RegularisedSol.Rd
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).
RegularisedSol(Kn, alphaReg, r,
regularization = c("Tikhonov", "LF", "cut-off"),
...)
numeric \(n \times n\) matrix.
regularisation parameter; numeric in ]0,1].
numeric vector of length
n.
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).
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.
the regularised solution, a vector of length n.
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.
## 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