The Empirical Assessment of the Convergence Rate for the Bootstrap Estimation in Design of Experiment Approach

Abstract:

Design of experiment (DoE) is a set of practical recipes and theoretical assumptions leading to the optimization of the technological process and/or the stabilization of its output quality. Practically, all the DoE approaches assume the normality of a random noise and the quasi-linearity of models taken from the general linear model (GLM) class. It allows to use traditional least-square methodology to identification of a model parameters and their confidence intervals. It gives usually sufficient results but completely fails if the model is not from GLM class or a random noise has not a normal distribution. The solution for such problems is the bootstrap approach, a resampling method based on Monte Carlo strategies. This paper tries to answer a question how many repetitions should be made to estimate parameters of the prediction model with sufficient accuracy.

• Periodical: Solid State Phenomena (Volume 235)
• Pages: 16-23
• DOI: https://doi.org/10.4028/www.scientific.net/SSP.235.16
• Online since: July 2015
• Authors: Przemysław Osocha*, Robert Ulewicz, Paweł Szataniak, Mariusz Pietraszek, Maciej Kołomycki, Norbert Radek, Łukasz Pasieczyński
• Keywords: Bootstrap Estimation, Convergence Rate, Design of Experiment, Parametric Model, Response Surface Model

References

[1] O. Kempthorne, K. Hinkelmann, Design and analysis of experiments. Vol. 1. Introduction to experimental design, John Wiley & Sons, Hoboken, NJ, USA, (2007).

[2] M.D. Grigoriu, Stochastic Systems - Uncertainty Quantification and Propagation, Springer-Verlag London Ltd., London, (2012).

[3] O. Christensen, K.L. Christensen, Approximation Theory - From Taylor Polynomials to Wavelets, Springer-Science+Business Media, New York, (2005).

[4] W. Feller, An Introduction to Probability Theory and Its Applications, John Wiley & Sons, Hoboken, (1968).

[5] D.C. Montgomery, Design and Analysis of Experiments, John Wiley & Sons, Inc., Hoboken, (2008).

[6] R.A. Fisher, The Design of Experiments, Oliver and Boyd Press, Edinburgh, (1935).

[7] G.E.P. Box, K.B. Wilson, On the Experimental Attainment of Optimum Conditions, J. Roy. Stat. Soc. Series B 13/1 (1951) 1-45.

[8] J. McElroy, Dr Taguchi - Japans Secret Weapon, Automot. Ind. 164/8 (1984) 1-18.

[9] M.S. Phadke, Quality Engineering Using Robust Design, Prentice Hall International, Inc., London, (1989).

[10] K.R. Bhote, A.K. Bhote, World Class Quality Second Edition - Using Design of Experiments to Make It Happen, AMACOM, New York (2000).

DOI: 10.2307/1269965

[11] Design and analysis of experiments. Vol. 3. Special Designs and Applications, K. Hinkelmann (Ed. ), John Wiley & Sons, Hoboken, NJ, USA (2012).

[12] S. de la Rosa de Sáa, M. Gil, M. García, M. Lubiano, Fuzzy Rating vs. Fuzzy Conversion Scales: An Empirical Comparison through the MSE, in: R. Kruse, M.R. Berthold, C. Moewes, M.Á. Gil, P. Grzegorzewski and O. Hryniewicz, (Eds. ), Synergies of Soft Computing and Statistics for Intelligent Data Analysis, Springer Berlin Heidelberg (2013).

DOI: 10.1007/978-3-642-33042-1_15

[13] R. Likert, A Technique for the Measurement of Attitudes, Arch Psychol 140 (1932) 1-55.

[14] J. Pietraszek, A. Goroshko, The Heuristic Approach to the Selection of Experimental Design, Model and Valid Pre-Processing Transformation of DoE Outcome, Adv Mater Res-Switz 874 (2014) 145-149.

DOI: 10.4028/www.scientific.net/amr.874.145

[15] J. Pietraszek, Response surface methodology at irregular grids based on Voronoi scheme with neural network approximator, Adv Soft Comp (2003) 250-255.

DOI: 10.1007/978-3-7908-1902-1_35

[16] A. Goroshko, V. Royzman, J. Pietraszek, Construction and practical application of hybrid statistically-determined models of multistage mechanical systems, Mechanika (2014) (5) 489-493.

DOI: 10.5755/j01.mech.20.5.8221

[17] J. Pietraszek, A. Gadek-Moszczak, T. Torunski, Modeling of Errors Counting System for PCB Soldered in the Wave Soldering Technology, Adv Mater Res-Switz 874 (2014) 139-143.

DOI: 10.4028/www.scientific.net/amr.874.139

[18] J. Shao, D. Tu, The Jackknife and Bootstrap, Springer, New York (1995).

[19] J. Pietraszek, A. Gądek-Moszczak, The Smooth Bootstrap Approach to the Distribution of a Shape in the Ferritic Stainless Steel AISI 434L Powders, 197 (2013) 162-167.

DOI: 10.4028/www.scientific.net/ssp.197.162

[20] A. Gadek-Moszczak, J. Pietraszek, B. Jasiewicz, S. Sikorska, L. Wojnar, The Bootstrap Approach to the Comparison of Two Methods Applied to the Evaluation of the Growth Index in the Analysis of the Digital X-ray Image of a Bone Regenerate, Stud Comput Intell 572 (2015).

DOI: 10.1007/978-3-319-10774-5_12

[21] N. Radek, J. Pietraszek, B. Antoszewski, The Average Friction Coefficient of Laser Textured Surfaces of Silicon Carbide Identified by RSM Methodology, Adv Mater Res-Switz 874 (2014) 29-34.

DOI: 10.4028/www.scientific.net/amr.874.29

[22] N. Radek, K. Bartkowiak, Performance properties of electro-spark deposited carbide-ceramic coatings modified by laser beam, Physcs Proc 5 (2010) 417-423.

DOI: 10.1016/j.phpro.2010.08.163

[23] N. Radek, B. Antoszewski, The influence of laser treatment on the properties of electro-spark deposited coatings, Kovove Mater 47 (2009) (1) 31-38.

[24] N. Radek, K. Bartkowiak, Laser Treatment of Cu-Mo Electro-Spark Deposited Coatings, Physcs Proc 12 (2011) 499-505.

DOI: 10.1016/j.phpro.2011.03.061

[25] N. Radek, K. Bartkowiak, Laser treatment of electro-spark coatings deposited in the carbon steel substrate with using nanostructured WC-Cu electrodes, Physcs Proc 39 (2012) 295-301.

DOI: 10.1016/j.phpro.2012.10.041

[26] N. Radek, J. Konstanty, Cermet Esd Coatings Modified by Laser Treatment, Arch Metall Mater 57 (2012) (3) 665-670.

DOI: 10.2478/v10172-012-0071-y

[27] R. Zieliński, Statistical Tables [in Polish], PWN, Warszawa (1972).

[28] Mathcad 15. 0 (M005 [MC15_M005_20101105]), Parametric Technology Corporation, Needham (2010).

[29] STATISTICA (data analysis software system), version 10., StatSoft, Inc., Tulsa, OK, USA (2011).

[30] A. Gadek-Moszczak, L. Wojnar, Objective, Quantitative and Automatic X-Ray Image. Analysis of the Bone Regenerate in the Ilizarov Method, (2009) 453-458.

[31] A. Gadek-Moszczak, S. Zmudka, Description of 3D microstructure of the composites with polypropylene (PP) matrix and Tuf particles fillers, Solid State Phenomen 197 (2013) 186-191.

DOI: 10.4028/www.scientific.net/ssp.197.186

[32] A. Gadek-Moszczak, N. Radek, S. Wronski, J. Tarasiuk, Application the 3D Image Analysis Techniques for Assessment the Quality of Material Surface Layer Before and After Laser Treatment, Adv Mater Res-Switz 874 (2014) 133-138.

DOI: 10.4028/www.scientific.net/amr.874.133

[33] A. Szczotok, R. Przeliorz, Phase transformations in CMSX-4 nickel-base superalloy, Iop Conf Ser-Mat Sci 35 (2012).

DOI: 10.1088/1757-899x/35/1/012005

[34] A. Szczotok, On gamma-gamma ' eutectic quantitative evaluation in the as-cast CMSX-4 nickel-based superalloy, Solid State Phenomen 197 (2013) 203-208.

DOI: 10.4028/www.scientific.net/ssp.197.203

[35] K. Trzewiczek, A. Szczotok, A. Gadek-Moszczak, Evaluation of the State for The Material of the Live Steam Superheater Pipe Coils of V Degree, Adv Mater Res-Switz 874 (2014) 35-42.

DOI: 10.4028/www.scientific.net/amr.874.35

[36] P. Osocha, P. Duda, B. Weglowski, Determining temperature and stress changes in thick-walled elements of steam lines, Inz Chem Procesowa 25 (2004) (4) 2249-2256.

[37] B. Weglowski, P. Osocha, Modelling of Creep for Y Pipe from Ferritic-Martensitic P91 Steel, Rynek Energii (2009) (6) 140-145.

[38] P. Duda, R. Dwornicka, Optimization of heating and cooling operations of steam gate valve, Struct Multidiscip O 40 (2010) (1-6) 529-535.

DOI: 10.1007/s00158-009-0370-8

[39] R. Dwornicka, The Impact of the Power Plant Unit Start-Up Scheme on the Pollution Load, Adv Mater Res-Switz 874 (2014) 63-69.

DOI: 10.4028/www.scientific.net/amr.874.63

[40] E. Skrzypczak-Pietraszek, A. Hensel, Polysaccharides from Melittis melissophyllum L. herb and callus, Pharmazie 55 (2000) (10) 768-771.

[41] E. Skrzypczak-Pietraszek, J. Pietraszek, Seasonal Changes of Flavonoid Content in Melittis melissophyllum L. (Lamiaceae), Chem Biodivers 11 (2014) (4) 562-570.

DOI: 10.1002/cbdv.201300148