[Master Index] [Index for PublicToolbox/regutools]

# l_curve

## (PublicToolbox/regutools/l_curve.m in BrainStorm 2.0 (Alpha))

### Function Synopsis

    [reg_corner,rho,eta,reg_param] = l_curve(U,sm,b,method,L,V)

### Help Text

    L_CURVE Plot the L-curve and find its "corner".

     [reg_corner,rho,eta,reg_param] =
                      l_curve(U,s,b,method)
                      l_curve(U,sm,b,method)  ,  sm = [sigma,mu]
                      l_curve(U,s,b,method,L,V)

     Plots the L-shaped curve of eta, the solution norm || x || or
     semi-norm || L x ||, as a function of rho, the residual norm
     || A x - b ||, for the following methods:
        method = 'Tikh'  : Tikhonov regularization   (solid line )
        method = 'tsvd'  : truncated SVD or GSVD     (o markers  )
        method = 'dsvd'  : damped SVD or GSVD        (dotted line)
        method = 'mtsvd' : modified TSVD             (x markers  )
     The corresponding reg. parameters are returned in reg_param.
     If no method is specified, 'Tikh' is default.

     If any output arguments are specified, then the corner of the L-curve
     is identified and the corresponding reg. parameter reg_corner is
     returned.  Use routine l_corner if an upper bound on eta is required.

     If the Spline Toolbox is not available and reg_corner is requested,
     then the routine returns reg_corner = NaN for 'tsvd' and 'mtsvd'.

### Cross-Reference Information

This function calls

    dsvd                               C:\BrainStorm_2001\PublicToolbox\regutools\dsvd.m
    l_corner                           C:\BrainStorm_2001\PublicToolbox\regutools\l_corner.m
    plot_lc                            C:\BrainStorm_2001\PublicToolbox\regutools\plot_lc.m
    tsvd                               C:\BrainStorm_2001\PublicToolbox\regutools\tsvd.m

This function is called by

    regudemo                           C:\BrainStorm_2001\PublicToolbox\regutools\regudemo.m

### Listing of function C:\BrainStorm_2001\PublicToolbox\regutools\l_curve.m

    function [reg_corner,rho,eta,reg_param] = l_curve(U,sm,b,method,L,V)
    %L_CURVE Plot the L-curve and find its "corner".
    %
    % [reg_corner,rho,eta,reg_param] =
    %                  l_curve(U,s,b,method)
    %                  l_curve(U,sm,b,method)  ,  sm = [sigma,mu]
    %                  l_curve(U,s,b,method,L,V)
    %
    % Plots the L-shaped curve of eta, the solution norm || x || or
    % semi-norm || L x ||, as a function of rho, the residual norm
    % || A x - b ||, for the following methods:
    %    method = 'Tikh'  : Tikhonov regularization   (solid line )
    %    method = 'tsvd'  : truncated SVD or GSVD     (o markers  )
    %    method = 'dsvd'  : damped SVD or GSVD        (dotted line)
    %    method = 'mtsvd' : modified TSVD             (x markers  )
    % The corresponding reg. parameters are returned in reg_param.
    % If no method is specified, 'Tikh' is default.
    %
    % If any output arguments are specified, then the corner of the L-curve
    % is identified and the corresponding reg. parameter reg_corner is
    % returned.  Use routine l_corner if an upper bound on eta is required.
    %
    % If the Spline Toolbox is not available and reg_corner is requested,
    % then the routine returns reg_corner = NaN for 'tsvd' and 'mtsvd'.

    % Reference: P. C. Hansen & D. P. O'Leary, "The use of the L-curve in
    % the regularization of discrete ill-posed problems", Report UMIACS-
    % TR-91-142, Dept. of Computer Science, Univ. of Maryland, 1991;
    % to appear in SIAM J. Sci. Comp.

    % Per Christian Hansen, UNI-C, 03/17/93.

    % Set defaults.
    if (nargin==3), method='Tikh'; end  % Tikhonov reg. is default.
    npoints = 100;  % Number of points on the L-curve for Tikh and dsvd.
    smin_ratio = 16*eps;  % Smallest regularization parameter.

    % Initialization.
    [m,n] = size(U); [p,ps] = size(sm);
    if (nargout > 0), locate = 1; else locate = 0; end
    beta = U'*b; beta2 = b'*b - beta'*beta;
    if (ps==1)
      s = sm; beta = beta(1:p);
    else
      s = sm(p:-1:1,1)./sm(p:-1:1,2); beta = beta(p:-1:1);
    end
    xi = beta(1:p)./s;

    if (method(1:4)=='Tikh' | method(1:4)=='tikh')

      eta = zeros(npoints,1); rho = eta; reg_param = eta; s2 = s.^2;
      reg_param(npoints) = max([s(p),s(1)*smin_ratio]);
      ratio = (s(1)/reg_param(npoints))^(1/(npoints-1));
      ratio = (s(1)/reg_param(npoints))^(1/(npoints-1));
      for i=npoints-1:-1:1, reg_param(i) = ratio*reg_param(i+1); end
      for i=1:npoints
        f = s2./(s2 + reg_param(i)^2);
        eta(i) = norm(f.*xi);
        rho(i) = norm((1-f).*beta(1:p));
      end
      if (m > n & beta2 > 0), rho = sqrt(rho.^2 + beta2); end
      marker = '-'; pos = .8; txt = 'Tikh.';

    elseif (method(1:4)=='tsvd' | method(1:4)=='tgsv')

      eta = zeros(p,1); rho = eta;
      eta(1) = xi(1)^2;
      for k=2:p, eta(k) = eta(k-1) + xi(k)^2; end
      eta = sqrt(eta);
      if (m > n)
        if (beta2 > 0), rho(p) = beta2; else rho(p) = eps^2; end
      else
        rho(p) = eps^2;
      end
      for k=p-1:-1:1, rho(k) = rho(k+1) + beta(k+1)^2; end
      rho = sqrt(rho);
      reg_param = [1:p]'; marker = 'o'; pos = .75;
      if (ps==1)
        U = U(:,1:p); txt = 'TSVD';
      else
        U = U(:,1:p); txt = 'TGSVD';
      end

    elseif (method(1:4)=='dsvd' | method(1:4)=='dgsv')

      eta = zeros(npoints,1); rho = eta; reg_param = eta;
      reg_param(npoints) = max([s(p),s(1)*smin_ratio]);
      ratio = (s(1)/reg_param(npoints))^(1/(npoints-1));
      for i=npoints-1:-1:1, reg_param(i) = ratio*reg_param(i+1); end
      for i=1:npoints
        f = s./(s + reg_param(i));
        eta(i) = norm(f.*xi);
        rho(i) = norm((1-f).*beta(1:p));
      end
      if (m > n & beta2 > 0), rho = sqrt(rho.^2 + beta2); end
      marker = ':'; pos = .85;
      if (ps==1), txt = 'DSVD'; else txt = 'DGSVD'; end

    elseif (method(1:4)=='mtsv')

      if (nargin~=6)
        error('The matrices L and V must also be specified')
      end
      [p,n] = size(L); rho = zeros(p,1); eta = rho;
      [Q,R] = qr(L*V(:,n:-1:n-p));
      for i=1:p
        k = n-p+i;
        Lxk = L*V(:,1:k)*xi(1:k);
        zk = R(1:n-k,1:n-k)\(Q(:,1:n-k)'*Lxk); zk = zk(n-k:-1:1);
        eta(i) = norm(Q(:,n-k+1:p)'*Lxk);
        if (i < p)
          rho(i) = norm(beta(k+1:n) + s(k+1:n).*zk);
        else
          rho(i) = eps;
        end
      end
      if (m > n & beta2 > 0), rho = sqrt(rho.^2 + beta2); end
      reg_param = [n-p+1:n]'; txt = 'MTSVD';
      U = U(:,reg_param); sm = sm(reg_param);
      marker = 'x'; pos = .7; ps = 2;  % General form regularization.

    else, error('Illegal method'), end

    % Locate the "corner" of the L-curve, if required.  If the Spline
    % Toolbox is not available, return NaN for reg_corner.
    if (locate)
      SkipCorner = ( (method(1:4)=='tsvd' | method(1:4)=='tgsv' | ...
                      method(1:4)=='mtsv') & exist('spdemos')~=2 );
      if (SkipCorner)
        reg_corner = NaN;
      else
        [reg_corner,rho_c,eta_c] = l_corner(rho,eta,reg_param,U,sm,b,method);
      end
    end

    % Make plot.
    plot_lc(rho,eta,marker,ps,reg_param);
    if (locate & ~SkipCorner)
      HoldState = ishold; hold on;
      loglog([min(rho)/100,rho_c],[eta_c,eta_c],'--',...
             [rho_c,rho_c],[min(eta)/100,eta_c],'--')
      title(['L-curve, ',txt,' corner at ',num2str(reg_corner)]);
      if (~HoldState), hold off; end
    end

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