Regularised Inverse

Regularised solution of the (ill-posed) problem \(K\phi = r\) where \(K\) is a \(n \times n\) matrix, \(r\) is a given vector of length n. Users can choose one of the 3 schemes described in Carrasco and Florens (2007).

Usage

RegularisedSol(Kn, alphaReg, r,
               regularization = c("Tikhonov", "LF", "cut-off"),
               ...)

Arguments

Kn

numeric \(n \times n\) matrix.

alphaReg

regularisation parameter; numeric in ]0,1].

r

numeric vector of length n.

regularization

regularization scheme to be used, one of "Tikhonov" (Tikhonov scheme), "LF" (Landweber-Fridmann) and "cut-off" (spectral cut-off). See Details.

...

the value of \(c\) used in the "LF" scheme. See Carrasco and Florens(2007).

Details

Following Carrasco and Florens(2007), the regularised solution of the problem \(K \phi=r\) is given by : $$\varphi_{\alpha_{reg}} = \sum_{j=1}^{n} q(\alpha_{reg},\mu_j)\frac{<r,\psi_j >}{\mu_j} \phi_j , $$ where \(q\) is a (positive) real function with some regularity conditions and \(\mu,\phi,\psi\) the singular decomposition of the matrix \(K\).

The regularization parameter defines the form of the function \(q\). For example, the "Tikhonov" scheme defines \(q(\alpha_{reg},\mu) = \frac{\mu^2}{\alpha_{reg}+\mu^2}\).

When the matrix \(K\) is symmetric, the singular decomposition is replaced by a spectral decomposition.

Value

the regularised solution, a vector of length n.

References

Carrasco M, Florens J and Renault E (2007). “Linear inverse problems in structural econometrics estimation based on spectral decomposition and regularization.” Handbook of econometrics, 6, pp. 5633–5751.

See also

solve

Examples

## Adapted from R examples for Solve 
## We compare the result of the regularized sol to the expected solution

hilbert <- function(n) { i <- 1:n; 1 / outer(i - 1, i, "+")}

K_h8 <- hilbert(8);
r8 <- 1:8

alphaReg_robust <- 1e-4
Sa8_robust <- RegularisedSol(K_h8,alphaReg_robust,r8,"LF")

alphaReg_accurate <- 1e-10
Sa8_accurate <- RegularisedSol(K_h8,alphaReg_accurate,r8,"LF")

## when pre multiplied by K_h8, the expected solution is 1:8
## User can check the influence of the choice of alphaReg