Mirza_Report_M4.pdf

Regularization 1 Module 4 Assignment — Regularization Muhammad U. Mirza College of Professional Studies, Northeastern University Toronto ALY6015 - Intermediate Analytics Dr. Matthew Goodwin February 4, 2024

Regularization 2 Introduction In this statistical analysis report, I explore the application of Ridge and LASSO regression techniques alongside stepwise selection to predict graduation rates using the College dataset from the ISLR package. Regularization methods like Ridge and LASSO help prevent overfitting by penalizing the magnitude of coefficients, while stepwise selection iteratively refines models by criteria such as the Akaike Information Criterion (AIC). Overall, this comprehensive approach aims to generate more precise and insightful predictions for graduation rates in the College dataset. Analysis Split the data into a train and test set The college dataset contains 777 observations and 18 variables. To predict graduation rates accurately, I divided the College dataset into a training set, which constitutes 70% (543 observations) of the data, and a test set, which makes up the remaining 30% (234 observations). This split, guided by the Feature_Selection_R.pdf document, is crucial for evaluating the model's performance on data it has not been trained on. I set a random seed for reproducibility, allowing for consistent results across multiple runs. For regression analysis in glmnet, the datasets were converted to matrix format using the model.matrix function. This step separated the predictor variables into train_x and test_x, and the response variable, Grad.Rate, into train_y and test_y. This transformation is essential, as glmnet requires numerical inputs and a clear delineation between predictors and response. This methodical preparation of the data ensures that the analysis is structured and poised for the modeling phase.

Regularization 3 Ridge Regression Ridge Regression combats multicollinearity in datasets with highly correlated predictors by incorporating an L2 regularization penalty into the loss function. This shrinkage of coefficient magnitudes helps mitigate overfitting, enhancing model interpretability and reducing the undue impact of any single predictor. Additionally, Ridge Regression stabilizes the model by improving its generalization capability, thereby decreasing the variability in the predictions it generates. Use the cv.glmnet function to estimate the lambda.min and lambda.1se values. Compare and discuss the values. To accurately determine the optimal regularization strength for our Ridge regression model, I employed the cv.glmnet function, utilizing a 10-fold cross-validation method. This technique involves dividing the dataset into ten parts, training the model on nine, and testing it on the tenth, repeatedly, to ensure robust estimation. The analysis yielded two critical lambda values: lambda.min and lambda.1se. The lambda.min represents the value that minimizes the prediction error, indicating the most regularized model that still provides the lowest loss. On the other hand, lambda.1se is a more conservative estimate, providing a simpler model within one standard error of the minimum error. The logged values of these lambdas show the balance we seek between model complexity and predictive accuracy, with lambda.min focusing on precision and lambda.1se on simplicity and robustness. Figure 1: Lambda min and 1se (Ridge Regression)

Regularization 4 The calculated logged values of these lambdas, -2.612328 for lambda.min and 0.2717177 for lambda.1se, suggest a spectrum of regularization from tight to more relaxed, balancing complexity against generalization. Plot the results from the cv.glmnet function provide an interpretation. What does this plot tell us? Figure 2: Ridge Regression Plot Figure 2 visually represents the model's mean squared error against varying levels of regularization (lambda). The plot indicates two significant lambda values: lambda.min and lambda.1se. At lambda.min, the model retains 16 predictors, a point which corresponds to the left dotted line and reflects the most complex model with the lowest error. The right dotted line at lambda.1se represents a balance between complexity and generalization, where the model retains 9 predictors, offering a more generalizable model at the cost of slightly higher error. This graphical analysis aids in selecting an optimal regularization parameter to ensure a model that generalizes well to new data.

Regularization 5 Fit a Ridge regression model against the training set and report on the coefficients. Is there anything interesting? Figure 3: Ridge regression model with lambda.min Figure 4: Ridge regression with lambda.1se Upon fitting the Ridge regression model to the training data, I evaluated the coefficients' magnitude and significance using both lambda.min and lambda.1se from the cross-validation process. Lambda.min, which is associated with the least prediction error, resulted in a model where all predictors were retained, and most had non-zero coefficients. This suggests a model complexity that tries to fit the training data as closely as possible without undue penalization.

Regularization 6 Figure 5: Ridge regression lambda.min model coefficients Figure 6: Ridge regression lambda.1se model coefficients Conversely, the model using lambda.1se, which is within one standard error of the minimum error, showed somewhat smaller coefficients, indicating a more regularized approach.

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