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Fungible Correlation Matrices: A Method for Generating Nonsingular, Singular, and Improper Correlation Matrices for
1a University of Minnesota.
Multivariate Behavioral Research
|June 21, 2016
Summary
This study introduces fungible correlation matrices, which allow for infinite variations in predictor correlations while maintaining fixed regression coefficients and R-squared. These matrices are useful for Monte Carlo research, particularly in comparing penalized regression and matrix-smoothing algorithms.
Area of Science:
- Statistics
- Computational Statistics
- Data Science
Background:
- Standardized regression coefficients and R-squared values can be satisfied by numerous predictor correlation matrices.
- These matrices, termed fungible correlation matrices, allow for flexibility in statistical modeling.
Purpose of the Study:
- To present an algorithm for generating positive definite (PD), positive semidefinite (PSD), or indefinite (ID) fungible correlation matrices.
- To demonstrate the utility of these matrices in Monte Carlo research.
Main Methods:
- An algorithm is described for generating fungible correlation matrices with specified smallest eigenvalues.
- The algorithm's underlying equations are explored from algebraic and geometric viewpoints.
- R code for generating these matrices is provided.
Main Results:
- Fungible correlation matrices can be generated in PD, PSD, or ID forms.
- Simulation studies confirm their utility in comparing penalized regression and matrix-smoothing algorithms.
Conclusions:
- Fungible correlation matrices offer a valuable tool for statistical research, particularly in simulation studies.
- The developed algorithm facilitates the generation of diverse correlation structures for robust model comparison.
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