Citation :

Kanaa Thomas, Ngongang Ludovic, Huisken Mejouyo Paul William, Pesdjock Mathieu Jean Pierre, Ndongue Esseme Emmanuel, Tonye Emmanuel, "Gram-Charlier Series for Enhanced Areal Roughness Assessment in End-Milled Low-Carbon Steel Surface Replication," International Journal of Engineering Trends and Technology (IJETT), vol. 74, no. 5, pp. 342-361, 2026. Crossref, https://doi.org/10.14445/22315381/IJETT-V74I5P123

Abstract

Areal roughness significantly determines surface functionality. Traditional contact methods, though standard, are limiting surface evaluation to single profiles, rendering that evaluation inadequate for multi-tooth processes like milling. Non-contact techniques, however, rely on contact-based regression. This study proposes a framework that applies Gram-Charlier Probability Density Functions (PDFs) to enhance contact measurement accuracy for optimizing surface replication in machining operations. The approach models areal roughness using both Gaussian and Gram-Charlier PDFs. Analysis of 170 milled surfaces under varied machining conditions involves three-dimensional contact measurements that employ Hermite polynomials and cumulants for Gram-Charlier functions construction. The methodology determines roughness values through PDF peak analysis. The methodology determines roughness values through PDF peak analysis. Validation demonstrates that Gram-Charlier models achieve R² values exceeding 0.99 across all extraction densities (four to twenty-one linear extractions per surface), confirming superior performance. Response Surface Methodology links this statistical model to machining parameters, enabling replication similarity assessment. Application of Minkowski and Shannon's entropy family distance metrics identifies optimal combinations of spindle speed, depth of cut, and feed per tooth, revealing the depth of cut as the most sensitive parameter, and the carbon steel with 0.15% as the best material over the three selected for achieving precise surface replication control.

Keywords

Contact Area Roughness, Gram-Charlier Series, Replicate Surfaces, Similarity/Distance, Response Surface Methodology.

References

[1] Rüstem Binali et al., “Machinability Investigations Based on Tool Wear, Surface Roughness, Cutting Temperature, Chip Morphology and Material Removal Rate during Dry and MQL-Assisted Milling of Nimax Mold Steel,” Lubricants, vol. 11, no. 3, pp. 1-14, 2023.
[CrossRef] [Google Scholar] [Publisher Link]

[2] Praveen Kumar Gandla et al., “Evaluation of Surface Roughness in Incremental Forming Using Image Processing-Based Methods,” Measurement, vol. 164, 2020.
[CrossRef] [Google Scholar] [Publisher Link]  

[3] Baofeng He, Siyuan Ding, and Zhaoyao Shi, “A Comparison between Profile and Areal Surface Roughness Parameters,” Metrology and Measurement Systems, vol. 28, no. 3, pp. 413-438, 2021.
[CrossRef] [Google Scholar] [Publisher Link]    

[4]  J. Schmidt et al., “Comparison of Areal and Profile Surface Measurement Methods for Evaluating Surface Properties of Machined Components,” Procedia CIRP, vol. 102, pp. 459-464, 2021.
[CrossRef] [Google Scholar] [Publisher Link]

[5] Alain Cornet, and Jean-Paul Deville, Surface Physics and Engineering, EDP Sciences, 1998.
[Google Scholar] [Publisher Link]

[6] Shi Hyoung Ryu, Deok Ki Choi, and Chong Nam Chu, “Roughness and Texture Generation on End Milled Surfaces,” International Journal of Machine Tools and Manufacture, vol. 46, no. 3-4, pp. 404-412, 2006.
[CrossRef] [Google Scholar] [Publisher Link] 

[7] V.F. Bez’’yazychnyi, and A.N. Sutyagin, “Influence of Vibration on Surface Roughness in Turning,” Russian Engineering Research, vol. 39, no. 7, pp. 612-616, 2019.
[CrossRef] [Google Scholar] [Publisher Link] 

[8] D.D. Trung, “Influence of Cutting Parameters on Surface Roughness during Milling AISI 1045 Steel,” Tribology in Industry, vol. 42, no. 4, pp. 658-665, 2020.
[CrossRef] [Google Scholar] [Publisher Link]

[9] John Henry Navarro-Devia et al., “Chatter Detection in Milling Processes - A Review on Signal Processing and Condition Classification,” The International Journal of Advanced Manufacturing Technology, vol. 125, no. 9-10, pp. 3943-3980, 2023.
[CrossRef] [Google Scholar] [Publisher Link]

[10] Richard Leach, Characterisation of Areal Surface Texture, 1^st ed., Springer Berlin, Heidelberg, 2013.
[CrossRef] [Google Scholar] [Publisher Link]

[11] Vladimir Lyukshin, Dmitry Shatko, and Pavel Strelnikov, “Methods and Approaches to the Surface Roughness Assessment,” Materials Today: Proceedings, vol. 38, pp. 1441-1444, 2020.
[CrossRef] [Google Scholar] [Publisher Link]

[12] Shilpa Karegoudra, and Vamsidhar Yendapalli, “A Systematic Review on Non-Contact Methods to Estimate the Surface Roughness,” Materials Today: Proceedings, vol. 100, pp. 75-92, 2024.
[CrossRef] [Google Scholar] [Publisher Link]

[13] Dana Kubátová, “Problems of Surface Roughness Measurement with a Focus on the Scanned Points,” Proceedings of the 29^th DAAAM International Symposium, Vienna, Austria, pp. 1256-1265, 2018.
[CrossRef] [Google Scholar] [Publisher Link]

[14] Szeréna-Krisztina Fecske et al., “Interdependence of Amplitude Roughness Parameters on Rough Gaussian Surfaces,” Tribology Letters, vol. 68, no. 43, 2020.
[CrossRef] [Google Scholar] [Publisher Link]

[15] Juraj Ružbarský, “The Difficulty of Measuring the Roughness of Glossy Surfaces Using the Triangulation Principle,” Applied Sciences, vol. 13, no. 8, pp. 1-14, 2023.
[CrossRef] [Google Scholar] [Publisher Link] 

[16] G. Farkas, and Á. Drégelyi-Kiss, “Measurement Uncertainty of Surface Roughness Measurement,” IOP Conference Series: Materials Science and Engineering, XXIII International Conference on Manufacturing, Kecskemét, Hungary, vol. 448, 2018.
[CrossRef] [Google Scholar] [Publisher Link] 

[17] Enhui Lu et al., “Designing Indices to Measure Surface Roughness Based on the Color Distribution Statistical Matrix (CDSM),” Tribology International, vol. 122, pp. 96-107, 2018.
[CrossRef] [Google Scholar] [Publisher Link] 

[18] Donna L. Mohr, William J. Wilson, and Rudolf J. Freund, Chapter 2 - Probability and Sampling Distributions, Statistical Methods, 4^th ed., Academic Press, pp. 65-122, 2022.
[CrossRef] [Publisher Link] 

[19] Yves Brunet-Moret, Distribution Functions in Limited Development, ORSTOM Notebooks, Hydrology Series, 1971. [Online]. Available: https://horizon.documentation.ird.fr/exl-doc/pleins_textes/pleins_textes_4/hydrologie/14871.pdf

[20] Chris Tsokos, and Rebecca Wooten, The Joy of Finite Mathematics: The Language and Art of Math, Academic Press, 2016.
[CrossRef] [Google Scholar] [Publisher Link]

[21] Tieyuan Shen, and Yiheng Xiang, “Optimization of Probability Density Functions Applicable for Hourly Rainfall,” Atmosphere, vol. 14, no. 7, pp. 1-23, 2023.
[CrossRef] [Google Scholar] [Publisher Link] 

[22] Harald Cramér, Mathematical Methods of Statistics, Princeton University Press, 1999.
[Google Scholar] [Publisher Link] 

[23] William Feller, An Introduction to Probability Theory and its Applications, Volume 2, 2^nd ed., Wiley, 1971.
[Google Scholar] [Publisher Link] 

[24] Singiresu S. Rao, Engineering Optimization: Theory and Practice, 4^th ed., Wiley, 2019.
[CrossRef] [Google Scholar] [Publisher Link] 

[25] S. Blinnikov, and R. Moessner, “Expansions for Nearly Gaussian Distributions,” Astronomy and Astrophysics Supplement Series, vol. 130, no. 1, pp. 193-205, 1998.
[CrossRef] [Google Scholar] [Publisher Link] 

[26] Junye Ma, and Lin Li, “Literature Review on Engineering Surface Modeling,” MethodsX, Vol. 10, 2023.
[CrossRef] [Google Scholar] [Publisher Link] 

[27] Tobias Grimm et al., “Surface Analysis in Additive Manufacturing: A Systematic Literature Review Regarding Powder Bed Fusion Processes,” Surface Topography: Metrology and Properties, vol. 13, no. 1, pp. 1-36, 2025.
[CrossRef] [Google Scholar] [Publisher Link]

[28] Carsten Engler et al., “New Method for Assessing the Repeatability of the Measuring System for Roughness Measurements,” International Journal of Metrology and Quality Engineering (IJMQE), vol. 155, pp. 1-15, 2024.
[CrossRef] [Google Scholar] [Publisher Link]

[29] Eric Jondeau, and Michael Rockinger, “Gram-Charlier Densities,” Journal of Economic Dynamics and Control, vol. 25, no. 10, pp. 1457-1483, 2001.
[CrossRef] [Google Scholar] [Publisher Link]

[30] Armando Pelliccioni, “The Gram-Charlier Method to Evaluate the Probability Density Function in Monodimensional Case,” Nuovo Cimento-Italian Physical Society Section C, pp. 432-452, 1996.
[Google Scholar]

[31] Maria Grazia Zoia, Gianmarco Vacca, and Laura Barbieri, “Modeling Multivariate Financial Series and Computing Risk Measures via Gram-Charlier-Like Expansions,” Risks, vol. 8, no. 4, pp. 1-21, 2020.
[CrossRef] [Google Scholar] [Publisher Link]

[32] Ángel León, and Trino-Manuel Ñíguez, “The Transformed Gram Charlier Distribution: Parametric Properties and Financial Risk Applications,” Journal of Empirical Finance, vol. 63, pp. 323-349, 2021.
[CrossRef] [Google Scholar] [Publisher Link]

[33] A. Pelliccioni A, and M. Angeloni, “A New Formulation of the Gram-Charlier Method: Performance for Fitting Non-Normal Distribution,” Nuovo Cimento-Italian Physical Society Section C, vol. 30, no, 4, pp. 381-394, 2007.
[Google Scholar]

[34] Wei Wang, Naoki Ikegaya, and Tsubasa Okaze, “Comparing Weibull Distribution Method and Gram-Charlier Series Method within the Context of Estimating Low-Occurrence Strong Wind Speed of Idealized Building Cases,” Journal of Wind Engineering and Industrial Aerodynamics, vol. 236, 2023.
[CrossRef] [Google Scholar] [Publisher Link]

[35] Grzegorz Tyl et al., “Population Balance Approach to Model Ostwald Ripening of Silica Using Gram-Charlier Series Expansion-Based Closure,” Chemical Engineering Research and Design, vol. 159, pp. 491-504, 2020.
[CrossRef] [Google Scholar] [Publisher Link]

[36] William Palin Elderton, Frequency Curves and Correlation, 4^th ed., Cambridge University Press, 2011.
[Google Scholar] [Publisher Link]

[37] Samuel Kotz, Distribution in Statistics: Continuous Univariate Distributions, John Wiley & Sons, 1970.
[Google Scholar]

[38] Alan Stuart, and Keith Ord, Kendall's Advanced Theory of Statistics, Volume 1, Distribution Theory, 6^th Ed., John Wiley & Sons, 2010.
[Google Scholar] [Publisher Link]

[39] R.A. Fisher, Statistical Methods for Research Workers, Breakthroughs in Statistics, Springer, New York, NY, pp. 66-70, 1992.
[CrossRef] [Google Scholar] [Publisher Link]

[40] Solomon Kullback, and Richard A. Leibler, “On Information and Sufficiency,” The Annals of Mathematical Statistics, vol. 22, no. 1, pp. 79-86, 1951.
[Google Scholar] [Publisher Link]

[41] L.N. Vaserstein, “Markov Processes Over Denumerable Products of Spaces Describing Large Systems of Automata,” Problems of Information Transfer, vol. 5, no. 3, pp. 64-73, 1969.
[Google Scholar] [Publisher Link]

[42] Alison L. Gibbs, and Francis Edward Su, “On Choosing and Bounding Probability Metrics,” International Statistical Review, vol. 70, no. 3, pp. 419-435, 2002.
[CrossRef] [Google Scholar] [Publisher Link]

[43] Jie Yu et al., “A New Study on Distance Metrics as Similarity Measurement,” 2006 IEEE International Conference on Multimedia and Expo, Toronto, ON, Canada, pp. 533-536, 2006.
[CrossRef] [Google Scholar] [Publisher Link]

[44] Sung-Hyuk Cha, “Comprehensive Survey on Distance/Similarity Measures between Probability Density Functions,” International Journal of Mathematical models and Methods in Applied Sciences, vol. 1, no. 4, pp. 300-307, 2007.
[CrossRef] [Google Scholar] [Publisher Link]

[45] M-J. Lesot, M. Rifqi, and H. Benhadda, “Similarity Measures for Binary and Numerical Data: A Survey,” International Journal of Knowledge Engineering and Soft Data Paradigms, vol. 1, no. 1, pp. 63-84, 2008.
[CrossRef] [Google Scholar] [Publisher Link]

[46] Seung-Seok Choi, Sung-Hyuk Cha, and Charles C. Tappert, “A Survey of Binary Similarity and Distance Measures,” Journal of Systemics, Cybernetics and Informatics, vol. 8, no. 1, pp. 43-48, 2010.
[Google Scholar] [Publisher Link]

[47] Shaker K. Ali, Zahoor M. Aydam, and Biadaa M. Rashed, “Similarity Metrics for Classification: A Review,” IOP Conference Series: Materials Science and Engineering: 2^nd International Scientific Conference of Al-Ayen University (ISCAU-2020), Thi-Qar, Iraq, vol. 928, pp. 1-18, 2020.
[CrossRef] [Google Scholar] [Publisher Link]

[48] Shuyuan Yang et al., “Progressive Neighbours’ Pursuit for Radar Images Classification,” Applied Soft Computing, vol. 109, 2021.
[CrossRef] [Google Scholar] [Publisher Link]

[49] Fischer Black, and Myron Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, vol. 81, no.  3, pp. 637-654, 1973.
[CrossRef] [Google Scholar] [Publisher Link]

[50] R.E. Kalman, “A New Approach to Linear Filtering and Prediction Problems,” Journal of Fluids Engineering, vol. 82, no. 1, pp. 35-45, 1960.
[CrossRef] [Google Scholar] [Publisher Link]

[51] Carl Edward Rasmussen, and Christopher K.I. Williams, Gaussian Processes for Machine Learning, MIT Press, 2005.
[CrossRef] [Google Scholar] [Publisher Link]

[52] Douglas A. Reynolds, Thomas F. Quatieri, and Robert B. Dunn, “Speaker Verification Using Adapted Gaussian Mixture Models,” Digital Signal Processing, vol. 10, no. 1-3, pp. 19-41, 2000.
[CrossRef] [Google Scholar] [Publisher Link]

[53] HáKon Gudbjartsson, and Samuel Patz, “The Rician Distribution of Noisy MRI Data,” Magnetic Resonance in Medicine, vol. 34, no. 6, pp. 910-914, 1995.
[CrossRef] [Google Scholar] [Publisher Link]

[54] R. Lyman Ott, and Michael T. Longnecker, An Introduction to Statistical Methods and Data Analysis, 7^th ed., Brooks/Cole, Cengage Learning, 2010.
[Google Scholar] [Publisher Link]

[55] Wei Wang, Koki Seta, and Naoki Ikegaya, “Modelling Probability Density Functions based on the Gram-Charlier Series with Higher-Order Statistics: Theoretical Derivation and Application,” Journal of Wind Engineering and Industrial Aerodynamics, vol. 231, 2022.
[CrossRef] [Google Scholar] [Publisher Link]   

[56] Jacques Goupy, Introduction to Experimental Design, 5^th ed., Dunod, 2017.
[Publisher Link]   

[57] P. Dagnelie, “Statistical Theory and Methods (vol. 2). Gembloux Agricultural Press, Variability of the Rainfall Regime of Non-Sahelian West Africa, 1975.
[Google Scholar]

[58] Wei Lin, and Jin E. Zhang, “The Valid Regions of Gram-Charlier Densities with High Order Cumulants,” Journal of Computational and Applied Mathematics, vol. 407, 2022.
[CrossRef] [Google Scholar] [Publisher Link]   

[59] Mário N. Berberan-Santos, “Expressing a Probability Density Function in Terms of Another PDF: A Generalized Gram-Charlier Expansion,” Journal of Mathematical Chemistry, vol. 42, no. 3, pp. 585-594, 2006.
[CrossRef] [Google Scholar] [Publisher Link]   

[60] D.E. Barton, and K.E. Dennis, “The Conditions under Which Gram-Charlier and Edgeworth Curves are Positive Definite and Unimodal,” Biometrika, vol. 39, no. 3-4, pp. 425-427, 1952.
[CrossRef] [Google Scholar] [Publisher Link]   

[61] Jean-Pierre Chateau, and Daniel Dufresne, “Gram-Charlier Processes and Equity-Indexed Annuities,” Centre for Actuarial Studies, University of Melbourne, Working Paper 227, 2012.
[Google Scholar]

[62] Oh Kang Kwon, “Analytic Expressions for the Positive Definite and Unimodal Regions of Gram-Charlier Series,” Communications in Statistics - Theory and Methods, vol. 51, no. 15, pp. 5064-5084, 2022.
[CrossRef] [Google Scholar] [Publisher Link]   

[63] Sai Lotfi et al., “The Effect of High-Speed Milling on Surface Roughness of 42CrMo4 Hardened Steel Using a Ball Nose End-Mill Cutter,” International Conference Design and Modeling of Mechanical Systems, Hammamet, Tunisia, vol. 1, pp. 375-381, 2019.
[CrossRef] [Google Scholar] [Publisher Link]

[64] Davide Chicco, Matthijs J. Warrens, and Giuseppe Jurman, “The Coefficient of Determination R-Squared is more Informative than SMAPE, MAE, MAPE, MSE and RMSE in Regression Analysis Evaluation,” PeerJ Computer Science, vol. 7, pp. 1-24, 2021.
[CrossRef] [Google Scholar] [Publisher Link]

[65] V.B. Bokov, “Surface Roughness in Final Turning: Development of an Adequate Empirical Model,” Russian Engineering Research, vol. 36, no. 7, pp. 559-564, 2016.
[CrossRef] [Google Scholar] [Publisher Link]

[66] David J. Whitehouse, Handbook of Surface Metrology, 1^st ed., CRC Press, 1994.
[CrossRef] [Google Scholar] [Publisher Link]

[67] ISO 4288:1996, Geometrical Product Specifications (GPS) - Surface Texture: Profile Method - Rules and Procedures for the Assessment of Surface Texture, ISO, 1996. [Online]. Available: https://www.iso.org/standard/2096.html

[68] Stanislaw Raczynski, Chapter Eleven - Uncertain Future: A Trip, Reachable Sets of Dynamic Systems, Academic Press, 2023.
[CrossRef] [Google Scholar] [Publisher Link]

[69] Bharat Bhushan, Principles and Applications of Tribology, John Wiley & Sons, 2013.
[CrossRef] [Google Scholar] [Publisher Link]

[70] Claudiu L. Giusca, Richard K. Leach, and Frank Helery, “Calibration of the Scales of Areal Surface Topography Measuring Instruments: Part 2 – Amplification Coefficient, Linearity and Squareness,” Measurement Science and Technology, vol. 23, no. 6, 2012.
[CrossRef] [Google Scholar] [Publisher Link]

[71] Tzu-Liang Tseng, Udayvarun Konada, and Yongjin Kwon, “A Novel Approach to Predict Surface Roughness in Machining Operations using Fuzzy Set Theory,” Journal of Computational Design and Engineering, vol. 3, no. 1, pp. 1-13, 2016.
[CrossRef] [Google Scholar] [Publisher Link]

[72] A.I. Germashev et al., “Influence of the Cut Axial Depth on Surface Roughness at High-Speed Milling of Thin-Walled Workpieces,” Science and Technology, vol. 20, no. 2, pp. 127-131, 2021.
[CrossRef] [Google Scholar] [Publisher Link]

[73]  Pradeep L. Menezes, Kishore, Satish V. Kailas, “Influence of Roughness Parameters on Coefficient of Friction under Lubricated Conditions,” Sadhana, vol. 33, no. 3, pp. 181-190, 2008.
        [CrossRef] [Google Scholar] [Publisher Link]