# Deep Learning Methods for Modeling Bitcoin Price

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## Abstract

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## 1. Introduction

## 2. Deep Learning Methods

#### 2.1. Deep Recurrent Convolution Neural Network (DRCNN)

_{xs}, W

_{ss}, and W

_{so}define the weights from the input layer x to the hidden layer s, by the biases of the hidden layer and output layer. Equation (3) points out σ and o as the activation functions.

_{h}is the weight and b

_{h}is the bias. The model calculates the residuals caused by the difference between the predicted and the actual observations in the training stage [20]. Stochastic gradient descent is applied for optimization to learn the parameters. Considering that the data at time t is r, the loss function is determined as shown in Equation (7).

#### 2.2. Deep Neural Decision Trees (DNDT)

_{d}with its NN f

_{d}(x

_{d}), we can determine all the final nodes of the DT as appears in Equation (10).

#### 2.3. Deep Learning Linear Support Vector Machines (DSVR)

_{i}, where i = 1, ..., 10; p

_{i}specifies a discrete probability distribution, ${\sum}_{i}^{10}{p}_{i}=1$.

#### 2.4. Sensitivity Analysis

_{ij}being the effect of interaction between two variables. The Sobol decomposition allows the estimation of a total sensitivity index, STi, which measures the sum of all the sensitivity effects involved in the independent variables.

## 3. Data and Variables

## 4. Results

#### 4.1. Descriptive Statistics

#### 4.2. Empirical Results

#### 4.3. Post-Estimations

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Variables | Description |
---|---|

(a) Demand and Supply | |

Transaction value | Value of daily transactions |

Number of Bitcoins | Number of mined Bitcoins currently circulating on the network |

Bitcoins addresses | Number of unique Bitcoin addresses used per day |

Transaction volume | Number of transactions per day |

Unspent transactions | Number of valid unspent transactions |

Blockchain transactions | Number of transactions on blockchain |

Blockchain addresses | Number of unique addresses used in blockchain |

Block size | Average block size expressed in megabytes |

Miners reward | Block rewards paid to miners |

Mining commissions | Average transaction fees (in USD) |

Cost per transaction | Miners’ income divided by the number of transactions |

Difficulty | Difficulty mining a new blockchain block |

Hash | Times a hash function can be calculated per second |

Halving | Process of reducing the emission rate of new units |

(b) Attractive | |

Forum posts | Number of new members in online Bitcoin forums |

Forum members | New posts in online Bitcoin forums |

(c) Macroeconomic and Financial | |

Texas oil | Oil Price (West Texas) |

Brent oil | Oil Price (Brent, London) |

Dollar exchange rate | Exchange rate between the US dollar and the euro |

Dow Jones | Dow Jones Index of the New York Stock Exchange |

Gold | Gold price in US dollars per troy ounce |

Variables | Obs | Mean | SD | Min | Max |
---|---|---|---|---|---|

Transaction value | 112 | 342,460,106,866,711.0000 | 143,084,554,727,531.0000 | 59,238,547,391,199.6000 | 735,905,260,141,564.0000 |

Number of bitcoins | 112 | 13,634,297.4824 | 3,709,010.0736 | 5,235,454.5455 | 18,311,982.5000 |

Bitcoins addresses | 112 | 285,034.2515 | 219,406.3874 | 1576.8333 | 849,668.1000 |

Transaction volume | 112 | 154,548.8041 | 117,104.3686 | 1105.5000 | 373,845.6000 |

Unspent transactions | 112 | 28,581,914.9054 | 22,987,595.3012 | 78,469.7273 | 66,688,779.9000 |

Blockchain transactions | 112 | 156,444,312.9120 | 161,252,448.1997 | 237,174.8889 | 520,792,976.5000 |

Blockchain addresses | 112 | 4,812,692.05 | 13,735,245.35 | −14,437,299.03 | 117,863,226.2 |

Block size | 112 | 0.4956 | 0.3638 | 0.0022 | 0.9875 |

Miners reward | 112 | 420,160,582,581,028.0000 | 174,396,895,338,462.0000 | 101,244,436,734,897.0000 | 796,533,076,376,536.0000 |

Mining commissions | 112 | 9,581,973,325,205.4400 | 42,699,799,790,392.8000 | 0.2591 | 315,387,506,596,395.0000 |

Cost per transaction | 112 | 155,354,364,458,705.0000 | 156,696,788,525,225.0000 | 0.1179 | 757,049,771,708,905.0000 |

Difficulty | 112 | 187,513,499,336,866.0000 | 195,421,886,528,251.0000 | 212,295,141,771.2000 | 836,728,509,520,663.0000 |

Hash | 112 | 110,434,372.2765 | 154,717,725.3881 | 0.5705 | 516,395,703.4338 |

Halving | 112 | 279,853,454,485,387.0000 | 162,806,469,642,875.0000 | 6,473,142,955,255.1700 | 804,437,327,302,638.0000 |

Forum posts | 112 | 9279.8844 | 8585.0583 | 455.0000 | 53132.0000 |

Forum members | 112 | 2432.2545 | 3394.4635 | 30.6364 | 14,833.3409 |

Texas Oil | 112 | 72.4878 | 23.7311 | 21.1230 | 135.6700 |

Brent Oil | 112 | 78.4964 | 26.5819 | 19.1900 | 139.3800 |

Dollar exchange rate | 112 | 1.3767 | 0.9604 | 1.0494 | 8.7912 |

Dow Jones | 112 | 15,926.7161 | 3324.8875 | 11,602.5212 | 22,044.8627 |

Gold | 112 | 1329.400847 | 244.4099259 | 739.15 | 1846.75 |

Sample | DRCNN | DNDT | DSVR | ||||||
---|---|---|---|---|---|---|---|---|---|

Acc. (%) | RMSE | MAPE | Acc. (%) | RMSE | MAPE | Acc. (%) | RMSE | MAPE | |

Training | 97.34 | 0.66 | 0.29 | 95.86 | 0.70 | 0.33 | 94.49 | 0.75 | 0.38 |

Validation | 96.18 | 0.71 | 0.34 | 95.07 | 0.74 | 0.37 | 93.18 | 0.81 | 0.43 |

Testing | 95.27 | 0.77 | 0.40 | 94.42 | 0.79 | 0.42 | 92.61 | 0.84 | 0.47 |

DRCNN | DNDT | DSVR |
---|---|---|

Transaction value | Transaction volume | Transaction value |

Transaction volume | Block size | Block size |

Block size | Blockchain transactions | Blockchain transactions |

Cost per transaction | Cost per transaction | Cost per transaction |

Difficulty | Difficulty | Difficulty |

Dollar exchange rate | Forum posts | Forum posts |

Dow Jones | Dow Jones | Dollar exchange rate |

Gold | Gold | Dow Jones |

Gold |

Horizon | DRCNN | DNDT | DSVR | ||||||
---|---|---|---|---|---|---|---|---|---|

Acc. (%) | RMSE | MAPE | Acc. (%) | RMSE | MAPE | Acc. (%) | RMSE | MAPE | |

t + 1 | 94.19 | 0.81 | 0.52 | 92.35 | 0.87 | 0.59 | 88.34 | 0.97 | 0.65 |

t + 2 | 91.37 | 0.92 | 0.63 | 89.41 | 1.03 | 0.67 | 85.76 | 1.10 | 0.78 |

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**MDPI and ACS Style**

Lamothe-Fernández, P.; Alaminos, D.; Lamothe-López, P.; Fernández-Gámez, M.A.
Deep Learning Methods for Modeling Bitcoin Price. *Mathematics* **2020**, *8*, 1245.
https://doi.org/10.3390/math8081245

**AMA Style**

Lamothe-Fernández P, Alaminos D, Lamothe-López P, Fernández-Gámez MA.
Deep Learning Methods for Modeling Bitcoin Price. *Mathematics*. 2020; 8(8):1245.
https://doi.org/10.3390/math8081245

**Chicago/Turabian Style**

Lamothe-Fernández, Prosper, David Alaminos, Prosper Lamothe-López, and Manuel A. Fernández-Gámez.
2020. "Deep Learning Methods for Modeling Bitcoin Price" *Mathematics* 8, no. 8: 1245.
https://doi.org/10.3390/math8081245