Applications of the Householder transformation to ridge-type estimation methods (Q1200229)

The author considers the model \(y=Hx+e\), where \(y\) is an \((m\times 1)\) vector of observation residuals, \(H\) is an \((m\times n)\) mapping matrix, \(x\) is an \((n\times 1)\) state vector of corrections to a reference solution, \(e\) is an \((m\times 1)\) vector of observation errors, \(m\) is the total number of observations and \(n\) is the number of model variables in the state vector. The observations are assumed to have a Gaussian distribution with zero mean and covariance equal to \(R\). Instead of estimating the unknown \(x\) by the classical least squares estimator \(\hat x=(H^ TR^{-1}H)^{-1}H^ TR^{-1}y\) the author proposes to use \(\hat x^*=(H^ TR^{-1}H+D_ R^{-1}KD^{-1}_ R)^{-1}H^ TR^{-1}y\), where \(D\) is an \((n\times n)\) diagonal matrix whose \(i\)th diagonal element is the reciprocal of the square root of the corresponding element of \(H^ TR^{-1}H\) and \(K\) is an \((n\times n)\) diagonal matrix of biasing parameters. The reason is to stabilize the computation in the case when \(H^ TR^{-1}H\) is ill-conditioned. The main result of the paper is a detailed algorithm describing Householder transformations which can be used in this modification in order to obtain desired \(QR\)-decomposition which can be useful in the modified situation.