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stepwiselm creates a linear model and automatically adds to or trims the model. To create a small model, start from a constant model. To create a large model, start with a model containing many terms. A large model usually has lower error as measured by the fit to the original data, but might not have any advantage in predicting new data.
stepwiselm can use all the name-value options from fitlm, with additional options relating to the starting and bounding models. In particular:
For a small model, start with the default lower bounding model: 'constant' (a model that has no predictor terms).
The default upper bounding model has linear terms and interaction terms (products of pairs of predictors). For an upper bounding model that also includes squared terms, set the Upper name-value pair to 'quadratic'.
This example shows how to compare models that stepwiselm returns starting from a constant model and starting from a full interaction model.
Load the carbig data and create a table from some of the data.
load carbig tbl = table(Acceleration,Displacement,Horsepower,Weight,MPG);
Create a mileage model stepwise starting from the constant model.
mdl1 = stepwiselm(tbl,'constant','ResponseVar','MPG')
1. Adding Weight, FStat = 888.8507, pValue = 2.9728e-103 2. Adding Horsepower, FStat = 3.8217, pValue = 0.00049608 3. Adding Horsepower:Weight, FStat = 64.8709, pValue = 9.93362e-15 mdl1 = Linear regression model: MPG ~ 1 + Horsepower*Weight Estimated Coefficients: Estimate SE tStat pValue (Intercept) 63.558 2.3429 27.127 1.2343e-91 Horsepower -0.25084 0.027279 -9.1952 2.3226e-18 Weight -0.010772 0.00077381 -13.921 5.1372e-36 Horsepower:Weight 5.3554e-05 6.6491e-06 8.0542 9.9336e-15 Number of observations: 392, Error degrees of freedom: 388 Root Mean Squared Error: 3.93 R-squared: 0.748, Adjusted R-Squared 0.746 F-statistic vs. constant model: 385, p-value = 7.26e-116
Create a mileage model stepwise starting from the full interaction model.
mdl2 = stepwiselm(tbl,'interactions','ResponseVar','MPG')
1. Removing Acceleration:Displacement, FStat = 0.024186, pValue = 0.8765 2. Removing Displacement:Weight, FStat = 0.33103, pValue = 0.56539 3. Removing Acceleration:Horsepower, FStat = 1.7334, pValue = 0.18876 4. Removing Acceleration:Weight, FStat = 0.93269, pValue = 0.33477 5. Removing Horsepower:Weight, FStat = 0.64486, pValue = 0.42245 mdl2 = Linear regression model: MPG ~ 1 + Acceleration + Weight + Displacement*Horsepower Estimated Coefficients: Estimate SE tStat pValue (Intercept) 61.285 2.8052 21.847 1.8593e-69 Acceleration -0.34401 0.11862 -2.9 0.0039445 Displacement -0.081198 0.010071 -8.0623 9.5014e-15 Horsepower -0.24313 0.026068 -9.3265 8.6556e-19 Weight -0.0014367 0.00084041 -1.7095 0.088166 Displacement:Horsepower 0.00054236 5.7987e-05 9.3531 7.0527e-19 Number of observations: 392, Error degrees of freedom: 386 Root Mean Squared Error: 3.84 R-squared: 0.761, Adjusted R-Squared 0.758 F-statistic vs. constant model: 246, p-value = 1.32e-117
mdl1 has four coefficients (the Estimate column), and mdl2 has six coefficients.
The adjusted R-squared of mdl1 is 0.746, which is slightly less (worse) than that of mdl2, 0.758.
Create a mileage model stepwise with a full quadratic model as the upper bound, starting from the full quadratic model:
mdl3 = stepwiselm(tbl,'quadratic',... 'ResponseVar','MPG','Upper','quadratic');
Compare the three model complexities by examining their formulas.
ans = MPG ~ 1 + Horsepower*Weight
ans = MPG ~ 1 + Acceleration + Weight + Displacement*Horsepower
ans = MPG ~ 1 + Weight + Acceleration*Displacement + Displacement*Horsepower + Acceleration^2
The adjusted R2 values improve slightly as the models become more complex:
RSquared = [mdl1.Rsquared.Adjusted, ... mdl2.Rsquared.Adjusted, mdl3.Rsquared.Adjusted]
RSquared = 0.7465 0.7580 0.7599
Compare residual plots of the three models.
subplot(3,1,1) plotResiduals(mdl1) subplot(3,1,2) plotResiduals(mdl2) subplot(3,1,3) plotResiduals(mdl3)
The models have similar residuals. It is not clear which fits the data better.
Interestingly, the more complex models have larger maximum deviations of the residuals:
Rrange1 = [min(mdl1.Residuals.Raw),max(mdl1.Residuals.Raw)]; Rrange2 = [min(mdl2.Residuals.Raw),max(mdl2.Residuals.Raw)]; Rrange3 = [min(mdl3.Residuals.Raw),max(mdl3.Residuals.Raw)]; Rranges = [range1;range2;range3]
Rranges = -10.7725 14.7314 -11.4407 16.7562 -12.2723 16.7927