mireabulletinRussian Technological JournalRussian Technological Journal2782-32102500-316XRTU MIREA10.32362/2500-316X-2024-12-3-93-103YSWUJGmireabulletin-923Research ArticleМАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕMATHEMATICAL MODELINGАнализ подходов к определению тренда в структуре временного рядаAnalysis of approaches to identification of trend in the structure of the time serieshttps://orcid.org/0009-0008-8756-7267МохнаткинаУ. С.MokhnatkinaU. S.

Мохнаткина Ульяна Станиславовна, студент

119454, Москва, пр-т Вернадского, д. 78

Ulyana S. Mokhnatkina, Student

78, Vernadskogo pr., Moscow, 119454 

atlantika@live.ruhttps://orcid.org/0009-0004-0905-3827ПарфеновД. В.ParfenovD. V.

Парфенов Денис Васильевич, к.т.н., доцент, доцент кафедры высшей математики, Институт искусственного интеллекта

119454, Москва, пр-т Вернадского, д. 78

Scopus Author ID 57217119805

Denis V. Parfenov, Cand. Sci. (Eng.), Associate Professor, Higher Mathematics Department, Institute of Artificial Intelligence

78, Vernadskogo pr., Moscow, 119454 

Scopus Author ID 57217119805

parfenov@mirea.ruhttps://orcid.org/0000-0001-5325-6198ПетрусевичД. А.PetrusevichD. A.

Петрусевич Денис Андреевич, к.ф.-м.н., доцент кафедры высшей математики, Институт искусственного интеллекта

119454, Москва, пр-т Вернадского, д. 78

Scopus Author ID 55900513600, ResearcherID AAA-6661-2020

Denis A. Petrusevich, Cand. Sci. (Phys.-Math.), Associate Professor, Higher Mathematics Department, Institute of Artificial Intelligence

78, Vernadskogo pr., Moscow, 119454 

Scopus Author ID 55900513600, ResearcherID AAA-6661-2020

petrusevich@mirea.ruМИРЭА – Российский технологический университетРоссияMIREA – Russian Technological UniversityRussian Federation20243105202412393103Copyright © Мохнаткина У.С., Парфенов Д.В., Петрусевич Д.А., 20242024Мохнаткина У.С., Парфенов Д.В., Петрусевич Д.А.Mokhnatkina U.S., Parfenov D.V., Petrusevich D.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.rtj-mirea.ru/jour/article/view/923Цели

Цели. Основная цель – сравнить качество прогнозирования моделей временных рядов, по-разному описывающих тренд, и сформировать заключение о применимости каждого подхода при описании тренда в зависимости от свойств временного ряда.

Методы

Методы. Тренд может рассматриваться как склонность рассматриваемой величины к возрастанию или убыванию в долгосрочной перспективе. Также встречается подход, при котором тренд является функцией некоторого вида, отражающей закономерности в поведении рассматриваемого временного ряда (речь идет о закономерностях, характеризующих поведение ряда для всего рассматриваемого периода, а не краткосрочные особенности). В работе рассматривается разложение STL, построение моделей ARIMA, использование моделей ACD (усредненного условного смещения) и другие подходы. Хотя разложение на тренд, сезонность, остаток и является общеупотребительной практикой, многие комбинации, представленные в вычислительном эксперименте, построены впервые (например, использование ряда Фурье для моделирования тренда, совмещение модели сезонности и модели тренда на основе алгоритма ACD). Во второй части работы представлен вычислительный эксперимент, в котором модели, использующие различные подходы к понятию тренда, его выделению и обработке, сравниваются по значению функции максимального правдоподобия и по прогнозу на тестовый период для динамических рядов макроэкономической статистики РФ; цены акций Сбербанка РФ на Московской бирже временного периода 2000–2021 гг.

Результаты

Результаты. Во всех экспериментах один из наиболее точных прогнозов сделан при помощи метода LOESS. Для сезонных рядов достаточно точные результаты показывает моделирование тренда на основе многочлена и сезонности на основе функций ARIMA, совмещение модели тренда на основе алгоритма ACD и сезонности на основе ETS и моделирование на основе ряда Фурье.

Выводы

Выводы. Метод LOESS для групп сезонных и несезонных рядов дает наилучший результат по всем показателям, поэтому можно рекомендовать именно этот метод для получения наиболее точных результатов для рядов различной природы. Моделирование тренда с помощью разложения в ряд Фурье приводит к достаточно точным результатам на временных рядах различной природы. Для сезонных рядов один из лучших результатов дает комбинация моделирования тренда на основе многочлена и сезонности в виде модели ARIMA.

Objectives

Objectives. The study set out to compare the forecasting quality of time series models that describe the trend in different ways and to form a conclusion about the applicability of each approach in describing the trend depending on the properties of the time series.

Methods

Methods. A trend can be thought of as the tendency of a given quantity to increase or decrease over the long term. There is also an approach in which a trend is viewed as some function, reflecting patterns in the behavior of the time series. In this case, we discuss the patterns that characterize the behavior of the series for the entire period under consideration, rather than short-term features. The experimental part involves STL decomposition, construction of ARIMA models (one of the stages of preparation for which includes differentiation, i.e., removal of the trend and transition to a weakly stationary series), construction of ACD models (average conditional displacement) and other approaches. Time-series models based on various trend models are compared with respect to the value of the maximum likelihood function. Many of the combinations have not been constructed before (Fourier series as a trend model, combination of ACD model for trend with seasonal models). Example forecasts of macroeconomic statistics of the Russian Federation and stock prices of Sberbank on the Moscow Exchange in the time range of 2000–2021 are presented.

Results

Results. In the experiments, The LOESS method obtained the best results. A combination of polynomial model for trend description and ARIMA for seasonally description and combination of ACD algorithm for trend and ETS for seasonal model obtained good forecasts in case of seasonal time series, while Fourier time series as a trend model also achieved close quality of prediction.

Conclusions

Conclusions. Since the LOESS method for groups of seasonal and non-seasonal series gives the best results for all indicators, this method can be recommended for obtaining the most accurate results for series of different nature. Trend modeling using Fourier series decomposition leads to quite accurate results for time series of different natures. For seasonal series, one of the best results is given by the combination of modeling a trend on the basis of a polynomial and seasonality in the form of the ARIMA model.

динамические рядымакроэкономическая статистикаARIMAACDвременные рядытрендфункция максимального правдоподобиямоделирование трендаdynamic seriesmacroeconomic statisticsARIMAACDtime seriestrendmaximum likelihood functiontrend modelingReferencesAllen R. Time series methods in the monitoring of intracranial pressure. Part 1: Problems, suggestion for a monitoring scheme and review of appropriate techniques. J. Biomed. Eng. 1983;5(1):5–18. https://doi.org/10.1016/0141-5425(83)90073-0Allen R. Time series methods in the monitoring of intracranial pressure. Part 1: Problems, suggestion for a monitoring scheme and review of appropriate techniques. J. Biomed. Eng. 1983;5(1):5–18. https://doi.org/10.1016/0141-5425(83)90073-0Blom J.A., Ruyter J.F., Saranummi F., Beneken J.W. Detection of trends in monitored variables. In: Carson E.R., Cramp D.G. (Eds.). Computer and Controls in Clinical Medicine. New York: Plenum; 1985. P. 153–174. https://doi.org/10.1007/978-14613-2437-9_6Blom J.A., Ruyter J.F., Saranummi F., Beneken J.W. Detection of trends in monitored variables. In: Carson E.R., Cramp D.G. (Eds.). Computer and Controls in Clinical Medicine. New York: Plenum; 1985. P. 153–174. https://doi.org/10.1007/978-14613-2437-9_6Challis R.E., Kitney R.I. Biomedical signal processing (in four parts). Part I: Time domain methods. Med. Biol. Eng. Comput. 1990;28(6):509–524. https://doi.org/10.1007/bf02442601Challis R.E., Kitney R.I. Biomedical signal processing (in four parts). Part I: Time domain methods. Med. Biol. Eng. Comput. 1990;28(6):509–524. https://doi.org/10.1007/bf02442601Haimowitz I.J., Kohane I.S. Automated trend detection with alternative temporal hypotheses. In: Proceedings of the 13th International Joint Conference of Artificial Intelligence IJCAI-93. 1993. P. 146–151.Haimowitz I.J., Kohane I.S. Automated trend detection with alternative temporal hypotheses. In: Proceedings of the 13th International Joint Conference of Artificial Intelligence IJCAI-93. 1993. P. 146–151.Helsel D.R., Hirsch R.M., Ryberg K.R., Archfield S.A. Statistical Methods in Water Resources. USGS Science Publishing Network, Reston Publishing Service Center; 2018. 458 p. ISBN 978-1-4113-4348-1. https://doi.org/10.3133/tm4a3Helsel D.R., Hirsch R.M., Ryberg K.R., Archfield S.A. Statistical Methods in Water Resources. USGS Science Publishing Network, Reston Publishing Service Center; 2018. 458 p. ISBN 978-1-4113-4348-1. https://doi.org/10.3133/tm4a3Ding H., Li Z., Ren Q., Chen H., Song M., Wang Y. Single-variable method for predicting trends in chlorophyll a concentration based on the similarity of time series. Ecological Indicators. 2022;14096):109027. https://doi.org/10.1016/j.ecolind.2022.109027Ding H., Li Z., Ren Q., Chen H., Song M., Wang Y. Single-variable method for predicting trends in chlorophyll a concentration based on the similarity of time series. Ecological Indicators. 2022;14096):109027. https://doi.org/10.1016/j.ecolind.2022.109027Yao J., Wang P., Wang G., Shrestha S., Xue B., Sun W. Establishing a time series trend structure model to mine potential hydrological information from hydrometeorological time series data. Sci. Total Environ. 2020;698:134227. https://doi.org/10.1016/j.scitotenv.2019.134227Yao J., Wang P., Wang G., Shrestha S., Xue B., Sun W. Establishing a time series trend structure model to mine potential hydrological information from hydrometeorological time series data. Sci. Total Environ. 2020;698:134227. https://doi.org/10.1016/j.scitotenv.2019.134227De Leo F., De Leo A., Besio G., Briganti R. Detection and quantification of trends in time series of significant wave heights: An application in the Mediterranean Sea. Ocean Eng. 2020;202:107155. https://doi.org/10.1016/j.oceaneng.2020.107155De Leo F., De Leo A., Besio G., Briganti R. Detection and quantification of trends in time series of significant wave heights: An application in the Mediterranean Sea. Ocean Eng. 2020;202:107155. https://doi.org/10.1016/j.oceaneng.2020.107155Hyndman R.J., Athanasopoulos G. Forecasting: Principles and Practice. 3rd ed. OTexts; 2021. 442 p. ISBN-13 978-0-98750713-6Hyndman R.J., Athanasopoulos G. Forecasting: Principles and Practice. 3rd ed. OTexts; 2021. 442 p. ISBN-13 978-0-98750713-6Mann H.B. Nonparametric tests against trend. Econometrica. 1945;13(3):2453–259. https://doi.org/10.2307/1907187Mann H.B. Nonparametric tests against trend. Econometrica. 1945;13(3):2453–259. https://doi.org/10.2307/1907187Kendall M.G. Rank Correlation Methods. 2nd ed. Hafner Publishing Co.; 1955. 196 p.Kendall M.G. Rank Correlation Methods. 2nd ed. Hafner Publishing Co.; 1955. 196 p.Kohns D., Bhattacharjee A. Nowcasting growth using Google Trends data: A Bayesian Structural Time Series model. Int. J. Forecast. 2022;39(3):1384–1412. https://doi.org/10.1016/j.ijforecast.2022.05.002Kohns D., Bhattacharjee A. Nowcasting growth using Google Trends data: A Bayesian Structural Time Series model. Int. J. Forecast. 2022;39(3):1384–1412. https://doi.org/10.1016/j.ijforecast.2022.05.002Yahyaoui H., Al-Daihani R. A novel trend based SAX reduction technique for time series. Expert Systems with Applications. 2019;130(C):113–123. https://doi.org/10.1016/j.eswa.2019.04.026Yahyaoui H., Al-Daihani R. A novel trend based SAX reduction technique for time series. Expert Systems with Applications. 2019;130(C):113–123. https://doi.org/10.1016/j.eswa.2019.04.026Xie Y., Liu S., Huang S., Fang H., Ding M., Huang C., Shen T. Local trend analysis method of hydrological time series based on piecewise linear representation and hypothesis test. J. Clean. Prod. 2022;339(1):130695. https://doi.org/10.1016/j.jclepro.2022.130695Xie Y., Liu S., Huang S., Fang H., Ding M., Huang C., Shen T. Local trend analysis method of hydrological time series based on piecewise linear representation and hypothesis test. J. Clean. Prod. 2022;339(1):130695. https://doi.org/10.1016/j.jclepro.2022.130695Vamoş C., Crăciun M. Automatic Trend Estimation. Dordrecht, Heidelberg, New York, London: Springer; 2013. 131 p. https://doi.org/10.1007/978-94-007-4825-5Vamoş C., Crăciun M. Automatic Trend Estimation. Dordrecht, Heidelberg, New York, London: Springer; 2013. 131 p. https://doi.org/10.1007/978-94-007-4825-5Feng Y., Zhou C. Forecasting financial market activity using a semiparametric fractionally integrated Log-ACD. Int. J. Forecast. 2015;31(2):349–363. http://doi.org/10.1016/j.ijforecast.2014.09.001Feng Y., Zhou C. Forecasting financial market activity using a semiparametric fractionally integrated Log-ACD. Int. J. Forecast. 2015;31(2):349–363. http://doi.org/10.1016/j.ijforecast.2014.09.001Allen D., Chan F., McAleer M., Peiris S. Finite sample properties of the QMLE for the Log-ACD model: Application to Australian stocks. J. Econometrics. 2008;147(1):163–185. https://doi.org/10.1016/j.jeconom.2008.09.020Allen D., Chan F., McAleer M., Peiris S. Finite sample properties of the QMLE for the Log-ACD model: Application to Australian stocks. J. Econometrics. 2008;147(1):163–185. https://doi.org/10.1016/j.jeconom.2008.09.020Epperson J. On the Runge example. The American Mathematical Monthly. 1987;94(4):329–341. https://doi.org/10.2307/2323093Epperson J. On the Runge example. The American Mathematical Monthly. 1987;94(4):329–341. https://doi.org/10.2307/2323093Drozdov I., Petrusevich D. Water pollution time series analysis. IOP Conf. Ser.: Mater. Sci. Eng. 2021;1047(1):012095. http://doi.org/10.1088/1757-899X/1047/1/012095Drozdov I., Petrusevich D. Water pollution time series analysis. IOP Conf. Ser.: Mater. Sci. Eng. 2021;1047(1):012095. http://doi.org/10.1088/1757-899X/1047/1/012095Petrusevich D. Review of missing values procession methods in time series data. J. Phys.: Conf. Ser. 2021;1889(3):032009. http://doi.org/10.1088/1742-6596/1889/3/032009Petrusevich D. Review of missing values procession methods in time series data. J. Phys.: Conf. Ser. 2021;1889(3):032009. http://doi.org/10.1088/1742-6596/1889/3/032009Wang P., Zheng X., Ai G., Liu D., Zhu B. Time series prediction for the epidemic trends of COVID-19 using the improved LSTM deep learning method: Case studies in Russia, Peru and Iran. Chaos, Solitons & Fractals. 2020;140:110214. https://doi.org/10.1016/j.chaos.2020.110214Wang P., Zheng X., Ai G., Liu D., Zhu B. Time series prediction for the epidemic trends of COVID-19 using the improved LSTM deep learning method: Case studies in Russia, Peru and Iran. Chaos, Solitons & Fractals. 2020;140:110214. https://doi.org/10.1016/j.chaos.2020.110214Kumar B., Sunil P., Yadav N. A novel hybrid model combining βSARMA and LSTM for time series forecasting. Appl. Soft Comput. 2023;134:110019. https://doi.org/10.1016/j.asoc.2023.110019Kumar B., Sunil P., Yadav N. A novel hybrid model combining βSARMA and LSTM for time series forecasting. Appl. Soft Comput. 2023;134:110019. https://doi.org/10.1016/j.asoc.2023.110019Abebe M., Noh Y., Kang Y.-J., Seo C., Kim D., Seo J. Ship trajectory planning for collision avoidance using hybrid ARIMA-LSTM models. Ocean Eng. 2022;256:111527. https://doi.org/10.1016/j.oceaneng.2022.111527Abebe M., Noh Y., Kang Y.-J., Seo C., Kim D., Seo J. Ship trajectory planning for collision avoidance using hybrid ARIMA-LSTM models. Ocean Eng. 2022;256:111527. https://doi.org/10.1016/j.oceaneng.2022.111527Arunkumar K.E., Kalaga D.V., Kumar M.S., Kawaji M., Brenza T.M. Comparative analysis of Gated Recurrent Units (GRU), long Short-Term memory (LSTM) cells, autoregressive Integrated moving average (ARIMA), seasonal autoregressive Integrated moving average (SARIMA) for forecasting COVID-19 trends. Alexandria Eng. J. 2022;61(10):7585–7603. https://doi.org/10.1016/j.aej.2022.01.011Arunkumar K.E., Kalaga D.V., Kumar M.S., Kawaji M., Brenza T.M. Comparative analysis of Gated Recurrent Units (GRU), long Short-Term memory (LSTM) cells, autoregressive Integrated moving average (ARIMA), seasonal autoregressive Integrated moving average (SARIMA) for forecasting COVID-19 trends. Alexandria Eng. J. 2022;61(10):7585–7603. https://doi.org/10.1016/j.aej.2022.01.011Ning Y., Kazemi H., Tahmasebi P. A comparative machine learning study for time series oil production forecasting: ARIMA, LSTM, and Prophet. Comput. Geosci. 2022;164:105126. https://doi.org/10.1016/j.cageo.2022.105126Ning Y., Kazemi H., Tahmasebi P. A comparative machine learning study for time series oil production forecasting: ARIMA, LSTM, and Prophet. Comput. Geosci. 2022;164:105126. https://doi.org/10.1016/j.cageo.2022.105126Anghinoni L., Zhao L., Ji D., Pan H. Time series trend detection and forecasting using complex network topology analysis. Neural Netw. 2019;117:295–306. https://doi.org/10.1016/j.neunet.2019.05.018Anghinoni L., Zhao L., Ji D., Pan H. Time series trend detection and forecasting using complex network topology analysis. Neural Netw. 2019;117:295–306. https://doi.org/10.1016/j.neunet.2019.05.018Box G., Jenkins G., Reinsel G.C. Time Series Analysis: Forecasting and Control. John Wiley and Sons; 2008. 784 p. ISBN-13 978-0470272848Box G., Jenkins G., Reinsel G.C. Time Series Analysis: Forecasting and Control. John Wiley and Sons; 2008. 784 p. ISBN-13 978-0470272848Petropoulos F., Hyndman R.J., Bergmeir C. Exploring the sources of uncertainty: Why does bagging for time series forecasting work? Eur. J. Oper. Res. 2018;268(2):545–554. https://doi.org/10.1016/j.ejor.2018.01.045Petropoulos F., Hyndman R.J., Bergmeir C. Exploring the sources of uncertainty: Why does bagging for time series forecasting work? Eur. J. Oper. Res. 2018;268(2):545–554. https://doi.org/10.1016/j.ejor.2018.01.045Грамович Я.В., Мусатов Д.Ю., Петрусевич Д.А. Применение беггинга в прогнозировании временных рядов. Russ. Technol. J. 2024;12(1):101–110. https://doi.org/10.32362/2500-316X-2024-12-1-101-110Gramovich I.V., Musatov D.Yu., Petrusevich D.A. Implementation of bagging in time series forecasting. Russ. Technol. J. 2024;12(1):101–110. https://doi.org/10.32362/2500-316X-2024-12-1-101-110]Zhao K., Wulder M.A., Hu T., Bright R., Wu Q., Qin H., Li Y., Toman E., Mallick B., Zhang X., Brown M. Detecting change-point, trend, and seasonality in satellite time series data to track abrupt changes and nonlinear dynamics: A Bayesian ensemble algorithm. Remote Sens. Environ. 2019;232:111181. https://doi.org/10.1016/j.rse.2019.04.034Zhao K., Wulder M.A., Hu T., Bright R., Wu Q., Qin H., Li Y., Toman E., Mallick B., Zhang X., Brown M. Detecting change-point, trend, and seasonality in satellite time series data to track abrupt changes and nonlinear dynamics: A Bayesian ensemble algorithm. Remote Sens. Environ. 2019;232:111181. https://doi.org/10.1016/j.rse.2019.04.034Li J., Li Z.-L., Wu H., You N. Trend, seasonality, and abrupt change detection method for land surface temperature time-series analysis: Evaluation and improvement. Remote Sens. Environ. 2022;280:113222. https://doi.org/10.1016/j.rse.2022.113222Li J., Li Z.-L., Wu H., You N. Trend, seasonality, and abrupt change detection method for land surface temperature time-series analysis: Evaluation and improvement. Remote Sens. Environ. 2022;280:113222. https://doi.org/10.1016/j.rse.2022.113222

The authors declare that there are no conflicts of interest present.