Tikhonovregularizáció

Tikhonov regularizáció, also known as ridge regression or Tikhonov regularization, is a method used in mathematics and statistics to address ill-posed problems, particularly in the context of linear regression. Ill-posed problems are those where a small change in the input data can lead to a very large change in the output solution, or where no unique solution exists. This often occurs when dealing with noisy data or when the system of equations is nearly singular.

The core idea of Tikhonov regularization is to add a penalty term to the objective function that is being minimized. In the standard least squares problem, we aim to minimize the sum of squared errors between the observed data and the model's predictions. Tikhonov regularization modifies this by adding a term that penalizes large coefficients. The most common form of the penalty is proportional to the sum of the squares of the coefficients, multiplied by a regularization parameter, often denoted by lambda (λ).

Mathematically, if the original problem is to find a vector x that minimizes ||Ax - b||^2, the Tikhonov regularized problem becomes minimizing ||Ax - b||^2 + λ||x||^2. The parameter λ controls the strength of the regularization. A larger λ imposes a stronger penalty on large coefficients, leading to a more stable but potentially less precise solution. A smaller λ approaches the standard least squares solution.

This technique is widely applied in various fields, including image processing, machine learning, inverse problems, and econometrics, where it helps to obtain more robust and meaningful solutions when faced with unstable or noisy data. It effectively shrinks the estimated coefficients towards zero, thereby reducing the variance of the estimator at the expense of a small increase in bias.