In this post, we learn about the role of eigenvalue multicollinearity detection. In the context of the detection of multicollinearity, eigenvalues are used to assess the degree of linear dependence among explanatory (regressors, independent) variables in a regression model. Therefore, by understanding the role of eigenvalue multicollinearity detection, one can take appropriate steps to improve the reliability and interpretability of the regression models.
Table of Contents
Decomposition of Eigenvalues and Eigenvectors
The pair-wise correlation matrix of explanatory variables is decomposed into eigenvalues and eigenvectors. Whereas Eigenvalues represent the variance explained by each Principal Component and Eigenvectors represent the directions of maximum variance.
The Decomposition Process
Firstly, compute the correlation coefficients between each pair of variables in the dataset.
Secondly, find the Eigenvalues and Eigenvectors: solve the following equation for each eigenvalue ($\lambda$) and eigenvector ($vV)
$$A v = \lambda v$$
where $A$ is the correlation matrix, $v$ is the eigenvector, and $\lambda$ is the eigenvalue.
The above equation essentially means that multiplying the correlation matrix ($A$) by the eigenvector ($v$) results in a scaled version of the eigenvector, where the scaling factor is the eigenvalue. This can be solved using various numerical methods, such as the power method or QR algorithm.
Interpreting Eigenvalue Multicollinearity Detection
A set of eigenvalues of relatively equal magnitudes indicates little multicollinearity (Freund and Littell 2000: 99). A small number of large eigenvalues suggests that a small number of component variables describe most of the variability of the original observed variables ($X$). Because of the score constraint, a number of large eigenvalues implies that there will be some small eigenvalues or some small variances of component variables.
A zero eigenvalue means perfect multicollinearity among independent/explanatory variables and very small eigenvalues imply severe multicollinearity. Conventionally, an eigenvalue close to zero (less than 0.01) or condition number greater than 50 (30 for conservative persons) indicates significant multicollinearity. The condition index, calculated as the ratio of the largest eigenvalue to the smallest eigenvalue $\left(\frac{\lambda_{max}}{\lambda_{min}}\right)$, is a more sensitive measure of multicollinearity. A high condition index (often above 30) signals severe multicollinearity.
The proportion of variances tells how much percentage of the variance of parameter estimate (coefficient) is associated with each eigenvalue. A high proportion of variance of an independent variable coefficient reveals a strong association with the eigenvalue. If an eigenvalue is small enough and some independent variables show a high proportion of variation with respect to the eigenvalues then one may conclude that these independent variables have significant linear dependency (correlation).
Presence of Multicollinearity in Regression Model
Since Multicollinearity is a statistical phenomenon where two or more independent/explanatory variables in a regression model are highly correlated, the existence/presence of multicollinearity may result in
- Unstable Coefficient Estimates: Estimates of regression coefficients become unstable in the presence of multicollinearity. A small change in the data can lead to large changes in the estimates of the regression coefficients.
- Inflated Standard Errors: The standard errors of the regression coefficients inflated due to the presence of multicollinearity, making it difficult to assess the statistical significance of the coefficients.
- Difficulty in Interpreting Coefficients: It becomes challenging to interpret the individual effects of the independent variables on the dependent variable when they are highly correlated.
How to Mitigate the Effects of Multicollinearity
If multicollinearity is detected, several strategies can be employed to mitigate the effects of multicollinearity. By examining the distribution of eigenvalues, researchers (statisticians and data analysts) can identify potential issues and take appropriate steps to address them, such as feature selection or regularization techniques.
- Feature Selection: Remove redundant or highly correlated variables from the model.
- Principal Component Regression (PCR): Transform the original variables into a smaller set of uncorrelated principal components.
- Partial Least Squares Regression (PLSR): It is similar to PCR but also considers the relationship between the independent variables and the dependent variable.
- Ridge Regression: Introduces a bias-variance trade-off to stabilize the coefficient estimates.
- Lasso Regression: Shrinks some coefficients to zero, effectively performing feature selection.
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