compute the "closest" hE-vector in reach of the given hC-representation Distinguishing confined/valid projections and the PCA approach is ensured via the construction of sampleProjetionData and/or the evaluation THIS IS NO USER FUNCTION
0001 function heMatrixProjected = estimateHeMatrix(heMatrix, ... 0002 projectionMatrix, sampleProjectionData) 0003 % compute the "closest" hE-vector in reach of the given hC-representation 0004 % 0005 % Distinguishing confined/valid projections and the PCA approach is ensured 0006 % via the construction of sampleProjetionData and/or the evaluation 0007 % 0008 % THIS IS NO USER FUNCTION 0009 0010 % The elk-library: convex geometry applied to crystallization modeling. 0011 % Copyright (C) 2014 Alexander Reinhold 0012 % 0013 % This program is free software: you can redistribute it and/or modify it 0014 % under the terms of the GNU General Public License as published by the 0015 % Free Software Foundation, either version 3 of the License, or (at your 0016 % option) any later version. 0017 % 0018 % This program is distributed in the hope that it will be useful, but 0019 % WITHOUT ANY WARRANTY; without even the implied warranty of 0020 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 0021 % General Public License for more details. 0022 % 0023 % You should have received a copy of the GNU General Public License along 0024 % with this program. If not, see <http://www.gnu.org/licenses/>. 0025 0026 0027 %% comments (based on confinement projection) 0028 % We assume appropriate confinement partitions exist so that the sample 0029 % points are approximated by shapes for which the same validity 0030 % projection is valid. The target confinement partition is concluded 0031 % directly from the sample vector. 0032 % If this assumption is not proper, we cannot simply ignore these points 0033 % when the error measure is calculated. Furthermore, we have to avoid 0034 % jumps in the objective function. 0035 % To resolve the first issue, a ratio is defined that describes, how 0036 % many columns of heMatrix contribute to the error: 0037 % ratio = sum(notProperVector(hcMatrix)) / nCol. 0038 % Invalid vectors in the hcMatrix are substituted by their original 0039 % values so that these cannot contribute to the error anymore. 0040 % To resolve the second issue, we have to switch continuously from the 0041 % proper state to the improper state. This is, fortunately relatively 0042 % simple when the distance to the constreints is evaluated accordingly. 0043 % A parameter "distanceLimit" controls this behavior, analogously to the 0044 % growth rate validity mapping. However, this time we use an approximate 0045 % of the outer distance so that the fade-out does start when the vectors 0046 % already have become invalid. (This outer distance approximate is NOT the 0047 % distance from the validity cone). 0048 % to additionally avoid that NO vectors are proper, a penalty scheme is 0049 % added at the very end. It fixes optimizations that do converge towards 0050 % "no vectors", but require a good knowledge or tuning to set the slope 0051 % according to the expected error values. This penalty scheme could also 0052 % very simply be used to optimize for "ensure that the existent facets 0053 % match for original and approximation for as much as posisble vectors". 0054 0055 % mapping matrices 0056 mappingFullToReduced = orth(projectionMatrix)'; 0057 mappingReducedToFull = pinv(mappingFullToReduced); 0058 0059 % function handle to convert distances to "smoothing-filter" values 0060 % smoothFilter = @(distanceVector)(1*(distanceVector <= 0) + ... 0061 % (distanceVector <= par.distanceLimit).*(distanceVector > 0).*... 0062 % (par.distanceLimit - distanceVector)/par.distanceLimit); 0063 0064 % we cycle through the groups of projections to generate the 0065 % approximated vectors 0066 heMatrixProjected = nan*heMatrix; 0067 % fadeOutScaling = zeros(1, size(heMatrix, 2)); 0068 for iProjection = 1:length(sampleProjectionData.list) 0069 heFilter = sampleProjectionData.assignedProjectionVector == iProjection; 0070 0071 % generate the closest vectors in the subspace given by the current 0072 % projection, assuming a confinement region where the 0073 % pre-calculated validity projection is valid 0074 % when the validityProjection is the unit matrix, we should use the 0075 % initial projectionMatrix instead of the mapping matrices for 0076 % accuracy reasons 0077 thisValidityProjection = sampleProjectionData.list(iProjection).validityProjection; 0078 if all(diag(thisValidityProjection) == 1) && ... 0079 all(all(abs(thisValidityProjection - diag(diag(thisValidityProjection))) <= eps)) 0080 heMatrixInSubspace = projectionMatrix * heMatrix(:, heFilter); 0081 else 0082 heMatrixInSubspace = mappingReducedToFull*pinv(... 0083 thisValidityProjection * mappingReducedToFull) * ... 0084 heMatrix(:, heFilter); 0085 end 0086 % projectionApplicableDistance = max(... 0087 % sampleProjectionData.list(iProjection).projectionApplicableConstraint * ... 0088 % heMatrixInSubspace, [], 1); 0089 % the projection is smoothened out while we assume this projection 0090 % is valid for a few MORE points than exact. 0091 % if isempty(projectionApplicableDistance) 0092 % projectionApplicableDistance = ones(1, sum(vectorFilter)); 0093 % else 0094 % projectionApplicableDistance = smoothFilter(projectionApplicableDistance); 0095 % end 0096 0097 % generate the corresponding valid hC-vectors 0098 heMatrixValid = sampleProjectionData.list(iProjection).validityProjection * ... 0099 heMatrixInSubspace; 0100 % facetsValidDistance = max(... 0101 % sampleProjectionData.list(iProjection).facetPresentConstraint * ... 0102 % heMatrixValid , [], 1); 0103 % if isempty(facetsValidDistance) 0104 % facetsValidDistance = ones(1, sum(vectorFilter)); 0105 % else 0106 % facetsValidDistance = smoothFilter(facetsValidDistance); 0107 % end 0108 0109 % join applicability 0110 % vectorSmoothFilter = min(projectionApplicableDistance, ... 0111 % facetsValidDistance); 0112 0113 % assign values 0114 % heMatrixProjected(:, vectorFilter) = ... 0115 % bsxfun(@times, heMatrixValid, vectorSmoothFilter) + ... 0116 % bsxfun(@times, heMatrix(:, vectorFilter), (1-vectorSmoothFilter)); 0117 heMatrixProjected(:, heFilter) = heMatrixValid; 0118 % fadeOutScaling(vectorFilter) = vectorSmoothFilter; 0119 end 0120 0121 if isnan(heMatrixProjected) 0122 error('elk:approximation:internalError', ['Estimated projection of ' ... 0123 'hE-matrix contains nan. This should not happen.']); 0124 end 0125 0126 % fadeOutScaling = ones(1, size(heMatrix, 2));