8 Journal of the Chinese Institute of Industrial Engineers, Vol. 4, No., pp. 8-89 (007) APPLYING PRINCIPAL COMPONEN ANALYSIS O A GR&R SUDY Fu-Kwun Wang* Departent of Industrial Manageent National aiwan University of Science and echnology 43 Keelung Road, Sec. 4, aipei, aiwan 06, R. O. C. Chih-Wen Yang Bureau of Standards, Metrology and Inspection Ministry of Econoic Affairs, aiwan ABSRAC he gauge repeatability and reproducibility (GR&R) study is typically conducted on a single quality characteristic. However, anufacturing tests in a GR&R study usually have ultiple characteristics with a ultivariate noral distribution. Principal coponent analysis (PCA) ethod can transfor the ultiple characteristics into one or a few irrelevant variables and provide sufficient inforation. hen, these irrelevant variables were analyzed using analysis of variance. wo coposite indices such as precision to tolerance (P/) ratio and easureent variation to total variation of easureent syste ratio ( σ gauge / σ ) cobining fro all variables were used to evaluate the adequacy for the easureent process. A real exaple was used to deonstrate the application of the proposed ethodology. Keyword: gauge repeatability and reproducibility, ultiple characteristics, principal coponent analysis. INRODUCION* * he purpose of easureent syste analysis (MSA) is to separate the variation aong devices being easured fro the error in the easureent syste. Here, the easureent syste error can be the cobination of gauge bias, repeatability, reproducibility, stability, and linearity []. In general, the easureent syste s variation can be characterized by location (stability, bias and linaraity) and width/spread (repeatability and reproducibility). In this study, we focused on the easureent syste capability study which ais to deterine how uch the total observed variability is due to the gauge. Several different operators, either for different set-ups or for a different tie period, use the gauge to obtain replicate easureents on units. In these types of studies, two coponents of the easureent syste variability are defined as repeatability and reproducibility. Repeatability is defined as the variation in easureents obtained with one gage when used several ties by one appraiser while easuring a characteristic on one part. Reproducibility is defined as the variation in the average of the easureents ade by different appraisers using the sae * Corresponding author: fukwun@ail.ntust.edu.tw gage when easuring a characteristic on one part. wo ethods coonly used in the analysis of a gauge repeatability and reproducibility study are: () an analysis of variance approach followed by estiation of the appropriate variance coponents; and () an X-bar and Range chart ethod to estiate the standard deviations of the coponents of gauge variability. Burdick et al. [3] provide a good review of ethods for conducting and analyzing easureent syste capability studies, which are based on the analysis of variance approach. Wang and Li [] present that the Bootstrap ethod can be used for obtaining the confidence intervals of the gauge variability when the control chart ethod is used for finding the point estiates. One real-life exaple is used to show the application of this control chart with Bootstrapping ethod and coparisons are ade with three experiental design procedures in ters of point estiates and confidence intervals for repeatability, reproducibility and total gauge variability. Voelkel [] proposes a new circle-diaeter easure to estiate the gauge variability for two-diensional data. Larsen [8] presents a study which is to extend the univariate single-instance case to coon anufacturing test scenarios where ultiple paraeters are tested on each device with a sequence of tests, which ay include retest, test, and repair steps. Although Pearson proposed the principal co-
Wang and Yang: Applying Principal Coponent Analysis to a GR&R Study 83 ponent analysis (PCA) in 90, Hotelling [6] did not present the general ipleentation procedure until 933. Principal coponent analysis is a statistical ethod for ultivariate data analysis that can be used in particular to reduce the data set being considered. It transfors nubers of original related easureent variables into a set of uncorrelated linear functions. he first principal coponent variable having displayed the axiu variation aong the objects cobines all of original variables linearly. Siilarly, the second, the third, and other principal coponent variables have the sae properties as the first principal coponent variable. Note that the second one represents the next largest variation, and so forth. In ost practical probles, analyzing a portion of coponent variables is enough to depict ost of the variation inforation of the process. Fortunately, this portion of principal coponent variables defines a lower diensional subspace where the proble doain is siple. Moreover, the inforation of the process in this lower diensional subspace can be used to attain process control effectively. Section presents the easureent error study. he procedure of obtaining gauge variability using principal coponents analysis is discussed in Section 3. One real world case will be provided to deonstrate the proposed ethodology in Section 4. Section 5 ade the conclusion.. MEASUREMEN ERROR SUDY An iportant assuption of any statistical process control (SPC) ipleentations is adequate capability of the gauge and the inspection syste. In any process involving easureent of anufactured products, soe of the observed variability will be due to variability in the product itself, and soe will be due to easureent error or gauge variability. Expressed atheatically, σ = σ + σ () total where product total easureent error σ is the total variance of the observed process, σ product is the coponent of variance due to the product, and σ easureenterror is the coponent of variance due to easureent error. he easureent error and the product easureent are assued to be independent of each other. Furtherore, the previous definitions of repeatability, reproducibility and total gauge variability are used. hus, the variability σ easureent error is the su of two variance coponents, say σ easureent error = σ gauge = σ repeatability + σ reproducibility () A traditional gauge study on a single quality characteristic eploy a rando two-factor design with parts and operators as factors. ypically, several operators (say b operators) are chosen at rando to conduct easureents on the randoly selected parts (say a parts) fro a anufacturing process. Each part is easured n ties by each operator. hus, the odel is given by X = µ + P + O + ( PO ) + E ijk i j i =,,, a j =,,, b (3) k =,,, n where µ is a constant, and Ρ i, O j, ( PO) ij, Eijk are jointly independent noral rando variables with eans of zero and variances σ P, σ O, σ PO and σ E, respectively. he analysis of variance for odel in equation (3) is shown in able. able. ANOVA for two-factor odel Source of Degree of Mean Expected variation freedo square ean square Parts a- MS P bnσ P + nσ PO + σ E Operators b- MS O anσ O nσ PO E Parts* Operators (a-)(b-) MSOP nσ PO + σ E Replications ab(n-) MS E σ E Furtherore, fro the able, we have the point estiates for the paraeters of interest which are given by ˆ σ repeatability = ˆ σ E = MS E ˆ reproducibility ˆ O ˆ σ = σ + σ PO = [ MSO + ( a ) MS PO ams E ] / an ˆ σ gauge = [ MSO + ( a ) MSPO + a( n ) MSE ]/ an ˆ σ = [ anms P+ MSO + ( a ) MS PO + a( n ) MS E ] / an If interaction effect is not significant, then the full odel can be reduced to Y ijk = µ + Pi + O j + Eijk When a gauge study is based on ultiple quality characteristics (say ), the odel in equation (3) can be extended to new odel and is given by X = µ + α + β + ( αβ ) + ε ijk i j ij ij ijk ijk
84 Journal of the Chinese Institute of Industrial Engineers, Vol. 4, No. (007) i =,,, a j =,,, b (4) k =,,, n µ where µ = is a constant vector, µ αi ~ N( 0, Σ α ), β j ~ N( 0, Σ β ), αβij ~ N( 0, Σ αβ ), ε ijk ~ N ( 0, Σ ε ) which all the rando vectors are jointly independent. he point estiates for the paraeters of interest and are given by Σrepeatabil ity = Σ ε Σreproducib ility = Σ β + Σ αβ Σ gauge = Σε + Σβ + Σαβ Σ = Σα + Σε + Σ β + Σαβ he coponents of variance are estiated using the standard MANOVA ethod of oents. 3. APPLYING PRINCIPAL COMPONEN ANALYSIS IN A R&R SUDY In this section, the application of principal coponent analysis (PCA) is deonstrated, and the procedure for deciding how any coponents to extract is discussed. he flow of procedures is shown in Figure. Once the quality easureents have been obtained, ultivariate norality should be exained prior to applying the PCA. Mardia [9] shows that the norality of ultivariate data is validated by using a univariate analog. A function, MVMM, of International ath eatics and statistics language (IMSL) coputes Mardia s ultivariate easureents for p values of the ultivariate skewness and kurtosis [7]. hese easureents are then used to exaine ultivariate norality. here are three types of statistical tests to base upon skewness statistics, kurtosis statistics, and onibus-test statistics. he onibus-test statistics is obtained fro cobining noral data of the skewness and kurtosis statistics. he approxiated expected value, asyptotic standard error, and asyptotic p-value for onibus-test statistics are coputed under the null hypothesis of the ultivariate noral distribution. hen, these values are noralized. hese scores are cobined into an asyptotic chi-squared statistic with two degrees of freedo. he chi-squared statistic ay be used for ultivariate norality test. A p-value of the chi-squared statistic can also be coputed. hus, the assuption of either ultivariate norality or ultivariate non-norality can be verified. Assuing that X is a n saple data atrix, where denotes the nuber of product quality characteristics observed fro a part and n represents the nuber of parts being easured. Also, X is the saple ean of the observation which is an -vector value, and S, a nonsingular syetric atrix, is the covariance between observations. Engineering specifications are given for each value, where LSL and USL are --vector values of the lower specification liits and upper specification liits, respectively. he vector represents the target values for the quality characteristics. In addition, the spectral decoposition can be used to obtain D = U SU, where D is a diagonal atrix. he diagonal eleents of D, λ, λ,, λ, are the eigenvalues of S and the coluns of U, u, u,, u, are the eigenvectors of S. Consequently, the ith principal coponent (PCi) which is also called as new variable ( Y i ) is given by Yi = PCi = u x, i =,,, (5) where x's are vectors of the observations on the original variables. With these choices, Var( Yi ) = λi, i =,,, and Cov( Yi, Yk ) = 0, i k. he engineering specifications of PCi s and their target values are LSLPCi = ui LSL, USLPCi = ui USL, i =,,, (6) PCi = ui he ratio of each eigenvalue to the suation of the eigenvalues (ie, λ i / λi, i =,,,. ) is the proportion of variability associated with each principal coponent variable. However, only soe principal coponents can contribute to the ost of syste s variability, e.g. 80% to 90%. By using this subset, the ultivariate quality characteristic proble can be reduced in diension. Anderson [] proposes a test for identifying the significant coponents. he test statistics is λ j j= k+ χ = ( n ) ln λ j + ( n )( k) ln j= k+ k (7) Where χ has r = ( / )( k)( k + ) degrees of freedo. Jackson [5] further applies to the hypothesis H0 : λ k + = = λ against the alternatives where at least one eigenvalue is different fro the others. Analyzing the loading and principal coponent requires that the principal coponents and their variables closely correspond to each other, i.e. the angle between vectors, representing in R,
Wang and Yang: Applying Principal Coponent Analysis to a GR&R Study 85 is sall. he correlation between the ith variable and the jth principal coponent is given by / λ j ρ ij = uij (8) Sii where u ij denotes the loading for the ith observation in the jth principal coponent variable, λ j hus, two coposite indices cobing fro all new variables are given by = k / k index I i ) = k / k index ( I i ) () Multi-noral ransforation No Collect Measureent Data & Specifications est Multivariate Norality? (Mardia SW statistic) represents the eigenvalue associated with that principal coponent, and S ii is the variance of the ith variable (see Cadia and Jolliffe [4]). he gauge study on the ultivariate quality Yes characteristics which are correlated to each other can be used by the PCA approach. hen, the point estiates of interest on all new variables are used by an ANOVA ethod. he purpose of a GR&R study is to Using PCA to obtain New variables New specifications deterine if the variability of the easureent syste is sall relative to the variability of the onitored process. In this study, two coon ratios (P/ Calculate each new variable P/ ratio & ratio [] and σ gauge / σ [0] in GR&R studies σ are used to deterine whether the easureent syste gauge / σ is adequate or not. he P/ ratio is a function of kσ gauge σ gauge expressed as P / = %00% Coposite indices USL LSL k / k where USL and LSL are specification liits and k is index = ( Ii ) either 5.5 or 6. he value k=6 corresponds to the k / k nuber of standard deviations between the natural index = ( Ii) tolerance liits of a noral process. he value k=5.5 corresponds to the liiting value of the nuber Figure. A flowchart of the analysis procedure of standard deviations between bounds of a 95% tolerance interval that contains at least 99% of a noral population. In general, the ratio value is less 4. ILLUSRAE EXAMPLE than 0% indicated the easureent syste is adequate. If the ratio value is between 0% and 0%, it In order to deonstrate the proposed ethodology, the data fro a real-world case (solderability indicates the easureent syste is oderate adequate. If the ratio value is between 0% and 30%, it tests) was used. en parts and three operators were taken to conduct this experient. Each operator indicates the easureent syste is inadequate. easured each of 0 parts in five consecutive trials. Furtherore, a easureent syste is unacceptable he easuring conditions are: the insertion speed is if the ratio value exceeds 30%. For each new variable, 0 /s, depth is 3, and tie is 5 seconds. hree we have the P/ ratio and theσ gauge / σ ratio values ( 0, and Fax) are recorded during the which are given by easuring process. 0 :denote the response tie Ii = 5.5σ gauge( y / olerance ( y i ) i ), (seconds) when the device begins to soldering with (9) Sn and the lower and upper liits are set at (0.3,(9).0). i =,,, k :denote the response tie (seconds) when the I i = σ / σ, solderability reaches to /3 axiu force and the gauge ( yi ) ( y i ) (0) lower and upper liits are set at (0.5,.). Fax: i =,,, k denote the axiu force (units of illinewtons, N) during the easuring process and the lower and upper liits are set at (.0,.). ( () he P-value for Mardia SW statistic is 0.3 () (Mardia, 980 and IMSL, 995). hus, the assuption of ultivariate norality can not be rejected at 95% confidence level. able shows the loading and eigenvalue of the new variables using the principal coponent analysis.
86 Journal of the Chinese Institute of Industrial Engineers, Vol. 4, No. (007) Operator Replicates Response Part able. Measureent data for the exaple A B C 3 4 5 3 4 5 3 4 5 0 0.45 0.4 0.38 0.48 0.46 0.46 0.4 0.45 0.39 0.48 0.4 0.49 0.45 0.47 0.44 0.79 0.75 0.73 0.85 0.84 0.77 0.74 0.79 0.7 0.85 0.78 0.89 0.8 0.88 0.85 Force F ax.079.07.075.073.08.08.07.068.075.08.065.085.079.073.07 0 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.63 0.6 0.59 0.6 0.6 0.6 0.6 0.65 0.97 0.99 0.96 0.9 0.97 0.9 0.94 0.97 0.94 0.88 0.93 0.95 0.97 0.96.0 Force F ax.03.05.096.03.09.03.0..08.093.093.05.09.0.9 3 0 0.55 0.53 0.54 0.55 0.5 0.54 0.55 0.54 0.5 0.56 0.5 0.54 0.54 0.58 0.53 0.9 0.83 0.89 0.9 0.86 0.86 0.88 0.9 0.8 0.9 0.85 0.88 0.9 0.95 0.9 Force F ax.036.07.04.05.04.05.05.037.046.034.035.06.046.038.043 4 0 0.64 0.63 0.64 0.65 0.64 0.63 0.66 0.64 0.65 0.6 0.66 0.64 0.64 0.67 0.6.0.0 0.96.05.0 0.96 0.99 0.98 0.97 0.9.0 0.97 0.99.0 0.97 Force F ax.078.08.063.08.079.09.079.08.068.079.09.08.079.08.076 5 0 0.68 0.67 0.67 0.66 0.69 0.68 0.67 0.68 0.66 0.69 0.7 0.67 0.68 0.69 0.66.04.0.07.0.08.05.0.03 0.99.07.09.03.0.0 0.98 Force F ax.083.078.099.09.088.09.094.098.086.06.098.09.094.096.083 6 0 0.57 0.56 0.58 0.56 0.57 0.58 0.56 0.57 0.56 0.59 0.56 0.58 0.57 0.55 0.59 0.88 0.9 0.9 0.87 0.93 0.94 0.87 0.89 0.9 0.95 0.9 0.9 0.94 0.9 0.97 Force F ax.4.098.9.3..35.5.8...6.4.3.8.3 7 0 0.6 0.6 0.63 0.6 0.63 0.6 0.65 0.63 0.6 0.63 0.6 0.63 0.6 0.6 0.67 0.96.0 0.98 0.97.03 0.94.03 0.99 0.98 0.97 0.95 0.96 0.95 0.97.03 Force F ax.063.054.057.06.048.069.054.06.058.048.046.058.06.05.069 8 0 0.49 0.5 0.5 0.49 0.5 0.5 0.5 0.5 0.53 0.49 0.49 0.5 0.5 0.5 0.53 0.84 0.85 0.83 0.8 0.86 0.86 0.8 0.85 0.88 0.8 0.8 0.87 0.85 0.86 0.9 Force F ax.074.089.09.087.078.085.08.09.094.087.076.093.088.09.095 9 0 0.58 0.59 0.6 0.6 0.59 0.57 0.6 0.59 0.6 0.59 0.59 0.6 0.58 0.6 0.6 0.9 0.94 0.94 0.98 0.95 0.9 0.95 0.94 0.99 0.93 0.94 0.93 0.9 0.94 0.97 Force F ax.067.075.06.068.057.058.06.068.076.065.076.069.056.067.06 0 0 0.7 0.75 0.73 0.74 0.7 0.7 0.75 0.7 0.75 0.73 0.75 0.7 0.74 0.7 0.78.05.06...06.03.07.06..09.09.04.08.06. Force F ax.085.098.097.098.088.086.097.088.098.095.098.086.096.09.06
Wang and Yang: Applying Principal Coponent Analysis to a GR&R Study 87 First, the hypothesis test, H 0 : λ = λ = λ3, produces a value of χ 5 = 8. 86, which is significant at the 95% confidence level. hat is, the hypothesis is rejected. hen, testing the hypothesis 0 : λ = λ H 3 produces a value of χ =. 8, which is also significant at the 95% confidence level. herefore, the first two new variables are used to evaluate the gauge variabiltiy at 98.7% total variability. he specification liits for the new variables Y and Y were set at [0.87,.833] and [0.5, 0.986], respectively. he results of the gauge study using the ANOVA ethod for the new variables Y and Y are shown in able 3. able. he results for exaple using PCA ethod Y Y Y 3 Loading Loading Loading Variables X 0.679-0.93 0.708 X 0.677-0. -0.706 X 3 0.85 0.958-0.0 eigenvalue.0543 0.9066 0.039 % explained of total variability 68.5 30..3 able 3. he results for the new variables Y Y Souce VarCop P / % Contribution VarCop P / % Contribution σ R 0.0049 7.99 0.0000957 6.58 σ O 0.000036 0.5 0.000008 0.9 σ gauge 0.0085 7.6 8.4 0.0000984 0.78 6.77 σ part 0.0304 58.8 9.76 0.0036 40.00 93.3 0.04389 6.4 00 0.0045 4.4 00 able 4. he results for the original variables X ( 0 ) X ( ) X 3 (F ax ) Souce VarCop P / % Contr. VarCop P / % Contri. VarCop P / % Contri. σ R 0.00034 4.38 0.003 3.95 0.000055 8.54 σ O 0.00003 0.5 0.00005 0.65 0.00000 0.7 σ gauge 0.000336 3.48 4.53 0.008 5.4 4.60 0.000056 9.4 8.7 σ part 0.00707 6.88 95.47 0.00689 6.06 85.40 0.000585 6.6 9.9 0.0074 63.33 00 0.00807 66.07 00 0.000640 65.7 00 Note: Contri. = Contribution. hus, two coposite indices cobing fro two new variables are / index = (7.6 0.78) = 3.78 / index = (8.4 6.77) = 7.647 If the gauge study does not consider the correlation aong the original variables, then the results for the original variables using the ANOVA ethod are shown in able 4. Fro the results in able 4, two coposite indices cobing fro all original variables are /3 index = (3.48 5.4 9.4) = 8.707 /3 index = (4.53 4.60 8.7) = 8.3 With respect to the coposite index for the P/ ratio, we found that the value by the traditional ethod is overestiated by the PCA approach about 8.707 3.78 35.75%, which is 00 = 35.75%. 3.78 Also, with respect to the coposite index for the σ gauge / σ ratio, we found that the value by the traditional ethod is overestiated by the PCA approach about.54%, which is
88 Journal of the Chinese Institute of Industrial Engineers, Vol. 4, No. (007) 8.3 7.46 00 =.54%. 7.46 5. CONCLUSION Currently, the ANOVA ethod for the gauge R&R study can only be applied to univariate data. However, ost of tie, the product quality has been easured in several characteristics. hat is, it is coon to deal with the ultivariate data while easuring process perforance. When these variables are correlated with one another (high diension proble doain), the PCA ethod can transfor the ultiple characteristics into one or a few irrelevant variables and provide sufficient inforation. hen, these irrelevant variables were analyzed by using analysis of variance. wo coposite indices such as precision to tolerance (P/) ratio and easureent variation to total variation of easureent syste ratio ( σ gauge / σ ) cobining fro all variables were used to evaluate the adequacy for the easureent process. Fro the case study, we found that two coposite indices (P/ and σ gauge / σ ) by the traditional ethod are overestiated by the PCA approach about 35.75% and.54%, respectively. hus, we ust be careful when conducting a GR&R study with a ultiple quality characteristics. ACKNOWLEDGEMEN he authors wish to gratefully acknowledge the referees of this paper who helped to clarify and iprove the presentation. REFERENCES. AIAG Editing Group, Measureent Systes Analysis, Autootive Industry Action Group, Detroit-MI, USA (998).. Anderson,. W., Asyptotic theory for principal coponent analysis, Annals of Matheatical Statistics, 34, -48 (963). 3. Burdick, R. K. and G. A. Larsen, Confidence intervals on easures of variability in R&R Studies, Journal of Quality echnology, 9, 6-73 (997). 4. Cadia, J. and I.. Jolliffe, Loading and correlation in the interpretation of principal coponents, Journal of Applied Statistics,, 03-4 (995). 5. Jackson, J. E., Principal coponent and factor analysis: part I principal coponents, Journal of Quality echnology,, 0-3 (980). 6. Hotelling, H., Analysis of a coplex of statistical variables into principal coponents, Journal of Educational Psychology, 4, 47-44 (933). 7. IMSL Stat Library, Microsoft Fortran Power Station, Version 4.0, Houston-X, USA (995). 8. Larsen, G., Measureent syste analysis in a production test environent with ultiple test paraeters, Quality Engineering, 6, 97-306 (003). 9. Mardia, K. V., ests for Univariate and Multivariate Norality. Handbook of Statistics. North-Holland, Asterda-New York, USA (980). 0. Stout, G., Measureent put to the test, Quality, 33, 4-48 (994).. Voelkel, J. O., Gauge R&R analysis for two-diensional data with circular tolerances, Journal of Quality echnology, 35, 53-67 (003).. Wang, F. K. and E. Y. Li, Confidence intervals in repeatability and reproducibility using bootstrap ethod, Quality Manageent and Business Excellence, 4, 34-354 (003). ABOU HE AUHORS Fu-Kwun Wang received his Ph.D. degree in Industrial Engineering fro Arizona State University, epe, Arizona, in 996. Currently, he is a professor in the Departent of Industrial Manageent at National aiwan University of Science & echnology, aiwan. His priary research interests are in quality & reliability, supply chain anageent, and siulation. He has published ore than 40 journal papers in these fields. Chih-Wen Yang received his M.S. degree in Industrial Engineering and Manageent fro Natioanl aipei University of echnology, aiwan, in 004. Currently, he is a chief 3 rd section in the Bureau of Standards, Metrology and Inspection Ministry of Econoic Affairs, aiwan. His priary research interest is quality, easureent and calibration. (Received July 005; revised January 006; accepted May 006)
Wang and Yang: Applying Principal Coponent Analysis to a GR&R Study 89 應用主成份分析法於量測重複性與再現性之研究 王福琨 * 台灣科技大學工業管理學系 06 台北市基隆路四段 43 號楊志文經濟部標準檢驗局 摘要 量測重複性與再現性研究一般均著重於單一品質特性, 但在製造測試方面量測重複性與再現性研究實際上是具有多變量常態分配與多重品質特性之因子 本研究利用主成份分析法 (Principal Coponents Analysis;PCA) 將有相關之多個品質特性轉換成一個或少數幾個不相關且可提供足夠資訊之變數, 並針對這些新的變數再利用變異數分析法 (ANOVA), 來探討多重品質特性量測能力的問題 而在探討量測能力指標方面有 : 精 密度 / 規格公差比 ( P 比 ), 及量測變異對量測系統總變異貢獻度 ( σ gauge / σ ) 等二個綜合指標, 以作為評估量測過程的適當性, 同時對所提研究方法之應用, 列舉一實例作為論證 關鍵詞 : 量測重複性與再現性, 多重品質特性, 主要成份分析 ( 聯絡人 : fukwun@ail.ntust.edu.tw)