. So, a multi-criteria-based evaluation was made to find the top normalization methods.

The MCDM methods have been applied in many decision-making systems of organizations involving automobiles and airlines [55,56]. TOPSIS is an efficient and more accurate MCDM technique to rank various models under study [56,57]. Additionally, the TOPSIS ordering utilizes both best and worst criteria for picking the best model, and so it is highly reliable. TOPSIS was also proved to be better in terms of accuracy and robustness when applied to prediction and classification machine learning models [7,58,59]. Therefore, this study uses TOPSIS to select the best normalization in a FLANN network. TOPSIS analysis was used to rank the 16 models generated by using four normalizations (N1, N3, N7, N10) in Legendre FLANN (L1, L3, L7, L10), Chebyshev FLANN (C1, C3, C7, C10), Laguerre FLANN (La1, La3, La7, La10) and trigonometric FLANN (T1, T3, T7, T10).

5. Results

To test the second hypothesis this study, we divided each of the 10 normalized datasets into a train dataset and test dataset in a 2:1 ratio. A Legendre FLANN with 5 × 1 input–output neurons was used for training. The network used a sliding window of size 5 to input five historic prices and generate the 6th price. This model uses the RMSE value as a measure of fitness to be minimized by the MCMA algorithm. With a population of 20, this model is iterated 100 times to forecast the daily Bitcoin price for all the datasets. The optimized weights generated for each model under study are used to predict the test prices. The efficiency of each model was tested using 15 error measures. The minimum error metric was used as the final error after running each model 10 times.

The minimum error values for each normalization method in the testing of LMCMA model using the Bitfinex dataset are shown in

Table 7. Error values in testing Legendre network for Bitfinex dataset.

	E1	E2	E3	E4	E5	E6	E7	E8	E9	E10	E11	E12	E13	E14	E15
N1	0.0311	0.0009	0.0252	0.0276	0.9783	1.8105	2.5190	5.7332	1.7232	0.4713	0.6861	7.6912	0.0032	−0.3069	0.1425
N2	0.0151	0.0002	0.0115	0.0192	0.9877	1.2899	1.1481	3.2831	1.2221	0.1102	0.3321	4.4434	0.0038	−0.0596	0.1102
N3	0.0016	2.5 × 10−6	0.0011	0.0223	0.9831	0.4303	0.1130	3.4503	1.3484	0.0011	0.0352	4.3821	−7.5 × 10−5	0.0331	0.1228
N4	0.0311	0.0010	0.0234	0.0266	0.9760	−0.7847	2.3361	4.1774	1.6791	0.4704	0.6863	5.3311	−0.0036	0.2726	0.1576
N5	0.0303	0.0009	0.02237	0.0259	0.9772	−1.1616	2.2373	3.9061	1.6082	0.4462	0.6682	4.8567	−0.0050	0.1839	0.1544
N6	0.0372	0.0014	0.0274	0.0456	0.9737	5.9091	2.7393	11.014	1.5871	0.6741	0.8210	9.8643	0.0088	−0.1576	0.1593
N7	0.0010	1.1 × 10−6	0.0008	0.0010	0.9268	−0.1359	0.0780	0.1537	2.9490	0.0005	0.0226	14.562	−0.0007	0.0015	0.2369
N8	0.0212	0.0005	0.0160	0.0156	0.9812	0.0288	1.5999	2.3942	1.6683	0.2188	0.4687	7.6121	0.0011	−0.4101	0.1405
N9	0.0498	0.0025	0.0379	0.0552	0.9742	15.8971	3.7911	26.414	1.9765	1.2040	1.0973	10.243	−0.0094	0.2647	0.1663
N10	0.0006	3.8 × 10−7	0.0004	0.0006	0.9738	−0.0293	0.0450	0.0898	1.7092	0.0002	0.0142	8.7222	−0.0001	0.0026	0.1595

.

From Table 7, it is observed that the N10-based model generated minimum values for error metrics E1, E2, E3, E4, E7, E8, E10, and E11. N2 gives optimal E5, E9, and E15. N3, N5, N8 and N9 give minimum E12, E6, E14, and E13, respectively.

The minimum error values for each normalization method in the testing of LMCMA model using the Binance dataset are shown in

Table 8. Error values in testing Legendre network for Binance dataset.

	E1	E2	E3	E4	E5	E6	E7	E8	E9	E10	E11	E12	E13	E14	E15
N1	0.0316	0.0010	0.0237	0.0278	0.9774	−1.2234	2.3736	4.5359	1.6217	0.4863	0.6974	6.6900	−0.0074	0.1353	0.1493
N2	0.0166	0.0003	0.0126	0.0211	0.9849	1.2521	1.2594	3.7527	1.3381	0.1348	0.3672	4.8957	0.0024	−0.1113	0.1190
N3	0.0017	3.1 × 10−6	0.0013	0.0249	0.9791	1.3151	0.1326	4.1768	1.5723	0.0014	0.0386	5.6469	0.0002	−0.0103	0.1397
N4	0.0347	0.0012	0.0260	0.0294	0.9700	−2.5919	2.6045	4.5319	1.8684	0.5870	0.7661	6.8161	−0.0137	0.0875	0.1721
N5	0.0341	0.0011	0.0257	0.0292	0.9709	−0.6340	2.5761	4.6825	1.8480	0.5683	0.7538	6.6877	−0.0034	0.3341	0.1713
N6	0.0331	0.0011	0.0218	0.0408	0.9791	2.8622	2.1801	7.4688	1.2608	0.5348	0.7313	4.2826	0.0057	−0.1918	0.1475
N7	0.0009	7.5 × 10−7	0.0006	0.0009	0.9474	−0.0445	0.0632	0.1251	2.3987	0.0003	0.0191	8.5862	−0.0002	0.0033	0.2221
N8	0.0211	0.0004	0.0161	0.0154	0.9812	−0.7394	1.6172	2.4077	1.6826	0.2178	0.4667	5.2296	−0.0036	0.1215	0.1405
N9	0.0503	0.0025	0.0385	0.0555	0.9735	8.6668	3.8503	25.056	2.0029	1.2316	1.1097	7.2794	−0.0129	0.1964	0.1665
N10	0.0006	4.2 × 10−7	0.0004	0.0006	0.9711	−0.0461	0.0472	0.0936	1.7937	0.0002	0.0142	6.6701	−0.0002	0.0018	0.1652

.

From Table 8, it is observed that the N10 based model generated minimum values for error metrics E1, E2, E3, E4, E7, E8, E10 and E11. N2 gives optimal values for E5 and E15. N4 gives minimum values for E6 and E13. N6 gives minimum values for E9, E12, and E14.

The minimum error values for each normalization method in the testing of LMCMA model using the Coinbase Pro dataset are shown in

Table 9. Error values in testing Legendre network for Coinbase Pro dataset.

	E1	E2	E3	E4	E5	E6	E7	E8	E9	E10	E11	E12	E13	E14	E15
N1	0.0374	0.0014	0.0254	0.0472	0.9457	6.3794	2.5391	8.7106	2.2163	0.6793	0.8242	9.3933	0.0165	−0.0847	0.2149
N2	0.0232	0.0005	0.0145	0.0397	0.9470	4.1691	1.4473	5.6788	2.0143	0.2606	0.5105	7.6474	0.0104	−0.0513	0.2245
N3	0.0023	5.3 × 10−6	0.0012	0.0466	0.9234	−2.3142	0.1179	4.8308	1.9779	0.0026	0.0509	5.7636	−0.0004	0.0122	0.2728
N4	0.0305	0.0009	0.0172	0.0348	0.9581	2.3601	1.7266	4.5906	1.6233	0.4516	0.6721	5.8495	0.0065	−0.1421	0.1922
N5	0.0337	0.0011	0.0221	0.0387	0.9488	4.3509	2.2102	6.2972	2.0780	0.5517	0.7428	8.6793	0.0131	−0.0869	0.2082
N6	0.0297	0.0008	0.0150	0.0631	0.9555	5.6726	1.5014	9.0864	1.5080	0.4301	0.6558	5.950	0.0077	−0.1135	0.2085
N7	0.0009	8.9 × 10−7	0.0004	0.0009	0.9012	−0.0389	0.0475	0.0946	2.2177	0.0004	0.0207	6.4549	−0.0002	0.0045	0.3061
N8	0.0287	0.0008	0.0169	0.0271	0.9582	1.6756	1.6923	3.4928	1.7587	0.4016	0.6337	6.8786	0.0065	−0.1262	0.1926
N9	0.0682	0.0046	0.0474	0.1155	0.9412	−17.713	4.7418	24.285	2.4638	2.2602	1.5034	10.814	0.0336	−0.1383	0.2204
N10	0.0008	7.6 × 10−7	0.0005	0.0008	0.9145	−0.0273	0.0551	0.1098	2.575	0.0003	0.0193	8.8424	−0.0001	0.0055	0.2519

.

From Table 9, it is observed that the N10-based model generated minimum values for error metrics E1, E2, E4, E10, and E11. N3 gives minimum E12 and E13. N4 gives minimum E14 and E15. N6 gives minimum values for E9. N7 gives minimum values for E3, E7, and E8. N8 and N9 give optimal values for E5 and E6, respectively.

As it is visualized that none of the normalizations performs best for all error metrics, a TOPSIS-based ranking is conducted to rank the 10 normalization types during the testing of the LMCMA model. The ranking for each of the datasets under study is shown in

Table 10. Ranking normalization in Legendre network for Bitfinex dataset.

Models	Relative Closeness	Ranks
N1	0.6866	8
N2	0.8294	4
N3	0.9073	3
N4	0.6952	7
N5	0.7120	6
N6	0.5809	9
N7	0.9179	2
N8	0.7609	5
N9	0.4622	10
N10	0.9677	1

,

Table 11. Ranking normalization in Legendre network for Binance dataset.

Models	Relative Closeness	Ranks
N1	0.6640	6
N2	0.7959	4
N3	0.8704	3
N4	0.6399	7
N5	0.5949	9
N6	0.6198	8
N7	0.8996	2
N8	0.7688	5
N9	0.3975	10
N10	0.9492	1

and

Table 12. Ranking normalization in Legendre network for Coinbase Pro dataset.

Models	Relative Closeness	Ranks
N1	0.6409	9
N2	0.7720	4
N3	0.8799	3
N4	0.7367	6
N5	0.6946	8
N6	0.7102	7
N7	0.9493	2
N8	0.7559	5
N9	0.4695	10
N10	0.9552	1

.

The ranking in all the three datasets indicates that normalization N10, N7, and N3 are the top three scalers in each of the datasets for the LMCMA model. To check the impact of our proposed normalization N10 on other networks, the original min–max normalization of LMCMA (N1) along with the top three ranked normalization (N10, N7, and N3) are used to test three other FLANNs.

The testing of four networks: Legendre FLANN (Network 1), Chebyshev FLANN (Network 2), Laguerre FLANN (Network 3), and trigonometric FLANN (Network 4) using four normalization types (N1, N3, N7, and N10) generate 16 MCMA-based Bitcoin predictor models: L1, L3, L7, L10, C1, C3, C7, C10, La1, La3, La7, La10, T1, T3, T7, and T10.

The value of 15 error metrics for 16 models under testing for the Bitfinex dataset is shown in

Table 13. Error values in testing 16 predictor models for Bitfinex dataset.

	E1	E2	E3	E4	E5	E6	E7	E8	E9	E10	E11	E12	E13	E14	E15
L1	0.03223	0.0012	0.0245	0.0282	0.9771	−1.8613	2.3648	4.3754	1.6172	0.5002	0.7076	5.5932	−0.0081	0.1294	0.1552
L3	0.0022	3.3 × 10−6	0.0013	0.0256	0.9779	0.3399	0.1322	4.0141	1.5714	0.0016	0.0398	6.5965	−9.6 × 10−7	3.3983	0.1434
L7	0.0012	1.2 × 10−6	0.0008	0.0011	0.9171	−0.0719	0.0818	0.1620	3.1089	0.0006	0.0241	18.5957	−0.0004	0.0033	0.2793
L10	0.0011	3.3 × 10−7	0.0004	0.0006	0.9774	−0.0349	0.0412	0.0814	1.5630	0.0002	0.0125	5.5168	−0.0002	0.0018	0.1433
C1	0.0331	0.0011	0.0251	0.0292	0.9754	−0.5367	2.5076	4.9508	1.7149	0.5339	0.7307	7.3606	−0.0056	0.1975	0.1541
C3	0.0021	4.1 × 10−6	0.0015	0.0286	0.9723	−0.6145	0.1479	4.4659	1.7579	0.0019	0.0446	7.3825	−0.0002	0.0217	0.1635
C7	0.0008	6.0 × 10−7	0.0006	0.0008	0.9585	−0.0697	0.0567	0.1121	2.1519	0.0003	0.0171	8.0139	−0.0004	0.0017	0.1972
C10	0.0006	3.4 × 10−7	0.0004	0.0006	0.9766	−0.0616	0.0437	0.0865	1.6602	0.0002	0.0128	5.5037	−0.0003	0.0011	0.1498
La1	0.0482	0.0023	0.0386	0.0427	0.9479	−5.3082	3.8589	8.2493	2.6389	1.1281	1.0621	12.8545	−0.0107	0.2174	0.2712
La3	0.0029	8.8 × 10−6	0.0023	0.0432	0.9407	4.8669	0.2269	6.9132	2.6963	0.0043	0.0652	14.4414	0.0018	−0.0048	0.2614
La7	0.0015	2.2 × 10−6	0.0011	0.0015	0.8478	0.0528	0.1090	0.2159	4.1412	0.0011	0.0327	30.6548	0.0003	−0.0081	0.4155
La10	0.0012	1.3 × 10−6	0.0009	0.0011	0.9085	0.1606	0.0928	0.1838	3.5245	0.0006	0.0254	18.6284	0.0008	−0.0016	0.3219
T1	0.0371	0.0013	0.0289	0.0327	0.9692	−3.7812	2.8949	5.7768	1.9797	0.6674	0.8169	8.1391	−0.0108	0.1268	0.1914
T3	0.0015	2.3 × 10−6	0.0011	0.0215	0.9845	−0.3987	0.1094	3.2920	1.2994	0.0011	0.0334	4.9647	−6.0 × 10−5	0.0382	0.1253
T7	0.0007	5.5 × 10−7	0.0005	0.0007	0.9619	−0.0045	0.0546	0.1081	2.0733	0.0003	0.0164	11.7515	−2.2 × 10−5	0.0253	0.1997
T10	0.0006	4.1 × 10−7	0.0005	0.0006	0.9714	0.0096	0.0471	0.0932	1.7875	0.0002	0.0142	10.4295	5.0 × 10−5	−0.0082	0.1758

.

Table 13 indicates that L10 minimizes E2, E3, E4, E7, E8, E10, and E11. L3 minimizes E13. C10 minimizes E1, E3, E4, and E10. La1 minimizes E6. T3 optimizes E5, E9, E12, and E15. T10 minimizes E1, E4, E10, and E14.

The value of 15 error metrics for 16 models under testing for the Binance dataset is shown in

Table 14. Error values in testing 16 predictor models for Binance dataset.

	E1	E2	E3	E4	E5	E6	E7	E8	E9	E10	E11	E12	E13	E14	E15
L1	0.0321	0.0010	0.0235	0.0284	0.9767	−1.0431	2.3513	4.4178	1.6065	0.5026	0.7090	5.7412	−0.0041	0.2516	0.1561
L3	0.0019	3.7 × 10−6	0.0015	0.0273	0.9752	2.2238	0.1470	4.6759	1.7426	0.0017	0.0422	7.0101	0.0006	−0.0061	0.1535
L7	0.0010	1 × 10−6	0.0008	0.0010	0.9278	−0.1170	0.0762	0.1508	2.8912	0.0005	0.0225	11.4573	−0.0006	0.0018	0.2499
L10	0.0007	5.3 × 10−7	0.0005	0.0007	0.9636	−0.0381	0.0536	0.1060	2.0319	0.0003	0.0159	7.7659	−0.0002	0.0027	0.1852
C1	0.0384	0.0015	0.0287	0.0339	0.9668	−0.3224	2.8653	5.7375	1.9577	0.7178	0.8472	7.9581	−0.0036	0.4078	0.1802
C3	0.0018	3.2 × 10−6	0.0013	0.0255	0.9781	0.1436	0.1314	3.9789	1.5578	0.0016	0.0397	5.2927	6.9 × 10−6	−0.0466	0.1471
C7	0.0005	3.4 × 10−7	0.0004	0.0006	0.9768	−0.0598	0.0431	0.0852	1.6333	0.0001	0.0128	5.9478	−0.0003	0.0011	0.1483
C10	0.0007	5.2 × 10−7	0.0005	0.0007	0.9642	−0.0167	0.0529	0.1046	2.0053	0.0002	0.0159	7.7449	−8.4 × 10−6	0.0062	0.1899
La1	0.0497	0.0024	0.0372	0.0444	0.9444	−3.4176	3.7193	7.5597	2.5411	1.2021	1.0963	10.0492	−0.0018	1.3895	0.2840
La3	0.0029	8.4 × 10−6	0.0021	0.0415	0.9429	0.3779	0.2140	6.4894	2.5372	0.0041	0.0639	11.7489	0.0005	−0.0183	0.2523
La7	0.0013	1.8 × 10−6	0.0010	0.0013	0.8747	0.1775	0.1029	0.2037	3.9032	0.0009	0.0296	16.5124	0.0009	−0.0020	0.4111
La10	0.0011	1.2 × 10−6	0.0008	0.0011	0.9175	−0.0655	0.0817	0.1618	3.0991	0.0006	0.0241	16.1539	−0.0003	0.0036	0.2896
T1	0.0438	0.0019	0.0341	0.0388	0.9569	−3.9265	3.4064	6.9419	2.3273	0.9303	0.9645	8.7909	−0.0072	0.2672	0.2388
T3	0.0022	4.9 × 10−6	0.0016	0.0313	0.9671	−0.7967	0.1619	4.9266	1.9186	0.0024	0.0486	7.3737	−5.5 × 10−5	0.0879	0.1861
T7	0.0007	4.9 × 10−7	0.0005	0.0007	0.9655	0.0872	0.0524	0.1037	1.9889	0.0002	0.0155	7.7186	0.0004	−0.0011	0.2029
T10	0.0005	3.2 × 10−7	0.0004	0.0005	0.9773	0.0045	0.0421	0.0833	1.5958	0.0001	0.0126	5.1768	2.4 × 10−5	−0.0135	0.1558

.

Table 14 indicates that C3 optimizes E5, E9, E14, and E15. C7 minimizes E3 and E10. T1 minimizes E6 and E13. T10 minimizes E1, E2, E3, E4, E7, E8, E10, E11, and E12.

The value of 15 error metrics for 16 models under testing for the Coinbase Pro dataset is shown in Table 14.

Table 15. Error values in testing 16 predictor models for Coinbase Pro dataset.

	E1	E2	E3	E4	E5	E6	E7	E8	E9	E10	E11	E12	E13	E14	E15
L1	0.0472	0.0022	0.0372	0.0600	0.9133	10.8897	3.7237	13.8801	3.2502	1.0845	1.0414	15.6050	0.0270	−0.0825	0.2538
L3	0.0022	4.8 × 10−6	0.0011	0.0454	0.9307	3.3401	0.1190	5.2996	1.9961	0.0023	0.0484	7.3677	0.0008	−0.0062	0.2646
L7	0.0007	4.7 × 10−7	0.0004	0.0007	0.9474	0.0463	0.0391	0.0782	1.8262	0.0002	0.0152	6.7215	0.0002	−0.0020	0.2209
L10	0.0007	5.1 × 10−7	0.0004	0.0007	0.9436	−0.0533	0.0396	0.0791	1.8519	0.0002	0.0157	6.0692	−0.0003	0.0019	0.2295
C1	0.0399	0.0016	0.0274	0.0504	0.9381	6.6542	2.7404	9.3251	2.3919	0.7739	0.8797	10.8759	0.0169	−0.0944	0.2274
C3	0.0017	3.2 × 10−6	0.0008	0.0365	0.9543	1.4778	0.0837	3.5740	1.4051	0.0015	0.0393	4.0326	0.0004	−0.0084	0.2157
C7	0.0009	7.5 × 10−7	0.0005	0.0009	0.9162	−0.0603	0.0498	0.0989	2.3229	0.0003	0.0191	7.3224	−0.0003	0.0025	0.2603
C10	0.0006	4.1 × 10−7	0.0004	0.0006	0.9537	−0.0591	0.0378	0.0754	1.7666	0.0002	0.0142	6.0341	−0.0003	0.0014	0.2049
La1	0.0419	0.0018	0.0315	0.0509	0.9319	−9.0786	3.1456	9.5432	2.7457	0.8518	0.9229	12.8032	−0.0262	0.0671	0.2667
La3	0.0028	7.7 × 10−6	0.0013	0.0574	0.8889	1.3329	0.1334	5.6234	2.2367	0.0038	0.0614	8.0032	0.0005	−0.0148	0.3528
La7	0.0008	6.8 × 10−7	0.0004	0.0008	0.9244	0.0185	0.0404	0.0806	1.8862	0.0003	0.0182	6.4136	9.5 × 10−5	−0.0072	0.3010
La10	0.0013	1.8 × 10−6	0.0008	0.0013	0.7978	0.0057	0.0779	0.1554	3.6347	0.0009	0.0297	15.3198	3.3 × 10−5	−0.0546	0.5625
T1	0.0388	0.0015	0.0258	0.0473	0.9413	−6.4802	2.5843	7.3223	2.2557	0.7349	0.8573	8.6843	−0.0207	0.0729	0.2364
T3	0.0019	3.4 × 10−6	0.0008	0.0380	0.9499	−0.7306	0.0849	3.5417	1.4234	0.0017	0.0412	3.3908	−5.7 × 10−5	0.0618	0.2286
T7	0.0010	9.3 × 10−7	0.0005	0.0010	0.8970	−0.0219	0.0494	0.0985	2.3046	0.0005	0.0212	7.3653	−0.0001	0.0086	0.3406
T10	0.0007	5.4 × 10−7	0.0004	0.0007	0.9398	0.0363	0.0406	0.0809	1.8955	0.0003	0.0162	6.6538	0.0002	−0.0029	0.2846

indicates that L3 and L7 minimize E3 and E10. C1 minimizes E14. C3 optimizes E5 and E9. C10 minimizes E1, E3, E4, E7, E8, E10, E11, and E15. La1 minimizes E13. La7 minimizes E3. T1, T3, T7, and T10 minimize E6, E12, E2, and E3, respectively.

As none of the models under study dominated the performance, a TOPSIS-based ranking is executed for the three datasets. The ranking for the three datasets is shown in

Table 16. Ranking models for Bitfinex dataset.

Models	Relative Closeness	Ranks
L1	0.6316	13
L3	0.6481	12
L7	0.8579	7
L10	0.9839	1
C1	0.6103	14
C3	0.8364	9
C7	0.9436	4
C10	0.9780	2
La1	0.5001	16
La3	0.6846	11
La7	0.7669	10
La10	0.8397	8
T1	0.5815	15
T3	0.8776	6
T7	0.9279	5
T10	0.9458	3

,

Table 17. Ranking models for Binance dataset.

Models	Relative Closeness	Ranks
L1	0.6392	13
L3	0.7818	11
L7	0.8947	6
L10	0.9571	4
C1	0.5634	14
C3	0.8567	7
C7	0.9811	2
C10	0.9577	3
La1	0.4445	16
La3	0.7611	12
La7	0.8073	10
La10	0.8535	8
T1	0.5569	15
T3	0.8184	9
T7	0.9505	5
T10	0.9936	1

and

Table 18. Ranking models for Coinbase Pro dataset.

Models	Relative Closeness	Ranks
L1	0.3433	16
L3	0.7919	10
L7	0.9486	3
L10	0.9552	2
C1	0.4558	15
C3	0.8535	8
C7	0.9254	6
C10	0.9591	1
La1	0.5122	14
La3	0.7706	11
La7	0.9343	5
La10	0.7691	12
T1	0.5366	13
T3	0.7996	9
T7	0.9087	7
T10	0.9384	4

.

Table 16 suggests that L10, C10, and T10 were the top three models in the Bitfinex dataset. The Binance dataset is best predicted by T10, C7, and C10 models as inferred from Table 17. Table 18 indicates that C10, L10, and L7 are the best predictor models for the Coinbase Pro dataset. The top three predictor models in each of the three datasets under study are represented in

Table 19. Top three predictor models in each dataset.

Rank of Models	Bitfinex	Binance	Coinbase Pro
1	L10	T10	C10
2	C10	C7	L10
3	T10	C10	L7

.

Table 19 indicates that the proposed dual normalization (N10) enhances the prediction capability of the FLANN networks under study.

To present the mapping capability of the proposed normalization, the training, testing, and actual output of the first ranked model for each of the three datasets is shown in Figure 6, Figure 7 and Figure 8.

The mapping plots in Figure 6, Figure 7 and Figure 8 indicate that the dual normalization improves the prediction capability of polynomial-based neural network models. This validates the second hypothesis of this research.

The convergence RMSE plot against the number of iterations for the training of the LMCMA model using N1, N3, N7, and the proposed N10 dataset for each of the three datasets is shown in Figure 9, Figure 10 and Figure 11.

The convergence plots in Figure 9, Figure 10 and Figure 11 clearly show that the proposed dual normalization minimizes the RMSE faster in comparison to other normalization under study. This also validates the second hypothesis of this study.

The RMSE gain percentage of the proposed normalization in comparison to the original min–max normalization used in the training of the LMCMA model is pictorially presented in Figure 12.

It is visualized in Figure 12 that the proposed dual normalization (N10) gives 97.8%, 97.49%, and 97.16% RMSE gain over the original min–max normalization (N1). It also shows that N10 predicts better than the basic N3 and N7 normalization used to generate the N10 technique.

The performance of the LMCMA model using the proposed dual normalization (L10) was further compared with two other models. The econometric Linear Regression (LR) model and the Support Vector Regression (SVR) model were used to predict the closing Bitcoin prices on the three datasets under study. The error measures for test data are presented in

Table 20. Performance comparison with different econometric models.

	E1	E2	E3	E4	E5	E6	E7	E8	E9	E10	E11	E12	E13	E14	E15
Bitfinex
LR	0.0011	6.8 × 10−7	0.0006	0.0008	0.9528	−0.007	0.0607	0.1203	2.31	0.0003	0.0181	14.95	−3.3 × 10−5	0.02	0.221
SVR	0.0009	9.1 × 10−7	0.0007	0.0009	0.9369	0.085	0.0714	0.1414	2.71	0.0004	0.0210	16.53	0.0004	−0.002	0.271
L10	0.0008	3.3 × 10−7	0.0004	0.0006	0.9774	−0.035	0.0412	0.0814	1.56	0.0002	0.0125	5.52	−0.0002	0.001	0.143
Binance
LR	0.0008	6.8 × 10−7	0.0006	0.0008	0.9527	−0.007	0.0608	0.1205	2.31	0.0003	0.0182	10.08	−3.3 × 10−5	0.0204	0.222
SVR	0.0009	8.8 × 10−7	0.0007	0.0009	0.9383	0.081	0.0705	0.1397	2.67	0.0004	0.0207	11.16	0.0004	−0.002	0.266
L10	0.0007	5.3 × 10−7	0.0005	0.0007	0.9636	−0.038	0.0536	0.1060	2.03	0.0003	0.0159	7.76	−0.0002	0.0027	0.185
Coinbase Pro
LR	0.0012	1.4 × 10−6	0.0006	0.0012	0.8504	−0.004	0.056	0.1118	2.61	0.0007	0.0255	8.35	−1.8 × 10−5	0.0723	0.399
SVR	0.0011	1.3 × 10−6	0.0006	0.0012	0.8512	−0.015	0.0607	0.1211	2.83	0.0006	0.0255	10.37	−7.1 × 10−5	0.0186	0.429
L10	0.0007	5.1 × 10−7	0.0004	0.0007	0.9436	−0.053	0.0396	0.0791	1.85	0.0002	0.0157	6.07	−0.0003	0.0019	0.229

.

As observed in Table 20, the L10 model performs better than the LR and the SVR model in terms of all the error measures. The RMSE of each of the three models for the three datasets under study is shown in Figure 13.

As observed from Figure 13, the LMCMA model using the proposed dual normalization (N10) outperformed the LR and SVR models.

The Bitcoin closing prices are highly volatile, and as logarithmic distribution is more normal as compared to raw prices, this study further analyzed the performance of the proposed dual normalization on Bitcoin log returns for the three datasets under study. The error measures for each dataset are presented in

Table 21. Error measures in predicting test log returns for the three datasets.

	E1	E2	E3	E4	E5	E6	E7	E8	E9	E10	E11	E12	E13	E14	E15
D1	0.0051	2.6 × 10−5	0.0038	0.0051	−0.279	−0.046	0.379	0.759	0.796	0.013	0.1119	2.6264	−0.0002	0.1358	2.1313
D2	0.0053	2.8 × 10−5	0.004	0.0053	−0.388	−0.041	0.401	0.803	0.842	0.014	0.1165	4.4413	−0.0001	0.1684	1.8386
D3	0.0052	2.7 × 10−5	0.0024	0.0052	−0.352	−0.093	0.249	0.495	0.845	0.012	0.1139	4.4612	−0.0004	0.0621	1.9166

D1: Bitfinex dataset, D2: Binance dataset, D3: Coinbase Pro dataset.

.

As indicated by Table 21, the LMCMA model using the new normalization technique gives an RMSE of 0.0051, 0.0053, and 0.0052 for the Bitfinex, Binance, and Coinbase Pro datasets, respectively. In addition, the model performs well in all three datasets. The actual Bitcoin log returns plotted against the predicted train and test log returns for all the three datasets are shown in Figure 14, Figure 15 and Figure 16.

6. Discussion

Recently, Bitcoin is assumed to be a safe-haven asset by some investors. However, the dynamic and volatile behavior of the Bitcoin dataset is the biggest challenge to the growth of the Bitcoin market. A Bitcoin predictor model with sufficient stability can act as fuel in increasing investors and stakeholders in the digital market. The emergence of Bitcoin Cash has already created a positive spur in the digitization of companies [60]. Many research activities are involved in analyzing the Bitcoin dataset to suggest a better model of regression or classification.

In this study, the Bitcoin prices of three different markets were normalized using different normalization types and were analyzed in the LMCMA model to obtain the top normalization for pricing. A dual normalization was proposed using the top two normalizations during the training of the LMCMA model. The stability of the normalization technique proposed here is validated by testing its efficiency in a different test dataset. Additionally, the top performers were evaluated in three other FLANN models. As the performance evaluation of these 16 models is not decisive, this study utilized the TOPSIS-based performance measurement with 15 errors. The proposed normalization bagged the first rank in each of the datasets.

As the datasets were divided into training and testing datasets which have no common data between them, the efficient prediction of the proposed normalization during testing and training of the FLANN models suggest the stable feature of this normalization. It can be visualized in Table 14, Table 15 and Table 16 that the proposed normalization (N10) performs best in each network separately for all the three datasets.

In the Bitfinex dataset, Legendre-based models L1, L3, L7, and L10 were ranked 13, 12, 7, and 1, respectively. Similarly, Chebyshev models C1, C3, C7, and C10 were ranked 14, 9, 4, and 2, respectively. Laguerre models La1, La3, La7, and La10 were ranked 16, 11, 10, and 8, respectively. Trigonometric models T1, T3, T7, and T10 were ranked 15, 6, 5, and 3, respectively. It indicates that N10 outperformed the other normalization techniques in each of the networks. The similar inference is also validated for Binance and Coinbase Pro datasets. This also validates the stability of the proposed normalization.

As inferred from Table 19, although N10 is the best normalization in each network, still, the overall model for best prediction is different for different datasets. So, the network architecture also plays a vital role in the modeling of predictors.

The output plots for the first ranked model in each dataset clearly show the efficient mapping of dual normalization in Bitcoin price prediction. Furthermore, the RMSE gain percent of N10 during training is 97.8%, 97.49%, and 97.16% for Bitfinex, Binance, and Coinbase Pro datasets, respectively. The RMSE gain percentage for testing is 81.37%, 98.13%, and 98.73% for Bitfinex, Binance, and Coinbase Pro datasets, respectively.

The novelty of the proposed dual normalization is compared with some hybrid normalization available in the literature. It is shown in

Table 22. Comparison of proposed normalization with available hybrid normalization.

Reference #	Proposed Hybrid Normalization	Area of Application	Evaluation Criteria	Presence of De-Normalization Methods
41	Linear max and Linear sum	Selection of engineering design materials	Elastic modulus, density and cost	No
43	Log (average of min–max, z-score and robust scaler)	Stock price movement prediction	Accuracy, Precision, F1-score and Recall	No
44	MaxAbs scaler and vector scaler	Medical diagnosis	Accuracy, Precision, F1-score and Recall	No
Proposed dual normalization	Tanh estimator with vector normalization	Bitcoin daily closing price prediction	TOPSIS-based evaluation using 15 error metrics	Yes

.

Table 22 suggests that our proposed dual normalization is novel and efficient work in the field of Bitcoin price prediction. It is more reliable, as it is evaluated using 15 error metrics. This normalization is also stable as data can be denormalized to actual ones for better mapping.

This study also compared the linear regression econometric model with the proposed model. The nonlinear regression SVR is also analyzed and compared to the L10 model. The results indicate that the proposed model is efficient for Bitcoin price prediction. Similar results were also observed in predicting the Bitcoin daily log returns. The highly volatile feature of Bitcoin price is also mapped efficiently by the predictor model L10.

7. Conclusions

Recently, Bitcoin has been dominating the crypto market. This paper proposes a new dual normalization technique to efficiently predict Bitcoin closing prices by handling the huge variations in the dataset. This is the first paper to study the impact of data scaling on Bitcoin prices. To validate the proposed normalization, it was compared with nine different normalizations applied to three different datasets for modeling four different FLANNs. Furthermore, the predicted and actual outputs were plotted. The mapping of outputs visualizes the efficiency of the proposed method. To further validate the dataset, distribution plots are used. The use of this normalization enhances the prediction capability of the LMCMA model by giving an RMSE gain percent of more than 90 over min–max normalization in the Bitfinex, Binance, and Coinbase Pro datasets.

This research can help many investors and stakeholders in the crypto-market to make appropriate decisions on Bitcoin pricing. In addition, the proposed normalization indicates that instead of having a fixed normalization for research, different normalizations can be analyzed to select the best normalization for a particular network. This can give research directions to any area of research which requires normalization of its input.

During COVID-19, the huge rise in Bitcoin prices prompted many investors to invest in the crypto-market. However, after the pandemic, the decline in Bitcoin prices left many investors in dismay. The Bitcoin price is highly volatile and is in its initial phase. So, many econometric models need to be evaluated along with the FLANN-based model, which provides higher accuracy. However, this study reveals that an appropriate normalization can help in better scaling of such huge volatility in Bitcoin prices. This in turn can help in positive decisions with profitable investment in the crypto-market.

Although this study presented a novel idea to analyze normalization before any regression or classification, it has a few shortcomings which can be addressed in the future. The new normalization has only been tested on FLANN models, a linear regression model, and a nonlinear SVR model. It can be further tested for different other networks and econometric models. Additionally, this normalization can be tested for several other datasets to explore its capability in different areas of research. The models can be fine-tuned further by using several other machine-learning algorithms. A close look at the normalized data analysis points out that the range of transformation may play a vital role in the efficiency of a model. This needs to be further analyzed. This model analyzes the prediction of the closing prices, which is not sufficient to address the high volatility. In addition, the log returns prediction needs to be analyzed for different other networks for a better prediction It can be further used to normalize technical indicators and combine them with normalization types. As Bitcoin is highly volatile, its price movement prediction needs to be explored. The effect of different other preprocessing techniques such as window size and the ratio of splitting the dataset can be analyzed in the future. Furthermore, this research can be extended by designing an econometric model using various technical, social, and economic indicators which may influence the Bitcoin price volatility.