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description=LinearModelFit attempts to model the input data using a linear combination of functions.;

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Text of the page (random words):
sin x y x 5 y 5 1 wolfram language code linearmodelfit data 1 sin x cos y x y properties 7 data fitted functions 1 fit a linear model wolfram language code data block i j table i randomreal 10 j randomreal 10 4i 2j randomreal 10 lm linearmodelfit data x y x y obtain a list of available properties for a linear model wolfram language code lm properties extract the original data wolfram language code lm data obtain and plot the best fit wolfram language code fit lm bestfit wolfram language code show plot3d fit x 0 10 y 0 10 plotrange all graphics3d pointsize 0 025 point data obtain the best fit at a specific point wolfram language code lm bestfit 2 5 obtain the fitted function as a pure function wolfram language code lm function get the design matrix and response vector for the fitting wolfram language code matrixform lm designmatrix response residuals 1 examine residuals for a fit wolfram language code lm linearmodelfit randomreal 10 100 3 x y x y wolfram language code fr sr1 sr2 lm fitresiduals standardizedresiduals studentizedresiduals visualize the raw fit residuals wolfram language code listplot fr visualize scaled residuals in stem plots wolfram language code map listplot filling 0 sr1 sr2 row plot the absolute differences between the standardized and studentized residuals wolfram language code listplot abs sr1 sr2 sums of squares 1 fit a linear model to some data wolfram language code seedrandom 1 data flatten table x y 2 3x 1y randomreal 1 1 x randomreal 5 10 y randomreal 5 10 1 lm linearmodelfit data x y x y extract the estimated error variance and coefficient of variation wolfram language code lm estimatedvariance coefficientofvariation obtain an analysis of variance table for the model wolfram language code lm anova get the f statistics from the table wolfram language code lm anova all fstatistics deletemissing normal parameter estimation diagnostics 1 obtain a formatted table of parameter information wolfram language code seedrandom 1 data table x 2 3x 1sin x randomreal 2 2 x randomreal 5 100 lm linearmodelfit data x sin x cos x x wolfram language code lm parameterestimates obtain the statistics of fitted parameters wolfram language code lm parameterestimates all tstatistic normal influence measures 1 fit some data containing extreme values to a linear model wolfram language code seedrandom 1 data table i 2 4i log i randomreal i randomreal 1 10 20 data 3 8 1 data 3 8 1 5 lm linearmodelfit data x log x x use single deletion variances to check the impact on the error variance of removing each point wolfram language code listplot lm singledeletionvariances plotrange 0 all filling 0 check cook distances to identify highly influential points wolfram language code listplot lm cookdistances plotrange 0 all filling 0 use dffits values to assess the influence of each point on the fitted values wolfram language code listplot lm fitdifferences plotrange all filling 0 use dfbetas values to assess the influence of each point on each estimated parameter wolfram language code mapthread listplot 1 plotrange 5 1 filling 0 plotlabel 2 transpose lm betadifferences subscriptbox β 1 subscriptbox β 2 subscriptbox β 3 row prediction values 1 fit a linear model wolfram language code seedrandom 1 data flatten table x y 3exp x 1 2y randomreal 1 1 x randomreal 5 3 y randomreal 5 3 1 lm linearmodelfit data exp x y x y plot the predicted values against the observed values wolfram language code listplot transpose lm response predictedresponse framelabel observed predicted frame true axes false obtain tabular results for the mean prediction confidence intervals wolfram language code lm meanpredictions obtain tabular results for the single prediction confidence intervals wolfram language code lm singlepredictions get the single prediction intervals from the table wolfram language code lm singlepredictions all confidenceinterval normal extract 99 mean prediction bands wolfram language code lm meanpredictionbands confidencelevel 99 compute the 99 mean prediction bands at a specific location wolfram language code lm meanpredictionbands 4 1 confidencelevel 99 goodness of fit measures 1 obtain a table of goodness of fit measures for a linear model wolfram language code data flatten table i j randomreal 10 30i 10j randomreal 5 i randomreal 10 5 j randomreal 10 5 1 wolfram language code lm linearmodelfit data x y z x y z wolfram language code grid transpose lm adjustedrsquared aic bic rsquared alignment left compute goodness of fit measures for all possible linear submodels wolfram language code sub table join i linearmodelfit data i x y z adjustedrsquared rsquared aic bic i subsets x y z rank the models by wolfram language code grid join model adjustedrsquared rsquared aic bic sortby sub 3 rank the models by adjusted which penalizes for adding terms wolfram language code grid join model adjustedrsquared rsquared aic bic sortby sub 2 generalizations extensions 1 perform other mathematical operations on the functional form of the model wolfram language code lm linearmodelfit randomreal 10 20 x log x x integrate symbolically and numerically wolfram language code integrate lm x x wolfram language code nintegrate lm x x 1 5 find a predictor value that gives a particular value for the model wolfram language code findroot lm x 5 5 x 10 options 11 confidencelevel 1 the default gives 95 confidence intervals wolfram language code data 0 1 1 0 3 2 5 4 wolfram language code lm linearmodelfit data x x wolfram language code lm parameterestimates all confidenceinterval normal use 99 intervals instead wolfram language code lm linearmodelfit data x x confidencelevel 99 wolfram language code lm parameterestimates all confidenceinterval normal set the level to 90 within fittedmodel wolfram language code lm parameterestimates confidencelevel 9 all confidenceinterval normal includeconstantbasis 1 fit a simple linear regression model wolfram language code data 0 1 1 0 3 2 5 4 wolfram language code linearmodelfit data x x normal fit the linear model with intercept zero wolfram language code linearmodelfit data x x includeconstantbasis false normal linearoffsetfunction 1 fit data to a linear model wolfram language code data 0 1 1 1 5 3 2 5 4 wolfram language code linearmodelfit data x x normal fit data to a linear model with a known sqrt x term wolfram language code linearmodelfit data x x linearoffsetfunction sqrt normal nominalvariables 1 fit data treating the first variable as a nominal variable wolfram language code data a 0 1 b 2 2 a 2 1 8 b 0 2 5 nom linearmodelfit data x y x y nominalvariables x wolfram language code normal nom treat both variables as nominal wolfram language code linearmodelfit data x y x y nominalvariables all normal varianceestimatorfunction 1 use the default unbiased estimate of error variance wolfram language code lm linearmodelfit range 10 2 x x wolfram language code lm estimatedvariance assume a known error variance wolfram language code lm estimatedvariance varianceestimatorfunction 20 estimate the variance by the mean squared error wolfram language code lm estimatedvariance varianceestimatorfunction mean 2 weights 5 fit a model using equal weights wolfram language code linearmodelfit range 10 2 x x normal give explicit weights for the data points wolfram language code linearmodelfit range 10 2 x x weights 1 range 10 normal use around values to give different weights to data points wolfram language code data around 1 0 2 around 2 0 1 around 4 2 fit linearmodelfit data x x weights automatic wolfram language code show plot fit x x 0 3 listplot data plotrange all find the weights that were used to account for the uncertainty in the data wolfram language code fit weights use around values in both the independent values and responses wolfram language code seedrandom 1 data table around x randomreal 0 25 around 2x 2 x 0 3 randomreal 0 1 0 1 randomreal 0 25 x 1 1 0 2 fit linearmodelfit data 1 x x 2 x weights automatic normal wolfram language code show plot fit x 1 1 listplot data plotrange all axesorigin 0 fit a model of more than one variable with around values wolfram language code seedrandom 1 data join table around x randomreal 0 25 around y randomreal 0 25 around 2x 2 x x y y 2 randomreal 0 1 0 1 randomreal 0 25 x 1 1 0 2 y 1 1 0 2 fit linearmodelfit data 1 x x 2 y y 2 x y x y weights automatic normal try the fixedpoint algorithm to find the weights for the model wolfram language code data linearmodelfit data 1 x x 2 y y 2 x y x y weights automatic method fixedpoint reduce the damping factor and increase the maxiterations to reach convergence wolfram language code linearmodelfit data 1 x x 2 y y 2 x y x y weights automatic method fixedpoint dampingfactor 0 25 maxiterations 200 tolerance 10 10 workingprecision 1 use workingprecision to get higher precision in parameter estimates wolfram language code data table x x 3 x 10 wolfram language code lm linearmodelfit data x x 2 x workingprecision 50 obtain the fitted function wolfram language code lm bestfit reduce the precision in property computations after the fitting wolfram language code lm bestfit workingprecision 20 applications 6 fit the first 100 primes to a linear model wolfram language code lm linearmodelfit array prime 100 x x visualize the fit wolfram language code show listplot lm data plotstyle orange plot lm x x 0 100 the systematic trend in the residuals violates the assumption of independent normal errors wolfram language code listplot lm fitresiduals fit a linear model of multiple variables wolfram language code data map 1 2 3 1 2 3 7 randomreal 1 1 1 2 2 23 4 3 randomreal 10 100 3 wolfram language code lm linearmodelfit data x y z x y z visually inspect the residuals by data point wolfram language code listplot lm fitresiduals frame true plotrange all plot the residuals against each predictor variable wolfram language code table listplot transpose data all i lm fitresiduals frame true framelabel x y z i residual i 3 row plot cook s distances to diagnose leverage wolfram language code cd lm cookdistances wolfram language code listplot cd filling 0 frame true axesorigin 0 0 plotrange all plotlabel cook s distances find the positions of distances above a given cutoff value wolfram language code position cd _ 05 extract the associated data points wolfram language code extract data use plots to check the assumption of normal errors wolfram language code lm linearmodelfit randomreal 10 50 4 x y z x y z compare standardized residuals to standard normal values wolfram language code quantileplot lm standardizedresiduals table inversecdf normaldistribution q q 1 100 99 100 1 50 do the comparison with studentized residuals wolfram language code quantileplot lm studentizedresiduals table inversecdf normaldistribution q q 1 100 99 100 1 50 simulate some data with a continuous and a nominal variable wolfram language code groups control treatment 1 treatment 2 treatment 3 data blockrandom seedrandom 123 block vals times rand vals randomchoice groups 100 times randominteger 10 100 rand randomreal 1 100 transpose vals times vals thread rule groups 16 34 57 1 1 05 times rand fit an analysis of covariance model to the data wolfram language code lm linearmodelfit data treatment time treatment time nominalvariables treatment obtain an analysis of variance table for the model wolfram language code lm anova group the data by treatment wolfram language code grps drop gatherby sort data first none none 1 visualize the grouped data and associated curves wolfram language code show listplot grps plotrange all plot evaluate map lm t groups t 0 10 plotlegends linelegend groups legendlayout reversedcolumn use properties to compute additional results wolfram language code seedrandom 1 data flatten table x y z 1 2x 3 4y 10z randomreal 10 x randomreal 10 3 y randomreal 10 3 z randomreal 10 3 2 wolfram language code lm linearmodelfit data x y z x y z extract the design matrix and residuals wolfram language code desmat resids lm designmatrix fitresiduals compute white s heteroskedasticity consistent covariance estimate wolfram language code wc with inv inverse transpose desmat desmat xresid resids desmat inv transpose xresid xresid inv matrixform compare with the covariance assuming homoskedasticity wolfram language code lm covariancematrix matrixform compare standard errors based on the two covariance estimates wolfram language code sqrt diagonal wc wolfram language code lm parameterestimates all standarderror normal perform a breusch pagan test wolfram language code data flatten table x y z 1 2x 3 4y 10z randomreal 10 x randomreal 10 3 y randomreal 10 3 z randomreal 10 3 2 fit a model wolfram language code lm1 linearmodelfit data x y z x y z fit the squared errors to a model with the same predictors wolfram language code lm2 linearmodelfit block newdata data newdata all 1 lm1 fitresiduals 2 newdata x y z x y z compute the breusch pagan test statistic wolfram language code bp with sqresids lm1 fitresiduals 2 variance sqresids length data 1 total lm2 fitresiduals 2 2 total sqresids length data 2 compute the value wolfram language code 1 cdf chisquaredistribution length lm1 bestfitparameters 1 bp properties relations 10 designmatrix constructs the design matrix used by linearmodelfit wolfram language code data table i randomreal i 5 wolfram language code designmatrix data x x matrixform wolfram language code lm linearmodelfit data x x wolfram language code lm designmatrix matrixform by default linearmodelfit and generalizedlinearmodelfit fit equivalent models wolfram language code data table i randomreal i 1 i i 10 wolfram language code linearmodelfit data x x wolfram language code generalizedlinearmodelfit data x x linearmodelfit fits linear models assuming normally distributed errors wolfram language code data table i randomreal i 1 i i 10 wolfram language code linearmodelfit data x 2 x nonlinearmodelfit fits nonlinear models assuming normally distributed errors wolfram language code nonlinearmodelfit data a exp b x a b x fit and linearmodelfit fit equivalent models wolfram language code data table i randomreal i 1 i i 10 wolfram language code fit data 1 x 2 x wolfram language code lm linearmodelfit data x 2 x linearmodelfit allows for extraction of additional information about the fitting wolfram language code lm fitresiduals fit a linear model to data wolfram language code data flatten table i j i j exp i j randomreal 2 i 5 j 5 1 wolfram language code linearmodelfit data x y exp x y x y bestfitparameters perform the same fitting using a design matrix and response vector wolfram language code dm designmatrix data x y exp x y x y resp data all 1 wolfram language code linearmodelfit dm resp bestfitparameters obtain the parameter estimates via leastsquares wolfram language code leastsquares dm resp linearmodelfit fits linear models wolfram language code data table i i i 10 wolfram language code linearmodelfit data x 2 sin x x bestfitparameters findfit gives parameter estimates for linear and nonlinear mode...