Analysis and application of partial least square regression in arc welding process

Vol.12 No. 4J. CENT. SOUTH UNIV. TECHNOL.Aug.2005Article ID :1005 - 9784( 2005 )04 - 0453 - 06Analysis and application of partial least squareregression in arc welding processYANG Hai-lan( 杨海澜),CAI Yan(蔡艳),BAO Ye-feng( 包晔峰) ZHOU Yun(周昀)( Institute of Weld Engineering , Shanghai Jiaotong University , Shanghai 200030 , China )Abstract : Because of the relativity among the parameters , partial least square regressior( PLSR )was applied to build themodel and get the regression equation. The improved algorithm simplified the calculating process greatly because of the re-duction of calculation. The orthogonal design was adopted in this experiment. Every sample had strong representation ,which could reduce the experimental time and obtain the overall test data. Combined with the formation problem of gas met-al arc weld with big current , the auxiliary analysis technique of PLSR was discussed and the regression equation of form fac-tors( i. e. surface width , weld penetration and weld reinforcement ) to process parameters( i. e. wire feed rate , wire exten-sion , welding speed , gas flow , welding voltage and welding current )was given. The correlativity structure among variableswas analyzed and there was certain correlation between independent variables matrix X and dependent variables matrix Y.The regression analysis shows that the welding speed mainly influences the weld formation while the variation of gas flow incertain range has lttle influence on formation of weld. The ftting plot of regression accuracy is given. The ftting quality ofregression equation is bascally satisfactory.Key words : PLSR ; regression modeling ; formation of weldCLC number : TG409Document code : A1 INTRODUCTIONdimension while modeling ，so the characteristics ofmultivariate data could be observed on 2-dimensionalAr+ CO2 gas shielded weld was one of the mainplane. After PLSR analysis , not only the regressionwelding methods in production. The problem was howmodel could be obtained , but also the correlation ato make sure the shape and size of weld beam meet re-mong variables and the similarity among sample pointsquirements. Generally , formation of weld was denotedcould be observed , which made the analytical contentby surface width , weld penetration and weld reinforce-of data system more plentiful and gave the more de-ment. There were too many factors , which influencedtailed and practical explanation to the regression mod-the formation of weld. In Ar + CO2 gas shielded weld ,elf6]. In this paper , we took the formation of gas metalit mainly depended on welding current , welding voltageare weld( GMAW ) welding with big current for exam-and welding speed ete1].ple , then the specific analysis and application processBecause of the strong coupling among process pa-of PLSR were described in detail. Finally , the regres-rameters and large numbers of random and uncertainsion model of form factors to process parameters wasfactors , it is diffcult to build the model about arcgiven.welding process. The partial least square regression( PLSR ) was a new multivariate statistical analysis2 EXPERIMENTALmethod , which emphasized on the regression modelingof multiple dependent variables to multiple independent2.1 Experimental procedurevariables23].Especially for the relativity problem a-Inversion power supply with constant voltage out-mong multiple variables , its analysis results were moreput , rated current of 1 500 A was used in the experi-reliable and integral. Besides , PLSR could implementment. The feed rate was adjustable from 0. 5 tothe comprehensive application of various multivariate50 m/ min. Bead-on-plate weld with d1. 2 mm weldingstatistical analysis methods. The regression modelingwire was taken on steel plate. Shielding gas was mainlydata simplification and relativity analysis were includedAr中国煤化工2. To get rid of the 0x-in single algorithm , which greatly facilitated the multi-ide:MHCNMHthe plates were sandedvariate system analysis45]. PLSR reduced theoy the scounng wileel Deiore weld. After weld , the①Received date. : 2004 -07 - 28 ; Accepted date :2004 - 10 -07Correspnde&带ANG Hai-lan , Doctoral candidate ;Tel : + 86-21 62932585 ; E-mail : yh12005@ sjtu. edu. cn， 454.Jourmal CSUT Vol. 12 No.4 2005form factors of weld ,i. e. surface width , weld penetra-Table 1 Original datasheettion and weld reinforcement , were measured and 54x_x2X3x4xsx6yYz Y:groups of experimental data were obtained. .40220.73047.564093272:32270.53044.551094)22223336350.93045.548010113152. 2 Modeling mechanism36220.54047.56101112622PLSR dealt with the relationship between two data20 18matrixes , which generally were parameter matrix and2350.74044.5470881617index matrix in process quality control. In a sense ,32220.95043.055024 19PLSR could be considered as such a way which made36270.75046.0530912118principal component analysis( PCA ) for the two matri-40 3:0.5 50 47.5 500 114 17 20xes respectively , then established the inner relationship040220.73046.0640922924of them. PLSR treated the co-linearity in the same way32 2).5 30 43.5 500124as PCA did. Besides the information extraction from136350.93044.5490961313independent variables , PLSR also noticed the interpre-tation of independent variables to dependent variables,40270.94047.5570831419which was ignored by PCA. Wentzell et alf7] and32 35 0.7 40 44.0 46016Kemsley8J found that fewer factors were adopted to a-1632220.95043.0570681819chieve the least square error( LSE ) in PLSR than in1736270.75047.5550902120PCA.1840350.55047.05201261524Suppose that there were p independent variables940220.73044.06407622224{x .. x},q dependent variables {y1 .. vq }and n).5 3042.0500 102 16 20samples. Then the data matrixes Xnxp and Ywere2136350.93044.0510951215obtained. Component 1( linear combination of x ...，xp ) and u( linear combination of y ... dyq ) were ex-40270.94044.5580721618tracted from X and Y under the following condi-232 352532220.95043.054063202ltior36270.75043.55309024191 )1 and u, should represent X and Y to the grea-2740350.55045.04001151421test extent.2840220.73046.0650802052.42 ) Relativity between t and u should be the293220.5 30 42.5 49098 24 23maximum.3036350.93044.0470791217After extraction of ty and uy , PLSR implemented6 22).54043.55901022625the regression of X totp and Y to 1p , respectively. Al-3240270.94044.5560661821gorithm was stopped if the accuracy was good enough. .Otherwise second extraction of components would be3432220.95041.0560651717implemented. Finally , if m components t ... sm were; 270.7 50 44.0 530 79 17obtained , regression equation of yk to original variables3640350.55045.0500981425x ...此p was established through transition fromt tox ,3740220.73048.0640972521here h=1.. q32270.53044.550011324233936350.93046.049010816102.3 Modeling of PLSR.5 40 47.0 610 113 30 22The original datasheet is listed in Table 1. In Ta-40270.94048.0560871218ble 1 ,x denotes wire feed rate此denotes wire exten-4332220.95047.059070142(sion ixz denotes welding speed x4 denotes gas flow xsdenotes welding voltage ; x denotes welding current ;4540350.55048.0520114222:y denotes surface width 况denotes weld penetration ;40220.73047.5700823123y; denotes weld reinforcement.4732.0.5 30 45.0 530 100 25 22The orthogonal experimental design was adopted中国煤化工5500 87 14 15in this example. Correlative coefficients of independentCTHCNMHG6301023223variables are listed in Table 2.0.9 40 48.0 600 82 20 19According to the Cross-V alidation test ，iteration0.7 40 43.5 4770 12was stopped if accuracy was perfect enough.5232220.95044.56507522205 2750350.55047.05201071926YANG Hai-lan ,et al : Analysis and application of partial least square regression in are welding processTable 2 Correlative coefficients ofAnd the accumulatively interpretive abilities of t,independent variablesto x; and X are written as follows :xx:x.x;xeRd(x;2... 1m)= E Rd(x; 2n,)(7)100.617 0. 37120.006 -0. 828Rd(X2...1m)= 2 Rd(X;t)(8)630.017 0. 068Eqns.( 9 )-( 12 ) are also obtained as the same40. 087 -0. 076for yk and Y.*s0. 302Rd(ye 7 )=r(y; 1)(9)Rd( Y 7)=二习Rd(y; 2)( 10)The formula of Cross-Validation is as follows 1] :Rd(yk 2 r m)=〉Rdy; 2) (11)DEQ:=1--FEn-1(1)Rd(Y2..1m)=2Rd(Y2%)(12)where Q% is defined as Cross-Validation ; DE, is dis-turbed error by h components iFEn-1 is ftting error byThe calculating results of accuracy analysis areh-1 components. If DE，is smaller than FEn-1 to cer-listed in Table 4.tain extent , component th is considered to be helpfulTable 4 Results of accuracy analysisfor the improvement of regression accuracy.Rd(.，)1l2t3t:16 AccumulationTable 3 shows the value of cross-validation whenm=1 ,2 ,3 respectively.0.146 0.014 0.614 0.075 0. 1480. 003According to the decision prineiple Qh≥0. 0975 ,0.570 0.320 0.072 0.001 0.001 0. 035when two components 1 and 12 were included , the re-x3 0.235 0.649 0.115 0 0gression performance was the best.x4 0.003 0.001 0.033 0.904 0.059 0Table 3 Value of cross-validationQThreshold .0.107 0.011 0.631 0.021 0.229 0. 0010.0570.0975x6 0.667 0.266 0.023 0 0.002 0. 0430.137X 0.288 0.210 0.248 0.167 0.073 0.014-0.1710.065 0.588 0.010 0.008 00. 671y2 0.734 0.003 0.028 0.016 0.004 00.785Here , the regression equations were listed as fol-lows :0.513 0.115 0.050 0.015 0.004 00. 697γ =78. 383 +0. 725x，+0. 422x2 - 69.198x3 -Y 0.437 0.235 0.029 0.013 0.003 00.7180.109x。+0. 946xs -0.027x。(2)y2=2.952 +0. 21x, -0.39x2 -13. 457x; -0.018x, +0. 279xs +0. 032x。(3)It should be noticed that excessive Rd(X2.. ,y; =12.324 +0.16x -0. 113x2 - 12.612x3 -tm ) is not necessary in accuracy analysis. Noise infor-0. 019x, +0. 211xs +0. 01x。(4)mation is often included when extracting too many com-ponents , which brings adverse influence on the system3 ANALYSIS AND DISCUSSIONmodeling 1235. For this example ,Rd( X 7 12 )is 49.8%.ThoughRd(X223)isupto74.6%when3.1 Accuracy analysisthree components are extracted. In fact , the interpre-According to accuracy estimation theory , the in-tive ability oftz to Y is only 2.9%. Sotz is a very poorterpretive ability of t, to x; is defined as :interpretive factor. According to regression theory ,ad-Rd(x; ith )=r(x; h)(5)ditiv中国煤化工rary reduces the regres-where 1( x; h ) is the correlative coefficient of tn andsionIYHCNMHG(Y2山)of67.2%xj.indicates that the designed set X could denote set Y toSo the integrally interpretive ability of th to X isa certain extent.written as follows :乃数搪)= D 2 Rd(x;1)(6)P台456.Jourmal CSUT Vol. 12 No.4 20053.2 Correlative analysis among variablesOn one hand ,th could be constructed from X mul-tiplied by weight wi ,on the other hand , Y could beconstructed from t, multiplied by regression coefficient56t=Xw“xY=tr'“0x=n*sSo ,if x; has great contribution to construction ofth ,which has important influence on interpreting Vhk ,it2should be thought that intimate relation exists betweenx; and y4.The plot of w↑r, vs w2°r2 is drawn for observing15correlation between x; and yk as shown in Fig. 1. Coor-dinates of x and Yk are( wi; 102; )and( rIk r2k),re-Fig. 2 Plot of correlative coefficientsspectively. In addition , the more precise method is a-dopted to draw plot of correlative coefficients , whichtakes( r( x; 1 )(x; 12)) as coordinates of x; and ris reflected. x。 has larger positive correlation with y2.(yk 1)r(y% 1t2 )) as coordinates of Yk as shown inx is tightly close to xs and they have positive correla-Fig.2. Table 5 shows the value of r( th ; ),wi; andtion with y1 and V3. x2 has the same important influ-ence on constructing tp as x3 does to t2. x4 is close to「.the center of correlative circle ,so it has a weaker cor-Table 5 Results of correlative analysisrelation with all the other variables.Variable R(1 ; )风口; ) wi°11w02 厂2In PLSR ,plot of t, vs u is used to observe thelinear relation between t and u where there are thx1-0.383-0.117 -0.192-0.168first components of X and Y , respectively , as shown in0.755-0.5660.517-0.371Fig.3. If obvious linear relation between tp and u is0.4850. 8050. 6400. 884observed in Fig.3 ,it indicates X has significant corre-0.0560.0320. 044lation with Y. In such a case the selection of PLSR tobuild the linear model between X and Y should be rea--0.328-0. 106 -0.153-0.131sonable 15 J6].-0.8170.516 -0. 5100. 318From Fig.3 ,it can be seen that t, has apparentlyY1-0.256 -0.767 -0.206 - 0. 683negative correlation with uμ。This also proves the con-clusion from our accuracy analysis.-0.8570. 058-0.689 0.052-0.716 -0.339 -0.576 - 0.302■x31■.0-1bxz”■中国煤化工3-15MHCNMHGwiηFig.3 The first components of X and YFig. 1 Correlation scatter diagram among variables■-Component points ;一 - - Regression lineCompg伊數据1 with Fig. 2 , the same tendency3.3 Regression analysisHistogram of regression coefficients is adopted toYANG Hai-lan ,et al : Analysis and application of partial least square regression in arc welding processobserve the marginal effects. Fig. 4 shows histogram oftribution to them.regression coefficients from the standardized data.V ariable importance in project( VIP ) can be usedFig. 4 indicates that x has important meaningto estimate the relationship between X and Y. Its defi-nition is written as follows :0.8 间_PVIP=NRdYI 5ZRdY2n)u;0.6(13)For p independent variables x(j = 1 .... p),ifthey have the same influence on interpreting Y ,all the0.4VIP of them should be equal to 1. Otherwise , variableswhose VIP; is bigger than 1 should have more importantinfluence on interpreting Y. Fig. 5 shows histogram of&0.2VIP.InFig.5,VIPofx3isthemaximumandVIPofx。.is the minimum. This corresponds with the conclusion0ufrom Fig. 4.x石xx4XsxSurface width2-0.8向i 0.x'2. x3 X4X6Standardized parameterPenetrationFig. 5 Histogram of VIPIn order to check the modeling accuracy of re-gression equations ，predicting plot of dependentvariables are drawn according to coordinates( y; y;) ,i=1.... ,n. Here y is the predicting value of y;. Inthis figure , if all the sample points distribute evenlynear the diagonal line , the fitting accuracy of regres-sion equations should be satisfactory. Predicting plot ofform factors is shown in Fig. 6. .In Fig. 6 , as for surface width ，sample pointswhose ftting error is less than 10% occupy about 70%x2Xx3x4xs5 xof I中国煤化工is up to 87% when fit-ReinforcementtingMHCNMHGetwonumbersare67%Fig.4 Histogram of regression coefficientsand 78% , respectively for weld penetration. As forfrom standardized dataweld reinforcement , they are 70% and 93% , respec-(a)- -Surface width;( b )- -Penetration ;( c )-Reiforcementtively.for all thr西府数掘ion equations while x4 has lttle con-458.Jourmal CSUT Vol. 12 No.4 200530120-昌100虽20其20F8C6(100 12020Predicted surface width/mmPredicted penetration/mmPredicted reinforcement/mmFig. 6 Predicted plot of form factors■-Component points ; 一一 Standard regression lineIndustry Press , 1999.( in Chinese )4 CONCLUSIONS[7 ] Wentzell P D , Montoto L V. 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