Harjausregressio

Harjausregressio, also known as brush regression or Lasso regression with L1 regularization, is a statistical method used for linear regression. It is particularly useful when dealing with datasets that have a large number of features, some of which may be irrelevant or redundant. The core idea behind harjausregressio is to simultaneously perform feature selection and regularization.

The method works by adding a penalty term to the standard ordinary least squares (OLS) cost function. This penalty term is proportional to the absolute value of the regression coefficients (L1 norm). This L1 penalty encourages some coefficients to become exactly zero. When a coefficient is zero, it effectively means that the corresponding feature is excluded from the model, thus performing feature selection.

This zeroing out of coefficients is a key advantage of harjausregressio over standard OLS or ridge regression (which uses L2 regularization). Ridge regression shrinks coefficients towards zero but rarely makes them exactly zero, so it doesn't perform explicit feature selection. Harjausregressio can simplify models by identifying and removing less important predictors, leading to more interpretable and potentially more robust models, especially in high-dimensional scenarios.

The optimization problem for harjausregressio is to minimize the sum of squared errors plus the L1 penalty. This optimization is typically solved using iterative algorithms, as the L1 penalty is non-differentiable at zero. The strength of the regularization is controlled by a tuning parameter, often denoted by lambda (λ), which dictates the magnitude of the penalty. Cross-validation is commonly used to find an optimal value for this parameter.