Tikhonovreglugerð

Tikhonov regularization, also known as ridge regression, is a mathematical technique used to address ill-posed problems, particularly in linear regression. An ill-posed problem is one where small changes in the input data can lead to large changes in the output, or where a unique solution does not exist or is not stable. In the context of linear regression, this often occurs when the predictor variables are highly correlated (multicollinearity) or when there are more predictors than observations.

The core idea of Tikhonov regularization is to modify the standard least squares method by adding a penalty term to the cost function. This penalty term is proportional to the sum of the squares of the regression coefficients, scaled by a regularization parameter, often denoted by lambda (λ) or alpha (α). By adding this penalty, the algorithm is discouraged from assigning excessively large values to the coefficients.

The effect of Tikhonov regularization is to shrink the estimated coefficients towards zero. This shrinkage reduces the variance of the estimates, making the model more stable and less sensitive to noise in the data. A larger regularization parameter leads to more shrinkage, effectively simplifying the model. Conversely, a smaller parameter results in less shrinkage, closer to the standard least squares solution. The choice of the regularization parameter is crucial and is typically determined through cross-validation or other model selection techniques to balance bias and variance. Tikhonov regularization is widely applied in various fields, including machine learning, image processing, and geophysics.