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 full-averaged 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 model-averaged coefficients for both full-averaged 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 full-averaged 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 full-averaged model and conditional- averaged model
|
Performances by Data sets
The tables and plots below show the estimate accuracy based on the hold-out 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 Niculescu-MizilEven 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.”
Debugging myself 