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Wikipedia / Ridge regression

Ridge regression (also known as Tikhonov regularization, named for Andrey Tikhonov) is a method of estimating the coefficients of multiple-regression models in scenarios where the variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. It is a widely used method of regularization of ill-posed problem inverse problems. It is particularly useful to mitigate the problem of multicollinearity in linear regression, which commonly occurs in models with large numbers of parameters. In general, the method provides improved efficiency in parameter estimation problems in exchange for a tolerable amount of bias (see bias–variance tradeoff).

The theory was first introduced by Hoerl and Kennard in 1970 in their Technometrics papers "Ridge regressions: biased estimation of nonorthogonal problems" and "Ridge regressions: applications in nonorthogonal problems".

Ridge regression was developed as a possible solution to the imprecision of least square estimators when linear regression models have some multicollinear (highly correlated) independent variables—by creating a ridge regression estimator (RR). This provides a more precise ridge parameters estimate, as its variance and mean square estimator are often smaller than the least square estimators previously derived.

Overview

In the ordinary least squares solution of

the problem of a near-singular moment matrix is alleviated by adding positive elements to the diagonals, thereby decreasing its condition number is the regressand is the design matrix is the identity matrix serves as the constant shifting the diagonals of the moment matrix. It can be shown that this estimator is the solution to the least squares problem subject to the constraint is nothing but the Lagrange multiplier of the constraint. heuristically or find it via additional data-fitting strategies, see Determination of the Tikhonov parameter below.

, the constraint eventually becomes non-binding, and the ridge estimator converges to the minimum-norm ordinary least squares:

denoting the pseudoinverse.

Determination of the Tikhonov parameter

is accomplished either via cross-validation, or via a plug-in procedure, as follows.

Generalized cross-validation estimator

is the minimizer of the cross-validation loss or its generalizations. For example, Grace Wahba proved that the optimal parameter, in the sense of generalized cross-validation minimizes

is the residual sum of squares is the effective number of degrees of freedom.

Plug-in estimator

. Then, consider the following choice for the Tikhonov regularization parameter:

. It can be shown. More precisely,

is test response).

, the most natural estimators for these parameters are the usual least-squares ones:

thus gives the so-called plug-in estimator.

Alternative approaches to the data-driven selection of the Tikhonov regularization parameter include the discrepancy principle, L-curve method, restricted maximum likelihood.

History

Tikhonov regularization was invented independently in many different contexts. It became widely known through its application to integral equations in the works of Andrey Tikhonov and David L. Phillips. Some authors use the term Tikhonov–Phillips regularization. The finite-dimensional case was expounded by Arthur E. Hoerl, who took a statistical approach, and by Manus Foster, who interpreted this method as a Wiener–Kolmogorov (Kriging) filter. Following Hoerl, it is known in the statistical literature as ridge regression, named after ridge analysis ("ridge" refers to the path from the constrained maximum).

Tikhonov regularization for linear equations

Suppose that for a known real matrix may even be non-square.

The standard approach is ordinary least squares linear regression. does—that is, the solution is not unique—the problem is said to be ill posed. In such cases, ordinary least squares estimation leads to an overdetermined, or more often an underdetermined system of equations. Most real-world phenomena have the effect of low-pass filters. Therefore, in solving the inverse-problem, the inverse mapping operates as a high-pass filter that has the undesirable tendency of amplifying noise (eigenvalues. Ordinary least squares seeks to minimize the sum of squared residuals is the Euclidean norm.

, for some suitably chosen Tikhonov matrix. In many cases, this matrix is chosen as a scalar multiple of the identity matrix), giving preference to solutions with smaller norms; this is known as regularization. In other cases, high-pass operators (e.g., a difference operator or a weighted Fourier operator) may be used to enforce smoothness if the underlying vector is believed to be mostly continuous. This regularization improves the conditioning of the problem, thus enabling a direct numerical solution. Treating it as an ordinary least squares problem with augmented matrices this reduces to the unregularized least-squares solution, provided that (ATA)−1 exists. Note that in case of a complex matrix has to be replaced by the Hermitian transpose.

regularization is used in many contexts aside from linear regression, such as classification with logistic regression or support vector machines, and matrix factorization.

Application to existing fit results

In the context of arbitrary likelihood fits, this is valid, as long as the quadratic approximation of the likelihood function is valid. This means that, as long as the perturbation from the unregularised result is small, one can regularise any result that is presented as a best fit point with a covariance matrix. No detailed knowledge of the underlying likelihood function is needed.

Generalized Tikhonov regularization

(compare with the Mahalanobis distance is the inverse covariance matrix is the expected value.

, the Tikhonov matrix then appears in the Cholesky factorization and is considered a whitening filter.

Lavrentyev regularization

, as proposed by Mikhail Lavrentyev. or, equivalently up to a constant term

can be better conditioned, i.e., have a smaller condition number

Regularization in Hilbert space

Typically discrete linear ill-conditioned problems result from discretization of integral equations as a compact operator on Hilbert spaces is then a self-adjoint bounded invertible operator.

Relation to singular-value decomposition and Wiener filter

, this least-squares solution can be analyzed in a special way using the singular-value decomposition and is zero elsewhere. This demonstrates the effect of the Tikhonov parameter on the condition number of the regularized problem. For the generalized case, a similar representation can be derived using a generalized singular-value decomposition.

Finally, it is related to the Wiener filter is the rank.

Relation to probabilistic formulation

The probabilistic formulation of an inverse problem representing the a priori representing the uncertainties on the observed parameters..

Bayesian interpretation

seems rather arbitrary, the process can be justified from a Bayesian point of view. Note that for an ill-posed problem one must necessarily introduce some additional assumptions in order to get a unique solution. Statistically, the prior probability is sometimes taken to be a multivariate normal distribution. For simplicity here, the following assumptions are made: the means are zero; their components are independent; the components have the same standard deviation are also assumed to be independent. Under these assumptions the Tikhonov-regularized solution is the most probable solution given the data and the a priori, according to Bayes' theorem.

If the assumption of normality is replaced by assumptions of homoscedasticity and uncorrelatedness of errors, and if one still assumes zero mean, then the Gauss–Markov theorem entails that the solution is the minimal unbiased linear estimator.