diagnostics_GMM_1d

% diagnostics 1d

 given all parameters of a linearized estimation problem
 determine the essential diagnostic values
 assuming the number N of observations is small enough
    (requires several NxN matrices)

 r_U    = index set for parameters to be estimated (others are nuisance)
 Am     = Jacobian
 Cov_ll = CovM of observations
 W_ll   = WeightM of observations
 Cov_xx = CovM of all estimated parameters
 W_xx   = WeightM of all estimated parameters
 vv     = residuals

 riv    = redundancy numbers
 zv     = standardized test statistics
 nabla_lv = lowe bound for detectable outliers
 muv      = influence factor for all parameters
 muv1     = influence factor for parameters specified by r_U
 uv1q     = diagonal elements of \bar U_1 (numerator of muv1^2)
 uv2      = diagonal elements of U2

 Wolfgang Förstner 10/2016
 wfoerstn@uni-bonn.de

0001 %% diagnostics 1d
0002 %
0003 % given all parameters of a linearized estimation problem
0004 % determine the essential diagnostic values
0005 % assuming the number N of observations is small enough
0006 %    (requires several NxN matrices)
0007 %
0008 % r_U    = index set for parameters to be estimated (others are nuisance)
0009 % Am     = Jacobian
0010 % Cov_ll = CovM of observations
0011 % W_ll   = WeightM of observations
0012 % Cov_xx = CovM of all estimated parameters
0013 % W_xx   = WeightM of all estimated parameters
0014 % vv     = residuals
0015 %
0016 % riv    = redundancy numbers
0017 % zv     = standardized test statistics
0018 % nabla_lv = lowe bound for detectable outliers
0019 % muv      = influence factor for all parameters
0020 % muv1     = influence factor for parameters specified by r_U
0021 % uv1q     = diagonal elements of \bar U_1 (numerator of muv1^2)
0022 % uv2      = diagonal elements of U2
0023 %
0024 % Wolfgang Förstner 10/2016
0025 % wfoerstn@uni-bonn.de
0026 
0027 function [riv,zv,nabla_lv,muv,muv1,uv1q,uv2] = ...
0028     diagnostics_GMM_1d(r_U,Am,Cov_ll,W_ll,Cov_xx,W_xx,vv)
0029 
0030 % non-centrality parameter for alpha_0 = 0.001, beta_0 = 0.8
0031 delta_0 = 4.13;
0032 
0033 [N,U]  = size(Am);
0034 %% observations: detectatbility, statistical test
0035 Rm     = eye(N)-Am*Cov_xx*Am'*W_ll;        % redundancy matrix
0036 riv    = diag(Rm);                         % redundancy numbers
0037 zv     = vv./sqrt(riv.*diag(Cov_ll)+eps);  % normalized residuals
0038 nabla_lv = delta_0 * sqrt(1./riv);
0039 
0040 %% sensitivity wrtall parameters
0041 muv    = sqrt((1-riv)./(riv+eps));         % sensitivity factors
0042 
0043 %% sensitivity wrt selected parameters
0044 if ~isempty(r_U)
0045     % ranges
0046     r_C= r_U;
0047     r_D= setdiff(1:U,r_U);
0048     % partitioned design matrix A=[C,D]
0049     Cm = Am(:,r_C);
0050     Dm = Am(:,r_D);
0051     % reduced design matrix
0052     Cmr = Cm - Dm * inv(W_xx(r_D,r_D)) * W_xx(r_D,r_C);  %#ok<*MINV> % C_reduced
0053     Cov_11 = Cov_xx(r_C,r_C);
0054     % effect onto parameters
0055     U1q     = Cmr * Cov_11        * Cmr' * W_ll;         % U_C_bar
0056     uv1q    = diag(U1q);                                 % u_C_bar
0057     U2      = Dm  * inv(W_xx(r_D,r_D)) * Dm'  * W_ll;    % U_D
0058     uv2     = diag(U2) ;                                 % u_D
0059     % sensitivity factors
0060     muv1    = sqrt(uv1q ./ riv);
0061     
0062     %     check = uv1q+uv2+riv;  % should be ones
0063 end