Piotr Kokoszka https://orcid.org/0000-0001-9979-6536 , Mengting Lin https://orcid.org/0009-0002-7712-9585 , Haonan Wang https://orcid.org/0000-0002-8892-6232 , Stephen Hayne https://orcid.org/0000-0002-9578-3364

© P. Kokoszka, M. Lin, H. Wang, S. Hayne. Article available under the CC BY-SA 4.0 licence


(English) PDF


We develop statistical methodology for the quantification of risk of source-destination pairs in an internet network. The methodology is developed within the framework of functional data analysis and copula modeling. It is summarized in the form of computational algorithms that use bidirectional source-destination packet counts as input. The usefulness of our approach is evaluated by an application to real internet traffic flows and via a simulation study.


Copula, Functional data, Internet traffic, Principal components, Risk quantification.


Ahmed, Mohiuddin, Mahmood, Abdun Naser and Hu, Jiankun, (2016). A survey of network anomaly detection techniques. Journal of Network and Computer Applications, 60, pp. 19–31.

Amovin-Assagba, Martial, Gannaz, Irene and Jacques, Julien, (2022). Outlier detection in multivariate functional data through a contaminated mixture model. Computational Statistics & Data Analysis, 174, 107496.

Awan, Mazhar Javed, Farooq, Umar, Babar, Hafiz Muhammad Aqeel, Yasin, Awais, Nobanee, Haitham, Hussain, Muzammil, Hakeem, Owais and Zain, Azlan Mohd, (2021).

Real-time DDoS attack detection system using big data approach. Sustainability, 13, no. 19, 10743.

Berrendero, José R, Justel, Ana and Svarc, Marcela, (2011). Principal components for multivariate functional data. Computational Statistics & Data Analysis, 55, no. 9, pp. 2619–2634.

Billor, Nedret, Hada, Ali and Velleman, Paul, (2000). BACON: blocked adaptive computationally efficient outlier nominators. Computational Statistics and Data Analysis, 34, pp. 272–298.

Bosq, Dennis, (2000). Linear Processes in Function Spaces. Springer.

Chiou, Jeng-Min, Chen, Yu-Ting and Yang, Ya-Fang, (2014). Multivariate functional principal component analysis: A normalization approach. Statistica Sinica, pp. 1571–1596.

Czado, Claudia, (2019). Analyzing Dependent Data with Vine Copulas: A Practical Guide with R. Springer.

Dai, Wenlin and Genton, Marc G., (2018). Multivariate functional data visualization and outlier detection. Journal of Computational and Graphical Statistics, 27, no. 4, pp. 923–934.

Demarta, Stefano and McNeil, Alexander, (2005). The t copula and related copulas. International Statistical Review, 73, pp. 111–129.

Dong, Shi and Sarem, Mudar, (2019). DDoS attack detection method based on improved KNN with the degree of DDoS attack in software-defined networks. IEEE Access, 8, pp. 5039–5048.

Ferraty, Frédérick and Vieu, Philippe, (2006). Nonparametric Functional Data Analysis: Theory and Practice. Springer.

Fouladi, Ramin Fadaei, Kayatas, Cemil Eren and Anarim, Emin, (2016). Frequency based DDoS attack detection approach using naive Bayes classification. In 2016 39th International Conference on Telecommunications and Signal Processing (TSP), pp. 104–107. IEEE.

Fouladi, Ramin Fadaei, Seifpoor, Tina and Anarim, Emin, (2013). Frequency characteristics of DoS and DDoS attacks. In 2013 21st Signal Processing and Communications Applications Conference (SIU), pp. 1–4. IEEE.

Genest, Christian and Nešlehová, Johanna, (2012). Copulas and copula models. In Encyclopedia of Environmetrics (eds El-Shaarawi A.H. and PiegorschW.W.), 2 edn, volume 2, pp. 541–553. Wiley, Chichester.

Górecki, Tomasz, Krzyśko, Mirosław, Waszak, Łukasz and Wołyński, Waldemar, (2018). Selected statistical methods of data analysis for multivariate functional data. Statistical Papers, 59, no. 1, pp. 153–182.

Happ, Clara and Greven, Sonja, (2018). Multivariate functional principal component analysis for data observed on different (dimensional) domains. Journal of the American Statistical Association, 113, number 522, pp. 649–659.

Hofert, Marius, Kojadinovic, Ivan, Mächler, Martin and Yan, Jun, (2018). Elements of Copula Modeling with R. Springer.

Horváth, Lajos and Kokoszka, Piotr, (2012). Inference for Functional Data with Applications, volume 200. Springer Science & Business Media.

Hubert, Mia, Rousseeuw, Peter J and Vanden Branden, Karlien, (2005). ROBPCA: a new approach to robust principal component analysis. Technometrics, 47, no. 1, pp. 64–79.

Hussain, Alefiya, Heidemann, John and Papadopoulos, Christos, (2003). A framework for classifying denial of service attacks. In Proceedings of the 2003 Conference on Applications, Technologies, Architectures, and Protocols for Computer Communications, pp. 99–110.

Jacques, Julien and Preda, Cristian, (2014). Model-based clustering for multivariate functional data. Computational Statistics & Data Analysis, 71, pp. 92–106.

Joe, Harry, (2015). Dependence Modeling with Copulas. Chapman & Hall.

Kokoszka, Piotr and Reimherr, Matthew, (2017). Introduction to Functional Data Analysis. Chapman and Hall/CRC.

Krzyśko, Mirosław and Smaga, Łukasz, (2020). Measuring and testing mutual dependence of multivariate functional data. Statistics in Transition, 21, no. 3, pp. 21–37.

Krzyśko, Mirosław and Smaga, Łukasz, (2021). Two-sample tests for functional data using characteristic functions. Austrian Journal of Statistics, 50, no. 4, pp. 53–64.

Liao, Hung-Jen, Lin, Chun-Hung Richard, Lin, Ying-Chih and Tung, Kuang-Yuan, (2013). Intrusion detection system: A comprehensive review. Journal of Network and Computer Applications, 36, no. 1, pp. 16–24.

Modi, Chirag, Patel, Dhiren, Borisaniya, Bhavesh, Patel, Hiren, Patel, Avi and Rajarajan, Muttukrishnan, (2013). A survey of intrusion detection techniques in Cloud. Journal of Network and Computer Applications, 36, no. 1, pp. 42–57.

Nelsen, Roger, (2006). An Introduction to Copulas. Springer.

Nishanth, N. and Mujeeb, A., (2020). Modeling and detection of flooding-based denialof- service attack in wireless ad hoc network using Bayesian inference. IEEE Systems Journal, 15, no. 1, pp. 17–26.

Peng, Chen, Xu, Maochao, Xu, Shouhuai and Hu, Taizhong, (2018). Modeling multivariate cybersecurity risks. Journal of Applied Statistics, 45, no. 15, pp. 2718–2740.

Ramsay, James and Silverman, Bernard, (2005). Functional Data Analysis. Springer.

Sambangi, Swathi and Gondi, Lakshmeeswari, (2020). A machine learning approach for DDoS (distributed denial of service) attack detection using multiple linear regression. In Proceedings, volume 63, p. 51. MDPI.

Soysal, Murat and Schmidt, Ece Guran, (2010). Machine learning algorithms for accurate flow-based network traffic classification: Evaluation and comparison. Performance Evaluation, 67, no. 6, pp. 451–467.

Wu, Zhijun, Yue, Meng, Li, Douzhe and Xie, Ke, (2015). SEDP-based detection of low-rate DoS attacks. International Journal of Communication Systems, 28, no. 11, pp. 1772–1788.

Xu, Maochao, Hua, Lei and Xu, Shouhuai, (2017). A vine copula model for predicting the effectiveness of cyber defense early-warning. Technometrics, 59, no. 4, pp. 508–520.

Back to top
© 2019–2024 Copyright by Statistics Poland, some rights reserved. Creative Commons Attribution-ShareAlike 4.0 International Public License (CC BY-SA 4.0) Creative Commons — Attribution-ShareAlike 4.0 International — CC BY-SA 4.0