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The computer-based bootstrap method as a tool to select a relevant surface roughness parameter
Wear 254 (2003) 450–460
The computer-based bootstrap method as a tool to select a relevant surface roughness parameter D. Najjar∗ , M. Bigerelle, A. Iost Laboratoire de Métallurgie Physique et Génie des Matériaux, UMR CNRS 8517, Ecole Nationale Supérieure d’Arts et Métiers, Equipe Surfaces et Interfaces, 8 Boulevard Louis XIV, 59046 Lille Cedex, France Received 14 May 2002; accepted 28 January 2003
Abstract The aim of this paper is to present how to make the most of the recent and powerful statistical computer-based bootstrap method (CBBM) in roughness studies. This work shows that this statistical method can help to determine quantitatively, and without preconception, the most relevant roughness parameter that characterises the surface morphology of a manufactured product as far as a correlation with a particular function, property or application is concerned. The efficiency of this statistical method is illustrated in this paper describing the relationships between the brightness level and the surface roughness of cold-rolled low carbon steel strips; the relevance of 100 or so roughness parameters was studied via a computer software we have been upgrading for a few years. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Bootstrap method; Surface parameters; Roughness; Brightness; Cold-rolling
1. Introduction The roughness of machined surfaces is of prime importance across a very wide spectrum of technical and scientific activities; including not only tribologists and production engineers but also highway and aircraft engineers, hydrodynamicists and even bioengineers [1–5]. Indeed, the morphology of machined surfaces often influences the properties that govern the application of the manufactured product. Consequently there still exists an increasing interest in developing reliable methodologies suitable for quality control stage processes of surface products in a manufacturing environment. Because of these various industrial and scientific interests, a proliferation of roughness parameters, possibly running into hundreds, has been triggered to describe the different kinds of surface morphology with regard to specific functions, properties or applications. In spite of this proliferation, termed by Whitehouse the “parameter rash”, there is still no complete comprehensive account for the relevance of these roughness parameters [1,6]. This probably comes from a lack of global methodology combined with the limits of the software presently on the market whose func∗ Corresponding author. Tel.: +33-320-622765; fax: +33-320-622957. E-mail address: [email protected] (D. Najjar).
tion is to characterise a surface morphology. In our opinion, the main objective of such a global methodology should be to determine quantitatively, and without preconception, the most relevant roughness parameter that characterises the surface morphology of a manufactured product with regard to a correlation with a particular function, property or application. The main purpose of the present work is to introduce a new methodology so as to leave aside the subjectivity sometimes encountered in the practical and/or theoretical use of roughness parameters. This methodology aims at the construction and the calculation of a performance index as well as a confidence interval in order to account for the relevance of roughness parameters. It is based on an original statistical treatment of experimental data using the bootstrap theory first introduced by Efron et al. [7–10]. Roughly speaking, this computer-based bootstrap method (CBBM), which will be described in this paper, allows the replacement of statistical inference assumptions (therefore limiting the risk to assert wrong conclusions) by intensive calculations while making the most of the power of modern personal computers. The efficiency of the proposed methodology will be emphasised in this work through the study of the relationships between the brightness level and the surface roughness of cold-rolled low carbon steel strips.
0043-1648/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0043-1648(03)00140-6
D. Najjar et al. / Wear 254 (2003) 450–460
2. Experimental procedure 2.1. Presentation of material and industrial process The samples investigated were extracted from a low carbon steel (0.05 wt.% of carbon) strip manufactured by SOLLAC Atlantique France to obtain a bright finish aspect of the surface product. After hot-rolling in an industrial plant, the strip underwent a cold-rolling in another plant, which included: a pickling line, a 4-stand tandem mill, a batch annealing and a skin-pass mill. The strip dimensions were approximately 1 km long, 1 m wide and 1 mm thick. The bright finish aspect of the strips produced is controlled during the routine industrial process both by visual inspection and by the means of a glossmeter. At the end of the manufacturing process, visual inspection sometimes enables to detect parallel bands with different brightness levels over the whole length of skin-passed strips. Such a kind of brightness heterogeneity is obviously detrimental to the subsequent applications needing a bright finish aspect of the manufactured product. In this study, a sample was taken from a skin-passed strip visually presenting five parallel bands with different brightness levels. That sample was noted LRS (for “low roughness sample”). A sample originating from the same strip, but taken just after the 4-stand mill operation, was also considered to assess the effect of the roughness level on the results. That sample was noted HRS (for “high roughness sample”). 2.2. Measurements and numerical treatment Thirty surface profiles of 1 mm long were recorded in a 50 cm×10 cm area included in each of the five parallel bands of the LRS sample taken after the skin-pass mill. These areas were noted respectively J1, J2, J3, J4 and J5. The measurements were performed perpendicularly to the rolling direction using a tactile profilometer (Perthometer). This device recorded and electrically amplified approximately 8000 points along the profile length. Then, thirty brightness level measurements were performed with the glossmeter REFO 60-M (DIN67530, ISO 2813) used in the control stage of the routine industrial process. This device uses a visible light beam with a 60◦ —incidence angle. A sensor records the scattered intensity for a 60◦ —reflection angle (specular direction) and the intensity ratio (i.e. the brightness level) is printed out on a screen. The same experimental procedure was also strictly applied to the HRS sample originating from the same strip but taken just after the 4-stand mill operation. To the authors, the currently marketed computer programs intended for the characterisation of surface morphology most often investigate only one data file and estimate only several roughness parameters. They are also often adapted to a specific measuring device and are inadequate for statistical analysis. Moreover, another important concern is the timelessness of the software. Due to these limits, our laboratory has been developing a specific computer software for a few
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years, so as to process numerically the recorded profiles, the brightness level measurements and more generally all kinds of 2D or 3D signals. This software is based on two algorithms: a first one using the TurboPascal language and a second one using the statistical analysis system (SAS) language. In comparison with the software available on the market, the first algorithm enables to process simultaneously numerous data files originating from different experimental devices and to estimate numerous profile parameters. For each profile, 100 or so roughness parameters can be estimated by this algorithm; for more details about these calculated parameters, see [11]. In this study, particular attention will be paid to: • The arithmetic average roughness parameter Ra and the number of peaks per inch CTE PC; parameters commonly used in metallurgical and mechanical industries during the quality control stage of the products’ surface morphology. • The root mean square roughness Rq , the autocorrelation length Lac , the mean slope Da , the root mean square slope Dq and the fractal dimension ; parameters used in the models dealing with the relationships between the morphology of a surface and an optical wave [3,12–15]. The second algorithm allows an unbiased quantitative determination of the most relevant roughness parameter as far as the correlation is concerned between the surface morphology and a property or a function of a metallurgical product. The quantitative relevance of each roughness parameter Ri (1 ≤ i ≤ 105 in this study) is estimated by this algorithm computing the values of a statistical index of performance. This index was defined by applying the CBBM to the linear regression method. Based on the mathematical resampling technique, the main principle of this CBBM consists in generating a high number K (10000 in this study) of simulated bootstrap samples from the experimental data points exploiting the power of a modern computer. A bootstrap sample of size n (30 in this study), referred by k (1 ≤ k ≤ K) and noted (t1k , t2k , . . . , tnk ), is a collection of n values obtained by randomly sampling with replacement from the experiExp. Exp. Exp. mental data points (t1 , t2 , . . . , tn ). The bootstrap data set thus consists of some elements of the original data set; some never appearing, others appearing once, others appearing twice, etc. The bootstrap method, applied in this study to the linear regression method, is summarised on Fig. 1 for a given roughness parameter Ri . Whatever the area j (j = J1, J2, J3, J4, J5) in which the measurements have been carried out, the experimental data set contains thirty valExp. Exp. Exp. ues (B1 , B2 , . . . , B30 )j of brightness level and Exp. Exp. Exp. thirty values (Ri,1 , Ri,2 , . . . , Ri,30 )j of the roughness parameter under consideration. The simulated bootstrap samples obtained by randomly sampling with replacement scores of the experimental data set are respectively noted k ) and (R k , R k , . . . , R k ) ; the su(B1k , B2k , . . . , B30 j i,1 i,2 i,30 j perscript k refers to the kth bootstrap simulation. The means
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Exp.
Exp.
Exp.
Fig. 1. Synoptic scheme of the different stages of the computer-based bootstrap method applied to the linear regression method. (B1 , B2 , . . . , B30 )J 1 Exp. Exp. Exp. and (Ri,1 , Ri,2 , . . . , Ri,30 )J 1 represent respectively the thirty experimental values recorded in the area J1 for the brightness level and for the arbitrarily selected roughness parameter Ri .
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of these newly simulated samples are then calculated for each area j and are respectively noted (B k )j and (Rik )j .
These means are reported on a graph (B k )j = f ((Rik )j ) from which a slope noted Slopeki can be determined if a linear correlation is assumed between the variables. Repeating this procedure K times (k = 1, 2, . . . , K), a set of K slope values {Slope1i , Slope2i , . . . , SlopeK i } can thereby be obtained to plot an empirical probability density function (PDF) related to the roughness parameter Ri under consideration. Then, a mean µi of the bootstrapped slope values {Slope1i , Slope2i , . . . , SlopeK i } can firstly be calculated from this empirical PDF. Secondly, the 5% and the 95% quantiles (noted inf 5% and sup95% respectively) can i i be extracted to assess a 90% confidence interval equal to − inf 5% sup95% i . Thirdly, these statistical estimates are used i to define an unscaled coefficient Pik related to the roughness parameter Ri and the kth bootstrap simulation:
Pik =
slopeki sup95% i
− inf 5% i
Let us note P¯i the mean value of Pik defined by: µi P¯i = 95% supi − inf 5% i
(1)
(2)
If sup95% > 0 and inf 5% < 0, i.e. µi approximates the i i null-value, then no correlation exists between the roughness parameter Ri and the brightness level. The calculation of P¯i for each studied roughness parameter allows the quantitative ranking of their relevance as far as a linear correlation versus the brightness level is concerned; the higher |P¯i | the more relevant the related roughness parameter Ri . This statistical index of performance |P¯i | will be called the correlation power.
3. Experimental results 3.1. Relationships between roughness parameters and brightness level: traditional analysis Fig. 2 presents, for each specific area (J1, J2, J3, J4 and J5) of the LRS sample, a typical profile and the related probability histogram of heights considering all profiles. All the histograms are roughly normal in shape and, whatever the area, both the typical profiles and the probability histograms of the profile heights look similar at first sight. It is, therefore, not possible at this stage to differentiate the surface morphologies of the various areas even though they visually have different brightness levels. Fig. 3 summarises for each area the means and the standard error values of the brightness level, the arithmetic average roughness Ra , the number of peaks par inch CTE PC, the root mean square Rq , the autocorrelation length Lac , the mean slope Da , the root mean square slope Dq and the fractal dimension ANAM . The superscript ANAM means that
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the fractal dimension was estimated using the average normalised autocorrelation method (ANAM) developed by the authors [16,17]. On the whole profile length, the ANAM mathematically consists in applying a double averaging operation rather than a single one as in the oscillation or in the structure methods. This mathematical operation actually increases the robustness of the fractal dimension determination. Moreover, this method does not require the self-affinity of the profile contrary to the spectral density function method. Even though the mean brightness levels of the considered areas are sometimes close, they are always significantly different since no overlap of the standard bar errors can be noticed in Fig. 3. A similar analysis of the evolution of the roughness parameters Ra , CTE PC, Rq and Lac shows that an overlap of standard bar errors is always observed between different areas under study (see the areas noted J1, J3 and J4 in the case of Ra or J2, J3 and J5 in the case of Lac for examples). Consequently it is impossible to differentiate the various surface morphologies while taking into consideration these parameters commonly used either practically in metallurgical and mechanical industries or theoretically in the models dealing with the relationships between the morphology of a surface and an optical wave. Moreover, no obvious simple correlation seems to exist between the evolutions of these roughness parameters and the brightness level of the different areas under analysis. Contrary to the conclusions related to the four roughness parameters cited above, the calculated means of the slope parameters Da , Dq and fractal dimension ANAM are always significantly different. Indeed, no overlap of the standard bar errors can be noticed for these roughness parameters for the areas under study. Moreover, at first sight, a simple linear correlation seems to exist between these roughness parameters and the brightness levels. This visual impression is confirmed by looking at Table 1 which summarises the slopes of the best-fit straight lines estimated by the traditional linear regression method as well as their related standard errors, t-values and P-levels [18]. The results reported in this table indicate that the strongest and most significant correlation is observed in the case of the fractal dimension ANAM ; roughness parameter which has the highest t-value Table 1 Results of the linear regression method for the main roughness parameters of interest under study Roughness parameters
Slope
Standard error of the slope
t-value
P-level
ANAM Ra Rq CTE PC Lac Da Dq
31.02 −56 −64 −0.00241 0.28 −514 −417
8.32 94 81 0.002 0.14 163 144
3.72 −0.6 −0.8 −0.87 2.03 −3.1 −2.89
0.03 0.58 0.48 0.44 0.13 0.06 0.0629
The t-value represents the magnitude of the correlation while the P-level represents its reliability (only the results that lead to a P-level ≤0.05 are customarily considered borderline statistically significant).
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Fig. 2. Typical roughness profile and associated histogram of height distributions (8024 points × 30 profiles) for each area of the LRS sample.
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Fig. 3. Means and standard error values of brightness level, ANAM , Ra , CTE PC, Rq , Lac , Da and Dq related to the areas under study in the case of the LRS sample.
and the lowest P-level. By using this traditional approach, only the results that lead to a P-level ≤0.05 are customarily considered borderline statistically significant. The results reported in the table thus show that only the fractal dimension ANAM is relevant at this confidence level. Among the
main roughness parameters studied, the fractal dimension ANAM thus seems the most relevant one to describe the surface morphology of a skin-passed low carbon steel strip as far as a simple linear correlation with the brightness level is concerned.
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3.2. Bootstrap analysis Two main limits can be outlined in the traditional statistical procedure presented above to study the linear correlation between the roughness parameters and the brightness level of a skin-passed low carbon steel strip. The first limit is inherent to the statistical linear regression method itself with regard to our experimental results. Since the recorded values of the roughness parameters and brightness levels come from different experimental devices (a profilometer and a Exp.
glossmeter), only the related means (B Exp. )j and (Ri )j (with j = J1–J5) of these variables can be considered to find Exp.
the best-fit straight line of the graph (B Exp. )j = f ((Ri
)j )
while the deviations of the experimental scores from the calculated means must be given up. In other terms, for the variables under study, the information about their respective variability is necessarily lost when applying the linear regression method. The second limit is related to the number of roughness parameters under study. Indeed, there is no proof that the fractal dimension ANAM will remain the most relevant roughness parameter compared with other non-studied parameters like skewness for example. In order to trespass these two limits, we have been upgrading for a few years a computer application based on the bootstrap theory. The bootstrap analysis of the experimental results obtained on the LRS sample is first illustrated in Fig. 4 where the fractal dimension ANAM is the
Fig. 4. Results of the bootstrap analysis applied to the linear regression method between the brightness level and the fractal dimension estimated by ANAM for the LRS sample: (a) graph of the obtained slopes considering three bootstrap simulations; (b) histogram of the bootstrap slopes considering 10000 simulations and presentation of the statistical estimators used in the definition of P¯ANAM .
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Fig. 5. Histogram of the values of Pi for the roughness parameters Ra , CTE PC, Rq , Lac , Da , Dq and ANAM for the LRS sample.
selected roughness parameter. Fig. 4a shows the result of three bootstrap simulations and Fig. 4b shows the associated empirical PDF if 10000 bootstrap samples are taken into account. In this figure, the statistical estimates used to calculate the value of the correlation power |P¯ANAM | are marked with arrows. Fig. 5 presents the histogram of Pi for the main roughness parameters of interest in this study. Note that all the empirical PDFs are fairly normal in shape and that the fractal dimension ANAM has the highest correlation power. Fig. 6a is a plot summarising all the correlation power values for the 100 or so computed roughness parameters. In this figure, the numbers indicated between brackets show the respective position of Ra , CTE PC, Rq , Lac , Da , Dq and ANAM in the ranking according to their respective correlation power value. The dotted curves represent the 90% confidence intervals calculated from the empirical PDFs of all the roughness parameters. Because of the obvious overlap of some confidence intervals, it can be seen that a number of roughness parameters have the same kind of influence with regard to the brightness level. This figure shows that the fractal dimension ANAM always has the highest correlation power even if the 100 or so computed roughness parameters are considered. This means that the fractal dimension ANAM must be considered as the most relevant roughness parameter as far as a linear correlation is assumed between the surface topography and the brightness level of a skin-passed low carbon steel strip. This conclusion is emphasised by the fact that the fractal dimensions estimated by other usual methods are also well located in the ranking (oscillation method: third position, structure method: fifth position, power spectral density method: twelfth position). However, the correlation power is always lower when the fractal dimension is estimated by these methods rather than by ANAM. Therefore, the CBBM brings here an experimental confirmation of the robustness of the ANAM in com-
parison with usual methods intended to estimate the fractal dimension. Fig. 6b presents the correlation power values obtained for the HRS sample strictly following the same experimental procedure. An example of profile is also inserted in this figure for comparison with a profile recorded on the LRS sample in an equivalent area. One must know that the probability histograms of heights of the five areas under study are also fairly normal in shape for the HRS sample. It first appears in Fig. 6b that there is always an overlap of the confidence intervals and the amplitude range of the correlation power values is lower than in Fig. 6a. This means that the relevant discrimination of roughness parameters with brightness level is more difficult for the sample taken just after the 4-stand mill in comparison with the sample taken after the skin-pass mill. For the HRS sample, the first positions in the ranking are now mainly occupied by amplitude parameters. The most relevant parameter is the arithmetic average roughness parameter Ra immediately followed by the root mean square roughness Rq . 4. Discussion Efron first introduced the computer-based bootstrap method to avoid the risk of asserting wrong conclusions when analysing experimental data that transgress inference assumptions of the traditional statistical theory [7–10]. A main reason for making parametric assumptions is to facilitate the derivation from textbook formulas for standard errors. Unfortunately, the traditional statistical theory does not provide any formula to assess the accuracy of most statistical estimates other than the mean. Since no formula are required when using the CBBM in non-parametric mode, restrictive and sometimes dangerous assumptions about the form of underlying populations can be avoided.
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Fig. 6. Graph of the relevance of the whole roughness parameters ranked as a function of their correlation power |P¯i | for (a) the LRS sample; (b) the HRS sample. The numbers between brackets for Ra , CTE PC, Rq , Lac , Da , Dq and ANAM show the respective position of these roughness parameters in the ranking.
Moreover, a standard error (thus an assessment of the accuracy) can be calculated for any computable statistical estimate using the constructed empirical PDF. The CBBM does not work alone but its efficiency is emphasised when applied to traditional statistical procedure as
shown in this study in the case of the linear regression method. In our opinion, the CBBM is all the more relevant as it is a part of the search for global methodology to study the relationships between roughness parameters X and a
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surface property Y. When assuming a linear correlation between these variables, the least-square method used to estimate the best-fit straight line is based on the equation: Y = aX + b + ε where a is the slope, b the intercept and ε a randomly distributed homoscedastic error. Such an equation supposes firstly that the values of X must be evaluated without any noise. In other words, the variance of X must be equal to zero. However, both measurements of the variables X and Y are usually performed with different experimental devices (for example, the roughness profiles have been recorded with a tactile profilometer and the brightness level with a scatterometer in this study). Hence both these variables get a variance different from zero. Secondly, the assumption of error homoscedasticity (i.e. the variance of the error must be the same for each class of X values) is not always true. Moreover, apart from these limits, it has to be remembered that the morphology of a surface can be characterised by numerous roughness parameters. Theoretically speaking, statistical inference assumptions on the underlying population thus have to be tediously verified for each parameter to be able to calculate a confidence interval on any statistical estimate. In a mere application of this traditional statistical theory, any transgression of the inference assumptions, referring either to the linear regression method or to the form of the underlying population, may lead to a systematic bias on the estimation of the slope and/or the associated confidence interval. Since the use of mathematical formulas is limited in the bootstrap method, systematic biases on the estimation of statistics are minimised as well as the risk of asserting wrong conclusions on the relevance of roughness parameters with regard to the considered surface property. Basing on the purely statistical advantages mentioned above, the CBBM has been taken into account to define a performance index (the correlation power) with a confidence interval to assess the relevance of 100 or so roughness parameters with regard to a linear relationship versus the brightness level of cold-rolled low carbon steel strips. For each roughness parameter, these indicators were deduced from an empirical PDF constructed from the experimental data points only by the means of a computer and without any inference assumptions. Thanks to our algorithm, they can be simply plotted on only one graph to obtain a visual ranking and a direct appreciation of the relevance of all computed roughness parameters. The most important information that can be extracted from such a graph is related to the most relevant parameter in the ranking. Indeed, this parameter has to be considered as a privileged candidate both in the assessment of the surface quality of manufactured products during the industrial control stage and in the models dealing with the relationships between the surface morphology and an optical wave. In this study, the fractal dimension ANAM has been shown to be the most relevant roughness parameter in the case of a skin-passed strip. However, for a rougher sample taken from the same strip just after the tandem mill operation, the arithmetic average roughness must be preferred. Another interesting piece of information comes
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from the overlap of the confidence intervals, which indicates that number of roughness parameters have the same influence as far as the brightness level of cold-rolled low carbon steel strips is concerned. This confirms the well-known fact pointed out by Whitehouse that some roughness parameters have the same meaning [1,6]. Finally, it must be outlined that our computer software is much upgradable. Since the core of this application is based on the SAS language, it allows to process a lot of data file formats. Therefore, other surface properties than the brightness level can be assessed by various experimental devices for any future application. The algorithm can likewise be easily modified to test: firstly the relevance of any new roughness parameter found in the literature or created for a particular application, secondly a measuring technique (2D or 3D tactile profilometer, confocal microscopy, interferometer, AFM etc.) or thirdly a current treatment applied to profiles before the calculation of roughness parameters (filtering, fitting, scale length, cut-off, orientation of measurements etc.). The higher the calculated correlation power, the more relevant the new roughness parameter, the measuring technique and/or the pre-treatment. For the sake of simplicity and tractability, a linear correlation was considered between the roughness parameter and the brightness level in this study. However, simple non-linear mathematical functions can also be analysed by applying variable transformations like x → log(x), x → x 2 , x → exp(−x) etc. in the computer algorithm. The efficiency of the variable transformation can be then assessed looking at the values of the correlation power.
5. Conclusions This paper emphasises the efficiency of the computerbased bootstrap method when applying to the study of the relationships between roughness parameters and the brightness level of cold-rolled low carbon steel strips. As claimed by Efron, this method aims to carry out familiar statistical calculation (mean, standard errors, biases, confidence intervals etc.) but in an unfamiliar way: by purely computational means, rather than mathematical formulas [10]. The most of the CBBM has been made to introduce a new methodology so as to leave aside the subjectivity sometimes encountered in the practical and/or theoretical use of roughness parameters. Thanks to this method, we have defined an index of performance (the correlation power) with a confidence interval which allow to assess the relevance of 100 or so roughness parameters. This index indicates from an unbiased quantitative point of view the most relevant roughness parameter that characterises the surface morphology of cold-rolled low carbon steel strips with regard to their brightness level. Thanks to our upgradable computer application, such a methodology presented in this work can be extended to the study of any simple correlation between a surface morphol-
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ogy manufactured by any kind of machining process and any measurable surface property.
Acknowledgements The authors would like to thank V. Hague for the language correctness of the original manuscript. References [1] T.R. Thomas, Rough Surfaces 2nd ed., Imperial College Press, London, 1999. [2] D.J. Whitehouse, Handbook of Surface Metrology, Institute of Physics, Bristol, 1994. [3] J.C. Russ, Fractal Surfaces, Plenum Press, New York, 1994. [4] U. Persson, Measurement of surface roughness on rough machined surfaces using spectral speckle correlation and image analysis, Wear 160 (1993) 221–225. [5] J.C. Le Bosse, G. Hansali, J. Lopez, J.C. Dumas, Characterization of surface roughness by laser scattering: diffusely scattered intensity measurement, Wear 224 (1999) 236–244. [6] D.J. Whitehouse, Parameter rash—is there a cure? Wear 83 (1982) 75–78. [7] B. Efron, Bootstrap methods: another look at the jacknife, Ann. Stat. 7 (1979) 1–26.
[8] B. Efron, R. Tibshirani, Bootstrap measures for standard errors, confidence intervals, and other measures of statistical accuracy, Stat. Sci. 1 (1986) 54–77. [9] B. Efron, R. Tibshirani, Statistical data analysis in the computer age, Science 253 (1991) 390–395. [10] B. Efron, R. Tibshirani, An Introduction to the Bootstrap, Chapman and Hall, New York, 1993. [11] M. Bigerelle, Caractérisation géométrique des surfaces et interfaces: applications en métallurgie, Ph.D. Thesis, Ecole Nationale Supérieure d’Arts et Métiers, Lille, France, 1999. [12] P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, Pergamon Press, London, 1963. [13] E. Jackeman, Non-gaussian statistical models for scattering calculations, in: J.C. Dainty, D. Maystre (Eds.), Waves in Random Media, vol. 1, An Institute of Physics Journal, 1991, No. 3, pp. S109– S119. [14] R. Botet, E.Y. Poliakov, V.M. Shalaev, V.A. Markel, Fractal-surfaceenhanced optical responses, in: J.L. Vehel, E. Lutton, C. Tricot (Eds.), Fractals in Engineering, Springer, London, 1997, pp. 237–251. [15] T.V. Vorburger, E. Marx, T.R. Lettieri, Regimes of surface roughness measurable with light scattering, Wear 32 (1993) 3401–3408. [16] M. Bigerelle, A. Iost, Calcul de la dimension fractale d’un profil par la méthode des autocorrélations moyennées normées (AMN), C.R. Acad. Sci., Série IIb 323 (1996) 669–675. [17] M. Bigerelle, A. Iost, A new method to calculate fractal dimension of a surface: Application to the human cell proliferation, Comput. Math. Appl. 42 (2001) 241–253. [18] R.L. Packett, Principles of Regression Analysis, Oxford University Press, 1960.
The computer-based bootstrap method as a tool to select a relevant surface roughness parameter D. Najjar∗ , M. Bigerelle, A. Iost Laboratoire de Métallurgie Physique et Génie des Matériaux, UMR CNRS 8517, Ecole Nationale Supérieure d’Arts et Métiers, Equipe Surfaces et Interfaces, 8 Boulevard Louis XIV, 59046 Lille Cedex, France Received 14 May 2002; accepted 28 January 2003
Abstract The aim of this paper is to present how to make the most of the recent and powerful statistical computer-based bootstrap method (CBBM) in roughness studies. This work shows that this statistical method can help to determine quantitatively, and without preconception, the most relevant roughness parameter that characterises the surface morphology of a manufactured product as far as a correlation with a particular function, property or application is concerned. The efficiency of this statistical method is illustrated in this paper describing the relationships between the brightness level and the surface roughness of cold-rolled low carbon steel strips; the relevance of 100 or so roughness parameters was studied via a computer software we have been upgrading for a few years. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Bootstrap method; Surface parameters; Roughness; Brightness; Cold-rolling
1. Introduction The roughness of machined surfaces is of prime importance across a very wide spectrum of technical and scientific activities; including not only tribologists and production engineers but also highway and aircraft engineers, hydrodynamicists and even bioengineers [1–5]. Indeed, the morphology of machined surfaces often influences the properties that govern the application of the manufactured product. Consequently there still exists an increasing interest in developing reliable methodologies suitable for quality control stage processes of surface products in a manufacturing environment. Because of these various industrial and scientific interests, a proliferation of roughness parameters, possibly running into hundreds, has been triggered to describe the different kinds of surface morphology with regard to specific functions, properties or applications. In spite of this proliferation, termed by Whitehouse the “parameter rash”, there is still no complete comprehensive account for the relevance of these roughness parameters [1,6]. This probably comes from a lack of global methodology combined with the limits of the software presently on the market whose func∗ Corresponding author. Tel.: +33-320-622765; fax: +33-320-622957. E-mail address: [email protected] (D. Najjar).
tion is to characterise a surface morphology. In our opinion, the main objective of such a global methodology should be to determine quantitatively, and without preconception, the most relevant roughness parameter that characterises the surface morphology of a manufactured product with regard to a correlation with a particular function, property or application. The main purpose of the present work is to introduce a new methodology so as to leave aside the subjectivity sometimes encountered in the practical and/or theoretical use of roughness parameters. This methodology aims at the construction and the calculation of a performance index as well as a confidence interval in order to account for the relevance of roughness parameters. It is based on an original statistical treatment of experimental data using the bootstrap theory first introduced by Efron et al. [7–10]. Roughly speaking, this computer-based bootstrap method (CBBM), which will be described in this paper, allows the replacement of statistical inference assumptions (therefore limiting the risk to assert wrong conclusions) by intensive calculations while making the most of the power of modern personal computers. The efficiency of the proposed methodology will be emphasised in this work through the study of the relationships between the brightness level and the surface roughness of cold-rolled low carbon steel strips.
0043-1648/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0043-1648(03)00140-6
D. Najjar et al. / Wear 254 (2003) 450–460
2. Experimental procedure 2.1. Presentation of material and industrial process The samples investigated were extracted from a low carbon steel (0.05 wt.% of carbon) strip manufactured by SOLLAC Atlantique France to obtain a bright finish aspect of the surface product. After hot-rolling in an industrial plant, the strip underwent a cold-rolling in another plant, which included: a pickling line, a 4-stand tandem mill, a batch annealing and a skin-pass mill. The strip dimensions were approximately 1 km long, 1 m wide and 1 mm thick. The bright finish aspect of the strips produced is controlled during the routine industrial process both by visual inspection and by the means of a glossmeter. At the end of the manufacturing process, visual inspection sometimes enables to detect parallel bands with different brightness levels over the whole length of skin-passed strips. Such a kind of brightness heterogeneity is obviously detrimental to the subsequent applications needing a bright finish aspect of the manufactured product. In this study, a sample was taken from a skin-passed strip visually presenting five parallel bands with different brightness levels. That sample was noted LRS (for “low roughness sample”). A sample originating from the same strip, but taken just after the 4-stand mill operation, was also considered to assess the effect of the roughness level on the results. That sample was noted HRS (for “high roughness sample”). 2.2. Measurements and numerical treatment Thirty surface profiles of 1 mm long were recorded in a 50 cm×10 cm area included in each of the five parallel bands of the LRS sample taken after the skin-pass mill. These areas were noted respectively J1, J2, J3, J4 and J5. The measurements were performed perpendicularly to the rolling direction using a tactile profilometer (Perthometer). This device recorded and electrically amplified approximately 8000 points along the profile length. Then, thirty brightness level measurements were performed with the glossmeter REFO 60-M (DIN67530, ISO 2813) used in the control stage of the routine industrial process. This device uses a visible light beam with a 60◦ —incidence angle. A sensor records the scattered intensity for a 60◦ —reflection angle (specular direction) and the intensity ratio (i.e. the brightness level) is printed out on a screen. The same experimental procedure was also strictly applied to the HRS sample originating from the same strip but taken just after the 4-stand mill operation. To the authors, the currently marketed computer programs intended for the characterisation of surface morphology most often investigate only one data file and estimate only several roughness parameters. They are also often adapted to a specific measuring device and are inadequate for statistical analysis. Moreover, another important concern is the timelessness of the software. Due to these limits, our laboratory has been developing a specific computer software for a few
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years, so as to process numerically the recorded profiles, the brightness level measurements and more generally all kinds of 2D or 3D signals. This software is based on two algorithms: a first one using the TurboPascal language and a second one using the statistical analysis system (SAS) language. In comparison with the software available on the market, the first algorithm enables to process simultaneously numerous data files originating from different experimental devices and to estimate numerous profile parameters. For each profile, 100 or so roughness parameters can be estimated by this algorithm; for more details about these calculated parameters, see [11]. In this study, particular attention will be paid to: • The arithmetic average roughness parameter Ra and the number of peaks per inch CTE PC; parameters commonly used in metallurgical and mechanical industries during the quality control stage of the products’ surface morphology. • The root mean square roughness Rq , the autocorrelation length Lac , the mean slope Da , the root mean square slope Dq and the fractal dimension ; parameters used in the models dealing with the relationships between the morphology of a surface and an optical wave [3,12–15]. The second algorithm allows an unbiased quantitative determination of the most relevant roughness parameter as far as the correlation is concerned between the surface morphology and a property or a function of a metallurgical product. The quantitative relevance of each roughness parameter Ri (1 ≤ i ≤ 105 in this study) is estimated by this algorithm computing the values of a statistical index of performance. This index was defined by applying the CBBM to the linear regression method. Based on the mathematical resampling technique, the main principle of this CBBM consists in generating a high number K (10000 in this study) of simulated bootstrap samples from the experimental data points exploiting the power of a modern computer. A bootstrap sample of size n (30 in this study), referred by k (1 ≤ k ≤ K) and noted (t1k , t2k , . . . , tnk ), is a collection of n values obtained by randomly sampling with replacement from the experiExp. Exp. Exp. mental data points (t1 , t2 , . . . , tn ). The bootstrap data set thus consists of some elements of the original data set; some never appearing, others appearing once, others appearing twice, etc. The bootstrap method, applied in this study to the linear regression method, is summarised on Fig. 1 for a given roughness parameter Ri . Whatever the area j (j = J1, J2, J3, J4, J5) in which the measurements have been carried out, the experimental data set contains thirty valExp. Exp. Exp. ues (B1 , B2 , . . . , B30 )j of brightness level and Exp. Exp. Exp. thirty values (Ri,1 , Ri,2 , . . . , Ri,30 )j of the roughness parameter under consideration. The simulated bootstrap samples obtained by randomly sampling with replacement scores of the experimental data set are respectively noted k ) and (R k , R k , . . . , R k ) ; the su(B1k , B2k , . . . , B30 j i,1 i,2 i,30 j perscript k refers to the kth bootstrap simulation. The means
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Exp.
Exp.
Exp.
Fig. 1. Synoptic scheme of the different stages of the computer-based bootstrap method applied to the linear regression method. (B1 , B2 , . . . , B30 )J 1 Exp. Exp. Exp. and (Ri,1 , Ri,2 , . . . , Ri,30 )J 1 represent respectively the thirty experimental values recorded in the area J1 for the brightness level and for the arbitrarily selected roughness parameter Ri .
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of these newly simulated samples are then calculated for each area j and are respectively noted (B k )j and (Rik )j .
These means are reported on a graph (B k )j = f ((Rik )j ) from which a slope noted Slopeki can be determined if a linear correlation is assumed between the variables. Repeating this procedure K times (k = 1, 2, . . . , K), a set of K slope values {Slope1i , Slope2i , . . . , SlopeK i } can thereby be obtained to plot an empirical probability density function (PDF) related to the roughness parameter Ri under consideration. Then, a mean µi of the bootstrapped slope values {Slope1i , Slope2i , . . . , SlopeK i } can firstly be calculated from this empirical PDF. Secondly, the 5% and the 95% quantiles (noted inf 5% and sup95% respectively) can i i be extracted to assess a 90% confidence interval equal to − inf 5% sup95% i . Thirdly, these statistical estimates are used i to define an unscaled coefficient Pik related to the roughness parameter Ri and the kth bootstrap simulation:
Pik =
slopeki sup95% i
− inf 5% i
Let us note P¯i the mean value of Pik defined by: µi P¯i = 95% supi − inf 5% i
(1)
(2)
If sup95% > 0 and inf 5% < 0, i.e. µi approximates the i i null-value, then no correlation exists between the roughness parameter Ri and the brightness level. The calculation of P¯i for each studied roughness parameter allows the quantitative ranking of their relevance as far as a linear correlation versus the brightness level is concerned; the higher |P¯i | the more relevant the related roughness parameter Ri . This statistical index of performance |P¯i | will be called the correlation power.
3. Experimental results 3.1. Relationships between roughness parameters and brightness level: traditional analysis Fig. 2 presents, for each specific area (J1, J2, J3, J4 and J5) of the LRS sample, a typical profile and the related probability histogram of heights considering all profiles. All the histograms are roughly normal in shape and, whatever the area, both the typical profiles and the probability histograms of the profile heights look similar at first sight. It is, therefore, not possible at this stage to differentiate the surface morphologies of the various areas even though they visually have different brightness levels. Fig. 3 summarises for each area the means and the standard error values of the brightness level, the arithmetic average roughness Ra , the number of peaks par inch CTE PC, the root mean square Rq , the autocorrelation length Lac , the mean slope Da , the root mean square slope Dq and the fractal dimension ANAM . The superscript ANAM means that
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the fractal dimension was estimated using the average normalised autocorrelation method (ANAM) developed by the authors [16,17]. On the whole profile length, the ANAM mathematically consists in applying a double averaging operation rather than a single one as in the oscillation or in the structure methods. This mathematical operation actually increases the robustness of the fractal dimension determination. Moreover, this method does not require the self-affinity of the profile contrary to the spectral density function method. Even though the mean brightness levels of the considered areas are sometimes close, they are always significantly different since no overlap of the standard bar errors can be noticed in Fig. 3. A similar analysis of the evolution of the roughness parameters Ra , CTE PC, Rq and Lac shows that an overlap of standard bar errors is always observed between different areas under study (see the areas noted J1, J3 and J4 in the case of Ra or J2, J3 and J5 in the case of Lac for examples). Consequently it is impossible to differentiate the various surface morphologies while taking into consideration these parameters commonly used either practically in metallurgical and mechanical industries or theoretically in the models dealing with the relationships between the morphology of a surface and an optical wave. Moreover, no obvious simple correlation seems to exist between the evolutions of these roughness parameters and the brightness level of the different areas under analysis. Contrary to the conclusions related to the four roughness parameters cited above, the calculated means of the slope parameters Da , Dq and fractal dimension ANAM are always significantly different. Indeed, no overlap of the standard bar errors can be noticed for these roughness parameters for the areas under study. Moreover, at first sight, a simple linear correlation seems to exist between these roughness parameters and the brightness levels. This visual impression is confirmed by looking at Table 1 which summarises the slopes of the best-fit straight lines estimated by the traditional linear regression method as well as their related standard errors, t-values and P-levels [18]. The results reported in this table indicate that the strongest and most significant correlation is observed in the case of the fractal dimension ANAM ; roughness parameter which has the highest t-value Table 1 Results of the linear regression method for the main roughness parameters of interest under study Roughness parameters
Slope
Standard error of the slope
t-value
P-level
ANAM Ra Rq CTE PC Lac Da Dq
31.02 −56 −64 −0.00241 0.28 −514 −417
8.32 94 81 0.002 0.14 163 144
3.72 −0.6 −0.8 −0.87 2.03 −3.1 −2.89
0.03 0.58 0.48 0.44 0.13 0.06 0.0629
The t-value represents the magnitude of the correlation while the P-level represents its reliability (only the results that lead to a P-level ≤0.05 are customarily considered borderline statistically significant).
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Fig. 2. Typical roughness profile and associated histogram of height distributions (8024 points × 30 profiles) for each area of the LRS sample.
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Fig. 3. Means and standard error values of brightness level, ANAM , Ra , CTE PC, Rq , Lac , Da and Dq related to the areas under study in the case of the LRS sample.
and the lowest P-level. By using this traditional approach, only the results that lead to a P-level ≤0.05 are customarily considered borderline statistically significant. The results reported in the table thus show that only the fractal dimension ANAM is relevant at this confidence level. Among the
main roughness parameters studied, the fractal dimension ANAM thus seems the most relevant one to describe the surface morphology of a skin-passed low carbon steel strip as far as a simple linear correlation with the brightness level is concerned.
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3.2. Bootstrap analysis Two main limits can be outlined in the traditional statistical procedure presented above to study the linear correlation between the roughness parameters and the brightness level of a skin-passed low carbon steel strip. The first limit is inherent to the statistical linear regression method itself with regard to our experimental results. Since the recorded values of the roughness parameters and brightness levels come from different experimental devices (a profilometer and a Exp.
glossmeter), only the related means (B Exp. )j and (Ri )j (with j = J1–J5) of these variables can be considered to find Exp.
the best-fit straight line of the graph (B Exp. )j = f ((Ri
)j )
while the deviations of the experimental scores from the calculated means must be given up. In other terms, for the variables under study, the information about their respective variability is necessarily lost when applying the linear regression method. The second limit is related to the number of roughness parameters under study. Indeed, there is no proof that the fractal dimension ANAM will remain the most relevant roughness parameter compared with other non-studied parameters like skewness for example. In order to trespass these two limits, we have been upgrading for a few years a computer application based on the bootstrap theory. The bootstrap analysis of the experimental results obtained on the LRS sample is first illustrated in Fig. 4 where the fractal dimension ANAM is the
Fig. 4. Results of the bootstrap analysis applied to the linear regression method between the brightness level and the fractal dimension estimated by ANAM for the LRS sample: (a) graph of the obtained slopes considering three bootstrap simulations; (b) histogram of the bootstrap slopes considering 10000 simulations and presentation of the statistical estimators used in the definition of P¯ANAM .
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Fig. 5. Histogram of the values of Pi for the roughness parameters Ra , CTE PC, Rq , Lac , Da , Dq and ANAM for the LRS sample.
selected roughness parameter. Fig. 4a shows the result of three bootstrap simulations and Fig. 4b shows the associated empirical PDF if 10000 bootstrap samples are taken into account. In this figure, the statistical estimates used to calculate the value of the correlation power |P¯ANAM | are marked with arrows. Fig. 5 presents the histogram of Pi for the main roughness parameters of interest in this study. Note that all the empirical PDFs are fairly normal in shape and that the fractal dimension ANAM has the highest correlation power. Fig. 6a is a plot summarising all the correlation power values for the 100 or so computed roughness parameters. In this figure, the numbers indicated between brackets show the respective position of Ra , CTE PC, Rq , Lac , Da , Dq and ANAM in the ranking according to their respective correlation power value. The dotted curves represent the 90% confidence intervals calculated from the empirical PDFs of all the roughness parameters. Because of the obvious overlap of some confidence intervals, it can be seen that a number of roughness parameters have the same kind of influence with regard to the brightness level. This figure shows that the fractal dimension ANAM always has the highest correlation power even if the 100 or so computed roughness parameters are considered. This means that the fractal dimension ANAM must be considered as the most relevant roughness parameter as far as a linear correlation is assumed between the surface topography and the brightness level of a skin-passed low carbon steel strip. This conclusion is emphasised by the fact that the fractal dimensions estimated by other usual methods are also well located in the ranking (oscillation method: third position, structure method: fifth position, power spectral density method: twelfth position). However, the correlation power is always lower when the fractal dimension is estimated by these methods rather than by ANAM. Therefore, the CBBM brings here an experimental confirmation of the robustness of the ANAM in com-
parison with usual methods intended to estimate the fractal dimension. Fig. 6b presents the correlation power values obtained for the HRS sample strictly following the same experimental procedure. An example of profile is also inserted in this figure for comparison with a profile recorded on the LRS sample in an equivalent area. One must know that the probability histograms of heights of the five areas under study are also fairly normal in shape for the HRS sample. It first appears in Fig. 6b that there is always an overlap of the confidence intervals and the amplitude range of the correlation power values is lower than in Fig. 6a. This means that the relevant discrimination of roughness parameters with brightness level is more difficult for the sample taken just after the 4-stand mill in comparison with the sample taken after the skin-pass mill. For the HRS sample, the first positions in the ranking are now mainly occupied by amplitude parameters. The most relevant parameter is the arithmetic average roughness parameter Ra immediately followed by the root mean square roughness Rq . 4. Discussion Efron first introduced the computer-based bootstrap method to avoid the risk of asserting wrong conclusions when analysing experimental data that transgress inference assumptions of the traditional statistical theory [7–10]. A main reason for making parametric assumptions is to facilitate the derivation from textbook formulas for standard errors. Unfortunately, the traditional statistical theory does not provide any formula to assess the accuracy of most statistical estimates other than the mean. Since no formula are required when using the CBBM in non-parametric mode, restrictive and sometimes dangerous assumptions about the form of underlying populations can be avoided.
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Fig. 6. Graph of the relevance of the whole roughness parameters ranked as a function of their correlation power |P¯i | for (a) the LRS sample; (b) the HRS sample. The numbers between brackets for Ra , CTE PC, Rq , Lac , Da , Dq and ANAM show the respective position of these roughness parameters in the ranking.
Moreover, a standard error (thus an assessment of the accuracy) can be calculated for any computable statistical estimate using the constructed empirical PDF. The CBBM does not work alone but its efficiency is emphasised when applied to traditional statistical procedure as
shown in this study in the case of the linear regression method. In our opinion, the CBBM is all the more relevant as it is a part of the search for global methodology to study the relationships between roughness parameters X and a
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surface property Y. When assuming a linear correlation between these variables, the least-square method used to estimate the best-fit straight line is based on the equation: Y = aX + b + ε where a is the slope, b the intercept and ε a randomly distributed homoscedastic error. Such an equation supposes firstly that the values of X must be evaluated without any noise. In other words, the variance of X must be equal to zero. However, both measurements of the variables X and Y are usually performed with different experimental devices (for example, the roughness profiles have been recorded with a tactile profilometer and the brightness level with a scatterometer in this study). Hence both these variables get a variance different from zero. Secondly, the assumption of error homoscedasticity (i.e. the variance of the error must be the same for each class of X values) is not always true. Moreover, apart from these limits, it has to be remembered that the morphology of a surface can be characterised by numerous roughness parameters. Theoretically speaking, statistical inference assumptions on the underlying population thus have to be tediously verified for each parameter to be able to calculate a confidence interval on any statistical estimate. In a mere application of this traditional statistical theory, any transgression of the inference assumptions, referring either to the linear regression method or to the form of the underlying population, may lead to a systematic bias on the estimation of the slope and/or the associated confidence interval. Since the use of mathematical formulas is limited in the bootstrap method, systematic biases on the estimation of statistics are minimised as well as the risk of asserting wrong conclusions on the relevance of roughness parameters with regard to the considered surface property. Basing on the purely statistical advantages mentioned above, the CBBM has been taken into account to define a performance index (the correlation power) with a confidence interval to assess the relevance of 100 or so roughness parameters with regard to a linear relationship versus the brightness level of cold-rolled low carbon steel strips. For each roughness parameter, these indicators were deduced from an empirical PDF constructed from the experimental data points only by the means of a computer and without any inference assumptions. Thanks to our algorithm, they can be simply plotted on only one graph to obtain a visual ranking and a direct appreciation of the relevance of all computed roughness parameters. The most important information that can be extracted from such a graph is related to the most relevant parameter in the ranking. Indeed, this parameter has to be considered as a privileged candidate both in the assessment of the surface quality of manufactured products during the industrial control stage and in the models dealing with the relationships between the surface morphology and an optical wave. In this study, the fractal dimension ANAM has been shown to be the most relevant roughness parameter in the case of a skin-passed strip. However, for a rougher sample taken from the same strip just after the tandem mill operation, the arithmetic average roughness must be preferred. Another interesting piece of information comes
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from the overlap of the confidence intervals, which indicates that number of roughness parameters have the same influence as far as the brightness level of cold-rolled low carbon steel strips is concerned. This confirms the well-known fact pointed out by Whitehouse that some roughness parameters have the same meaning [1,6]. Finally, it must be outlined that our computer software is much upgradable. Since the core of this application is based on the SAS language, it allows to process a lot of data file formats. Therefore, other surface properties than the brightness level can be assessed by various experimental devices for any future application. The algorithm can likewise be easily modified to test: firstly the relevance of any new roughness parameter found in the literature or created for a particular application, secondly a measuring technique (2D or 3D tactile profilometer, confocal microscopy, interferometer, AFM etc.) or thirdly a current treatment applied to profiles before the calculation of roughness parameters (filtering, fitting, scale length, cut-off, orientation of measurements etc.). The higher the calculated correlation power, the more relevant the new roughness parameter, the measuring technique and/or the pre-treatment. For the sake of simplicity and tractability, a linear correlation was considered between the roughness parameter and the brightness level in this study. However, simple non-linear mathematical functions can also be analysed by applying variable transformations like x → log(x), x → x 2 , x → exp(−x) etc. in the computer algorithm. The efficiency of the variable transformation can be then assessed looking at the values of the correlation power.
5. Conclusions This paper emphasises the efficiency of the computerbased bootstrap method when applying to the study of the relationships between roughness parameters and the brightness level of cold-rolled low carbon steel strips. As claimed by Efron, this method aims to carry out familiar statistical calculation (mean, standard errors, biases, confidence intervals etc.) but in an unfamiliar way: by purely computational means, rather than mathematical formulas [10]. The most of the CBBM has been made to introduce a new methodology so as to leave aside the subjectivity sometimes encountered in the practical and/or theoretical use of roughness parameters. Thanks to this method, we have defined an index of performance (the correlation power) with a confidence interval which allow to assess the relevance of 100 or so roughness parameters. This index indicates from an unbiased quantitative point of view the most relevant roughness parameter that characterises the surface morphology of cold-rolled low carbon steel strips with regard to their brightness level. Thanks to our upgradable computer application, such a methodology presented in this work can be extended to the study of any simple correlation between a surface morphol-
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ogy manufactured by any kind of machining process and any measurable surface property.
Acknowledgements The authors would like to thank V. Hague for the language correctness of the original manuscript. References [1] T.R. Thomas, Rough Surfaces 2nd ed., Imperial College Press, London, 1999. [2] D.J. Whitehouse, Handbook of Surface Metrology, Institute of Physics, Bristol, 1994. [3] J.C. Russ, Fractal Surfaces, Plenum Press, New York, 1994. [4] U. Persson, Measurement of surface roughness on rough machined surfaces using spectral speckle correlation and image analysis, Wear 160 (1993) 221–225. [5] J.C. Le Bosse, G. Hansali, J. Lopez, J.C. Dumas, Characterization of surface roughness by laser scattering: diffusely scattered intensity measurement, Wear 224 (1999) 236–244. [6] D.J. Whitehouse, Parameter rash—is there a cure? Wear 83 (1982) 75–78. [7] B. Efron, Bootstrap methods: another look at the jacknife, Ann. Stat. 7 (1979) 1–26.
[8] B. Efron, R. Tibshirani, Bootstrap measures for standard errors, confidence intervals, and other measures of statistical accuracy, Stat. Sci. 1 (1986) 54–77. [9] B. Efron, R. Tibshirani, Statistical data analysis in the computer age, Science 253 (1991) 390–395. [10] B. Efron, R. Tibshirani, An Introduction to the Bootstrap, Chapman and Hall, New York, 1993. [11] M. Bigerelle, Caractérisation géométrique des surfaces et interfaces: applications en métallurgie, Ph.D. Thesis, Ecole Nationale Supérieure d’Arts et Métiers, Lille, France, 1999. [12] P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, Pergamon Press, London, 1963. [13] E. Jackeman, Non-gaussian statistical models for scattering calculations, in: J.C. Dainty, D. Maystre (Eds.), Waves in Random Media, vol. 1, An Institute of Physics Journal, 1991, No. 3, pp. S109– S119. [14] R. Botet, E.Y. Poliakov, V.M. Shalaev, V.A. Markel, Fractal-surfaceenhanced optical responses, in: J.L. Vehel, E. Lutton, C. Tricot (Eds.), Fractals in Engineering, Springer, London, 1997, pp. 237–251. [15] T.V. Vorburger, E. Marx, T.R. Lettieri, Regimes of surface roughness measurable with light scattering, Wear 32 (1993) 3401–3408. [16] M. Bigerelle, A. Iost, Calcul de la dimension fractale d’un profil par la méthode des autocorrélations moyennées normées (AMN), C.R. Acad. Sci., Série IIb 323 (1996) 669–675. [17] M. Bigerelle, A. Iost, A new method to calculate fractal dimension of a surface: Application to the human cell proliferation, Comput. Math. Appl. 42 (2001) 241–253. [18] R.L. Packett, Principles of Regression Analysis, Oxford University Press, 1960.