l1-penalised

 

l1-penalised likelihood estimator 

 

DESCRIPTION

 

Estimate the intensity function as the solution of a convex programming problem.

 

USAGE

 

f = GBP(signal,lambda,distribution,wtrans,wname,lev,max_IP_iter, IP_optimal_tol,feasible_tol,max_CG_iter,CG_tol,x_save,y_save, z_save,eta_save,rho_save,history_print)
 

REQUIRED ARGUMENTS

 

 

 

signal

 

1-d Noisy signal, length(signal)= 2^J

lambda

 

Smoothing parameter. It is level-dependent and can be estimated explicitly. The choice  lambda=0 set lambda to its universal value.

distribution

 

'gaussian' for Gaussian distribution

 'poisson' for Poisson distribution

 

wtrans

 

'wavedec'  for multilevel 1D wavelet decomposition performed by the Wavelet toolbox

'FWT_PO'  for 1d orthogonal wavelet transform performed by WaveLab802

'wavedec2'  for multilevel 2D wavelet decomposition performed by the Wavelet toolbox

'FWT2_PO'  for 2d orthogonal wavelet transform performed by WaveLab802

 

wname

 

'ha' for Haar wavelet

'db' for Daubechies wavelets

'co' for Coiflets wavelets

'sy' for Symmlets wavelets

 

lev

 

Coarsest resolution level

max_IP_iter  

 

Max number of iteration for the method

IP_optimal_tol

 

Optimal tolerance to check convergence of the method

feasible_tol

 

Feasible tolerance to check convergence of the method.

max_CG_iter

 

Max number of iteration for the Conjugate Gradient method

CG_tol

 

Tolerance for the Conjugate Gradient method

x_save

 

Initial value for Newton direction deltaX

y_save

 

Initial value for Newton direction deltaY

z_save

 

Initial value for Newton direction deltaZ

eta_save

 

Initial value for Newton direction deltaETA

rho_save

 

Initial value for the log-barrier parameter deltaRHO

hystory_print

 

Integer value to print the history of iterations

 

 

 

VALUE

 

 

 

f

 

Estimated intensity function  

 

BACKGROUND

 

The estimation of the intensity function is obtained by solving a convex programming problem in the wavelet domain by first deriving its dual and then developing a primal-dual log barrier internal point  method. Sardy, Antoniadis & Tseng (2003) proved that the appropriate smoothing parameter lambda is actually level-dependent and they gave an explicit expression for it.

 

REFERENCES

 

Sardy, S. Antoniadis, A. & Tseng, P. (2003). Automatic smoothing with wavelets for a wide class of distributions. J. Comp. Graph. Statist., 12 (to appear).

ACKNOWLEDGEMENT 

The GBP function is based on Matlab routines kindly provided by Sylvain Sardy.