Editing Ridge regression (section)

===Generalized Tikhonov regularization===
For general multivariate normal distributions for <math>\mathbf x</math> and the data error, one can apply a transformation of the variables to reduce to the case above. Equivalently, one can seek an <math>\mathbf x</math> to minimize
<math display="block">\left\|A \mathbf x - \mathbf b\right\|_P^2 + \left\|\mathbf x - \mathbf x_0\right\|_Q^2,</math>
where we have used <math>\left\|\mathbf{x}\right\|_Q^2</math> to stand for the weighted norm squared <math>\mathbf{x}^\mathsf{T} Q \mathbf{x}</math> (compare with the [[Mahalanobis distance]]). In the Bayesian interpretation <math>P</math> is the inverse [[covariance matrix]] of <math>\mathbf b</math>, <math>\mathbf x_0</math> is the [[expected value]] of <math>\mathbf x</math>, and <math>Q</math> is the inverse covariance matrix of <math>\mathbf x</math>. The Tikhonov matrix is then given as a factorization of the matrix <math>Q = \Gamma^\mathsf{T} \Gamma</math> (e.g. the [[Cholesky factorization]]) and is considered a [[Whitening transformation|whitening filter]].

This generalized problem has an optimal solution <math>\mathbf x^*</math> which can be written explicitly using the formula
<math display="block">\mathbf x^* = \left(A^\mathsf{T} PA + Q\right)^{-1} \left(A^\mathsf{T} P \mathbf{b} + Q \mathbf{x}_0\right),</math>
or equivalently, when ''Q'' is '''not''' a null matrix:
<math display="block">\mathbf x^* = \mathbf x_0 + \left(A^\mathsf{T} P A + Q \right)^{-1} \left(A^\mathsf{T} P \left(\mathbf b - A \mathbf x_0\right)\right).</math>