For those 4 classification problems, we also did the same processes as we mentioned in part I. Instead of going over the details, we would only show the model selection result of the four classification problems as following:
Table 1. Model Selection Results by Data Sets  
Wdbc  Ionosphere  Hypothyroid  
Gradient Boosting(n.trees, depth, n.minobsinnode)  500,9,10  450,9,15  300,4,10 
Random Forest(mtry)  6  16  18 
Neural Networks(num. of neurons in the hidden layer)  10  5  3 
SVM(type, param (degree/sigma),cost)  Linear,1,0.32  Radial, 5,1  Linear,1,1.32 
Ridge regression (lambda)  0.0001  0.0001  100000 
Logistic regression  No parameter to tune  
Model Averaging (logistic regression)
(AIC for the 3 model) 
3 logistic model
(AIC:31.0;35.9;41.3) 
3 logistic model
(AIC: 217.2; 231.6; 248.8) 
3 logistic model
(AIC: 176.28; 176.64; 177.79 ) 
Model Averaging
To improve the performances of linear ridge regression and logistic regression, we used the R packages (glmulti and MuMIn) for model averaging. For each data set, we chose the best 3 or 5 models to do model averaging to see if it will improve the performance of linear ridge regression (for regression) and logistic regression (for classification).
1.Regression Problem (Boston Housing)
We selected the top 5 ridge regression models based on their AIC scores:
weightable(avg.model) model aicc weights 1 y ~ 1 + CRIM + NOX + RM + AGE + DIS + TAX + PTRATIO + B + LSTAT 1327.777772 0.056535233839 2 y ~ 1 + CRIM + ZN + NOX + RM + AGE + DIS + RAD + TAX + PTRATIO + B + LSTAT 1328.092560 0.048301873324 3 y ~ 1 + CRIM + ZN + NOX + RM + AGE + DIS + TAX + PTRATIO + B + LSTAT 1328.147654 0.046989452154 4 y ~ 1 + CRIM + NOX + RM + AGE + DIS + RAD + TAX + PTRATIO + B + LSTAT 1329.091435 0.029313049215 5 y ~ 1 + CRIM + NOX + RM + AGE + DIS + TAX + PTRATIO + B 1329.205842 0.027683297814
By averaging the top 5 models, the results shows the variables {CRIM, RM, AGE, DIS, TAX, PTRATIO, B}are all significant in both fullaveraged model and conditional averaged model
2.Classification Problem
For classification problem, we choose 3 logistic model based on their AIC scores, and do model averaging. The results are shown as below:
 Wdbc
Table 3.1 Model Averaging Results of Wdbc  
Model 1  y ~ V4 + V8 + V9 + V15 + V22 + V23 + V29 + V30 + V32 
Model 2  y ~ V3 + V4 + V8 + V9 + V15 + V22 + V23 + V29 + V30 + V32 
Model 3  y ~ V3 + V4 + V7 + V8 + V9 + V15 + V22 + V23 + V29 + V30 + V32 
Variable importance  V15 V22 V23 V29 V30 V32 V4 V8 V9 V3 V7
Importance: 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.31 0.08 N containing models: 3 3 3 3 3 3 3 3 3 2 1 
Note  The modelaveraged coefficients for both fullaveraged model and conditional averaged model, no variables are significant. 
 Ionosphere
Table 3.2 Model Averaging Results of Ionosphere  
Model 1  y ~ V3 + V4 + V5 + V6 + V8 + V9 + V10 + V11 + V13 + V14 + V15 + V16 + V18 + V22 + V23 + V26 + V27 + V30 + V31 
Model 2  y ~ V3 + V4 + V5 + V6 + V8 + V9 + V10 + V11 + V13 + V14 + V15 + V16 + V18 + V22 + V23 + V24 + V26 + V27 + V28 + V30 + V31 
Model 3  y ~ V3 + V4 + V5 + V6 + V8 + V9 + V10 + V11 + V13 + V14 + V15 + V16 + V18 + V22 + V23 + V24 + V26 + V27 + V30 + V31 
Variable importance  V10 V11 V13 V14 V15 V16 V18 V22 V23 V26 V27 V3 V30 V31 V4 V5 V6 V8 V9 V24 V28
Importance: 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.53 0.23 N containing models: 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 1

Note  The results shows the variables { V14 V15 V16 V22 V23 V26 V27 V3 V30 V31 V4 V5 V9 }are all significant in both fullaveraged model and conditional averaged model

 Hypothyroid
Table 3.3 Model Averaging Results of Hypothyroid  
Model 1  y~ V4 + V6 + V14 + V15 + V16 + V24 
Model 2  y ~V4 + V6 + V7 + V14 + V15 + V16 + V24 
Model 3  y~ V4 + V6 + V7 + V14 + V15 + V16 + V17 + V24 
Variable importance  Relative variable importance:
V14 V15 V16 V24 V4 V6 V7 V17 Importance: 1.00 1.00 1.00 1.00 1.00 1.00 0.57 0.20 N containing models: 3 3 3 3 3 3 2 1

Note  The results shows the variables { V16 V24 }are all significant in both fullaveraged model and conditional averaged model

Performances by Data sets
The tables and plots below show the estimate accuracy based on the holdout test data set. The best performing model for each data set is boldfaced while the worst one is italic.
1.Boston Housing
Below shows the MSE of test data set for Boston Housing data, the Random Forest model has the lowest MSE 32.64723. The ridge regression performs the worst, whose MSE is 303.57214. But we can improve the ridge regression model by using model averaging (whose MSE is 264.95846) and feature selection (whose MSE is 55.51494).
Table 4. MSE(test) for Boston Housing Data  
Data sets  MSE (test)  Note 
Gradient Boosting  33.30788  
Random Forest  32.64723  
Neural Networks  87.52996  Used scaled data set 
SVM  97.98975  Used scaled data set 
Ridge regression  303.57214  1. Used scaled data set
2. The test error can be reduced to 55.51494 with feature selection 
Logistic regression  0.69170  
Model Averaging
(linear ridge regression) 
264.95846 
2. Classification Problems’ MSE (test) by Data Sets
In this table, we show the MSE of test data sets for classification problems, random forests performs the best on both Wdbc and Hypothyroid data sets; while the best model on Ionosphere is SVM.
Overall, the gradient boosting has the best average performance. In addition, the linear ridge regression has the poorest performance, due to it’s not quite suitable for classification problems. Also, logistic regression model has very bad average performance, but it performs very well on Hypothyroid. Just as No Free Lunch Theorem said there is no best learning algorithm for all data sets.
The model averaging models of Wdbc and Hypothyroid have improves the performance of original logistic regression, while the model of Ionosphere doesn’t.
Table 5. Classification Problems’ MSE (test) by Data Sets  
Wdbc  Ionosphere  Hypothyroid  Average performance  
Gradient Boosting  0.033457249070632  0.0610687022900763  0.0167548500881834  0.0370936004829639 
Random Forest  0.0260223048327138  0.0763358778625954  0.0149911816578483  0.0391164547843858 
Neural Networks  0.0446096654275093  0.0687022900763359  0.027336860670194  0.0468829387246797 
SVM  0.0371747211895911  0.0534351145038168  0.027336860670194  0.039315565454534 
Ridge regression  0.5278810409  0.1832061069  0.5987654321  0.436617526633333 
Logistic regression  0.0594795539  0.1297709924  0.02557319224  0.0716079128466667 
Model Averaging
(logistic regression) 
0.04832713755  0.1374045802  0.02469135802  0.0701410252566667 
3. ROC plots for Classification Problems
For the 3 classification problems, SVM, random forests, and gradient boosting have excellent performance on the area under the ROC. While Logistic regression and ridge regression performs the poorest. Comparing these plots, Wdbc’s models tend to have larger areas under the ROC curve than those of Ionosphere. These can be also shown by MSE of test data (in Table 5).
Conclusion
With the excellent performance on all 4 data sets, gradient boosting and random forests are the best algorithms overall.
Table 6. Difference Between the Algorithms  
Problem Type  Training Speed  Prediction Speed  Data need scaling?  Handle lots of features well?  
Gradient Boosting  Either  Slow  Fast  No  Yes 
Random Forest  Either  Slow  Moderate  No  Yes 
Neural Networks  Either  Slow  Moderate  Yes  Yes 
SVM (with kernel)  Either  Fast  Fast  Yes  Yes 
Ridge regression  Regression  Fast  Fast  Yes  No(need feature selection) 
Logistic regression  Classification  Fast  Fast  No (unless regularized)  No(need feature selection) 
For regression problems, the traditional linear ridge regression can be improved with model selection and model averaging. While using logistic regression to classify the response is not a good idea. For classification problems, SVM, gradient boosting and random forests perform very well. And the model averaging doesn’t improve all the logistic regression performance in each data set.
Just as Rich Caruana and Alexandru NiculescuMizilEven mentioned in their great paper:
“The best models sometimes perform poorly, and models with poor average An Empirical Comparison of Supervised Learning Algorithms performance occasionally perform exceptionally well.”