# Neural Networks for Financial Time Series Forecasting

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data

#### 2.2. Time Series Analysis

#### 2.3. Artificial Neural Networks

- Input: It represents a matrix array X of several data types such as image, audio signal, data frame etc. These input values are then processed as a linear combination of the latter with a weight matrix W and a bias vector. The weight is the additional parameter of a neural network that transforms the input data in the hidden layers of the network. It decides the speed at which the activation function will fire. The bias is a parameter of the neural network that helps the model to best fit the given data.
- Activation function: The process input is fed into an activation function defined as $\sigma (WX+b)$ which triggers a signal whenever a condition is satisfied. The four common types of activation functions are: the Rectified Linear Unit (ReLU), Sigmoid, Tangent hyperbolic (tanh), and Softmax. They are defined, respectively, in Equations (1)–(4).$$\begin{array}{ccc}\hfill ReLU\left(k\right)& =& max\left(0,k\right)\hfill \end{array}$$$$\begin{array}{ccc}\hfill Sigmoid\left(k\right)& =& \frac{1}{1+{e}^{-k}}\hfill \end{array}$$$$\begin{array}{ccc}\hfill tanh\left(k\right)& =& {\displaystyle \frac{{e}^{k}-{e}^{-k}}{{e}^{k}+{e}^{-k}}}\hfill \end{array}$$$$\begin{array}{ccc}\hfill Softma{x}_{i}\left(k\right)& =& {\displaystyle \frac{{e}^{{k}_{i}}}{{\sum}_{j}{e}^{{k}_{j}}}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall i=1,2,\cdots ,n.\hfill \end{array}$$
- Output: It is the numerical result after fitting the input into the activation function.

#### 2.4. Recurrent Neural Networks (RNNs)

#### 2.5. Accuracy Measures

## 3. Results and Discussion

#### 3.1. Preliminary Analysis

#### 3.2. Model Fit

#### 3.2.1. Data Splitting

#### 3.2.2. Features Selection

#### 3.2.3. Data Pre-Processing

#### 3.2.4. Parameters Tuning

#### 3.2.5. The Complete Algorithm and the Results for Model Fit

#### 3.3. Model Forecasting

## 4. Conclusions and Further Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Univaritate and multivariate closing price forecasting for the Shangai Stock Exchange (SSE) composite index, respectively, using simple RNN, LSTM and GRU models.

**Figure A2.**Univaritate and multivariate closing price forecasting for the EUR/USD exchange rate, respectively, using simple RNN, LSTM and GRU models.

**Table A1.**Mean absolute error for the univariate RNN, LSTM and GRU models, in sample (train and validation sets) and out-of-sample forecasting (train set) of the eight stock market indexes.

MAE | RNN | LSTM | GRU | ||||||
---|---|---|---|---|---|---|---|---|---|

Train | Valid | Test | Train | Valid | Test | Train | Valid | Test | |

IXIC | 0.0024 | 0.0038 | 0.0164 | 0.0029 | 0.0036 | 0.02569 | 0.0033 | 0.0043 | 0.0099 |

NYA | 0.0064 | 0.0063 | 0.0115 | 0.0071 | 0.0050 | 0.0104 | 0.0093 | 0.0143 | 0.0216 |

N100 | 0.0096 | 0.0072 | 0.0116 | 0.0102 | 0.0075 | 0.0106 | 0.0106 | 0.0109 | 0.0131 |

GDAXI | 0.0068 | 0.0079 | 0.0100 | 0.0081 | 0.0081 | 0.0113 | 0.0074 | 0.0095 | 0.0106 |

NGE | 0.0091 | 0.0035 | 0.0048 | 0.0100 | 0.0042 | 0.0051 | 0.0178 | 0.0091 | 0.0038 |

J580.JO | 0.0112 | 0.0138 | 0.0157 | 0.0134 | 0.0173 | 0.0214 | 0.0168 | 0.0120 | 0.0160 |

N225 | 0.0066 | 0.0066 | 0.0102 | 0.0071 | 0.0082 | 0.0101 | 0.0079 | 0.0079 | 0.0104 |

000001.SS | 0.0101 | 0.0058 | 0.0071 | 0.0097 | 0.0058 | 0.0057 | 0.0097 | 0.0189 | 0.0106 |

**Table A2.**Mean absolute error for the multivariate RNN, LSTM and GRU models, in sample (train and validation sets) and out-of-sample forecasting (train set) of the eight stock market indexes.

MAE | RNN | LSTM | GRU | ||||||
---|---|---|---|---|---|---|---|---|---|

Train | Valid | Test | Train | Valid | Test | Train | Valid | Test | |

IXIC | 0.00189 | 0.0034 | 0.0292 | 0.0025 | 0.01139 | 0.0922 | 0.0026 | 0.0074 | 0.0182 |

NYA | 0.0033 | 0.0028 | 0.0065 | 0.0032 | 0.0054 | 0.0153 | 0.0062 | 0.0080 | 0.0072 |

N100 | 0.0066 | 0.0055 | 0.0083 | 0.0054 | 0.0089 | 0.0149 | 0.0083 | 0.0113 | 0.0073 |

GDAXI | 0.0047 | 0.0090 | 0.0098 | 0.0062 | 0.0057 | 0.0106 | 0.0058 | 0.0075 | 0.0074 |

NGE | 0.0075 | 0.0036 | 0.0066 | 0.0069 | 0.0073 | 0.0066 | 0.0181 | 0.0034 | 0.0037 |

J580.JO | 0.0075 | 0.0179 | 0.0132 | 0.0050 | 0.0067 | 0.0097 | 0.0143 | 0.0050 | 0.0106 |

N225 | 0.0045 | 0.0050 | 0.0092 | 0.0041 | 0.0054 | 0.01169 | 0.0052 | 0.0052 | 0.0070 |

000001.SS | 0.0058 | 0.0055 | 0.0052 | 0.0054 | 0.0024 | 0.0037 | 0.0068 | 0.0074 | 0.0036 |

**Table A3.**Mean absolute error for the univariate RNN, LSTM and GRU models, in sample (train and validation sets) and out-of-sample forecasting (train set) of the six currency exchange rates.

MAE | RNN | LSTM | GRU | ||||||
---|---|---|---|---|---|---|---|---|---|

Train | Valid | Test | Train | Valid | Test | Train | Valid | Test | |

ZAR/USD | 0.0102 | 0.0062 | 0.0051 | 0.0115 | 0.0079 | 0.0061 | 0.0171 | 0.0077 | 0.0187 |

NGN/USD | 0.0031 | 0.0108 | 0.0092 | 0.0051 | 0.0109 | 0.0075 | 0.0074 | 0.0103 | 0.00071 |

GBP/USD | 0.0091 | 0.0105 | 0.0099 | 0.0109 | 0.0181 | 0.0167 | 0.0145 | 0.01141 | 0.0097 |

EUR/USD | 0.0143 | 0.0082 | 0.0070 | 0.0151 | 0.0087 | 0.0078 | 0.0172 | 0.0179 | 0.0103 |

RMB/USD | 0.0163 | 0.0187 | 0.0324 | 0.0161 | 0.0213 | 0.0297 | 0.0174 | 0.0173 | 0.0156 |

JPY/USD | 0.0121 | 0.0073 | 0.0053 | 0.0139 | 0.0089 | 0.0069 | 0.0174 | 0.0176 | 0.0064 |

**Table A4.**Mean absolute error for the multivariate RNN, LSTM and GRU models, in sample (train and validation sets) and out-of-sample forecasting (train set) of the six currency exchange rates.

MAE | RNN | LSTM | GRU | ||||||
---|---|---|---|---|---|---|---|---|---|

Train | Valid | Test | Train | Valid | Test | Train | Valid | Test | |

ZAR/USD | 0.0071 | 0.0060 | 0.0072 | 0.0113 | 0.0067 | 0.0054 | 0.0167 | 0.0110 | 0.0314 |

NGN/USD | 0.0087 | 0.0129 | 0.0154 | 0.0091 | 0.0381 | 0.0533 | 0.0151 | 0.0076 | 0.1532 |

GBP/USD | 0.0069 | 0.0109 | 0.0118 | 0.0069 | 0.0183 | 0.02095 | 0.0123 | 0.0063 | 0.0084 |

EUR/USD | 0.0117 | 0.0100 | 0.01029 | 0.0117 | 0.0074 | 0.0063 | 0.0164 | 0.0134 | 0.0119 |

RMB/USD | 0.0185 | 0.0143 | 0.0227 | 0.0146 | 0.0217 | 0.0436 | 0.0174 | 0.0164 | 0.0250 |

JPY/USD | 0.0166 | 0.0096 | 0.0069 | 0.0163 | 0.0118 | 0.0102 | 0.0247 | 0.0233 | 0.0129 |

## References

- Ake, B. The role of stock market development in economic growth: Evidence from some Euronext countries. Int. J. Financ. Res.
**2010**, 1, 14–20. [Google Scholar] - Sulandari, W.; Suhartono, S.; Rodrigues, P.C. Exponential Smoothing on Modeling and Forecasting Multiple Seasonal Time Series: An Overview. Fluct. Noise Lett.
**2021**, 20, 2130003. [Google Scholar] [CrossRef] - Ariyo, A.A.; Adewumi, A.O.; Ayo, C.K. Stock price prediction using the ARIMA model. In Proceedings of the 2014 UKSim-AMSS 16th International Conference on Computer Modelling and Simulation, Cambridge, UK, 26–28 March 2014; Volume 1, pp. 106–112. [Google Scholar]
- Merh, N.; Saxena, V.P.; Pardasani, K.R. A comparison between hybrid approaches of ANN and ARIMA for Indian stock trend forecasting. Bus. Intell. J.
**2010**, 3, 23–43. [Google Scholar] - Adebiyi, A.A.; Adewumi, A.O.; Ayo, C.K. Comparison of ARIMA and artificial neural networks models for stock price prediction. J. Appl. Math.
**2014**, 2014, 614342. [Google Scholar] [CrossRef][Green Version] - Sezer, O.B.; Gudelek, M.U.; Ozbayoglu, A.M. Financial time series forecasting with deep learning: A systematic literature review: 2005–2019. Appl. Soft Comput.
**2020**, 90, 106181. [Google Scholar] [CrossRef][Green Version] - Torres, D.G.; Qiu, H. Applying Recurrent Neural Networks for Multivariate Time Series Forecasting of Volatile Financial Data; KTH Royal Institute of Technology: Stockholm, Sweden, 2018. [Google Scholar]
- Siami-Namini, S.; Tavakoli, N.; Namin, A.S. A comparison of ARIMA and LSTM in forecasting time series. In Proceedings of the 2018 17th IEEE International Conference on Machine Learning and Applications (ICMLA), Orlando, FL, USA, 17–20 December 2018; pp. 1394–1401. [Google Scholar]
- Shahi, T.B.; Shrestha, A.; Neupane, A.; Guo, W. Stock Price Forecasting with Deep Learning: A Comparative Study. Mathematics
**2020**, 8, 1441. [Google Scholar] [CrossRef] - Sulandari, W.; Subanar, S.; Lee, M.H.; Rodrigues, P.C. Time series forecasting using singular spectrum analysis, fuzzy systems and neural networks. MethodsX
**2020**, 7, 101015. [Google Scholar] [CrossRef] - Sulandari, W.; Subanar, S.; Suhartono, S.; Utami, H.; Lee, M.H.; Rodrigues, P.C. SSA-based hybrid forecasting models and applications. Bull. Electr. Eng. Inform.
**2020**, 9, 2178–2188. [Google Scholar] [CrossRef] - Sulandari, W.; Lee, M.H.; Rodrigues, P.C. Indonesian electricity load forecasting using singular spectrum analysis, fuzzy systems and neural networks. Energy
**2020**, 190, 116408. [Google Scholar] [CrossRef] - Rodrigues, P.C.; Awe, O.O.; Pimentel, J.S.; Mahmoudv, R. Modelling the Behaviour of Currency Exchange Rates with Singular Spectrum Analysis and Artificial Neural Networks. Stats
**2020**, 3, 137–157. [Google Scholar] [CrossRef] - Li, A.W.; Bastos, G.S. Stock Market Forecasting Using Deep Learning and Technical Analysis: A Systematic Review. IEEE Access
**2020**, 8, 185232–185242. [Google Scholar] [CrossRef] - Kumar, G.; Jain, S.; Singh, U.P. Stock Market Forecasting Using Computational Intelligence: A Survey. Arch. Computat. Methods Eng.
**2021**, 28, 1069–1101. [Google Scholar] [CrossRef] - Basilio, M.P.; de Freitas, J.G.; Kämpffe, M.G.F.; Bordeaux Rego, R. Investment portfolio formation via multicriteria decision aid: A Brazilian stock market study. J. Model. Manag.
**2018**, 13, 394–417. [Google Scholar] [CrossRef] - Radojičić, D.; Radojičić, N.; Kredatus, S. A Multicriteria Optimization Approach for the Stock Market Feature Selection. Comput. Sci. Inf. Syst.
**2021**, 18, 749–769. [Google Scholar] [CrossRef] - Peng, H.G.; Xiao, Z.; Wang, J.Q.; Li, J. Stock Selection Multicriteria Decision-making Method Based on Elimination and Choice Translating Reality I with Z-numbers. Int. J. Intell. Syst.
**2021**, 36, 6440–6470. [Google Scholar] [CrossRef] - Wei, H.; Lai, K.K.; Nakamori, Y.; Wang, S. Forecasting foreign exchange rates with artificial neural networks: A review. Int. J. Inf. Technol. Decis. Mak.
**2004**, 3, 145–165. [Google Scholar] - Dadabada, P.; Vadlamani, R. Soft computing hybrids for FOREX rate prediction: A comprehensive review. Comput. Oper. Res.
**2018**, 99, 262–284. [Google Scholar] [CrossRef] - Ni, L.; Li, Y.; Wang, X.; Zhang, J.; Yu, J.; Qi, C. Forecasting of Forex Time Series Data Based on Deep Learning. Procedia Comput. Sci.
**2019**, 147, 647–652. [Google Scholar] [CrossRef] - Hochreiter, S.; Schmidhuber, J. Long Short-Term Memory. Neural Comput.
**1997**, 9, 1735–1780. [Google Scholar] [CrossRef] - Cho, K.; Van Merriënboer, B.; Gulcehre, C.; Bahdanau, D.; Bougares, F.; Schwenk, H.; Bengio, Y. Learning phrase representations using RNN encoder-decoder for statistical machine translations. arXiv
**2014**, arXiv:1406.1078. [Google Scholar] - Muller, F. 2020. Available online: https://www.relataly.com/stock-market-prediction-with-multivariate-time-series-in-python/1815/ (accessed on 26 April 2022).
- Nina, G.; Vladimir, N.; Zhigljavsky, A.A. Analysis of Time Series Structure: SSA and Related Techniques; CRC Press: Boca Raton, FL, USA, 2001. [Google Scholar]
- Rodrigues, P.C.; Pimentel, J.; Messala, P.; Kazemi, M. The decomposition and forecasting of mutual investment funds using singular spectrum analysis. Entropy
**2020**, 22, 83. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rodrigues, P.C.; Mahmoudvand, R. The benefits of multivariate singular spectrum analysis over the univariate version. J. Frankl. Inst.
**2018**, 355, 544–564. [Google Scholar] [CrossRef]

**Figure 1.**The input data $({x}_{1},{x}_{2},\dots {x}_{n})$ are processed as a linear combination with weights $({W}_{1},{W}_{2},\dots {W}_{n})$ and bias b. Then, the inputs are fed into an activation function in order to get the final result (predicted output).

**Figure 2.**The input layer transmits the initial data to the hidden layers through the connections (the black arrows) between them. After processing, the hidden layers transmit the result with the help of connections to the output layer.

**Figure 3.**Unlike other neural networks, the RNN uses the same parameters for the input entries because it does the same task on all inputs to produce the output. This reduces the complexity of the parameters.

**Figure 4.**Closing price of the stock market indices per day. From left to right: American indices (IXIC and NYA) and African indices (J580.JO and NGE) [top], and European indices (N100 and GDAXI), and Asian indices (000001.SS and N225) [bottom].

**Figure 5.**Closing price of the currency exchange rate (related to the USD) per day. From left to right: Euro, British Pound and Japanese Yen [top], and Nigerian Naira, Chinese Renmindi and South Africa Rand [bottom].

**Figure 8.**Feature selection considering the numeric variables: High, Low, Open, Closed and Volume, used to train the RNN models.

Index | Count | Mean | Minimum | Maximum | Standard Deviation |
---|---|---|---|---|---|

IXIC NYA N100 GDAXI NGE J580.J0 N225 000001.SS | 3375 3375 3424 3375 2054 2324 3275 3255 | 5005.10 10,046.22 838.54 10,046.22 28.15 36,946.57 15,942.95 2859.56 | 1268.64 4226.31 434.61 4226.31 7.88 20,716.48 7054.98 1706.70 | 14,138.78 16,590.43 1248.14 16,590.43 65.48 48,467.67 30,467.75 5497.90 | 2897.71 2457.78 179.82 2457.78 17.79 6700.48 5638.76 574.05 |

Currency | Count | Mean | Minimum | Maximum | Standard Deviation |
---|---|---|---|---|---|

ZAR/USD NGN/USD GBP/USD EUR/USD RMB/USD JPY/USD | 3488 3487 3487 3474 584 3474 | 0.097 0.0063 1.49 1.25 0.14 0.01 | 0.05 0.002 1.14 1.03 0.13 0.007 | 0.15 1.00 2.03 1.59 0.15 0.01 | 0.028 0.02 0.18 0.13 0.004 0.001 |

**Table 3.**Full specification of the parameters of the simple RNN, LSTM and GRU models for the stock market indexes and currency exchange rates.

Parameters | Simple RNN | LSTM | GRU |
---|---|---|---|

Learning Rate Number of neurons Number of layer Batch_size Activation function Number of epoch Recurrent_dropout Optimizer Weight Decay Dropout | 0.0016 16 3 64 tanh 100 0.3 Adam None None | 0.02 32 3 300 tanh 150 None Adam None None | 0.0114 128 3 300 tanh 150 None Adam None 0.1801 |

**Table 4.**Root mean square error for the univariate RNN, LSTM and GRU models, in sample (train and validation sets) and out-of-sample forecasting (train set) of the eight stock market indexes.

RMSE | RNN | LSTM | GRU | ||||||
---|---|---|---|---|---|---|---|---|---|

Train | Valid | Test | Train | Valid | Test | Train | Valid | Test | |

IXIC | 0.0032 | 0.0049 | 0.0217 | 0.0037 | 0.0050 | 0.0363 | 0.0044 | 0.0042 | 0.0131 |

NYA | 0.0089 | 0.0072 | 0.0167 | 0.0096 | 0.0071 | 0.0155 | 0.0122 | 0.0178 | 0.0241 |

N100 | 0.01297 | 0.0098 | 0.0164 | 0.0137 | 0.0104 | 0.0161 | 0.0141 | 0.0141 | 0.0172 |

GDAXI | 0.0094 | 0.0103 | 0.0148 | 0.0106 | 0.0097 | 0.0158 | 0.0099 | 0.0126 | 0.0152 |

NGE | 0.0128 | 0.0045 | 0.0057 | 0.0144 | 0.0054 | 0.0067 | 0.0244 | 0.0071 | 0.0049 |

J580.JO | 0.0154 | 0.0181 | 0.0218 | 0.0187 | 0.0224 | 0.0311 | 0.0232 | 0.0140 | 0.0227 |

N225 | 0.0092 | 0.0096 | 0.0140 | 0.0098 | 0.0111 | 0.0151 | 0.0105 | 0.0100 | 0.0138 |

000001.SS | 0.0071 | 0.0084 | 0.0101 | 0.0148 | 0.0083 | 0.0101 | 0.0143 | 0.0245 | 0.0133 |

**Table 5.**Root mean square error for the multivariate RNN, LSTM and GRU models, in sample (train and validation sets) and out-of-sample forecasting (train set) of the eight stock market indexes.

RMSE | RNN | LSTM | GRU | ||||||
---|---|---|---|---|---|---|---|---|---|

Train | Valid | Test | Train | Valid | Test | Train | Valid | Test | |

IXIC | 0.0027 | 0.0047 | 0.0407 | 0.0034 | 0.0154 | 0.1233 | 0.0036 | 0.0049 | 0.0222 |

NYA | 0.0048 | 0.0041 | 0.0094 | 0.0048 | 0.0068 | 0.0196 | 0.0084 | 0.0038 | 0.01050 |

N100 | 0.0094 | 0.0079 | 0.0122 | 0.0080 | 0.0106 | 0.0179 | 0.0113 | 0.0082 | 0.0111 |

GDAXI | 0.0064 | 0.0111 | 0.0130 | 0.0078 | 0.0079 | 0.0149 | 0.0078 | 0.0073 | 0.0106 |

NGE | 0.0103 | 0.0044 | 0.0071 | 0.0099 | 0.0082 | 0.0074 | 0.0246 | 0.0052 | 0.0047 |

J580.JO | 0.0101 | 0.0191 | 0.0165 | 0.0066 | 0.0082 | 0.0142 | 0.0198 | 0.0161 | 0.0128 |

N225 | 0.0059 | 0.0064 | 0.0126 | 0.0053 | 0.0067 | 0.0170 | 0.0069 | 0.0067 | 0.0089 |

000001.SS | 0.0093 | 0.0064 | 0.0072 | 0.0084 | 0.0036 | 0.0058 | 0.0103 | 0.0106 | 0.0057 |

**Table 6.**Root mean square error for the univariate RNN, LSTM and GRU models, in sample (train and validation sets) and out-of-sample forecasting (train set) of the six currency exchange rates.

RMSE | RNN | LSTM | GRU | ||||||
---|---|---|---|---|---|---|---|---|---|

Train | Valid | Test | Train | Valid | Test | Train | Valid | Test | |

ZAR/USD | 0.0139 | 0.0079 | 0.0064 | 0.0158 | 0.0102 | 0.0079 | 0.0224 | 0.0183 | 0.0199 |

NGN/USD | 0.0322 | 0.0249 | 0.0092 | 0.0319 | 0.0319 | 0.0075 | 0.0253 | 0.0548 | 0.0007 |

GBP/USD | 0.0123 | 0.0133 | 0.0119 | 0.0145 | 0.0215 | 0.0204 | 0.0191 | 0.0101 | 0.0123 |

EUR/USD | 0.0236 | 0.0104 | 0.0091 | 0.0228 | 0.0112 | 0.0102 | 0.0266 | 0.0189 | 0.0127 |

RMB/USD | 0.0219 | 0.0237 | 0.0363 | 0.0216 | 0.0266 | 0.0333 | 0.0225 | 0.0199 | 0.0199 |

JPY/USD | 0.0194 | 0.0096 | 0.0074 | 0.0202 | 0.0118 | 0.0097 | 0.0245 | 0.014 | 0.0087 |

**Table 7.**Root mean square error for the multivariate RNN, LSTM and GRU models, in sample (train and validation sets) and out-of-sample forecasting (train set) of the six currency exchange rates.

RMSE | RNN | LSTM | GRU | ||||||
---|---|---|---|---|---|---|---|---|---|

Train | Valid | Test | Train | Valid | Test | Train | Valid | Test | |

ZAR/USD | 0.0096 | 0.0074 | 0.0083 | 0.0145 | 0.0087 | 0.0070 | 0.0216 | 0.015 | 0.0336 |

NGN/USD | 0.0247 | 0.0152 | 0.0159 | 0.0201 | 0.0422 | 0.0536 | 0.0266 | 0.0273 | 0.1534 |

GBP/USD | 0.0094 | 0.0134 | 0.0136 | 0.0089 | 0.0211 | 0.0232 | 0.0165 | 0.0071 | 0.0107 |

EUR/USD | 0.0213 | 0.0118 | 0.0116 | 0.0202 | 0.0097 | 0.0083 | .0250 | 0.0189 | 0.0141 |

RMB/USD | 0.0248 | 0.0188 | 0.0278 | 0.0195 | 0.0269 | 0.0479 | 0.0227 | 0.0178 | 0.0283 |

JPY/USD | 0.0166 | 0.0096 | 0.0069 | 0.0163 | 0.0118 | 0.0102 | 0.0247 | 0.0233 | 0.01292 |

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## Share and Cite

**MDPI and ACS Style**

Sako, K.; Mpinda, B.N.; Rodrigues, P.C. Neural Networks for Financial Time Series Forecasting. *Entropy* **2022**, *24*, 657.
https://doi.org/10.3390/e24050657

**AMA Style**

Sako K, Mpinda BN, Rodrigues PC. Neural Networks for Financial Time Series Forecasting. *Entropy*. 2022; 24(5):657.
https://doi.org/10.3390/e24050657

**Chicago/Turabian Style**

Sako, Kady, Berthine Nyunga Mpinda, and Paulo Canas Rodrigues. 2022. "Neural Networks for Financial Time Series Forecasting" *Entropy* 24, no. 5: 657.
https://doi.org/10.3390/e24050657