RegularisedSol {StableEstim} | R Documentation |
Regularised Inverse
Description
Regularised solution of the (ill-posed) problem where
is a
matrix,
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 |
alphaReg |
regularisation parameter; numeric in ]0,1]. |
r |
numeric vector of |
regularization |
regularization scheme to be used, one of |
... |
the value of |
Details
Following Carrasco and Florens(2007), the regularised solution of the
problem is given by :
where is a (positive) real function with some regularity
conditions and
the singular decomposition of the
matrix
.
The regularization
parameter defines the form of the function
. For example, the
"Tikhonov"
scheme defines
.
When the matrix 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
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