LASSO (Least Absolute Shrinkage and Selection Operator) is also called L1 regularization, and Ridge is also called L2 regularization. Elastic Net is the combination of LASSO and Ridge. All three are techniques commonly used in machine learning to correct overfitting.

In this tutorial, we will cover

- What’s the difference between LASSO (L1), Ridge (L2), and Elastic Net?
- How to run LASSO for classification model using Python
`sklearn`

? - How to run Ridge for classification model using Python
`sklearn`

? - How to run Elastic Net for classification model using Python
`sklearn`

? - How to compare the performance of LASSO, Ridge, and Elastic Net?

**Resources for this post:**

- Python code is at the end of the post. Click here for the notebook.
- More video tutorials on Overfitting Correction
- More blog posts on Overfitting Correction
- If you prefer the video version of the tutorial, please check out the video on YouTube.

### Step 0: LASSO (L1) vs Ridge (L2) vs. Elastic Net

In step 0, we will talk about the differences between LASSO, Ridge, and elastic net.

LASSO and Ridge regularization correct overfitting by shrinking the coefficient of the model. During the model training process, instead of minimizing the model training error, they minimize the model training error plus a penalty term. LASSO and Ridge have different calculation algorithms for the penalty term. Elastic net’s penalty term is a combination of the algorithm from LASSO and Ridge.

- The penalty term has a parameter called lambda. It controls the strength of the penalty.
- When lambda equals 0, the penalty term equals 0. So the model is a model with no regularization.
- When lambda increases, the penalty term value increases, and the model coefficients values decrease.
- When lambda goes infinity, the model coefficients shrink to nearly 0. The model is left with only the intercept, which predicts the average value for every data point.

- LASSO’s penalty term is the penalty parameter lambda multiplied by the sum of the absolute value of the coefficients. Because LASSO’s coefficients may shrink to zeros, it can be used for automatic feature selection.
- Ridge shrinks the model coefficients based on the sum of the squared coefficients. Ridge does not shrink the model coefficients to zero.
- Elastic net’s penalty term is a combination of LASSO and Ridge regression’s penalty term. It sets some coefficients to zeros, but the number is smaller than LASSO.

### Step 1: Import Libraries

In the first step, let’s import the Python libraries needed for this tutorial.

We will use the breast cancer dataset for this tutorial, so `datasets`

from `sklearn`

needs to be imported. `pandas`

and `numpy`

are imported for data processing. `matplotlib`

is for visualization, and `StandardScaler`

is for data standardization.

For model training, we imported train_test_split and LogisticRegression.

For model performance evaluation, we imported `plot_confusion_matrix`

, `classification_report`

, `log_loss`

, `roc_curve`

, and `roc_auc_score`

.

# Dataset from sklearn import datasets # Data processing import pandas as pd import numpy as np # Visualization import matplotlib.pyplot as plt # Standardize the data from sklearn.preprocessing import StandardScaler # Model and performance evaluation from sklearn.model_selection import train_test_split from sklearn.linear_model import LogisticRegression from sklearn.metrics import plot_confusion_matrix, classification_report, log_loss, roc_curve, roc_auc_score

### Step 2: Read In Data

In the second step, the breast cancer data from `sklearn`

library is loaded and transformed into a pandas dataframe.

# Load the breast cancer dataset data = datasets.load_breast_cancer() # Put the data in pandas dataframe format df = pd.DataFrame(data=data.data, columns=data.feature_names) df['target']=data.target # Check the data information df.info()

The information summary shows that the dataset has 569 records and 31 columns.

<class 'pandas.core.frame.DataFrame'> RangeIndex: 569 entries, 0 to 568 Data columns (total 31 columns): # Column Non-Null Count Dtype --- ------ -------------- ----- 0 mean radius 569 non-null float64 1 mean texture 569 non-null float64 2 mean perimeter 569 non-null float64 3 mean area 569 non-null float64 4 mean smoothness 569 non-null float64 5 mean compactness 569 non-null float64 6 mean concavity 569 non-null float64 7 mean concave points 569 non-null float64 8 mean symmetry 569 non-null float64 9 mean fractal dimension 569 non-null float64 10 radius error 569 non-null float64 11 texture error 569 non-null float64 12 perimeter error 569 non-null float64 13 area error 569 non-null float64 14 smoothness error 569 non-null float64 15 compactness error 569 non-null float64 16 concavity error 569 non-null float64 17 concave points error 569 non-null float64 18 symmetry error 569 non-null float64 19 fractal dimension error 569 non-null float64 20 worst radius 569 non-null float64 21 worst texture 569 non-null float64 22 worst perimeter 569 non-null float64 23 worst area 569 non-null float64 24 worst smoothness 569 non-null float64 25 worst compactness 569 non-null float64 26 worst concavity 569 non-null float64 27 worst concave points 569 non-null float64 28 worst symmetry 569 non-null float64 29 worst fractal dimension 569 non-null float64 30 target 569 non-null int64 dtypes: float64(30), int64(1) memory usage: 137.9 KB

The target variable distribution shows 63% of ones and 37% of zeros in the dataset. Therefore, one means the patient has breast cancer, and 0 represents the patient does not have breast cancer.

# Check the target value distribution df['target'].value_counts(normalize=True)

1 0.627417 0 0.372583 Name: target, dtype: float64

### Step 3: Train Test Split

In step 3, we split the dataset into 80% training and 20% testing dataset. `random_state`

makes the random split results reproducible.

# Train test split X_train, X_test, y_train, y_test = train_test_split(df[df.columns.difference(['target'])], df['target'], test_size=0.2, random_state=42) # Check the number of records in training and testing dataset. print(f'The training dataset has {len(X_train)} records.') print(f'The testing dataset has {len(X_test)} records.')

The training dataset has 455 records, and the testing dataset has 114 records.

The training dataset has 455 records. The testing dataset has 114 records.

### Step 4: Standardization

Standardization is to rescale the features to the same scale. It is calculated by extracting the mean and divided by the standard deviation. After standardization, each feature has zero mean and unit standard deviation.

Standardization should be fit on the training dataset only to prevent test dataset information from leaking into the training process. Then, the test dataset is standardized using the fitting results from the training dataset.

There are different types of scalers. `StandardScaler`

and `MinMaxScaler`

are most commonly used. For a dataset with outliers, we can use `RobustScaler`

.

In this tutorial, we will use `StandardScaler`

.

# Initiate scaler sc = StandardScaler() # Standardize the training dataset X_train_transformed = pd.DataFrame(sc.fit_transform(X_train),index=X_train.index, columns=X_train.columns) # Standardized the testing dataset X_test_transformed = pd.DataFrame(sc.transform(X_test),index=X_test.index, columns=X_test.columns) # Summary statistics after standardization X_train_transformed.describe().T

We can see that after using `StandardScaler`

, all the features have zero mean and unit standard deviation.

Let’s get the summary statistics for the training data before standardization as well, and we can see that the mean and standard deviation can be very different in scale. For example, the area error has a mean value of 40 and a standard deviation of 47. On the other hand, the compactness error has a mean of about 0.023 and a standard deviation of 0.019.

# Summary statistics before standardization X_train.describe().T

### Step 5: Logistic Regression With No Regularization

In step 5, we will create a logistic regression with no regularization as the baseline model.

Logistic regression in `sklearn`

uses Ridge regularization by default. When checking the default hyperparameter values of the LogisticRegression(), we see that `penalty='l2'`

, meaning that L2 regularization is used.

# Check default values LogisticRegression()

LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True, intercept_scaling=1, l1_ratio=None, max_iter=100, multi_class='auto', n_jobs=None, penalty='l2', random_state=None, solver='lbfgs', tol=0.0001, verbose=0, warm_start=False)

We need to change the `penalty`

from `l2`

to ‘none’ to get the model with no regularization. After running the baseline logistic regression model, we also predicted the testing dataset using `.predict`

and calculated the predicted probabilities using `.predict_proba`

.

# Run model logistic = LogisticRegression(penalty='none', random_state=0).fit(X_train_transformed, y_train) # Make prediction logistic_prediction = logistic.predict(X_test_transformed) # Get predicted probability logistic_pred_Prob = logistic.predict_proba(X_test_transformed)[:,1]

After getting the predicted value and predicted probability, we are ready to check the model performance!

First, let’s check the ROC curve. We got the area under curve value of 0.9818.

# Get the false positive rate and true positive rate fpr,tpr, _=roc_curve(y_test,logistic_pred_Prob) # Get auc value auc=roc_auc_score(y_test,logistic_pred_Prob) # Plot the chart plt.plot(fpr,tpr,label="area="+str(auc)) plt.legend(loc=4)

The log loss value for the model is 2.02.

# Caclulate log loss log_loss(y_test,logistic_pred_Prob)

2.0150517046435854

The confusion matrix shows 1 false positives and 6 false negatives.

# Confusion matrix plot_confusion_matrix(logistic, X_test_transformed, y_test)

A total of 7 incorrect predictions correspond to an accuracy of 0.939.

We do not want to miss any actual cancer patient for the breast cancer prediction and do not mind having a few false positives, so recall would be the metric to pay most attention to. The recall is the true positive rate. It measures the percentage of actual cancer patients captured by the model. We can see that the logistic regression gave us the recall value of 0.915, meaning that 91.5% of the actual cancer patients are captured.

# Performance report print(classification_report(y_test, logistic_prediction, digits=3))

precision recall f1-score support 0 0.875 0.977 0.923 43 1 0.985 0.915 0.949 71 accuracy 0.939 114 macro avg 0.930 0.946 0.936 114 weighted avg 0.943 0.939 0.939 114

Next, let’s check the coefficients of the model. Based on their absolute values, I ranked the model coefficients from high to low, and we can see the top variables have coefficients in a few hundred.

# Model coefficients LogisticCoeff = pd.concat([pd.DataFrame(X_test_transformed.columns), pd.DataFrame(np.transpose(logistic.coef_))], axis = 1) LogisticCoeff.columns=['Variable','Coefficient'] LogisticCoeff['Coefficient_Abs']=LogisticCoeff['Coefficient'].apply(abs) LogisticCoeff.sort_values(by='Coefficient_Abs', ascending=False)

### Step 6: LASSO

In step 6, LASSO model is used to run the same analysis.

`penalty='l1'`

means LASSO regularization is applied.

`solver`

is an algorithm to use in the optimization problem. There are different types of solvers. For small datasets, â€˜liblinearâ€™ is a good choice, whereas â€˜sagâ€™ and â€˜sagaâ€™ are faster for large datasets.

# Run model lasso = LogisticRegression(penalty='l1', solver='liblinear', random_state=0).fit(X_train_transformed, y_train) # Make prediction lasso_prediction = lasso.predict(X_test_transformed) # Get predicted probability lasso_pred_Prob = lasso.predict_proba(X_test_transformed)[:,1]

The ROC/AUC value is 0.9967 for LASSO, higher than the logistic regression value of 0.9818.

# Get the false positive rate and true positive rate fpr,tpr, _= roc_curve(y_test,lasso_pred_Prob) # Get auc value auc = roc_auc_score(y_test,lasso_pred_Prob) # Plot the chart plt.plot(fpr,tpr,label="area="+str(auc)) plt.legend(loc=4)

The log loss decreased from baseline logistic regression’s 2.015 to 0.0685. That’s a significant improvement!

# Calculate log loss log_loss(y_test,lasso_pred_Prob)

0.06846705785516008

The confusion matrix shows the same false positive count of 1, but the false negative count decreased from 6 to 2.

# Confusion matrix plot_confusion_matrix(lasso, X_test_transformed, y_test)

Because of the decrease in false negative count, the accuracy increased from 0.939 to 0.974. And the recall value increased from 0.915 to 0.972.

# Performance report print(classification_report(y_test, lasso_prediction, digits=3))

precision recall f1-score support 0 0.955 0.977 0.966 43 1 0.986 0.972 0.979 71 accuracy 0.974 114 macro avg 0.970 0.974 0.972 114 weighted avg 0.974 0.974 0.974 114

LASSO’s coefficients decreased a lot compared to the logistic regression coefficients. As a result, about half of the features have a coefficient of zero.

So LASSO gives us a simpler model with better performance.

# Model coefficients lassoCoeff = pd.concat([pd.DataFrame(X_test_transformed.columns), pd.DataFrame(np.transpose(lasso.coef_))], axis = 1) lassoCoeff.columns=['Variable','Coefficient'] lassoCoeff['Coefficient_Abs']=lassoCoeff['Coefficient'].apply(abs) lassoCoeff.sort_values(by='Coefficient_Abs', ascending=False)

### Step 7: Ridge

In step 7, we will run Ridge regression by changing the penalty to l2.

# Run model ridge = LogisticRegression(penalty='l2', random_state=0).fit(X_train_transformed, y_train) # Make prediction ridge_prediction = ridge.predict(X_test_transformed) # Get predicted probability ridge_pred_Prob = ridge.predict_proba(X_test_transformed)[:,1] # Get the false positive rate and true positive rate fpr,tpr, _= roc_curve(y_test,ridge_pred_Prob) # Get auc value auc = roc_auc_score(y_test,ridge_pred_Prob) # Plot the chart plt.plot(fpr,tpr,label="area="+str(auc)) plt.legend(loc=4)

The ROC/AUC value is 0.9974, which is slightly higher than 0.9967 for LASSO.

The log loss decreased from LASSO regression’s 0.0685 to 0.0601. That’s an improvement too.

# Calculate log loss log_loss(y_test,ridge_pred_Prob)

0.06014109569918717

The confusion matrix shows the same false positive count of 2, and the false negative count of 1.

Although the total number of incorrect predictions is 3, the same as LASSO regression, Ridge regression has better performance because the false negative decreased by 1.

# Confusion matrix plot_confusion_matrix(ridge, X_test_transformed, y_test)

Because the number of incorrect predictions is the same between LASSO and Ridge, the accuracy is the same at 0.974. Because the false negative count decreased from 2 to 1, the recall value increased from 0.972 to 0.986.

# Performance matrix print(classification_report(y_test, ridge_prediction, digits=3))

precision recall f1-score support 0 0.976 0.953 0.965 43 1 0.972 0.986 0.979 71 accuracy 0.974 114 macro avg 0.974 0.970 0.972 114 weighted avg 0.974 0.974 0.974 114

Ridge’s coefficients decreased slightly compared to the LASSO regression coefficients. Some of the features have a coefficient close to zero, but none of them equals zero.

# Model coefficients ridgeCoeff = pd.concat([pd.DataFrame(X_test_transformed.columns), pd.DataFrame(np.transpose(ridge.coef_))], axis = 1) ridgeCoeff.columns=['Variable','Coefficient'] ridgeCoeff['Coefficient_Abs']=ridgeCoeff['Coefficient'].apply(abs) ridgeCoeff.sort_values(by='Coefficient_Abs', ascending=False)

Based on the performance comparison, we can see that Ridge has slightly better performance than LASSO, but LASSO has a simpler model than Ridge.

### Step 8: Elastic Net

In step 8, we will run elastic net regression by changing the penalty to `'elasticnet'`

. `l1_ratio=0.5`

means that the elastic net uses 50% LASSO and 50% Ridge.

# Run model elasticNet = LogisticRegression(penalty='elasticnet', solver='saga', l1_ratio=0.5, random_state=0).fit(X_train_transformed, y_train) # Make prediction elasticNet_prediction = elasticNet.predict(X_test_transformed) # Get predicted probability elasticNet_pred_Prob = elasticNet.predict_proba(X_test_transformed)[:,1] # Get the false positive rate and true positive rate fpr,tpr, _ = roc_curve(y_test,elasticNet_pred_Prob) # Get auc value auc = roc_auc_score(y_test,elasticNet_pred_Prob) # Plot the chart plt.plot(fpr,tpr,label="area="+str(auc)) plt.legend(loc=4)

The ROC/AUC value for the elastic net is 0.9974, which is about the same as the Ridge regression.

The log loss decreased from Ridge regression’s 0.0601 to 0.0597. That’s a slight improvement.

# Calculate log loss log_loss(y_test,elasticNet_pred_Prob)

0.05970787798591165

The confusion matrix shows the same false negative count of 1, and the false positive count decreased from 2 to 1.

# Confusion matrix plot_confusion_matrix(elasticNet, X_test_transformed, y_test)

Because the number of incorrect predictions decreased compared to LASSO and Ridge, the accuracy increased from 0.974 to 0.982. The recall value is the same at 0.986.

# Performance report print(classification_report(y_test, elasticNet_prediction, digits=3))

precision recall f1-score support 0 0.977 0.977 0.977 43 1 0.986 0.986 0.986 71 accuracy 0.982 114 macro avg 0.981 0.981 0.981 114 weighted avg 0.982 0.982 0.982 114

Ridge’s coefficients were set to zeros for some features, but the number of features with 0 coefficient is smaller than LASSO.

# Model coefficients elasticNetCoeff = pd.concat([pd.DataFrame(X_test_transformed.columns), pd.DataFrame(np.transpose(elasticNet.coef_))], axis = 1) elasticNetCoeff.columns=['Variable','Coefficient'] elasticNetCoeff['Coefficient_Abs']=elasticNetCoeff['Coefficient'].apply(abs) elasticNetCoeff.sort_values(by='Coefficient_Abs', ascending=False)

### Step 9: Put All Code Together

###### Step 1: Import Libraries # Dataset from sklearn import datasets # Data processing import pandas as pd import numpy as np # Visualization import matplotlib.pyplot as plt # Standardize the data from sklearn.preprocessing import StandardScaler # Model and performance evaluation from sklearn.model_selection import train_test_split from sklearn.linear_model import LogisticRegression from sklearn.metrics import plot_confusion_matrix, classification_report, log_loss, roc_curve, roc_auc_score ###### Step 2: Read In Data # Load the breast cancer dataset data = datasets.load_breast_cancer() # Put the data in pandas dataframe format df = pd.DataFrame(data=data.data, columns=data.feature_names) df['target']=data.target # Check the data information df.info() # Check the target value distribution df['target'].value_counts(normalize=True) ###### Step 3: Train Test Split # Train test split X_train, X_test, y_train, y_test = train_test_split(df[df.columns.difference(['target'])], df['target'], test_size=0.2, random_state=42) # Check the number of records in training and testing dataset. print(f'The training dataset has {len(X_train)} records.') print(f'The testing dataset has {len(X_test)} records.') ###### Step 4: Standardization # Initiate scaler sc = StandardScaler() # Standardize the training dataset X_train_transformed = pd.DataFrame(sc.fit_transform(X_train),index=X_train.index, columns=X_train.columns) # Standardized the testing dataset X_test_transformed = pd.DataFrame(sc.transform(X_test),index=X_test.index, columns=X_test.columns) # Summary statistics after standardization X_train_transformed.describe().T # Summary statistics before standardization X_train.describe().T ###### Step 5: Logistic Regression With No Regularization # Check default values LogisticRegression() # Run model logistic = LogisticRegression(penalty='none', random_state=0).fit(X_train_transformed, y_train) # Make prediction logistic_prediction = logistic.predict(X_test_transformed) # Get predicted probability logistic_pred_Prob = logistic.predict_proba(X_test_transformed)[:,1] # Get the false positive rate and true positive rate fpr,tpr, _=roc_curve(y_test,logistic_pred_Prob) # Get auc value auc=roc_auc_score(y_test,logistic_pred_Prob) # Plot the chart plt.plot(fpr,tpr,label="area="+str(auc)) plt.legend(loc=4) # Caclulate log loss log_loss(y_test,logistic_pred_Prob) # Confusion matrix plot_confusion_matrix(logistic, X_test_transformed, y_test) # Performance report print(classification_report(y_test, logistic_prediction, digits=3)) # Model coefficients LogisticCoeff = pd.concat([pd.DataFrame(X_test_transformed.columns),pd.DataFrame(np.transpose(logistic.coef_))], axis = 1) LogisticCoeff.columns=['Variable','Coefficient'] LogisticCoeff['Coefficient_Abs']=LogisticCoeff['Coefficient'].apply(abs) LogisticCoeff.sort_values(by='Coefficient_Abs', ascending=False) ###### Step 6: LASSO # Run model lasso = LogisticRegression(penalty='l1', solver='liblinear', random_state=0).fit(X_train_transformed, y_train) # Make prediction lasso_prediction = lasso.predict(X_test_transformed) # Get predicted probability lasso_pred_Prob = lasso.predict_proba(X_test_transformed)[:,1] # Get the false positive rate and true positive rate fpr,tpr, _= roc_curve(y_test,lasso_pred_Prob) # Get auc value auc = roc_auc_score(y_test,lasso_pred_Prob) # Plot the chart plt.plot(fpr,tpr,label="area="+str(auc)) plt.legend(loc=4) # Calculate log loss log_loss(y_test,lasso_pred_Prob) # Confusion matrix plot_confusion_matrix(lasso, X_test_transformed, y_test) # Performance report print(classification_report(y_test, lasso_prediction, digits=3)) # Model coefficients lassoCoeff = pd.concat([pd.DataFrame(X_test_transformed.columns),pd.DataFrame(np.transpose(lasso.coef_))], axis = 1) lassoCoeff.columns=['Variable','Coefficient'] lassoCoeff['Coefficient_Abs']=lassoCoeff['Coefficient'].apply(abs) lassoCoeff.sort_values(by='Coefficient_Abs', ascending=False) ###### Step 7: Ridge # Run model ridge = LogisticRegression(penalty='l2', random_state=0).fit(X_train_transformed, y_train) # Make prediction ridge_prediction = ridge.predict(X_test_transformed) # Get predicted probability ridge_pred_Prob = ridge.predict_proba(X_test_transformed)[:,1] # Get the false positive rate and true positive rate fpr,tpr, _= roc_curve(y_test,ridge_pred_Prob) # Get auc value auc = roc_auc_score(y_test,ridge_pred_Prob) # Plot the chart plt.plot(fpr,tpr,label="area="+str(auc)) plt.legend(loc=4) # Calculate log loss log_loss(y_test,ridge_pred_Prob) # Confusion matrix plot_confusion_matrix(ridge, X_test_transformed, y_test) # Performance matrix print(classification_report(y_test, ridge_prediction, digits=3)) # Model coefficients ridgeCoeff = pd.concat([pd.DataFrame(X_test_transformed.columns),pd.DataFrame(np.transpose(ridge.coef_))], axis = 1) ridgeCoeff.columns=['Variable','Coefficient'] ridgeCoeff['Coefficient_Abs']=ridgeCoeff['Coefficient'].apply(abs) ridgeCoeff.sort_values(by='Coefficient_Abs', ascending=False) ###### Step 8: Elastic Net # Run model elasticNet = LogisticRegression(penalty='elasticnet', solver='saga', l1_ratio=0.5, random_state=0).fit(X_train_transformed, y_train) # Make prediction elasticNet_prediction = elasticNet.predict(X_test_transformed) # Get predicted probability elasticNet_pred_Prob = elasticNet.predict_proba(X_test_transformed)[:,1] # Get the false positive rate and true positive rate fpr,tpr, _ = roc_curve(y_test,elasticNet_pred_Prob) # Get auc value auc = roc_auc_score(y_test,elasticNet_pred_Prob) # Plot the chart plt.plot(fpr,tpr,label="area="+str(auc)) plt.legend(loc=4) # Calculate log loss log_loss(y_test,elasticNet_pred_Prob) # Confusion matrix plot_confusion_matrix(elasticNet, X_test_transformed, y_test) # Performance report print(classification_report(y_test, elasticNet_prediction, digits=3)) # Model coefficients elasticNetCoeff = pd.concat([pd.DataFrame(X_test_transformed.columns),pd.DataFrame(np.transpose(elasticNet.coef_))], axis = 1) elasticNetCoeff.columns=['Variable','Coefficient'] elasticNetCoeff['Coefficient_Abs']=elasticNetCoeff['Coefficient'].apply(abs) elasticNetCoeff.sort_values(by='Coefficient_Abs', ascending=False)

### Summary

In this tutorial, we covered

- What’s the difference between LASSO (L1), Ridge (L2), and Elastic Net?
- How to run LASSO for the classification model?
- How to run Ridge for classification model?
- How to run Elastic Net for classification model?
- How to compare the performance of LASSO, Ridge, and Elastic Net?

We can see that regularization improved the model performance significantly than not using regularization.

Elastic net has the best performance among the three regularization algorithms, followed by Ridge and LASSO regression. However, this may not be true for all the datasets. Therefore, I suggest trying all three algorithms for your project, doing hyperparameter tuning, and choosing the algorithm that works best for your dataset.

For more information about data science and machine learning, please check out myÂ YouTube channelÂ andÂ Medium PageÂ or follow me onÂ LinkedIn.

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Michael

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