RegularisedSol {StableEstim}R Documentation

Regularised Inverse

Description

Regularised solution of the (ill-posed) problem Kϕ=rK\phi = r where KK is a n×nn \times n matrix, rr 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×nn \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 cc used in the "LF" scheme. See Carrasco and Florens(2007).

Details

Following Carrasco and Florens(2007), the regularised solution of the problem Kϕ=rK \phi=r is given by :

φαreg=j=1nq(αreg,μj)<r,ψj>μjϕj,\varphi_{\alpha_{reg}} = \sum_{j=1}^{n} q(\alpha_{reg},\mu_j)\frac{<r,\psi_j >}{\mu_j} \phi_j ,

where qq is a (positive) real function with some regularity conditions and μ,ϕ,ψ\mu,\phi,\psi the singular decomposition of the matrix KK.

The regularization parameter defines the form of the function qq. For example, the "Tikhonov" scheme defines q(αreg,μ)=μ2αreg+μ2q(\alpha_{reg},\mu) = \frac{\mu^2}{\alpha_{reg}+\mu^2}.

When the matrix KK 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

[Package StableEstim version 2.2 Index]