Math 185 Final Project Due December 8: Problem 1 - The Baseb

Math 185 Final Project Due December 8problem 1the Baseball Dataset C

The baseball dataset consists of the statistics of 263 players in Major League Baseball in the season 1986. The dataset (hitters.csv) consists of 20 variables: Number of times at bat, hits, home runs, runs, runs batted in, walks, years in major leagues, career at-bats, career hits, career home runs, career runs, career RBI, career walks, league, division, putouts, assists, errors, and 1987 annual salary. The goal is to perform variable selection with predictors measuring players' performance and the response variable being 1987 salary, using different methods including stepwise selection with BIC and ridge regression, and compare the results with sparse modeling via the Gradient Hard Thresholding Pursuit algorithm.

Paper For Above instruction

The analysis of baseball player statistics provides a rich context for applying various statistical methods in regression modeling, especially when dealing with multiple predictors and seeking parsimonious models that balance interpretability and predictive performance. In this paper, we explore variable selection techniques and regularization methods—namely best subset selection with BIC, ridge regression, and a sparsity-constrained method using Gradient Hard Thresholding Pursuit (GraHTP)—applied to the 1986 baseball dataset with the goal of modeling the 1987 salary.

Introduction

The dataset under review contains comprehensive statistical measures for Major League Baseball players from the 1986 season, along with corresponding career data and identifiers for league and division. Our primary response variable is the 1987 salary, expressed in thousands of dollars, which can be influenced by a multitude of performance metrics. The challenge lies in selecting an optimal subset of predictors that effectively explain salary variability, while avoiding overfitting and ensuring interpretability. Similarly, regularization techniques like ridge regression provide a means to handle multicollinearity and high-dimensional predictors without explicit variable exclusion. The ultimate goal is to compare and contrast these methodologies and evaluate their effectiveness in the context of this dataset.

Variable Selection via Best Subset Selection with BIC

Best subset selection aims to identify the combination of predictors that yields the best possible model according to a specified criterion—here, the Bayesian Information Criterion (BIC). The BIC balances model fit and complexity, penalizing models with excessive variables to discourage overfitting. We implement stepwise algorithms to navigate the model space effectively.

Forward Stepwise Selection

Starting from a null model containing no predictors, the forward stepwise selection iteratively adds the predictor that most improves the BIC at each step, until no further reduction in BIC is possible or all variables are included. This process can be implemented by defining a function in R that evaluates models with increasing numbers of predictors, calculating their BICs, and ultimately selecting the model with the lowest BIC. The plot of BIC versus the number of variables visualizes the model selection process, guiding us to the most parsimonious yet effective model.

Backward Stepwise Selection

Conversely, backward stepwise selection starts with the full model including all predictors. At each iteration, it removes the predictor whose exclusion most decreases the BIC, continuing until further removal would worsen the BIC. This process yields a sequence of nested models, with the optimal one determined by the minimum BIC. Plotting BIC against the number of variables helps in distinctly identifying the best model.

Comparison of Selected Models

Upon completing the forward and backward procedures, we compare the resulting models. An agreement indicates robustness in the variable selection, whereas divergence suggests model uncertainty. The model with the lowest BIC among the two provides our most recommended predictor set for salary prediction.

R Implementation of Variable Selection

The R functions formalize the above procedures, utilizing the 'leaps' package for efficient subset selection and BIC computation. The forward and backward algorithms iterate through model sizes, computing BICs, and selecting optimal models accordingly. Visualization aids interpretation, and the selected models are presented with their BIC values for direct comparison.

Regularization via Ridge Regression

Ridge regression offers an alternative approach, especially suitable when predictors are collinear. Here, variables are shrunk towards zero through an L2 penalty, which stabilizes estimates but retains all variables. The analysis begins with standardizing variables to ensure uniform penalty application, followed by evaluating a grid of tuning parameters (λ) spanning from very small to large, effectively altering the regularization intensity.

Estimating Ridge Coefficients and Cross-Validation

For each λ, ridge regression coefficients are computed using the function in R, stored over a matrix. To choose an optimal λ, we implement 10-fold cross-validation, assessing the mean squared prediction error. The λ minimizing this error is selected, and the coefficient estimates are refitted on the entire dataset. The resulting coefficients illustrate the degree of regularization, with none being zero due to ridge's nature, which does not promote variable exclusion but shrinks coefficients.

Comparative Analysis

The comparison between models obtained via stepwise selection and ridge regression reveals different philosophies: the former aims for sparse, interpretable models, while the latter balances bias and variance through shrinkage. Ridge's propensity to retain all variables, albeit with reduced magnitude, can be contrasted with the variable exclusion inherent in the subset selection models, providing insights into model complexity and predictive accuracy.

Sparse Approximation via GraHTP

The Gradient Hard Thresholding Pursuit (GraHTP) provides a method for solving sparsity-constrained least squares problems. The algorithm iteratively updates the coefficient vector by gradient steps followed by hard thresholding to enforce sparsity, aiming to recover a sparse model with a predefined sparsity level (number of non-zero coefficients). The application involves normalizing the data, initializing coefficients, and executing the iterative procedure until convergence.

Implementation and Model Selection

The functional implementation of GraHTP in R captures the optimization process, enabling us to generate models with varying sparsity levels (from 1 to 19 predictors). For each sparsity level, BIC serves as the criterion to select the best model. Subsequent comparison with models from previous methods allows us to evaluate the relative effectiveness of this sparsity-promoting approach versus classical stepwise and regularization techniques.

Conclusion

This comprehensive analysis leverages multiple statistical methodologies to elucidate the relationships between baseball players' performance metrics and their subsequent salaries. The variable selection via BIC identifies a sparse model balancing simplicity and explanatory power, while ridge regression offers a robust alternative in the presence of multicollinearity. The GraHTP-based sparse modeling provides an additional perspective, promoting interpretability through explicit sparsity. Comparing these models fosters understanding of the trade-offs inherent in each approach: interpretability, stability, and predictive performance. The insights gained not only enhance understanding of the dataset but also exemplify broader principles in statistical modeling and regularization.

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