Article citation information:Dogan, E. Analysis and comparison of long short-term memory networks short-term traffic prediction performance. Scientific Journal of Silesian University of Technology. Series Transport. 2020, 107, 19-32. ISSN: 0209-3324. DOI: https://doi.org/10.20858/sjsutst.2020.107.2. Erdem DOGANANALYSIS AND COMPARISON OF LONG SHORT-TERM MEMORY NETWORKS SHORT-TERM TRAFFIC PREDICTION PERFORMANCESummary. Long short-term memory networks (LSTM) produces promising results in the prediction of traffic flows. However, LSTM needs large numbers of data to produce satisfactory results. Therefore, the effect of LSTM training set size on performance and optimum training set size for short-term traffic flow prediction problems were investigated in this study. To achieve this, the numbers of data in the training set was set between 480 and 2800, and the prediction performance of the LSTMs trained using these adjusted training sets was measured. In addition, LSTM prediction results were compared with nonlinear autoregressive neural networks (NAR) trained using the same training sets. Consequently, it was seen that the increase in LSTM's training cluster size increased performance to a certain point. However, after this point, the performance decreased. Three main results emerged in this study: First, the optimum training set size for LSTM significantly improves the prediction performance of the model. Second, LSTM makes short-term traffic forecasting better than NAR. Third, LSTM predictions fluctuate less than the NAR model following instant traffic flow changes.Keywords: deep learning, traffic flow, short-term, prediction, LSTM, nonlinear autoregressive, training set size1. INTRODUCTIONNowadays, the number of vehicles and travel demands are increasing rapidly. This increase is responsible for delays, fuel loss and high emissions globally. For this reason, the efficiency of road capacities should be increased by directing and controlling road traffic with intelligent transport systems (ITS). However, ITS needs information about the current status of traffic variables and future estimates of this information (for example, volume, speed, travel time, etc.). For ITS to be more efficient, it is important that traffic parameters be accurately estimated, especially in the short term. Thus, ITS can make fast and accurate decisions for future traffic situations. For this reason, studies on predicting the short-term future situation of traffic become important. Researchers are working to make these predictions more accurate by developing new methods. Especially as deep learning has proven itself in many areas, the use of deep learning in short term traffic prediction has accelerated. Therefore, there is a need for research that better demonstrates the potentials of deep learning in this regard. The first study on short-term traffic flow estimation was performed using the Box-Jenkins method ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.3141/2024-03","ISBN":"0309029724","ISSN":"03611981","abstract":"\"Transportation Research Board [and] Commission on Sociotechnical Systems, National Research Council.\" \"Reports of the 58th annual meeting of the Transportation Research Board\"--Cataloging in Publication data [p. ii]. Papers sponsored by Committee on Urban System Operations, Committee on Parking and Terminals, and Committee on Freeway Operations. Analysis of freeway traffic time-series data by using Box-Jenkins techniques / Mohamed S. Ahmed, Allen R. Cook -- Automobile diversion : a strategy for reducing traffic in sensitive areas / Ronald H. Borowski -- Development and application of a freeway priority-lane model / Matthys P. Cilliers, Adolf D. May, Reed Cooper -- Safety considerations in the use of on-street parking / Jack B. Humphreys (and 3 others) -- Delay, time saved, and travel time information for freeway traffic management / R. Dale Hutchingson, Conrad L. Dudek -- Empirical analysis of the interdependence of parking restrictions and modal use / Curtis C. Lueck, Edward A. Beimborn -- Incident-detection algorithms. Part 1, Off-line evaluation / Moshe Levin, Gerianne M. Krause; Part 2, On-line evaluation / Moshe Levin, Gerianne Krause, James A. Budrick -- Development of a transport system management planning process in the Delaware Valley Region / Rasin K. Mufti, James J. Schwarzwalder -- FREFLO : a macroscopic simulation model of freeway traffic / Harold J. Payne -- Evaluation of the I-35 redesignation in San Antonio / William R. Stockton, Conrad L. Dudek, Donald B. Hatcher -- Improved air quality through transportation sysem management / John H. Suhrbier, Terry J. Atherton, Elizabeth A. Deakin -- Results of implementing low-cost freeway incident-management techniques / Gary L. Urbanek, Samuel C. Tignor, Gary C. Price -- High-occupancy vehicle considerations on an arterial corridor in Pensacola, Florida / Cecil O. Willis, Jr. -- Planning rail station parking : approach and application / L, K. Carpenter, E.M. Whitlock -- Development of freeway incident-detection algorithms by using pattern-recognition techniques / J. Tsai, E.R. Case.","author":[{"dropping-particle":"","family":"Ahmed","given":"Mohamed S","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Cook","given":"A. R.","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportation Research Record","id":"ITEM-1","issued":{"date-parts":[["1979"]]},"title":"Analysis of Freeway Traffic Time-Series Data by Using Box-Jenkins Techniques","type":"article-journal"},"uris":["http://www.mendeley.com/documents/?uuid=ab18910c-fb84-4f20-8688-fde662225f4d"]}],"mendeley":{"formattedCitation":"[1]","plainTextFormattedCitation":"[1]","previouslyFormattedCitation":"(1)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[1]. Time series methods were used to estimate traffic flow in other studies. ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"ISSN":"0209-3324","author":[{"dropping-particle":"","family":"Dogan","given":"Erdem","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Korkmaz","given":"Ersin","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Akgungor","given":"Ali Payidar","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Zeszyty Naukowe. 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ABSTRACT FROM AUTHOR","author":[{"dropping-particle":"","family":"Williams","given":"Billy M.","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Hoel","given":"Lester a.","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Journal of Transportation Engineering","id":"ITEM-2","issue":"6","issued":{"date-parts":[["2003"]]},"page":"664-672","title":"Modeling and Forecasting Vehicular Traffic Flow as a Seasonal ARIMA Process: Theoretical Basis and Empirical Results","type":"article-journal","volume":"129"},"uris":["http://www.mendeley.com/documents/?uuid=a6a80c38-630e-4877-a317-7f975ec68cde"]},{"id":"ITEM-3","itemData":{"DOI":"10.1109/PEITS.2008.135","ISBN":"9780769533421","abstract":"According to the complexity of the traffic historical data and the randomness of a lot of uncertain factors influence, a hybrid predicting model that combines both autoregressive integrated moving average (ARIMA) and multilayer artificial neural network (MLANN) is proposed in this paper. ARIMA is suitable for linear prediction and MLFNN is suitable for nonlinear prediction. This paper also investigates the issue on how to effectively model short term traffic flow time series with a new algorithm, which estimates the weights of the MLFNN and the parameters of ARMA model. Experimental results with real data sets indicate that the combined model can be an effective way to improve forecasting accuracy achieved by either of the models used separately.","author":[{"dropping-particle":"","family":"Zeng","given":"Dehuai","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Xu","given":"Jianmin","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Gu","given":"Jianwei","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Liu","given":"Liyan","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Xu","given":"Gang","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Proceedings - 2008 Workshop on Power Electronics and Intelligent Transportation System, PEITS 2008","id":"ITEM-3","issued":{"date-parts":[["2008"]]},"title":"Short term traffic flow prediction using hybrid ARIMA and ANN models","type":"paper-conference"},"uris":["http://www.mendeley.com/documents/?uuid=ec1799fa-5217-4e07-bc9c-daf5585ceb1e"]},{"id":"ITEM-4","itemData":{"DOI":"10.1109/ICMLC.2009.5212785","ISBN":"978-1-4244-3702-3","abstract":"Traffic flow data are in the form of spatial time series and are collected at specific locations at constant intervals of time. Space-time autoregressive time series modeling is a promising inductive method that uses a small number of parameters and can be used for online monitoring and prediction. In this paper, we develop space-time autoregressive models for urban traffic flow network scenarios. We evaluate the ability of the space-time autoregressive models to model the spatial and temporal correlations in the traffic network and show that the space-time model performs well.","author":[{"dropping-particle":"","family":"Lin","given":"Shu-Lan Lin Shu-Lan","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Huang","given":"Hong-Qiong Huang Hong-Qiong","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Zhu","given":"Da-Qi Zhu Da-Qi","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Wang","given":"Tian-Zhen Wang Tian-Zhen","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"2009 International Conference on Machine Learning and Cybernetics","id":"ITEM-4","issue":"July","issued":{"date-parts":[["2009"]]},"page":"12-15","title":"The application of space-time ARIMA model on traffic flow forecasting","type":"article-journal","volume":"6"},"uris":["http://www.mendeley.com/documents/?uuid=029d2b01-5578-4de3-a5cc-2d86c26b3a10"]},{"id":"ITEM-5","itemData":{"DOI":"10.1109/IVS.2011.5940418","ISBN":"9781457708909","ISSN":"1931-0587","abstract":"Short-time traffic flow prediction is a significant interest in transportation study, and it is essential in congestion control and traffic network management. In this paper, we propose an Autoregressive Integrated Moving Average with Generalized Autoregressive Conditional Heteroscedasticity (ARIMA-GARCH) model for traffic flow prediction. The model combines linear ARIMA model with nonlinear GARCH model, so it can capture both the conditional mean and conditional heteroscedasticity of traffic flow series. The model is calibrated, validated and used for prediction based on PeMS single loop detector data. The performance of the hybrid model is compared with that of standard ARIMA model. The results show that the introduction of conditional heteroscedasticity cannot bring satisfactory improvement to prediction accuracy, in some cases the general GARCH(1,1) model may even deteriorate the performance. Thus for ordinary traffic flow prediction, the standard ARIMA model is sufficient.","author":[{"dropping-particle":"","family":"Chen","given":"Chenyi","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Hu","given":"Jianming","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Meng","given":"Qiang","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Zhang","given":"Yi","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"IEEE Intelligent Vehicles Symposium, Proceedings","id":"ITEM-5","issue":"Iv","issued":{"date-parts":[["2011"]]},"page":"607-612","title":"Short-time traffic flow prediction with ARIMA-GARCH model","type":"article-journal","volume":"100084"},"uris":["http://www.mendeley.com/documents/?uuid=12267854-5a34-47f8-afd0-49cd4250624f"]},{"id":"ITEM-6","itemData":{"DOI":"10.1109/NGI.2006.1678242","ISBN":"0780394550","abstract":"The predictability of Internet traffic is a significant interest in many domains such as adaptive applications, congestion control, admission control, and network management. In this paper, we propose a new traffic prediction model called autoregressive integrated moving average with generalized autoregressive conditional heteroscedasticity (ARIMA/GARCH), which can capture traffic burstiness and exhibit self-similarity and long-range dependence (LRD). We discuss network traffic predictability related to different prediction applications and measure methods. We validate our prediction model by comparing with other models, includes non-model-based minimum mean square error (MMSE), pure self-similar fractional ARIMA (FARIMA). We use the real network traces to evaluate models. The results show that MMSE computation is simplest and fastest and can apply for online prediction applications. The results also show that FARIMA predictability relies on strong degree of self-similarity, our proposed ARIMA/GARCH model get the best adaptability and accuracy. Therefore ARIMA/GARCH model can be used for exact prediction applications","author":[{"dropping-particle":"","family":"Zhou","given":"Bo","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"He","given":"Dan","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Sun","given":"Zhili","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"2006 2nd Conference on Next Generation Internet Design and Engineering, NGI 2006","id":"ITEM-6","issue":"c","issued":{"date-parts":[["2006"]]},"page":"200-207","title":"Traffic predictability based on ARIMA/GARCH model","type":"article-journal","volume":"00"},"uris":["http://www.mendeley.com/documents/?uuid=718ec9c9-c4e7-4680-be52-b0fbdc5b133f"]},{"id":"ITEM-7","itemData":{"DOI":"10.3141/2024-14","ISSN":"0361-1981","author":[{"dropping-particle":"","family":"Shekhar","given":"Shashank","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Williams","given":"Billy M.","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportation Research Record: Journal of the Transportation Research Board","id":"ITEM-7","issue":"-1","issued":{"date-parts":[["2008"]]},"page":"116-125","title":"Adaptive Seasonal Time Series Models for Forecasting Short-Term Traffic Flow","type":"article-journal","volume":"2024"},"uris":["http://www.mendeley.com/documents/?uuid=75c563f5-ecc9-42f4-838f-25824b62cee7"]},{"id":"ITEM-8","itemData":{"DOI":"10.16984/saufenbilder.308997","ISSN":"1301-4048","abstract":"Günümüzde trafik kontrol sistemlerinin verimli çalışabilmesi için kısa dönemli trafiğin tahmin edilmesi gerekli olmaktadır. Bu çalışmada, Kırıkkale İl sınırlarındaki (D-200, E88) karayoluna     ait kısa dönemli trafik tahmin modellerinin geliştirilmesi için dönemsel otoregresif bileşik hareketli ortalama (SARIMA) tekniği kullanılmıştır.. Modellemede kullanılan veri seti, yolun 29+100 km’ sinde belirlenen kesiminde, yaklaşık 2 ay süre boyunca 15 dk.’ lık zaman dilimleri şeklinde yapılan sayımlardan her bir şerit için ayrı olarak elde edilmiştir. Elde edilen trafik akımı verilerinin %80’ i SARIMA modellerinin oluşturulması için, kalan kısmı ise modelin tahminlerinin test edilmesi için kullanılmıştır. Farklı tiplerdeki SARIMA modelleri iki şeritli karayolunun sağ ve sol şeritleri için oluşturulmuştur. Analizler sonucunda SARIMA(1,0,1)(0,1,1) 672 her iki şerit için en düşük tahmin hatalarını üretmiştir. Modellere ait R 2 değerleri ise sağ şerit için 0,92, sol şerit için 0,88 seviyelerinde olduğu hesaplanmıştır. Modellerin yüksek R2 değerleri ve düşük hatalar üretmesi nedeniyle geliştirilen SARIMA modellerinin Türkiye karayollarının kesintisiz akım koşullarını sağlayan yol kesiminde Türkiye'de kullanılabileceğini anlaşılmıştır.","author":[{"dropping-particle":"","family":"Doğan","given":"Erdem","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Sakarya University Journal of Science","id":"ITEM-8","issue":"2","issued":{"date-parts":[["2018","4","1"]]},"page":"1-1","title":"Developing Short-Term Traffic Forecasting Models Using Seasonal ARIMA Method for D-200 Highway","type":"article-journal","volume":"22"},"uris":["http://www.mendeley.com/documents/?uuid=61fcdc78-8213-3b85-b0ee-65a1e7fd4807"]}],"mendeley":{"formattedCitation":"[2–9]","plainTextFormattedCitation":"[2–9]","previouslyFormattedCitation":"(2–9)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[2-8] However, when artificial intelligence approaches and time series methods were compared, it was observed that artificial intelligence predicted short term traffic flow better ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"author":[{"dropping-particle":"","family":"Lint","given":"J W C","non-dropping-particle":"Van","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Hinsbergen","given":"CPIJ","non-dropping-particle":"Van","parse-names":false,"suffix":""}],"container-title":"Artificial Intelligence Applications to Critical Transportation Issues","id":"ITEM-1","issued":{"date-parts":[["2012"]]},"page":"22-41","title":"Short-term traffic and travel time prediction models","type":"article-journal","volume":"22"},"uris":["http://www.mendeley.com/documents/?uuid=77be33d9-7979-4ea1-ba49-34061fd08a6b"]}],"mendeley":{"formattedCitation":"[10]","plainTextFormattedCitation":"[10]","previouslyFormattedCitation":"(10)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[9]. Therefore, in this study, traffic flow estimation models were developed by using artificial intelligence and deep networks approaches and the size of the training sets were discussed ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"ISSN":"1648-4142","author":[{"dropping-particle":"","family":"Kumar","given":"Kranti","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Parida","given":"Manoranjan","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Katiyar","given":"Vinod Kumar","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transport","id":"ITEM-1","issue":"4","issued":{"date-parts":[["2015"]]},"page":"397-405","publisher":"Taylor & Francis","title":"Short term traffic flow prediction in heterogeneous condition using artificial neural network","type":"article-journal","volume":"30"},"uris":["http://www.mendeley.com/documents/?uuid=c31a9d37-9511-4a0c-a669-7e90a71a3e94"]}],"mendeley":{"formattedCitation":"[11]","plainTextFormattedCitation":"[11]","previouslyFormattedCitation":"(11)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[10]. Short-term traffic flow estimation was performed with ANNs in earlier times from deep learning approach. For instance, the dynamic wavelet ANN model was used to estimate traffic flow ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1061/(ASCE)0733-947X(2005)131:10(771)","ISSN":"0733-947X","author":[{"dropping-particle":"","family":"Jiang","given":"Xiaomo","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Adeli","given":"Hojjat","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Asce","given":"Hon M","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Journal of Transportation Engineering","id":"ITEM-1","issue":"10","issued":{"date-parts":[["2005","10","1"]]},"note":"From Duplicate 2 (Dynamic Wavelet Neural Network Model for Traffic Flow Forecasting - Jiang, X; Adeli, H)\n\ndoi: 10.1061/(ASCE)0733-947X(2005)131:10(771)","page":"771-779","publisher":"American Society of Civil Engineers","title":"Dynamic Wavelet Neural Network Model for Traffic Flow Forecasting","type":"article-journal","volume":"131"},"uris":["http://www.mendeley.com/documents/?uuid=66f85c5c-e353-4d89-93e7-bc69f4df43d6"]}],"mendeley":{"formattedCitation":"[12]","plainTextFormattedCitation":"[12]","previouslyFormattedCitation":"(12)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[11]. Dynamic traffic flow modelling is another approach to determine the amount of traffic flow ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1080/19427867.2016.1271761","ISSN":"1942-7867","author":[{"dropping-particle":"","family":"Fulari","given":"Shrikant","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Thankappan","given":"Ajitha","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Vanajakshi","given":"Lelitha","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Subramanian","given":"Shankar","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportation Letters","id":"ITEM-1","issue":"1","issued":{"date-parts":[["2019","1","2"]]},"note":"doi: 10.1080/19427867.2016.1271761","page":"43-53","publisher":"Taylor & Francis","title":"Traffic flow estimation at error prone locations using dynamic traffic flow modeling","type":"article-journal","volume":"11"},"uris":["http://www.mendeley.com/documents/?uuid=35c14e9a-fecd-4d16-8e96-5d209ce7392b"]}],"mendeley":{"formattedCitation":"[13]","plainTextFormattedCitation":"[13]","previouslyFormattedCitation":"(13)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[12]. ANN and K-NN were used together to estimate traffic flow ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"ISSN":"1848-4069","author":[{"dropping-particle":"","family":"Doğan","given":"Erdem","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Promet-Traffic&Transportation","id":"ITEM-1","issue":"1","issued":{"date-parts":[["2020"]]},"page":"65-78","title":"Short-term Traffic Flow Prediction Using Artificial Intelligence with Periodic Clustering and Elected Set","type":"article-journal","volume":"32"},"uris":["http://www.mendeley.com/documents/?uuid=5d58a83a-705d-4c12-966e-a1835b7e1ee2"]}],"mendeley":{"formattedCitation":"[14]","plainTextFormattedCitation":"[14]","previouslyFormattedCitation":"(14)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[13]. In another study, multiscale analysis-based intelligent ensemble modelling was used to predict airway traffic ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"ISSN":"1942-7867","author":[{"dropping-particle":"","family":"Xiao","given":"Yi","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Liu","given":"John J","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Xiao","given":"Jin","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Hu","given":"Yi","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Bu","given":"Hui","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Wang","given":"Shouyang","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportation Letters","id":"ITEM-1","issue":"2","issued":{"date-parts":[["2015"]]},"page":"73-79","publisher":"Taylor & Francis","title":"Application of multiscale analysis-based intelligent ensemble modeling on airport traffic forecast","type":"article-journal","volume":"7"},"uris":["http://www.mendeley.com/documents/?uuid=fe0305d6-265a-404d-a1fe-653eb3a6e3a4"]}],"mendeley":{"formattedCitation":"[15]","plainTextFormattedCitation":"[15]","previouslyFormattedCitation":"(15)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[14]. The traffic flow was modelled for the city of Istanbul using different time resolutions and the results were accurate despite the limited data ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"author":[{"dropping-particle":"","family":"Çetiner","given":"B","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Sari","given":"Murat","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Borat","given":"Oğuz","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Mathematical and Computational Applications","id":"ITEM-1","issue":"2","issued":{"date-parts":[["2010"]]},"page":"269-278","publisher":"Association for Scientific Research","title":"A neural network based traffic-flow prediction model","type":"article-journal","volume":"15"},"uris":["http://www.mendeley.com/documents/?uuid=26d8e735-a0fa-43c7-9308-10499550de7c"]}],"mendeley":{"formattedCitation":"[16]","plainTextFormattedCitation":"[16]","previouslyFormattedCitation":"(16)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[15] and some others ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1080/18128600802591681","ISBN":"1812860080","ISSN":"1812-8602","author":[{"dropping-particle":"","family":"Sheu","given":"Jiuh-Biing","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Lan","given":"Lawrence W.","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Huang","given":"Yi-San","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportmetrica","id":"ITEM-1","issue":"1","issued":{"date-parts":[["2009"]]},"page":"59-83","title":"Short-term prediction of traffic dynamics with real-time recurrent learning algorithms","type":"article-journal","volume":"5"},"uris":["http://www.mendeley.com/documents/?uuid=fcbcfeca-1586-4672-a733-a1521fe85791"]},{"id":"ITEM-2","itemData":{"author":[{"dropping-particle":"","family":"Çetiner","given":"B","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Sari","given":"Murat","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Borat","given":"Oğuz","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Mathematical and Computational Applications","id":"ITEM-2","issue":"2","issued":{"date-parts":[["2010"]]},"page":"269-278","publisher":"Association for Scientific Research","title":"A neural network based traffic-flow prediction model","type":"article-journal","volume":"15"},"uris":["http://www.mendeley.com/documents/?uuid=26d8e735-a0fa-43c7-9308-10499550de7c"]},{"id":"ITEM-3","itemData":{"DOI":"10.1111/j.1468-0394.2010.00567.x","ISSN":"02664720","abstract":"This paper proposes alternative approaches for the\\nprediction of short-term traffic flow using three\\nbranches of computational intelligence techniques,\\nnamely linear genetic programming (LGP), multilayer\\nperceptron (MLP) and fuzzy logic (FL). Different LGP,\\nMLP and FL models are developed for estimating the 5-\\nand 30-min traffic flow rates. New LGP- and MLP-based\\nprediction equations are derived for the traffic flow\\nrates in the five and thirty minute time intervals. The\\nmodels are established upon extensive databases of the\\ntraffic flow records obtained from Iran's Rasht-Qazvin\\nhighway. The results indicate that the proposed models\\nare effectively capable of predicting the target\\nvalues. The LGP-based models are found to be simple,\\nstraightforward and more practical for predictive\\npurposes compared with the other derived models.","author":[{"dropping-particle":"","family":"Zargari","given":"Shahriar Afandizadeh","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Siabil","given":"Salar Zabihi","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Alavi","given":"Amir Hossein","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Gandomi","given":"Amir Hossein","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Expert Systems","id":"ITEM-3","issue":"2","issued":{"date-parts":[["2010"]]},"page":"no-no","title":"A computational intelligence-based approach for short-term traffic flow prediction","type":"article-journal","volume":"29"},"uris":["http://www.mendeley.com/documents/?uuid=288de2e9-83b6-4334-aacf-940cb7d333a8"]},{"id":"ITEM-4","itemData":{"DOI":"10.1061/(ASCE)TE.1943-5436.0000337","ISSN":"0733-947X","abstract":"Predictions of fundamental traffic variables in the short-term or near-term future are vital for any successful dynamic traffic management application. Univariate short-term traffic flow prediction algorithms are popular in literature. However, to facilitate the operationalities of advanced adaptive traffic management systems, there is a necessity of developing multivariate traffic condition prediction algorithms. A new multivariate short-term traffic flow and speed prediction methodology is proposed in this paper where the traffic flow and speed observations from uncongested (or linear) and congested (or nonlinear) regimes are regime-adjusted to ensure consistent system dynamics. The prediction methodology is developed by using artificial neural networks (ANN) algorithms in conjunction with adaptive learning rules. These learning rules demonstrate significantly improved accuracy and simultaneous reduction in computation times. Additionally, the paper attempts to identify the most suitable adaptive learning rule from a chosen pool of rules. The validation of the prediction methodology is performed by using traffic data from multiple locations in the United Kingdom (U.K.). The results indicate that the proposed multivariate forecasting algorithm is effective and computationally parsimonious to simultaneously predict traffic flow and speed in freeway or highway networks. © 2012 American Society of Civil Engineers.","author":[{"dropping-particle":"","family":"Dunne","given":"Stephen","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Ghosh","given":"Bidisha","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Journal of Transportation Engineering","id":"ITEM-4","issue":"April","issued":{"date-parts":[["2012"]]},"page":"455-466","title":"Regime-Based Short-Term Multivariate Traffic Condition Forecasting Algorithm","type":"article-journal","volume":"138"},"uris":["http://www.mendeley.com/documents/?uuid=921cce92-0b07-4f2b-a76c-5c2d9b34e868"]}],"mendeley":{"formattedCitation":"[16–19]","plainTextFormattedCitation":"[16–19]","previouslyFormattedCitation":"(16–19)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[15-18]. Deep learning has recently gained interest in the prediction of various traffic parameters. Long short-term memory (LSTM) is in the sub-branch of deep learning. Previous studies on LSTM have evidence that deep learning and the performance of other methods were compared. For example, LSTM was compared with regression models ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.trc.2017.02.024","ISBN":"0968-090X","ISSN":"0968090X","abstract":"We develop a deep learning model to predict traffic flows. The main contribution is development of an architecture that combines a linear model that is fitted using ℓ1 regularization and a sequence of tanh layers. The challenge of predicting traffic flows are the sharp nonlinearities due to transitions between free flow, breakdown, recovery and congestion. We show that deep learning architectures can capture these nonlinear spatio-temporal effects. The first layer identifies spatio-temporal relations among predictors and other layers model nonlinear relations. We illustrate our methodology on road sensor data from Interstate I-55 and predict traffic flows during two special events; a Chicago Bears football game and an extreme snowstorm event. Both cases have sharp traffic flow regime changes, occurring very suddenly, and we show how deep learning provides precise short term traffic flow predictions.","author":[{"dropping-particle":"","family":"Polson","given":"Nicholas G.","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Sokolov","given":"Vadim O.","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportation Research Part C: Emerging Technologies","id":"ITEM-1","issued":{"date-parts":[["2017"]]},"page":"1-17","title":"Deep learning for short-term traffic flow prediction","type":"article-journal","volume":"79"},"uris":["http://www.mendeley.com/documents/?uuid=4b466b55-f922-4aa7-a47c-884b0df7420e"]}],"mendeley":{"formattedCitation":"[20]","plainTextFormattedCitation":"[20]","previouslyFormattedCitation":"(20)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[19]. As a result, LSTM has generally made better predictions, except in some cases. Researchers developed a model using LSTM to predict the short-term traffic flow in exceptional traffic conditions. In addition, the authors studied the characteristics of traffic data ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"author":[{"dropping-particle":"","family":"Yu","given":"Rose","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Li","given":"Yaguang","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Shahabi","given":"Cyrus","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Demiryurek","given":"Ugur","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Liu","given":"Yan","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Proceedings of the 2017 SIAM International Conference on Data Mining","id":"ITEM-1","issued":{"date-parts":[["2017"]]},"page":"777-785","publisher":"SIAM","title":"Deep learning: A generic approach for extreme condition traffic forecasting","type":"paper-conference"},"uris":["http://www.mendeley.com/documents/?uuid=10d4e725-369b-4e97-99c9-5d01d84789d9"]}],"mendeley":{"formattedCitation":"[21]","plainTextFormattedCitation":"[21]","previouslyFormattedCitation":"(21)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[20]. In another study, LSTM and recurrent ANN models were compared with ARIMA models ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"ISBN":"150904423X","author":[{"dropping-particle":"","family":"Fu","given":"Rui","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Zhang","given":"Zuo","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Li","given":"Li","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"2016 31st Youth Academic Annual Conference of Chinese Association of Automation (YAC)","id":"ITEM-1","issued":{"date-parts":[["2016"]]},"page":"324-328","publisher":"IEEE","title":"Using LSTM and GRU neural network methods for traffic flow prediction","type":"paper-conference"},"uris":["http://www.mendeley.com/documents/?uuid=009b3053-3966-41d2-8d6c-00f576adcd29"]}],"mendeley":{"formattedCitation":"[22]","plainTextFormattedCitation":"[22]","previouslyFormattedCitation":"(22)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[21]. As a result, researchers mentioned that artificial intelligence models work better. In the other study, LSTM and recurrent ANN and regression models were compared with LSTM obtaining better results ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"author":[{"dropping-particle":"","family":"Dai","given":"Xingyuan","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Fu","given":"Rui","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Lin","given":"Yilun","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Li","given":"Li","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Wang","given":"Fei-Yue","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"arXiv preprint arXiv:1707.03213","id":"ITEM-1","issued":{"date-parts":[["2017"]]},"title":"Deeptrend: A deep hierarchical neural network for traffic flow prediction","type":"article-journal"},"uris":["http://www.mendeley.com/documents/?uuid=4e5f09fe-ad7c-49e6-838f-ca2df0ec5288"]}],"mendeley":{"formattedCitation":"[23]","plainTextFormattedCitation":"[23]","previouslyFormattedCitation":"(23)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[22].LSTM and short-term traffic flow were reviewed in the literature, but so far, there was no study on the effect of training set size on LSTM performance. Therefore, in this study, LSTM and nonlinear autoregressive neural networks (NAR) were trained with different training sets size and the optimum size was determined for the problem. In addition, two models were compared, and their results discussed. Thus, the results of this study will help to determine the size of the training set for future studies.This article consists of introduction, methodology and conclusion. The subject and importance of this study are discussed, and the related literature is summarised in the introduction section of this article. The data used and the estimation of missing data are presented in the methodology section. Then, LSTM and NAR approaches were briefly explained, and the parameters of the models used in the study were introduced. Thereafter, NAR and LSTM estimates were tested by hypothesis testing and the results were discussed. Finally, the conclusions of the study were recalled in the conclusion section and recommendations were made for future studies.METHODOLOGYData collection and missing dataTraffic flow data were collected from the D200 highway. This highway connects the major cities of Turkey. The main road traffic is not interrupted 20 km forward and backward from the counted section. Therefore, there are uninterrupted conditions in the counting section. Data collection was performed with NC-350 traffic counters ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"URL":"http://www.mhcorbin.com/portable-traffic-analyzer/","id":"ITEM-1","issued":{"date-parts":[["0"]]},"title":"Mccorbin Web Site","type":"webpage"},"uris":["http://www.mendeley.com/documents/?uuid=95e13f58-b0c0-4e34-9b8f-978b981875fb"]}],"mendeley":{"formattedCitation":"[24]","plainTextFormattedCitation":"[24]","previouslyFormattedCitation":"(24)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[23]. The counting was conducted with traffic counting devices placed separately for the left and right lanes. The devices were set to record data every 15 minutes. Devices were counted for 47 days and 4,512 traffic flow data were collected.In the counting process, data cannot be recorded at some time intervals and this is very common. This data is called missing data. This is often the result of faults in the device or the limitations of the counting device. After counting operations, it was found that approximately 1% of the total data was missing. Autocorrelation reveals the degree of relationship of time series points with each other. The points with high autocorrelation are used in making future predictions. To complete the missing data, traffic data with high autocorrelation were used with missing data. The results of the autocorrelation calculation result are given in Fig. 1. Autocorrelation was high at point 672. Each counting operation has 15 min intervals. In other words, every point in the time series is related to the point 7 days (672 / (24 * 4) previous. This is a very common pattern in traffic flows. Fig. 1. Completion of missing dataIn this case, the missing data can be completed with the value at the point 672 interval before the missing point. Let X ∊ ℤ be traffic data with missing values and xt ∊ X indicates the traffic flow data at time t. Also, let QUOTE ∊ ℤ denote the missing data in the series and at time t. According to these definitions, the missing data is completed as in Equation 1. QUOTE (1)After completing the missing data, the data set was standardised with Equation 2 before training the models. QUOTE (2)where,xstd standardised data,x raw data, QUOTE mean of the dataset,sx standard deviation of the dataset.Long short-term memory Long short-term memory network (LSTM) is an advanced type of recurrent neural networks (RNNs) that can overcome the long-term dependence problem. RNNs produced successful results in sequence prediction tasks. However, it is often difficult for RNNs to learn long-term patterns ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"https://doi.org/10.1016/j.neucom.2018.08.067","ISSN":"0925-2312","abstract":"Traffic flow prediction plays a key role in intelligent transportation systems. However, since traffic sensors are typically manually controlled, traffic flow data with varying length, irregular sampling and missing data are difficult to exploit effectively. To overcome this problem, we propose a novel approach that is based on Long Short-Term Memory (LSTM) in this paper. In addition, the multiscale temporal smoothing is employed to infer lost data and the prediction residual is learned by our approach. We demonstrate the performance of our approach on both the Caltrans Performance Measurement System (PeMS) data set and our own traffic flow data set. According to the experimental results, our approach obtains higher accuracy in traffic flow prediction compared with other approaches.","author":[{"dropping-particle":"","family":"Tian","given":"Yan","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Zhang","given":"Kaili","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Li","given":"Jianyuan","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Lin","given":"Xianxuan","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Yang","given":"Bailin","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Neurocomputing","id":"ITEM-1","issued":{"date-parts":[["2018"]]},"page":"297-305","publisher":"Elsevier B.V.","title":"LSTM-based traffic flow prediction with missing data","type":"article-journal","volume":"318"},"uris":["http://www.mendeley.com/documents/?uuid=b479a485-5f3b-4b6e-a971-0d5d99d359bb"]}],"mendeley":{"formattedCitation":"[25]","plainTextFormattedCitation":"[25]","previouslyFormattedCitation":"(25)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[24]. LSTM can understand short- or long-term dependencies with the help of units that learn when to forget and when to update the information. Let xt be the input vector, ht be the output of the LSTM unit and Ct be the cell state at time t. In the first step, how much of the information in the Ct-1 will be forgotten is determined by forget gate. The forget gate is a layer that uses sigmoid function and uses ht-1 and xt to generate values between “0” and “1”. Therefore, ft in Fig. 2 can be written as: QUOTE (3)The next step is to identify new information that will be stored in the cell state. This step consists of two sub-steps: The first step is the input gate, which determines what information to update. The second step determines the vector containing the candidate values. In Fig. 2, the output value of the input gate is represented by it, while the output value of the second section is indicated by QUOTE . The it and QUOTE can be written as: QUOTE and QUOTE (4)After these steps, the old state vector (Ct-1) is updated to reveal the new state vector (Ct). The update process can be written as: QUOTE (5)Fig. 2. Long short-term memory network unitThe last step is to determine the hidden state (ht): QUOTE (6)The output gate (ot) is the process that determines which parts of the cell state will be in the output and can be written as: QUOTE (7)where σ() is the sigmoid function, W(f,i,c,o) matrices are the network parameters, b(f,i,c,o) is the bias matrices. And ⊙ denotes the product operation. LSTM can successfully overcome the exploding/vanishing gradients problem with these processes and gates ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"ISSN":"0218-4885","author":[{"dropping-particle":"","family":"Hochreiter","given":"Sepp","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems","id":"ITEM-1","issue":"02","issued":{"date-parts":[["1998"]]},"page":"107-116","publisher":"World Scientific","title":"The vanishing gradient problem during learning recurrent neural nets and problem solutions","type":"article-journal","volume":"6"},"uris":["http://www.mendeley.com/documents/?uuid=c0bcf99c-2650-45cb-9c2a-01031b5510b8"]}],"mendeley":{"formattedCitation":"[26]","plainTextFormattedCitation":"[26]","previouslyFormattedCitation":"(26)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[25].Nonlinear auto-regressive neural networksNonlinear autoregressive neural networks (NAR) are a customised neural network (ANN) model for time series. NAR predicts the future value by using the past data of the time series. NAR needs a training set like other ANNs. Let X ∊ ℤ be the traffic flow data and xt ∊ X denotes the traffic flow value at time t. In this case, the future traffic flow value will be: x ̂_(t+1)=f(x_t,x_(t-1),…x_(t-d)). Where, x ̂_(t+1) is the prediction value of the NAR, f(x) expresses the NAR black-box function and the d is the delay value. Backpropagation algorithm ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"https://doi.org/10.1016/B978-0-12-741252-8.50010-8","ISBN":"978-0-12-741252-8","abstract":"Publisher Summary This chapter presents a survey of the elementary theory of the basic backpropagation neural network architecture, covering the areas of architectural design, performance measurement, function approximation capability, and learning. The survey includes a formulation of the backpropagation neural network architecture to make it a valid neural network and a proof that the backpropagation mean squared error function exists and is differentiable. Also included in the survey is a theorem showing that any L2 function can be implemented to any desired degree of accuracy with a three-layer backpropagation neural network. An appendix presents a speculative neurophysiological model illustrating the way in which the backpropagation neural network architecture might plausibly be implemented in the mammalian brain for corticocortical learning between nearby regions of cerebral cortex. One of the crucial decisions in the design of the backpropagation architecture is the selection of a sigmoidal activation function.","author":[{"dropping-particle":"","family":"HECHT-NIELSEN","given":"ROBERT","non-dropping-particle":"","parse-names":false,"suffix":""}],"editor":[{"dropping-particle":"","family":"Wechsler","given":"Harry B T - Neural Networks for Perception","non-dropping-particle":"","parse-names":false,"suffix":""}],"id":"ITEM-1","issued":{"date-parts":[["1992"]]},"page":"65-93","publisher":"Academic Press","title":"III.3 - Theory of the Backpropagation Neural Network**Based on “nonindent” by Robert Hecht-Nielsen, which appeared in Proceedings of the International Joint Conference on Neural Networks 1, 593–611, June 1989. © 1989 IEEE.","type":"chapter"},"uris":["http://www.mendeley.com/documents/?uuid=8d945aa0-a62c-450e-b27b-2c8a72466394"]}],"mendeley":{"formattedCitation":"[27]","plainTextFormattedCitation":"[27]","previouslyFormattedCitation":"(27)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[26] and Levenberg-Marquardt method ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1017/CBO9781107415324.004","ISBN":"9788578110796","ISSN":"1098-6596","PMID":"25246403","abstract":"Predicting the binding mode of flexible polypeptides to proteins is an important task that falls outside the domain of applicability of most small molecule and protein−protein docking tools. Here, we test the small molecule flexible ligand docking program Glide on a set of 19 non-α-helical peptides and systematically improve pose prediction accuracy by enhancing Glide sampling for flexible polypeptides. In addition, scoring of the poses was improved by post-processing with physics-based implicit solvent MM- GBSA calculations. Using the best RMSD among the top 10 scoring poses as a metric, the success rate (RMSD ≤ 2.0 Å for the interface backbone atoms) increased from 21% with default Glide SP settings to 58% with the enhanced peptide sampling and scoring protocol in the case of redocking to the native protein structure. This approaches the accuracy of the recently developed Rosetta FlexPepDock method (63% success for these 19 peptides) while being over 100 times faster. Cross-docking was performed for a subset of cases where an unbound receptor structure was available, and in that case, 40% of peptides were docked successfully. We analyze the results and find that the optimized polypeptide protocol is most accurate for extended peptides of limited size and number of formal charges, defining a domain of applicability for this approach.","author":[{"dropping-particle":"","family":"Levenberg","given":"K","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Quarterly Journal of Applied Mathematics","id":"ITEM-1","issue":"2","issued":{"date-parts":[["1944"]]},"page":"164-168","title":"A method for the solution of certain non-linear problems in least squares","type":"article-journal","volume":"2"},"uris":["http://www.mendeley.com/documents/?uuid=d6f3c750-ed2d-47eb-b816-1270fd30b7a3"]},{"id":"ITEM-2","itemData":{"DOI":"10.1137/0111030","ISBN":"0368-4245","ISSN":"0368-4245","PMID":"17746741","abstract":"Most algorithms for the least-squares estimation of non-linear parameters have centered about either of two approaches. On the one hand, the model may be expanded as a Taylor series and corrections to the several parameters calculated at each iteration on the assumption of local lincarity. On the other hand, various modifications of the method of steepest-descent have been used. Both methods not infrequently run aground, the Taylor series method because of divergence of the successive iterates, the steepest-descent (or gradient) methods because of agonizingly slow convergence after the first few iterations. In this paper a maximum neighborhood method is developed which, in effect, performs an optimum interpolation between the Taylor series method and the gradient method, the interpolation being based upon the maximum neighborhood in which the truncated Taylor series gives an adequate representation of the nonlinear model. The results are extended to the problem of solving a set of nonlinear algebraic equations.","author":[{"dropping-particle":"","family":"Marquardt","given":"Donald W.","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Journal of the Society for Industrial and Applied Mathematics","id":"ITEM-2","issue":"2","issued":{"date-parts":[["1963"]]},"page":"431-441","title":"An Algorithm for Least-Squares Estimation of Nonlinear Parameters","type":"article-journal","volume":"11"},"uris":["http://www.mendeley.com/documents/?uuid=f829bbed-1a04-494c-94ab-4c6910c0d0aa"]}],"mendeley":{"formattedCitation":"[28,29]","plainTextFormattedCitation":"[28,29]","previouslyFormattedCitation":"(28, 29)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[28,29] were used for training. The connections of the NAR with the hidden and the output layers are shown in Fig. 3. The model uses a delay parameter QUOTE to estimate the traffic flow QUOTE at time t + 1. In this study, in the hidden layer tangent hyperbolic and in the output layer linear function were used as activation functions. To determine the appropriate NAR architecture, the number of hidden layer neurons was tested from 5 to 35. Then, the RMSE of different NAR architectures were analysed and it was decided that 3-10-1 was the appropriate NAR architecture.In this section, we first introduced the creation of training and test sets. Then, the effect of the size of the training sets on the predictions of NAR and LSTM was examined and finally, the prediction results of the two methods were evaluated by statistical tests.Fig. 3. Nonlinear autoregressive neural network architectureTraffic flow vector (X) consists of 47 daily traffic flow data. The data sets to be used for training were selected in six different sizes from 5 to 30 days. Let ej ⊂ X be a training set vector and the number of data in ej will be QUOTE and j=1,2,…,6. The last 17 days of the X vector were selected for testing the models. Let tm ⊂ X be a test set vector and m=1,2,…,17. The pseudo-code for the creation of training and test sets with these representations is as follows:1. Start2. Let, n := |X|, r :=|tm|, p :=|ej|,3. m = 1,4. j = 1,5. ej = {xt | (t>(n-(j*r+p) ⋀ t≤(n-j*R)}6. tm={{xt | (t>(n-(j*r) ⋀ t≤(n-(( j-1)*r)),7. If j < p and m<r Then,j = j + 1 and turn back to Step 4If j = p and m<r Then,m= m + 1 and turn back to Step 4If j = p and m = r Then, Stop.The delay parameter QUOTE or lag value was kept equal in the LSTM and NAR models, and this value was set to QUOTE . Thus, regardless of parameter d, the effect of data set size on performance was compared.NAR and LSTM models with training set size 480 QUOTE were named NAR5 and LSTM5 and the test results were given in Fig. 4 using box-plot. In Figs. 4 and 5, the outliers were shown with the (+) sign. If these (+) signs are counted from Figs. 4 and 5, it is understood that while LSTM produces ten outliers, the NAR has four outliers. This result indicates that LSTM predictions are rarely more than expected. When the median values were examined, it was seen that the value of NAR5 was higher than the value of LSTM5. In addition, it was observed that the range of LSTM5 was smaller than NAR5 with the examination of the upper/lower whiskers. Simply put, the LSTM approach was able to produce better results than the NAR with the smallest training set size examined. Fig. 4. Comparison of errors for trained models with different-sized training setsFig. 5 shows the RMSE values produced by the models as a result of the use of training set sizes between 10 and 30 days. It can be read from the median lines depicted in Fig. 5 that the LSTM produces lower RMSE for all training set sizes. It was observed that NAR error values were oscillated by increasing the size of the training set, but no clear decrease was observed. Furthermore, it is understood from Fig. 5 that the LSTM error values tend to decrease clearly for the same training set size increase. Thus, following examination of the average RMSE values of the models, it was found that the lowest error was in QUOTE for NAR and LSTM. Based on this, the error values of the models due to their training with QUOTE , training set size were examined more closely. Fig. 5 shows that the maximum RMSE value of NAR25 is 17 veh. However, the maximum prediction error value of LSTM25 was about 13 veh.Fig. 5. Comparison of errors for trained models with different-sized training setsTo observe the prediction of the models in more detail, the 17th test day was examined in Fig. 6. And to observe the estimations of the models in more detail, the 17th test day was examined in Fig. 6. The results of the remaining test days are presented in Appendix 1 for the reader's review. The coefficients of estimation of the two models were calculated and it was determined that both models produced high R2 values. The calculated R2 values for the remaining days can be examined in Appx 2 and 3. Like the RMSE values examined in the previous figures, LSTM predictions produced R2 values higher than NAR predictions for all test days. A remarkable situation was seen during the comparison of the models on the line graph. In Fig. 6, the prediction line of NAR makes high fluctuations to approach the actual value. On the other hand, the fluctuation of LSTM was less than NAR. The same examination was performed for the other test days and the same result was reached. In the light of these results, it was concluded that LSTM was less affected by instant traffic flow changes than NAR model. Although the LSTM was found to be more accurate than NAR, the statistical significance of this result was tested by t-test. The established hypothesis statements were established as follows:H0: If LSTM is used instead of NAR, the mean RMSE does not change. (μLSTM =μNAR)H1: If LSTM is used instead of NAR, the mean RMSE is decreased. (μNAR> μLSTM)where, μNAR and μLSTM represent the mean of the estimation errors of NAR and LSTM, respectively.Fig. 6.Comparison of NAR25 and LSTM25 short-term traffic flow predictions with real values. (t-1) of the 15 minThe results of the paired t-test are summarised in Tab. 1. Tab. 1 shows that the mean difference values (μ) of the two models are positive for all ej's. This indicates that the LSTM as less mean prediction errors than NAR. The confidence level of the hypothesis test was 95% (α = 0.05). The p-value was examined from the table and it was seen that p <0.05 was found for the other training set sizes except for the 5-day training set. In the light of these results, except for 5-day training set, H0 was rejected and H1 was accepted.Tab. 1Paired t-test results (α = 0.05)Training set sizeNumber of data/Number of days|ej|μσσx̅LowerUppertdfp-valueResult480 / 5 days4,2113,483,27-2,7311,141,29160,217Ho Accept960 / 10 days1,460,700,171,101,828,56160,000Ho Reject1440 / 15 days2,714,381,060,464,972,55160,021Ho Reject1920 / 20 days1,741,050,251,202,286,85160,000Ho Reject2400 / 25 days1,731,010,241,212,257,08160,000Ho Reject2880 / 30 days1,580,730,181,201,958,92160,000Ho RejectThe statistical analysis confirmed that the LSTM model usually predicted traffic flow more accurately than the NAR model for 15-min data. In addition, the improvement in the predictive performance of the NAR model was not observed by increasing the size of the training set. However, the improvement in the predictive performance of the LSTM model was clearly observed by increasing the size of the training set. However, it was determined that the increase in the size of the training set should be at certain levels. For this study, it was found that this size should have 2400 data (25 days) number for both models.3. CONCLUSIONAccurate short-term traffic forecasts will improve the decision-making capabilities of traffic control systems. Thus, traffic flow and traffic safety will reach better levels. In this study, training sets of different sizes were created. Then, the effects of these clusters on the predictive performance of LSTM and NAR models were examined. In terms of short-term traffic estimation, it was understood from the analysis results and statistical tests that LSTM models have better predictions than NAR models. The conclusions of this study are as follows:This study showed that a large amount of training set does not increase performance. For this reason, the optimum training set size of the new deep learning approaches should be determined.The larger training set size does not always mean better performance for LSTM and NAR.Improvement in LSTM estimation performance is observed towards optimum training set size. However, the same feature cannot be mentioned for NAR.LSTM is less affected by instant traffic flow changes than the NAR model. Therefore, LSTM produces stable results from NAR for short-term traffic prediction.Statistically, the LSTM approach performs better than that of NAR when the training set size is greater than 480.It was observed that LSTM produced more outliers than NAR. Therefore, in rare cases, LSTM is likely to make high errors.In this study, the size of the LSTM training set was discussed in the context of the prediction of traffic flow. The effects of other parameters of LSTM will be investigated in future studies. For this study, tests were performed for a time interval of 15 minutes, which is commonly used in the literature. In addition, smaller time intervals can be investigated in future studies. Another limitation of this study is the use of only one data set. Future studies will be enriched with different data sets from different regions.ITS will be an indispensable tool in the future traffic control of cities. This will make future traffic flow forecasts much more important. Therefore, it can easily be foreseen that the studies will continue for more effective use of deep learning in road traffic prediction.ReferencesAhmed M.S., A.R. Cook. 1979. „Analysis of freeway traffic time-series data by using box-jenkins techniques”. Transportation Research Record. DOI: https://doi.org/10.3141/2024-03.Çetiner B., M. Sari, Borat O. 2010. „A neural network based traffic-flow prediction model”. Mathematical and Computational Applications 15(2): 269-278.Chen C., J. Hu, Q. Meng, Y. 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Expert Systems 29(2). DOI: https://doi.org/10.1111/j.1468-0394.2010.00567.x.Zeng D., J. Xu, J. Gu, L. Liu, G. Xu. 2008. „Short term traffic flow prediction using hybrid ARIMA and ANN models”. Proceedings 2008 Workshop on Power Electronics and Intelligent Transportation System. PEITS 2008. DOI: https://doi.org/10.1109/PEITS.2008.135.Zhou B., D. He, Z. Sun. 2006. „Traffic predictability based on ARIMA/GARCH model”. 2nd Conference on Next Generation Internet Design and Engineering, NGI 2006: 200-207. DOI: https://doi.org/10.1109/NGI.2006.1678242.Appendix 1. Comparison of LSTM and NAR models with actual valuesAppendix 1 (contd). Comparison of LSTM and NAR models with actual valuesReceived 02.04.2020; accepted in revised form 29.05.2020Scientific Journal of Silesian University of Technology. Series Transport is licensed under a Creative Commons Attribution 4.0 International License PAGE 32E. DoganDepartment of Civil Engineering, Faculty of Engineering, Kirikkale University, Yahsihan, 71451, Kirkkale, Turkey. Email: edogan@kku.edu.trAnalysis and comparison of long short-term memory… PAGE 31.Scientific Journal of Silesian University of Technology. Series TransportZeszyty Naukowe Politechniki Śląskiej. 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