A Comparison Analysis of Fuzzy and Bayesian Linear Model Parameter Estimates for Replicated Response Measures

Abstract

It is possible to define functional relationship between replicated response measures and input variables by using fuzzy and Bayesian modeling approaches. The main aim of the study is to present the alternative usability of fuzzy modeling approach to Bayesian modeling approach with defining a proper alpha-cut level among the many alpha-cut levels. In this study, the uncertainty of estimated model parameters were compared by transforming the estimated parameter values to intervals. Interval valued parameter estimates were obtained through alpha-cut level presentation and credible intervals for fuzzy and Bayesian approaches, respectively. Thus, it was achieved to model the replicated response measured (RRM) data set without making any probabilistic modeling assumptions which were hard to satisfy for small sized RRM data set. To compare the interval valued model parameter estimates in the proposed study, midpoint, width, radius and Hausdorff metrics were used. And also, interval type residuals were calculated to see the performance of predicted fuzzy and Bayesian models for making clear comparison. Two data sets from the literature, which were called Roman Catapult and Printing Ink, were used and the obtained results were discussed in application part.

Keywords

• Replicated response measures
• Fuzzy modeling
• Bayesian modeling
• Interval arithmetic

References

1. [1] Khuri, A. I., Mukhopadhyay, S., “Response surface methodology”, WIREs Computational Statistics, 2:128-149, (2010).
2. [2] Zadeh, L. A., “The concept of a Linguistic variable and applications to approximate reasoning-part-I, II, III”, Information Science, 8:199-249, (1975).
3. [3] Lai, Y. J., Chang, S., “A fuzzy approach for multi response optimization: An off-line quality engineering problem”, Fuzzy Sets and Systems, 63:117−129, (1994).
4. [4] Akbar, M. S., Otok, B. W., Prastyo, D. D., “Fuzzy modeling approach and global optimization for dual response surface”, Jurnal Teknik Industri, 9:102−111, (2007)
5. [5] Xie, H., Lee, Y. C., “Process optimization using a fuzzy logic response surface method”, IEEE Transactions on Components, Packaging, and Manufacturing Technology-Part A, 17 (1994).
6. [6] Prasad, K., Nath, N., “Comparison of sugarcane juice based beverage optimisation using response surface methodology with fuzzy method”, Sugar Tech., 4:109−115, (2002).
7. [7] Xu, R., Dong, Z., “Fuzzy modeling in response surface method for complex computer model based design optimization”, Proc. 2nd IEEE/ASME Int. Conf. Mechatronic and Embedded Systems and Applications, 1−6, (2006).
8. [8] Bashiri, M., Hosseininezhad, S. J., “A fuzzy programming for optimizing multi-response surface in robust designs”, Journal of Uncertain Systems, 3:163−173, (2009).

9. [9] Bashiri, M., Hosseininezhad, S. J., “Fuzzy development of multiple response optimization”, Group Decision and Negotiation 21:417-438, (2012).
10. [10] Türkşen, Ö., Güler, N., “Comparison of fuzzy logic based models for the multi-response surface problems with replicated response measures”, Applied Soft Computing, 37:887-896, (2015).
11. [11] Türkşen, Ö., “Analysis of Response Surface Model Parameters with Bayesian Approach and Fuzzy Approach”, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 24:109-122, (2016).
12. [12] Türkşen, Ö., “A Fuzzy Modeling Approach for Replicated Response Measures Based on Fuzzification of Replications with Descriptive Statistics and Golden Ratio”, Süleyman Demirel University Journal of Natural and Applied Sciences, 22:153-159, (2018).
13. [13] Steinberg, D. M., “A Bayesian Approach to Flexible Modeling of Multivariable Response Functions”, Journal of Multivariate Analysis, 34:157-172, (1990).
14. [14] Chen, Y., “Bayesian Hierarchical Modelling of Dual Response Surfaces”, Ph.D. thesis, Virginia Polytechnic Institute and State University, (2005).
15. [15] Chen, Y., Ye, K., “Bayesian Hierarchical Modelling on Dual Response Surfaces in Partially Replicated Designs”, Quality Technology & Quantitative Management, 6:371-389, (2009).
16. [16] Chen, Y., Ye, K., “A Bayesian hierarchical approach to dual response surface modelling”, Journal of Applied Statistics, 38:1963-1975, (2011).
17. [17] Peterson, J. J., Quesada, G. M., Castillo, E., “A Bayesian Reliability Approach to Multiple Response Optimization with Seemingly Unrelated Regression Models”, Quality Technology & Quantitative Management, 6:353-369, (2009).
18. [18] Quesada, G. M., Castillo, E., Peterson, J. J., “A Bayesian Approach for Multiple Response Surface Optimization in the Presence of Noise Variables”, 1-24, (2002).
19. [19] Kozan, E., Köksoy, O., “A Bayesian Parameter Estimation Approach to Response Surface Optimization in Quality Engineering”, Sakarya University Journal of Science, 23:767-774, (2019).
20. [20] Wang, L., Xiong, C., Yang, Y., “A novel methodology of reliability-based multidisciplimary design optimization under hybrid interval and fuzzy uncertainties”, Computer Methods in Applied Mechanics and Engineering, 337:439-457, (2018).
21. [21] Wang, L., Xiong, C., Wang, X., Xu, M., Li, Y., “A dimesion-wise method and its improvement for multidisciplinary interval uncertainty analysis”, Applied Mathematical Modelling, 59:680-695, (2018).
22. [22] Wang, L., Wang, X., Su, H., Lin, G., “Reliability estimation of fatigue crack growth prediction via limited measured data”, International Journal of Mechanical Sciences, 121:44-57, (2017).
23. [23] Moore, R. E., Kearfott, R. B., Cloud, M. J., “Introduction to Interval Analysis”, SIAM, (2009).
24. [24] Palumbo, F., Irpino, A. “Multidimensional Interval-Data: Metrics and Factorial Analysis”, Multivariate Statistical and Visualization Methods to Analyze, to Summarize, and to Evaluate Performance Indicators, 689-698, (2005).
25. [25] Diamond, P. “Fuzzy least squares”, Information Sciences, 46:141-157, (1998).
26. [26] Paulino, C. D., Turkman, M. A. A., Murteira, B., Silva, G. L. “Estatistica Bayesiana”, Fundaçao Calouste Gulbenkian, (2018).
27. [27] Chachi, J., Taheri, S. M. “A Least-Absolutes Regression Model for Imprecise Response Based on the Generalized Hausdorff- Metric”, Journal of Uncertain Systems, 7:265-276, (2013).
28. [28] Galdino, S. “Interval-valued Data Clustering Based on the Range City Block Metric”, 2016 IEEE International Conference on Systems, Man, and Cybernetics-SMC 2016, 228-234, (2016).
29. [29] Luner, J.J. “Achieving continuous improvement with the dual response approach: A demonstration of the Roman catapult”, Quality Engineering ,6:691-705, (1994).
30. [30] Box, G.E.P., Draper, N.R. “Response Surface Mixtures and Ridge Analysis”, John Wiley and Sons, (2007).