Custom Grid Search for Ridge, Lasso, and Elastic Net Models

Model Fitting with Custom Grid Search:

To conduct a custom grid search to find a model that performs better than the one presented in the lecture materials, you can use scikit-learn library in Python. Here is an example code for fitting the Ridge, Lasso, and Elastic Net models individually:

from sklearn.linear_model import Ridge, Lasso, ElasticNet
from sklearn.model_selection import GridSearchCV
from sklearn.datasets import load_boston
from sklearn.metrics import mean_squared_error

# Load the Boston Housing dataset for demonstration
boston = load_boston()
X = boston.data
y = boston.target

# Define the parameter grid with candidate values for each model
ridge_param_grid = {'alpha': [0.1, 1.0, 10.0]}
lasso_param_grid = {'alpha': [0.1, 1.0, 10.0]}
enet_param_grid = {'alpha': [0.1, 1.0, 10.0], 'l1_ratio': [0.2, 0.5, 0.8]}

# Create Ridge, Lasso, and Elastic Net models
ridge = Ridge()
lasso = Lasso()
enet = ElasticNet()

# Perform grid search with cross-validation for each model
ridge_grid_search = GridSearchCV(estimator=ridge, param_grid=ridge_param_grid, cv=5)
ridge_grid_search.fit(X, y)

lasso_grid_search = GridSearchCV(estimator=lasso, param_grid=lasso_param_grid, cv=5)
lasso_grid_search.fit(X, y)

enet_grid_search = GridSearchCV(estimator=enet, param_grid=enet_param_grid, cv=5)
enet_grid_search.fit(X, y)

# Print the best hyperparameters found for each model
print('Best hyperparameters for Ridge: ', ridge_grid_search.best_params_)
print('Best hyperparameters for Lasso: ', lasso_grid_search.best_params_)
print('Best hyperparameters for Elastic Net: ', enet_grid_search.best_params_)

This code uses the Boston Housing dataset as an example and performs a grid search with different values of alpha for Ridge, Lasso, and Elastic Net models. The param_grid dictionary defines the candidate values for alpha for each model. The GridSearchCV class is then used to perform the grid search with 5-fold cross-validation. Finally, the best hyperparameters found for each model are printed.

Grid Search Configuration:

The grid search configuration used in the above code includes the candidate values for the hyperparameters of each model. Specifically, the param_grid dictionary contains the candidate values for alpha for Ridge, Lasso, and Elastic Net models. In the code example, the candidate values used are [0.1, 1.0, 10.0] for alpha in each model.

Optimized Hyperparameter Values:

The optimized hyperparameter values can be obtained from the best_params_ attribute of each grid search object (ridge_grid_search.best_params_, lasso_grid_search.best_params_, enet_grid_search.best_params_). These values will vary depending on the dataset and problem being solved. In this case, the best hyperparameters found through the grid search for each model will be printed as 'Best hyperparameters for Ridge: ', 'Best hyperparameters for Lasso: ', and 'Best hyperparameters for Elastic Net: ' followed by the values.

Calculate measures to check the fitness of the model:

To evaluate the fitness of the model, you can use metrics such as mean squared error (MSE) and R-squared. These metrics can be calculated using the mean_squared_error and r2_score functions from the scikit-learn library, respectively. Here is an example code:

from sklearn.metrics import r2_score

# Predict the target variable using the optimized models
ridge_y_pred = ridge_grid_search.predict(X)
lasso_y_pred = lasso_grid_search.predict(X)
enet_y_pred = enet_grid_search.predict(X)

# Calculate mean squared error
ridge_mse = mean_squared_error(y, ridge_y_pred)
lasso_mse = mean_squared_error(y, lasso_y_pred)
enet_mse = mean_squared_error(y, enet_y_pred)

# Calculate R-squared
ridge_r2 = r2_score(y, ridge_y_pred)
lasso_r2 = r2_score(y, lasso_y_pred)
enet_r2 = r2_score(y, enet_y_pred)

# Print the metrics
print('Ridge MSE: ', ridge_mse)
print('Ridge R-squared: ', ridge_r2)
print('Lasso MSE: ', lasso_mse)
print('Lasso R-squared: ', lasso_r2)
print('Elastic Net MSE: ', enet_mse)
print('Elastic Net R-squared: ', enet_r2)

In this code, the predict function is used to generate predictions for the target variable using the optimized models. The mean_squared_error function is then used to calculate the mean squared error, and the r2_score function is used to calculate the R-squared value. These metrics provide insights into the model's performance, with lower mean squared error and higher R-squared indicating better performance.

Comparison of Actual and Predicted Outcomes:

To create a graph comparing the actual and predicted outcome variables, you can use the matplotlib library in Python. Here is an example code:

import matplotlib.pyplot as plt

# Plot the actual and predicted outcomes for Ridge model
plt.scatter(range(len(y)), y, label='Actual')
plt.scatter(range(len(ridge_y_pred)), ridge_y_pred, label='Predicted')
plt.xlabel('Sample')
plt.ylabel('Outcome')
plt.legend()
plt.title('Ridge Model')
plt.show()

# Plot the actual and predicted outcomes for Lasso model
plt.scatter(range(len(y)), y, label='Actual')
plt.scatter(range(len(lasso_y_pred)), lasso_y_pred, label='Predicted')
plt.xlabel('Sample')
plt.ylabel('Outcome')
plt.legend()
plt.title('Lasso Model')
plt.show()

# Plot the actual and predicted outcomes for Elastic Net model
plt.scatter(range(len(y)), y, label='Actual')
plt.scatter(range(len(enet_y_pred)), enet_y_pred, label='Predicted')
plt.xlabel('Sample')
plt.ylabel('Outcome')
plt.legend()
plt.title('Elastic Net Model')
plt.show()

In this code, scatter plots are created where the x-axis represents the sample index and the y-axis represents the outcome variable. The actual outcomes are plotted as blue dots, and the predicted outcomes are plotted as orange dots. The legend and titles are also added for clarity.

By comparing the graphs, you can visually assess the accuracy of the predictions. If the predicted outcomes closely align with the actual outcomes, it indicates a high accuracy of the model's predictions.

Comparison with Lecture Note Results:

To compare your model's results with the Support Vector Machines results presented in the lecture notes, you need to refer to the lecture notes and extract the relevant information about the SVM model used, including the hyperparameters and performance measures reported.

You can then compare your optimized hyperparameter values (from question 3) with those reported in the lecture notes. If they are similar or close, it indicates that your model has found similar optimal values for the hyperparameters.

Similarly, you can compare the performance measures obtained from your models (from question 4) with those reported in the lecture notes. If they are similar or close, it indicates that your models' performance is comparable to the one presented in the lecture notes.

By comparing the results, you can gain insights into the similarities and differences between your models and the ones presented in the lecture notes, which can help you assess the effectiveness of your approach and identify areas for improvement.

Perform feature selection:

To perform feature selection using the Lasso model, you can analyze the coefficients of the selected features. In the Lasso model, the regularization term encourages sparsity, allowing the model to select only the most important features.

Here is an example code to perform feature selection using the Lasso model:

# Fit the Lasso model with the optimized hyperparameters
lasso_model = Lasso(alpha=lasso_grid_search.best_params_['alpha'])
lasso_model.fit(X, y)

# Get the coefficients of the predictors
lasso_coefs = lasso_model.coef_

# Get the indices of the important predictors (non-zero coefficients)
important_predictors = [i for i, coef in enumerate(lasso_coefs) if coef != 0]

In this code, the Lasso model is fitted using the optimized hyperparameters. The coefficients of the predictors are then obtained using the coef_ attribute of the Lasso model object. Finally, the indices of the important predictors (non-zero coefficients) are extracted.

The choice of important predictors depends on the problem and the specific dataset. In general, it is recommended to select predictors with non-zero coefficients as they contribute to the model's predictions. However, it is important to consider domain knowledge and the context of the problem when interpreting and selecting predictors.