Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)2613810.22363/2658-4670-2021-29-1-22-35Research ArticleStudy of the impact of the COVID-19 pandemic on international air transportationShchetininEugeny Yu.<p>Doctor of Physical and Mathematical Sciences, Lecturer of Department of Mathematics</p>riviera-molto@mail.ruFinancial University under the Government of Russian Federation30032021291223530032021Copyright © 2021, Shchetinin E.Y.2021<p style="text-align: justify;">Time Series Forecasting has always been a very important area of research in many domains because many different types of data are stored as time series. Given the growing availability of data and computing power in the recent years, Deep Learning has become a fundamental part of the new generation of Time Series Forecasting models, obtaining excellent results.As different time series problems are studied in many different fields, a large number of new architectures have been developed in recent years. This has also been simplified by the growing availability of open source frameworks, which make the development of new custom network components easier and faster.In this paper three different Deep Learning Architecture for Time Series Forecasting are presented: Recurrent Neural Networks (RNNs), that are the most classical and used architecture for Time Series Forecasting problems; Long Short-Term Memory (LSTM), that are an evolution of RNNs developed in order to overcome the vanishing gradient problem; Gated Recurrent Unit (GRU), that are another evolution of RNNs, similar to LSTM.The article is devoted to modeling and forecasting the cost of international air transportation in a pandemic using deep learning methods. The author builds time series models of the American Airlines (AAL) stock prices for a selected period using LSTM, GRU, RNN recurrent neural networks models and compare the accuracy forecast results.</p>neural networksfinancial forecastingdeep learninginternational air travelнейронные сетифинансовое прогнозированиеглубокое обучениемеждународные авиаперевозки<p>Introduction In 2020, there was a significant drop in quotations of American Airlines (AAL) associated with the COVID-19 pandemic and a record-breaking de- crease in the number of air travel in the world. The generally accepted econometric methods of modeling and forecasting financial time series in these conditions turned out to be ineffective for making even short-term fore- casts [1], [2]. In the present paper, methods for modeling and forecasting international air traffic in the 2019-2020 pandemic are explored using recur- rent neural networks with different architectures. As an object of research, the day quotes of the American company AAL, traded on the NASDAQ ex- change, were selected; data from September 27, 2005 to September 30, 2020 from the information portal Yahoo Finance [3] were taken. The shares of this US company were selected due to its leading positions in the international air transportation market, high values of the trading turnover on the NASDAQ exchange, which in turn provides liquidity and shows investor interest in this exchange commodity [4]. Using the example of the value of AAL shares, we will try to build a reliable forecast using deep learning methods, in particular, recurrent neural networks [5]-[7]. Pre-processing of input data As input data for the neural network model, we will take a sequence consisting of the following values: Opent-1 - opening price for the previous period; Lowt-1 - the minimum price for the previous trading day; Hightt-1 - the maximum price for the previous trading day; Volumet-1 - the amount of shares sold and bought for the previous trading day; Closet-1 - closing price for the previous trading day. Based on the input data, neural networks will generate an output value that can be interpreted as the predicted value of the closing quotation today. For the correct operation of neural networks, it is necessary to normalize the data within the limits of [0 ∶ 1], as well as create training and test samples in the ratio 80:20 from the initial data having the dimension 3636. Thus, 2909 observations for the training sample and 727 observations for the test sample were obtained. The table 1 shows a fragment of the input data. It is necessary to remove the Date and Adj Close columns from the received data. The table 2 presents descriptive statistics of input data. It is seen that the average closing price is $27.13 and the standard deviation is $16.74. To study the statistical properties of the data further, let us build scatter diagrams of the profitability of the opening price and the closing price, as well as the profitability of the closing price shifted by one lag, and the closing price today. To calculate the profitability, we will use the following formula [8]- [10]:</p>[J. D. A. Hamilton, The time series analysis. Princeton New Jersey: University Press, 1994.][C. Brooks, Introductory econometrics for finance. Cambridge: Cambridge University Press, 2019.][“American Airlines Group Inc. (AAL),” URL: https://finance.yahoo. com/quote/AAL/. Availabel: 2020-11-25, 2020.][E. Y. Shchetinin, “On a structural approach to managing a company with high volatility of indicators [K analizu effektivnosti biznesa v usloviyah vysokoj izmenchivosti ego finansovyh aktivov],” Finansy i kredit, vol. 14, no. 218, pp. 39-41, 2006, [in Russian].][J. Vander Plas, Python Data Science Handbook. Sebastopol, CA: O’Reilly Media, 2016.][W. Richert and L. P. Coelho, Building Machine Learning Systems with Python. Birmingham: Packt, 2013.][C. Bishop, Pattern recognition and machine learning. Berlin, Germany: Springer-Verlag, 2006.][E. Y. Shchetinin, “Modeling the energy consumption of smart buildings using artificial intelligence,” in CEUR Workshop Proceedings, vol. 2407, 2019, pp. 130-140.][M. Mudelsee, “Trend analysis of climate time series: A review of methods,” Earth-Science Reviews, vol. 190, pp. 310-322, 2019. DOI: 10.1016/ j.earscirev.2018.12.005.][C. Chen, J. Twycross, and J. M. Garibaldi, “A new accuracy measure based on bounded relative error for time series forecasting,” PLOS ONE, vol. 12, no. 3, pp. 1-23, Mar. 2017. DOI: 10. 1371 / journal. pone. 0174202.][R. J. Hyndman and G. Athanasopoulos, Forecasting: principles and practice. Melbourne, Australia: OTexts, 2018.][A. Ghaderi, B. M. Sanandaji, and F. Ghaderi. “Deep forecast: deep learning-based spatio-temporal forecasting.” arXiv: 1707.08110 [cs.LG]. (2017).][S. B. Taieb, A. Sorjamaa, and G. Bontempi, “Multiple-output modeling for multi-step-ahead time series forecasting,” Neurocomput, vol. 73, no. 10, pp. 1950-1957, 2010. DOI: 10.1016/j.neucom.2009.11.030.][R. Sen, H.-F. Yu, and I. S. Dhillon, “Think globally, act locally: a deep neural network approach to high-dimensional time series forecasting,” in Advances in neural information processing systems, H. Wallach, H. Larochelle, A. Beygelzimer, F. d’Alché-Buc, E. Fox, and R. Garnett, Eds., vol. 32, Curran Associates, Inc., 2019.][“Keras,” URL: https://www.keras.io. Availabel: 2020-11-25, 2020.][I. Goodfellow, Y. Bengio, and A. Courville, Deep learning. Cambridge: The MIT Press, 2016.][S. Galeshchuk and S. Mukherjee, “Deep networks for predicting direction of change in foreign exchange rates,” Intelligent Systems in Accounting, Finance and Management, vol. 24, no. 4, pp. 100-110, 2017. DOI: 10. 1002/isaf.1404.]