l1-penalised |
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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 |
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signal |
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1-d Noisy signal, length(signal)= 2^J |
lambda |
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Smoothing parameter. It is level-dependent and can be estimated explicitly. The choice lambda=0 set lambda to its universal value. |
distribution |
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'gaussian' for Gaussian distribution 'poisson' for Poisson distribution |
wtrans |
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'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 |
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'ha' for Haar wavelet 'db' for Daubechies wavelets 'co' for Coiflets wavelets 'sy' for Symmlets wavelets |
lev |
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Coarsest resolution level |
max_IP_iter |
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Max number of iteration for the method |
IP_optimal_tol |
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Optimal tolerance to check convergence of the method |
feasible_tol |
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Feasible tolerance to check convergence of the method. |
max_CG_iter |
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Max number of iteration for the Conjugate Gradient method |
CG_tol |
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Tolerance for the Conjugate Gradient method |
x_save |
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Initial value for Newton direction deltaX |
y_save |
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Initial value for Newton direction deltaY |
z_save |
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Initial value for Newton direction deltaZ |
eta_save |
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Initial value for Newton direction deltaETA |
rho_save |
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Initial value for the log-barrier parameter deltaRHO |
hystory_print |
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Integer value to print the history of iterations |
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VALUE |
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f |
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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.