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Epsilon'Epsilon in the Standard Model Theoretical Update

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a r X i v :h e p -p h /0410215v 1 14 O c t 2004ε′/εIN THE STANDARD MODEL:THEORETICAL UPDATEA.PICHDepartament de F´ısica Te`o rica,IFIC,Universitat de Val`e ncia–CSIC,E-46071Val`e ncia,Spain.E-mail:Antonio.Pich@ific.uv.esA complete analysis of isospin breaking in K →2πamplitudes,including both strong (m u =m d )andelectromagnetic corrections at next-to-leading order in chiral perturbation theory,has been achieved recently.1We discuss the implication of these effects,together with the previously knownchiral loop corrections,2on the direct CP-violating ratio ε′/ε.One finds Re (ε′/ε)= 19+11−9·10−4.1IntroductionThe CP–violating ratio ε′/εconstitutes afundamental test for our understanding of flavour–changing phenomena.The exper-imental status has been clarified by the KTEV,3Re (ε′/ε)=(20.7±2.8)·10−4,and NA48,4Re (ε′/ε)=(14.7±2.2)·10−4,mea-surements.The present world average,3,4,5,6Re (ε′/ε)=(16.7±1.6)·10−4,(1)demonstrates the existence of direct CP vio-lation in K decays.The CP violating signal is generated through the interference of two different K 0→ππdecay amplitudes,ε′√Re A 2−Im A 0√Below the resonance region one can use symmetry considerations to define another effectivefield theory in terms of the QCDGoldstone bosons.Chiral perturbation the-ory(χPT)describes13,14the pseudoscalar–octet dynamics,through a perturbative ex-pansion in powers of momenta and quarkmasses over the chiral symmetry breakingscaleΛχ∼1GeV.Chiral symmetryfixes the allowed operators.At lowest order,the mostgeneral effective bosonic Lagrangian with thesame SU(3)L⊗SU(3)R transformation prop-erties as L∆S=1effcontains three terms:L∆S=12=G8L8+G27L27+G ew L ew.(4) L ew gives the low–energy realization of Q8, while Q6is included in the octet term.L∆S=12determines the K→ππampli-tudes at O(p2).The calculation of the chi-ral couplings G i from the short–distance La-grangian(3)requires to perform the match-ing between the two effective theories.This can be done in the limit of an infinite num-ber of quark colours,because the four–quark operators factorize into currents which have well–known chiral realizations.This is equiv-alent to the standard large–N C evaluations of Q i .Therefore,up to minor variations on some input parameters,the correspond-ingε′/εprediction,obtained at lowest order in both the1/N C andχPT expansions,repro-duces the published results of the Munich11 and Rome12groups.3Chiral CorrectionsThe large–N C limit is only applied to the matching between the3–flavour quark the-ory andχPT.The evolution from the elec-troweak scale down toµ<m c has to be done without any unnecessary expansion in powers of1/N C;otherwise,one would miss large corrections of the form1M WW,Z,γ,gτ,µ,e,νit,b,c,s,d,uStandardModel<∼m cγ,g;µ,e,νis,d,u Ln f=3QCD,L∆S=1,2effM Kγ;µ,e,νiπ,K,ηχPT?OPEN C→∞Figure1.Evolution from M W to M K.16mic corrections,through chiral loops which are of higher order in both the momen-tum and1/N C expansions.2These next-to-leading contributions,which give rise to the large S–wave strong phasesδI(δI=0at N C→∞),were overlooked in thefirst large–N C predictions11,12giving a too smallε′/ε. Larger values ofε′/εwere in fact obtained within models containing some kind of pion rescattering.17TheχPT framework allows us to incorporate rigorously these corrections in a model independent way.The one–loopχPT analyses1,2,18of K→2πshow indeed that pion loops provide an important enhancement of the A0amplitude, associated with large infrared logarithms in-volving the light pion mass,and a sizeable reduction of A2.These chiral corrections de-stroy the numerical cancellation between the Q6and Q8contributions,generating a large enhancement of theε′/εprediction.2A complete one–loop calculation,includ-ing electromagnetic and isospin violation cor-rections,has been achieved recently.1The loop contributions are fully determined by chiral symmetry.The local corrections gen-erated by the different higher-order chiral la-grangians have been computed at leading or-der in the1/N C expansion.To account for isospin breaking,we can writeε′√Re A(0)0(1−Ωeff)−Im A emp200311(EURIDICE),MCYT(FPA-2001-3031)and Generalitat Valenciana(GRU-POS03/013and GV04B-594).References1.V.Cirigliano,G.Ecker,H.Neufeld andA.Pich,Eur.Phys.J.C33(2004)369;Phys.Rev.Lett.91(2003)162001.2.E.Pallante, A.Pich and I.Scimemi,Nucl.Phys.B617(2001)441;E.Pal-lante and A.Pich,Phys.Rev.Lett.84(2000)2568;Nucl.Phys.B592(2000)294.3.KTeV collab.,Phys.Rev.D67(2003)012005;Phys.Rev.Lett.83(1999)22.4.NA48collab.,Phys.Lett.B544(2002)97;Eur.Phys.J.C22(2001)231;Phys.Lett.B465(1999)335.5.NA31collab.,Phys.Lett.B206(1988)169;Phys.Lett.B317(1993)233.6.E731collab.,Phys.Rev.Lett.70(1993)1203.7.F.J.Gilman and M.B.Wise,Phys.Rev.D20(1979)2392;D21(1980)3150.8.A.J.Buras,hep-ph/9806471.9.A.J.Buras,M.Jamin and ut-enbacher,Nucl.Phys.B408(1993)209;Phys.Lett.B389(1996)749.10.M.Ciuchini et al.,Phys.Lett.B301(1993)263;Z.Phys.C68(1995)239. 11.S.Bosch et al.,Nucl.Phys.B565(2000)3;A.J.Buras et al.,Nucl.Phys.B592(2001)55; A.J.Buras and M.Jamin,JHEP01(2004)048.12.M.Ciuchini et al.,hep-ph/9910237.13.S.Weinberg,Physica96A(1979)327.14.J.Gasser and H.Leutwyler,Nucl.Phys.B250(1985)456;517;539.15.W.A.Bardeen, A.J.Buras and J.-M.G´e rard,Nucl.Phys.B293(1987)787;Phys.Lett.B192(1987)138,B180(1986)133.16.A.Pich,hep-ph/9806303.17.S.Bertolini et al.,Rev.Mod.Phys.72(2000)65;T.Hambye et al.,Nucl.Phys.B564(2000)391.18.J.Kambor et al.,Nucl.Phys.B346(1990)17;Phys.Lett.B261(1991) 496;Phys.Rev.Lett.68(1992)1818;J.Bijnens,E.Pallante and J.Prades, Nucl.Phys.B521(1998)305;E.Pal-lante,JHEP01(1999)012.19.V.Cirigliano,J.F.Donoghue,and E.Golowich,Eur.Phys.J.C18(2000)83.20.G.Ecker,G.M¨u ller,H.Neufeld andA.Pich,Phys.Lett.B477(2000)88.21.E.G´a miz et al.,JHEP01(2003)060;hep-ph/0408044;M.Jamin,J.A.Oller and A.Pich,Eur.Phys.J.C24(2002) 237;K.Maltman and J.Kambor,Phys.Rev.D65(2002)074013;S.M.Chen et al.,Eur.Phys.J.C22(2001)31;H.Wittig,hep-lat/0210025; C.Aubinet al.,Phys.Rev.D70(2004)031504;M.G¨o ckeler et al.,hep-lat/0409312. 22.V.Cirigliano et al.,Phys.Lett.B555(2003)71,B522(2001)245,B475 (2000)351;Phys.Rev.D65(2002) 054014;J.F.Donoghue and E.Golowich, Phys.Lett.B478(2000)172.23.M.Knecht,S.Peris and E.de Rafael,Phys.Lett.B508(2001)117,B457 (1999)227;S.Peris and E.de Rafael, Phys.Lett.B490(2000)213.24.S.Narison,Nucl.Phys.B593(2001)3.25.J.Bijnens et al.,hep-ph/0309216;JHEP10(2001)009,06(2000)035,01(1999) 023.26.J.I.Noaki et al.,Phys.Rev.D68(2003)014501;T.Blum et al.,Phys.Rev.D68 (2003)114506;D.Be´c irevi´c et al.,Nucl.Phys.B(Proc.Suppl.)119(2003)619.27.S.Peris,hep-ph/0310063;T.Hambye,S.Peris and E.de Rafael,JHEP05 (2003)027.28.M.Golterman and E.Pallante,Phys.Rev.D69(2004)074503;JHEP10 (2001)037,08(2001)023.29.T.Bhattacharya et al.,hep-lat/0409046;D.Pekurovsky and G.Kilcup,Phys.Rev.D64(2001)074502.。

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0112-湍流模型介绍

0112-湍流模型介绍


2
Outline
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Turbulence models introduction Boussinesq hypothesis Eddy viscosity concept Zero equation model One equation model Two equation models Algebraic stress model Reyolds stress model Comparison Applications Developments Conclusion
ij
T
U i U j ui ' u j ' t x xi j






A new quantity appears: the turbulent viscosity or eddy viscosity (νt ). The second term is added to make it applicable to normal turbulent stress. The turbulent viscosity depends on the flow, i.e. the state of turbulence. The turbulent viscosity is not homogeneous, i.e. it varies in space.
Turbulence Models and Their Applications
Turbulence models
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A turbulence model is a procedure to close the system of mean flow equations. For most engineering applications it is unnecessary to resolve the details of the turbulent fluctuations. Turbulence models allow the calculation of the mean flow without first calculating the full time-dependent flow field. We only need to know how turbulence affected the mean flow. In particular we need expressions for the Reynolds stresses. For a turbulence model to be useful it:
sas线性回归分析案例（Case...

sas线性回归分析案例（Case...

sas线性回归分析案例（Case study of SAS linear regressionanalysis）linear regression20094788 Chen Lei calculates 2Southwest Jiao Tong UniversitySouthWest JiaoTong University-------------------------------------------------------------------Linear regression is divided into single linear regression and multiple linear regression.The model of unary linear regression isY=..0+..1X+ epsilon,HereXIndependent variable,YDependent variable,Epsilon is a random error term.It is usually assumed that the mean of the random error isZeroThe variance is(..2..2>0),..2 andXValue independent. If further assumptionsRandom errorThe difference follows a normal distribution, which is called a normal linear model. In general, withKAn independent variable and a dependent variable, dependent variableThe value can be broken down into two parts: part is due to theinfluence of the independent variable, that is to sayFunction as an argumentAmong them, the function form is alreadyKnow, but contain some unknown parameters; another part is due to other UN considered factors and random effects, that is, random errors.When a function is a linear function of unknown parameters, it is called a linear regression analysis model.If there are multiple dependent variables, the regression model is:Y=..0+..1X1+..2X2+.+..IXi+..Due to the linear dieThe model contains random errors, so the regressionThe straight line reflected by the model is uncertain. The main purpose of regression analysis is to derive from theseIn the uncertain straight line, find a line which can best fit the original data information and describe it as a regression modelRelationship between independent variables,The straight line is called the regression equation.throughOften in regression analysis, yesEpsilon has the most commonly used classical assumptions.1. The expected value of epsilon isZero2, epsilon for allXFor example, it has the same variance.3, epsilon obeys normal distribution and is independent of each otherVariable.Explanation of linear regression,This paperBased on examples.In the following example, there is a one element regression analysis, and another twoMeta regression analysis.Examples(Data analysis method_exercises2.4_page79)A company manager who knows about the monthly sales of a cosmetics in a cityY(unit: box) with the middle of the cityThe number of people who use the cosmetics..1 (unit: thousand persons) and their per capita monthly income..2 (unit: yuan) betweenIn a certain monthFifteenThree cities were surveyed to obtain the above views Measured values, such as tableTwo point one twoAs shown.surfaceTwo point one twoCosmetics sales dataCitySales volume (y)Number of people (x1)Income (x2)CitySales volume (y)Number of people (x1)Income (x2)OneOne hundred and sixty-twoTwo hundred and seventy-fourTwo thousand four hundred and fiftyNineOne hundred and sixteenOne hundred and ninety-fiveTwo thousand one hundred and thirty-seven TwoOne hundred and twentyOne hundred and eightyThree thousand two hundred and fifty-four TenFifty-fiveFifty-threeTwo thousand five hundred and sixtyThreeTwo hundred and twenty-threeThree hundred and seventy-fiveThree thousand eight hundred and two ElevenTwo hundred and fifty-twoFour hundred and thirtyFour thousand and twentyFourOne hundred and thirty-oneTwo hundred and fiveTwo thousand eight hundred and thirty-eight TwelveTwo hundred and thirty-twoThree hundred and seventy-twoFour thousand four hundred and twenty-seven FiveSixty-sevenEighty-sixTwo thousand three hundred and forty-seven ThirteenOne hundred and forty-fourTwo hundred and thirty-sixTwo thousand six hundred and sixtySixOne hundred and sixty-nineTwo hundred and sixty-fiveThree thousand seven hundred and eighty-twoFourteenOne hundred and threeOne hundred and fifty-sevenTwo thousand and eighty-eight SevenEighty-oneNinety-eightThree thousand and eightFifteenTwo hundred and twelveThree hundred and seventyTwo thousand six hundred and five EightOne hundred and ninety-twoThree hundred and thirtyTwo thousand four hundred and fiftyhypothesisYand..1,Linear regression relation is found between..2 ....=..0+..1....1+..2....2+..,..=1,2,... 15.amongIndependent and identically distributed... (0,..2)(One)Coefficient of linear regression..0,..1,Least squares estimation and error variance of..2..2 estimates, writes regression equations, and...Regression coefficientInterpret;(Two)The ANOVA table was used to explain the significance of linear regression test. Square of the coefficient of the complex correlation..2valueAnd explain its meaning;(ThreeSeparately seek..1 andThe confidence of..2 is95%Confidence interval;(Four)YesThe number of people tested by alpha =0.05 ..1 and income..2Sales volumeYIs the effect significant?Regression coefficientTest of general hypothesis test method ..1 andThe interaction of..2 (i.e...1..2) yesYIs the effect significant?;Data importEdit window inputThis questionTheData import code:TitleData analysis method_exercises2.4_page79"; / *Title, omission does not affect analysis results * /DataMylib.ch2_2_4;*First, a new logical library,Logical LibrariesMylibCreate data setCh2_2_4*/Input y X1 x2 @ @ /*@@; Represents a continuous input,YDependent variable,X1,X2Independent variable* /Cards; / *Start input data* /1622742450120180, 32542233753802131205283867862347, 1692653782819830081923302450, 1161952137Fifty-five532560252430402023, 37244271442362660103157, 20882123702605;*Missing data"."Otherwise, the corresponding set of data will be automatically deleted* /Run/*runStatement is used to illustrate all rows before the statement in the current procedure step* /PressF8After run,Open logical libraryMylibYou can see the new data setCh2_2_4.SASA variety of imports are provided According to the manner, for example: One,Read data from file,INFILEF:\Mylib\CH2_2_4.txt";TwoAnd the use of established data sets,Proc reg data=mylib.ch2_2_4;You can also import directly from outsideExcelOther ways. The program above is entered directly in the editbox.procedure callThe procedure to call in this questionyesProc regProcess.Proc regProcess isSASsystemMany regression analysis process of the system in the Except that it can fit the general linear regression model,A variety of optimal model selection methods and model checking methods are also provided.Among themOneTwo)ThreeThe results of multivariate linear regression analysis are mainly used. (Four) will use a linear regression analysisResults.(I)Yand..,Linear regression analysisProcReg*transferRegProcess use* /MOdel y=x1 x2;*Dependent variableYThe independent variable is X1,X2*/Run;ModelStatement: used to define the model's dependent variables, arguments, model options, and output options.Common options areSelection=,Specifies the variable selection method:FORWARD(forward input method),BACKWARDXiang HoushanDivision),STEPWISE(stepwise regression),ADJRSQ(modified multiple correlation coefficient criterion),CP(Cp criterionEtc..NOINTSaid, is often included in the modelNumber item;STBThe regression coefficient, output standard;CLIThe output of single predictive value, confidence interval; RResidual scores are performedAnalysis of results of the analysis and output; IOutput(XTX).1matrix.Format:MODELDependent variable name=Argument rankingTheseoption]Cases:Model y=x1 / x2 selection=stepwise / *; stepwise regression* /After running the program, get the results Parameter estimation table(One)Least squares estimation:= = (0,... 1,... 2) = (3.45261,0.49600,0.00920) Regression equation:Y=3.45261+0.49600..1+0.00920..2ANOVA table(TwoError variance estimate:... 2=MSE=4.74040Multiple correlation coefficientSquares:..2=0.9989(R-Square)Significance: from the value of the complex correlation coefficient, it can be seen that it is highly significantand..1,..2)Multiple correlation coefficient SquaresCan also passBy calculation:..2=SSR/SST=53845/53902=0.9989 (Three)Confidence interval:K+....t1..2 (N.P) s...)...0.975 (12) =2.17881 (via check) T distribution table obtained) You can also pass the functionY=TINV(P,DFObtain...1=0.496+/-2.179*0.00605Draw (Zero point four eight two eight ,Zero point five zero nine two )..2=0.0092+/-2.179*0.00096811,DrawZero point zero zero seven one ,Zero point zero one one three )(Two)YandLinear regression analysisProcRegData=mylib.ch2_2_4; / *Direct reference data set* /Model y=x1;Run;(FourThe coefficient of multiple correlation is: Zero point nine nine one zeroX1YesYSignificant influence(Three)YandLinear regression analysis ProcRegData=mylib.ch2_2_4; / * Direct reference data set * /Model y=x2;Run;(Four)The coefficient of quadratic correlation is square: Zero point four zero eight seven,X2YesYThe effect is not significant(Four)YandLinear regression analysis of... Data mylib.ch2_2_4;Set mylib.ch2_2_4;*Read data set* /Z=x1*x2;*New argumentZ*/Run;Proc reg;Model y=z;*Argument isZ*/Run;(Four)The square of the complex correlation coefficient is: Zero point nine zero three zero,X1X2YesYSignificant impactLinear regression analysis using modules (I)Linear regression analysisstart-upSASSystem, and click "solution" in turn"->"Analysis"->"Analysts"And then click "file""->Open, open the data set"Ch2_2_4.sas7bdat",FigureVariable listindependent variable dependent variableThe value of confidence a Click "Statistics" in turn" ->"Regression"->"Simple" pop-up dialog boxOne)Variable settingsOn the left hand side of the variables listCentral ElectionYClick"DependentThe button is set as dependent variable ;SelectedX2Click"Explanatory"Button, set it as an argument."ModelIn the settings bar, select by default" Linear"" means linear regression.(Two)TestsSet upClick"TestsButton to eject the dialog boxConfidence defaults toZero Point Zero FiveMay change.Click"OK".(Three)PlotsSet upClick"Plots"Button" pops up the plotting Options dialog boxChoice"ResidulTab."Studentized"Represents a student residual," Normal quantile-Quantile plot"Stands for normality."QQGraph check.Settings as shownResidual columnNormal inspectionTest barVariable columnvariance analysisparameter estimationClick"OK"And click on the main settings dialog box "OK",ThereforeAnd get resultsregression equationClick"Analysis (new, project) "Dialog box""Plot of RSTUDENTVsX2"" pops up the residual graph Dialog boxClick again"Plot of RSTUDENTVsNQQ"Pop upQQchartThe normal state of the residual by the studentQQIt can be seen that the model error term is approximately normal distribution.Independent variable selection(two) manyLinear regression analysisstart-upSASSystem, click "solution" in turnResolution"->"Analysis">"Analyst", and then click "file"" ->Open, open the data set"Ch2_2_4.sas7bdat".Click "Statistics" in turn"->"Regression"->"Linear" pop-up dialog boxSelect argumentX1,X2Dependent variableY. Click"ModelButton to eject the dialog boxIn"Selection method"Column" provides independent variable selection, such as: Stepwise selectionExpressStep regression method;Adjusted R-SquareIndicates the modified multiple correlation coefficient criterion. This example selectsStepwise regression method. Click"OK".PlotsThe setting is similar to the one element regression analysis. Last click"OK".Multivariate linear analysis:Residual plotQQIn addition, click"Analysis (New project)"Dialog box"Code"Pop-up program dialog box.The above process is mainly explained by linear regressionSASThe use of the system, and therefore less analysis of the results. For example: byQQAs can be seen from the graphspotApproach a straight line, IndicateError termApproximatejuststateDistribution.。

The Standard Model Higgs boson as the inflaton

The Standard Model Higgs boson as the inflaton


is to put M to zero and consider the “induced” gravity
[10, 11, 12, 13, 14], in which the electroweak symmetry
bhraepapkeinnsgifg√enξer∼ate1s/(th√eGPNlaMncWk
1 In our notations the conformal coupling is ξ = −1/6.
4 December 2007
MP = (8πGN )−1/2 = 2.4 × 1018 GeV. This model has “good” particle physics phenomenology but gives
“badhe Higgs field
is too large and matter fluctuations are many orders of
magnitude larger than those observed. Another extreme
Abstract
We argue that the Higgs boson of the Standard Model can lead to inflation and produce cosmological perturbations in accordance with observations. An essential requirement is the non-minimal coupling of the Higgs scalar field to gravity; no new particle besides already present in the electroweak theory is required.
lmodel2模型II线性回归包说明说明书

lmodel2模型II线性回归包说明说明书

Package‘lmodel2’October13,2022Type PackageTitle Model II RegressionVersion1.7-3Date2018-02-05Author Pierre LegendreMaintainer Jari Oksanen<********************>Depends R(>=2.14.0)Description Computes model II simple linear regression using ordinaryleast squares(OLS),major axis(MA),standard major axis(SMA),andranged major axis(RMA).License GPL-2LazyLoad yesNeedsCompilation noRepository CRANDate/Publication2018-02-0522:59:16UTCR topics documented:lmodel2 (1)mod2ex1 (5)Index6 lmodel2Model II regressionDescriptionThis function computes model II simple linear regression using the following methods:ordinary least squares(OLS),major axis(MA),standard major axis(SMA),and ranged major axis(RMA).The model only accepts one response and one explanatory variable.1Usagelmodel2(formula,data=NULL,range.y=NULL,range.x=NULL,nperm=0)Argumentsformula A formula specifying the bivariate model,as in lm and aov.data A data frame containing the two variables specified in the formula.range.y,range.xParametres for ranged major axis regression(RMA).If range.y=NULL andrange.x=NULL,RMA will not be computed.If only one of them is NULL,the program will stop.If range.y="relative":variable y has a true zero(relative-scale variable).If range.y="interval":variable y possibly includesnegative values(interval-scale variable).If range.x="relative":variable xhas a true zero(relative-scale variable).If range.x="interval":variable xpossibly includes negative values(interval-scale variable)nperm Number of permutations for the tests.If nperm=0,tests will not be computed. DetailsModel II regression should be used when the two variables in the regression equation are random,i.e.not controlled by the researcher.Model I regression using least squares underestimates theslope of the linear relationship between the variables when they both contain error.Ordinary least squares(OLS)is,however,appropriate in some cases as a model II regression model;see the “Model II User’s guide,R edition”which you can read using command vignette("mod2user").The model II regression methods of ordinary least squares(OLS),major axis(MA),standard major axis(SMA),and ranged major axis(RMA)are described in Legendre and Legendre(1998,Section10.3.2).OLS,MA,and SMA are also described in Sokal and Rohlf(1995).The PDF document“Model II User’s guide,R edition”provided with this function contains a tutorial for model II regression,and can be read with command vignette("mod2user").The plot function plots the data points together with one of the regression lines,specified by method="OLS",method="MA"(default),method="SMA",or method="RMA",and its95percent con-fidence interval.ValueThe default output provides the regression output.It draws information from a list,produced by function lmodel2,which contains the following elements:y The response variable.x The explanatory variable.regression.resultsA table with rows corresponding to the four regression methods.Column1gives the method name,followed by the intercept and slope estimates,the anglebetween the regression line and the abscissa,and the permutational probability(one-tailed,for the tail corresponding to the sign of the slope estimate).confidence.intervalsA table with rows corresponding to the four regression methods.The methodname is followed by the parametric95the intercept and slope estimates.eigenvalues Eigenvalues of the bivariate dispersion,computed during major axis regression.H The H statistic used for computing the confidence interval of the major axisslope.Notation following Sokal and Rohlf(1995).n Number of objects.r Correlation coefficient.rsquare Coefficient of determination(R-square)of the OLS regression.P.param2-tailed parametric P-value for the test of r and the OLS slope.theta Angle between the two OLS regression lines,lm(y~x)and lm(x~y).nperm Number of permutations for the permutation tests.epsilon Any value smaller than epsilon is considered to be zero.info.slope Information about the slope notation when r=0.info.CI Information about the confidence limits notation when the slope is infinite.call Call of the function.NoteThe package exports only the main functions lmodel2,plot.lmodel2and lines.lmodel2.Much of the work is done by internal functions which are not directly visible,but you can use triple colon to see or directly use these functions(e.g.,lmodel2:::print.lmodel2).Internal functions that perform essential parts of the analysis are MA.reg,SMA.reg,CLma,CLsma and permutest.lmodel2.Author(s)Pierre Legendre,Departement de Sciences Biologiques,Universite de MontrealReferencesLegendre,P.and L.Legendre.1998.Numerical ecology,2nd English edition.Elsevier Science BV, Amsterdam.Sokal,R.R.and F.J.Rohlf.1995.Biometry–The principles and practice of statistics in biological research.3rd edition.W.H.Freeman,New York.See AlsoA tutorial(file“Model II User’s guide,R edition”)is provided with this function,and can be readwithin R session using command vignette("mod2user",package="lmodel2").Examples##The example data files are described in more detail in the##\dQuote{Model II User s guide,R edition}tutorial.##Example1(surgical unit data)data(mod2ex1)Ex1.res<-lmodel2(Predicted_by_model~Survival,data=mod2ex1,nperm=99)Ex1.resplot(Ex1.res)##Example2(eagle rays and Macomona)data(mod2ex2)Ex2.res<-lmodel2(Prey~Predators,data=mod2ex2,"relative","relative",99) Ex2.resop<-par(mfrow=c(1,2))plot(Ex2.res,"SMA")plot(Ex2.res,"RMA")par(op)##Example3(cabezon spawning)op<-par(mfrow=c(1,2))data(mod2ex3)Ex3.res<-lmodel2(No_eggs~Mass,data=mod2ex3,"relative","relative",99)Ex3.resplot(Ex3.res,"SMA")plot(Ex3.res,"RMA")par(op)##Example4(highly correlated random variables)op<-par(mfrow=c(1,2))data(mod2ex4)Ex4.res<-lmodel2(y~x,data=mod2ex4,"interval","interval",99)Ex4.resplot(Ex4.res,"OLS")plot(Ex4.res,"MA")par(op)#Example5(uncorrelated random variables)data(mod2ex5)Ex5.res<-lmodel2(random_y~random_x,data=mod2ex5,"interval","interval",99) Ex5.resop<-par(mfrow=c(2,2))plot(Ex5.res,"OLS")plot(Ex5.res,"MA")plot(Ex5.res,"SMA")plot(Ex5.res,"RMA")par(op)##Example6where cor(y,x)=0by construct(square grid of points)y0=rep(c(1,2,3,4,5),5)x0=c(rep(1,5),rep(2,5),rep(3,5),rep(4,5),rep(5,5))plot(x0,y0)Ex6=as.data.frame(cbind(x0,y0))zero.res=lmodel2(y0~x0,data=Ex6,"relative","relative")print(zero.res)op<-par(mfrow=c(1,2))plot(zero.res,"OLS")plot(zero.res,"MA")par(op)mod2ex15 mod2ex1Example Data Sets for Model II RegressionDescriptionThese example data sets for model II regression are discussed in the vignette“Model II Regression User Guide”.Usagedata(mod2ex1)FormatSee vignette“Model II Regression User Guide”.Examplesdata(mod2ex1)Index∗datasetsmod2ex1,5∗modelslmodel2,1∗regressionlmodel2,1aov,2CLma(lmodel2),1CLsma(lmodel2),1lines.lmodel2(lmodel2),1lm,2lmodel2,1MA.reg(lmodel2),1mod2ex1,5mod2ex2(mod2ex1),5mod2ex3(mod2ex1),5mod2ex4(mod2ex1),5mod2ex5(mod2ex1),5permutest.lmodel2(lmodel2),1plot.lmodel2(lmodel2),1print.lmodel2(lmodel2),1SMA.reg(lmodel2),1vignette,56。