Automatic Analysis Pipeline

Linear Regression Model

Author

Dominik Laa

Published

June 14, 2026

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1 Overview

This pipeline utilizes linear regression modeling to predict properties of groups outside the training set. A validation set is utilized to check the predictions. Results are dependent on the before used filter method in Data_Overview.qmd.

2 Model Creation

2.1 Model Definitions

Overview Linear Regression Models
Model-Id	Description	Formula
M01_RoM_Minimal	Rule of Mixtures: Mat × Vol	Material_Hard * Lattice_Vol_Sc
M02_RoM_CellSize	Rule of Mixtures + Cell Size	Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc
M03_RoM_Structure	Rule of Mixtures + Structure	Material_Hard * Lattice_Vol_Sc + Structure
M04_RoM_Full	Rule of Mixtures + Structure + Cell Size	Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc
M05_StrVol_Full	Structure × Volume Interaction	Material_Hard * Lattice_Vol_Sc + Structure * Lattice_Vol_Sc
M06_StrVol_Nested	Structure–Volume (Nested)	Material_Hard * Lattice_Vol_Sc + Structure:Lattice_Vol_Sc
M07_MatCell	Material × Cell Size	Material_Hard * Lattice_Vol_Sc + Material_Hard * Cell_Size_Sc + Structure
M08_PolyVol	Polynomial Volume (Quadratic)	Material_Hard * (Lattice_Vol_Sc + I(Lattice_Vol_Sc^2)) + Structure
M09_PolyCellSize	Polynomial Cell Size (Quadratic)	Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc + I(Cell_Size_Sc^2)
M10_GibsonAshby	Gibson–Ashby Scaling (Vol^1.5)	Material_Hard * I(Lattice_Vol_Sc^1.5) + Structure
M11_GA_StrVol	Gibson–Ashby + Structure × Vol^1.5	Material_Hard * I(Lattice_Vol_Sc^1.5) + Structure * I(Lattice_Vol_Sc^1.5) + Cell_Size_Sc
M12_Slenderness	Strut Slenderness (Vol / Cell Size)	Material_Hard * Lattice_Vol_Sc + I(Lattice_Vol_Sc / Cell_Size_Sc) + Structure
M13_Slenderness_MS	Strut Slenderness + Mat × Structure	Material_Hard * Lattice_Vol_Sc + Material_Hard * Structure + I(Lattice_Vol_Sc / Cell_Size_Sc)
M14_PhysHybrid	Physics-Informed Hybrid (GA + Slenderness)	Material_Hard * I(Lattice_Vol_Sc^1.5) + Structure:I(Lattice_Vol_Sc^1.5) + I(Lattice_Vol_Sc / Cell_Size_Sc)
M15_MatStrVol	Three-Way: Mat × Structure × Volume	Material_Hard * Structure * Lattice_Vol_Sc
M16_MatStr_Add	Mat × Structure + Mat × Vol (Additive)	Material_Hard * Structure + Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc
M17_PolyVol_MatStr	Polynomial Volume + Mat × Structure	Material_Hard * (Lattice_Vol_Sc + I(Lattice_Vol_Sc^2)) + Material_Hard * Structure

2.2 Preliminary Models

Table Preliminary Model Building
Target	Model	Formula	R²	R² SD	RMSE	RMSE SD
Epsilon_b	Rule of Mixtures + Cell Size	E_Break ~ Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc	0.139	0.128	5.311	0.374
Epsilon_b	Strut Slenderness (Vol / Cell Size)	E_Break ~ Material_Hard * Lattice_Vol_Sc + I(Lattice_Vol_Sc / Cell_Size_Sc) + Structure	0.124	0.205	5.292	0.619
Epsilon_b	Rule of Mixtures + Structure + Cell Size	E_Break ~ Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc	0.108	0.173	5.397	0.503
Epsilon_m	Rule of Mixtures + Cell Size	E_Max ~ Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc	0.475	0.058	0.510	0.028
Epsilon_m	Polynomial Cell Size (Quadratic)	E_Max ~ Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc + I(Cell_Size_Sc^2)	0.423	0.062	0.535	0.029
Epsilon_m	Rule of Mixtures + Structure + Cell Size	E_Max ~ Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc	0.377	0.104	0.555	0.046
Sigma_b	Strut Slenderness (Vol / Cell Size)	S_Break ~ Material_Hard * Lattice_Vol_Sc + I(Lattice_Vol_Sc / Cell_Size_Sc) + Structure	0.668	0.081	4.724	0.571
Sigma_b	Rule of Mixtures: Mat × Vol	S_Break ~ Material_Hard * Lattice_Vol_Sc	0.626	0.060	5.037	0.395
Sigma_b	Rule of Mixtures + Structure + Cell Size	S_Break ~ Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc	0.619	0.057	5.081	0.376
Sigma_m	Rule of Mixtures: Mat × Vol	S_Max ~ Material_Hard * Lattice_Vol_Sc	0.734	0.035	3.422	0.218
Sigma_m	Strut Slenderness (Vol / Cell Size)	S_Max ~ Material_Hard * Lattice_Vol_Sc + I(Lattice_Vol_Sc / Cell_Size_Sc) + Structure	0.730	0.053	3.441	0.332
Sigma_m	Rule of Mixtures + Structure	S_Max ~ Material_Hard * Lattice_Vol_Sc + Structure	0.692	0.040	3.684	0.234

2.3 Final Models (Complete Training Data)

Formula	Model	Target	R² Training	RMSE
E_Break ~ Material_Hard * Structure * Lattice_Vol_Sc	Three-Way: Mat × Structure × Volume	Epsilon_b	0.749	2.876
E_Break ~ Material_Hard * I(Lattice_Vol_Sc^1.5) + Structure * I(Lattice_Vol_Sc^1.5) + Cell_Size_Sc	Gibson–Ashby + Structure × Vol^1.5	Epsilon_b	0.683	5.596
E_Break ~ Material_Hard * I(Lattice_Vol_Sc^1.5) + Structure:I(Lattice_Vol_Sc^1.5) + I(Lattice_Vol_Sc / Cell_Size_Sc)	Physics-Informed Hybrid (GA + Slenderness)	Epsilon_b	0.683	3.586
E_Max ~ Material_Hard * Structure * Lattice_Vol_Sc	Three-Way: Mat × Structure × Volume	Epsilon_m	0.734	0.364
E_Max ~ Material_Hard * Structure + Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc	Mat × Structure + Mat × Vol (Additive)	Epsilon_m	0.698	0.388
E_Max ~ Material_Hard * (Lattice_Vol_Sc + I(Lattice_Vol_Sc^2)) + Material_Hard * Structure	Polynomial Volume + Mat × Structure	Epsilon_m	0.662	0.410
S_Break ~ Material_Hard * Structure * Lattice_Vol_Sc	Three-Way: Mat × Structure × Volume	Sigma_b	0.885	2.802
S_Break ~ Material_Hard * (Lattice_Vol_Sc + I(Lattice_Vol_Sc^2)) + Material_Hard * Structure	Polynomial Volume + Mat × Structure	Sigma_b	0.858	3.106
S_Break ~ Material_Hard * Structure + Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc	Mat × Structure + Mat × Vol (Additive)	Sigma_b	0.851	3.183
S_Max ~ Material_Hard * Structure * Lattice_Vol_Sc	Three-Way: Mat × Structure × Volume	Sigma_m	0.897	2.139
S_Max ~ Material_Hard * (Lattice_Vol_Sc + I(Lattice_Vol_Sc^2)) + Material_Hard * Structure	Polynomial Volume + Mat × Structure	Sigma_m	0.873	2.368
S_Max ~ Material_Hard * Lattice_Vol_Sc + Structure * Lattice_Vol_Sc	Structure × Volume Interaction	Sigma_m	0.861	2.480

3 Predictions

3.1 Generate Predictions

3.2 Compare Predictions

Validation summary created

Overall entries: 64

3.2.1 Target: \(\sigma_m\)

Model	Formula	R2_Training	R2_Validation
Structure–Volume (Nested)	S_Max ~ Material_Hard * Lattice_Vol_Sc + Structure:Lattice_Vol_Sc	0.830	0.757
Rule of Mixtures: Mat × Vol	S_Max ~ Material_Hard * Lattice_Vol_Sc	0.797	0.727
Rule of Mixtures + Cell Size	S_Max ~ Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc	0.804	0.714

3.2.2 Target: \(\epsilon_m\)

Model	Formula	R2_Training	R2_Validation
Rule of Mixtures + Structure + Cell Size	E_Max ~ Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc	0.651	0.623
Polynomial Cell Size (Quadratic)	E_Max ~ Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc + I(Cell_Size_Sc^2)	0.651	0.623
Material × Cell Size	E_Max ~ Material_Hard * Lattice_Vol_Sc + Material_Hard * Cell_Size_Sc + Structure	0.651	0.622

3.2.3 Target: \(\sigma_b\)

Model	Formula	R2_Training	R2_Validation
Rule of Mixtures: Mat × Vol	S_Break ~ Material_Hard * Lattice_Vol_Sc	0.730	0.708
Structure–Volume (Nested)	S_Break ~ Material_Hard * Lattice_Vol_Sc + Structure:Lattice_Vol_Sc	0.775	0.699
Rule of Mixtures + Cell Size	S_Break ~ Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc	0.748	0.689

3.2.4 Target: \(\epsilon_b\)

Model	Formula	R2_Training	R2_Validation
Rule of Mixtures + Cell Size	E_Break ~ Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc	0.422	0.319
Rule of Mixtures: Mat × Vol	E_Break ~ Material_Hard * Lattice_Vol_Sc	0.351	0.224
Material × Cell Size	E_Break ~ Material_Hard * Lattice_Vol_Sc + Material_Hard * Cell_Size_Sc + Structure	0.600	-0.283

Plot saved: C:/Users/Dominik Laa/TU_Cloud/2026 Science/Sebastian_Paper/Review1/Auswertung_v3/output/Validation_vs_Prediciton.png
  Size: 8.0 × 8.0 cm (W:50% × H:50%)
  DPI: 600

Modells ued in validation plot:

\(\sigma_m\): M06_StrVol_Nested — Structure–Volume (Nested)
\(\epsilon_m\): M04_RoM_Full — Rule of Mixtures + Structure + Cell Size
\(\sigma_b\): M01_RoM_Minimal — Rule of Mixtures: Mat × Vol
\(\epsilon_b\): M02_RoM_CellSize — Rule of Mixtures + Cell Size

3.3 All Predictions Plot

Plot saved: C:/Users/Dominik Laa/TU_Cloud/2026 Science/Sebastian_Paper/Review1/Auswertung_v3/output/S_Max.png
  Size: 16.0 × 8.0 cm (W:100% × H:50%)
  DPI: 600

Plot saved: C:/Users/Dominik Laa/TU_Cloud/2026 Science/Sebastian_Paper/Review1/Auswertung_v3/output/E_Max.png
  Size: 16.0 × 8.0 cm (W:100% × H:50%)
  DPI: 600

Plot saved: C:/Users/Dominik Laa/TU_Cloud/2026 Science/Sebastian_Paper/Review1/Auswertung_v3/output/S_Break.png
  Size: 16.0 × 8.0 cm (W:100% × H:50%)
  DPI: 600

Plot saved: C:/Users/Dominik Laa/TU_Cloud/2026 Science/Sebastian_Paper/Review1/Auswertung_v3/output/E_Break.png
  Size: 16.0 × 8.0 cm (W:100% × H:50%)
  DPI: 600

3.4 Validation Error Plot

Plot saved: C:/Users/Dominik Laa/TU_Cloud/2026 Science/Sebastian_Paper/Review1/Auswertung_v3/output/experimental_validation_error.png
  Size: 16.0 × 8.0 cm (W:100% × H:50%)
  DPI: 600

`````{=html}

Standardisierte Koeffizienten (β) — * = n.s.
Predictor	σ_m	ε_m	σ_b	ε_b
Material_Hard	-0.616	0.539	-0.641	0.52
Lattice_Vol_Sc	-0.302	0.1*	-0.189	0.024*
StructureFCCXYZ	NA	-0.3*	NA	NA
StructureNeovius	NA	0.081*	NA	NA
StructureSchoen-Gyroid	NA	0.382	NA	NA
StructureSchwarz Primitive	NA	0.326	NA	NA
Cell_Size_Sc	NA	-0.296	NA	-0.27


## DELETE THIS LATER

::: {.cell tbl-cap='Modellvergleich – R² Prediction vs. Training je Zielgröße'}
`````{=html}
<table class="table table-striped table-hover table-condensed" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
<tr>
<th style="empty-cells: hide;border-bottom:hidden;" colspan="1"></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; font-weight: bold; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">σ<sub>m</sub>
</div></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; font-weight: bold; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">ε<sub>m</sub>
</div></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; font-weight: bold; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">σ<sub>b</sub>
</div></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; font-weight: bold; " colspan="2"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">ε<sub>b</sub>
</div></th>
</tr>
  <tr>
   <th style="text-align:left;"> Model </th>
   <th style="text-align:center;"> R<sup>2</sup><sub>Pred</sub> </th>
   <th style="text-align:center;"> R<sup>2</sup><sub>Train</sub> </th>
   <th style="text-align:center;"> R<sup>2</sup><sub>Pred</sub> </th>
   <th style="text-align:center;"> R<sup>2</sup><sub>Train</sub> </th>
   <th style="text-align:center;"> R<sup>2</sup><sub>Pred</sub> </th>
   <th style="text-align:center;"> R<sup>2</sup><sub>Train</sub> </th>
   <th style="text-align:center;"> R<sup>2</sup><sub>Pred</sub> </th>
   <th style="text-align:center;"> R<sup>2</sup><sub>Train</sub> </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;font-weight: bold;"> M01 </td>
   <td style="text-align:center;"> 0.727 </td>
   <td style="text-align:center;"> 0.797 </td>
   <td style="text-align:center;"> 0.533 </td>
   <td style="text-align:center;"> 0.504 </td>
   <td style="text-align:center;"> 0.708 </td>
   <td style="text-align:center;"> 0.730 </td>
   <td style="text-align:center;"> 0.224 </td>
   <td style="text-align:center;"> 0.351 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> M02 </td>
   <td style="text-align:center;"> 0.714 </td>
   <td style="text-align:center;"> 0.804 </td>
   <td style="text-align:center;"> 0.591 </td>
   <td style="text-align:center;"> 0.594 </td>
   <td style="text-align:center;"> 0.689 </td>
   <td style="text-align:center;"> 0.748 </td>
   <td style="text-align:center;"> 0.319 </td>
   <td style="text-align:center;"> 0.422 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> M03 </td>
   <td style="text-align:center;"> 0.591 </td>
   <td style="text-align:center;"> 0.833 </td>
   <td style="text-align:center;"> 0.588 </td>
   <td style="text-align:center;"> 0.565 </td>
   <td style="text-align:center;"> 0.449 </td>
   <td style="text-align:center;"> 0.799 </td>
   <td style="text-align:center;"> -0.638 </td>
   <td style="text-align:center;"> 0.518 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> M04 </td>
   <td style="text-align:center;"> 0.582 </td>
   <td style="text-align:center;"> 0.839 </td>
   <td style="text-align:center;"> 0.623 </td>
   <td style="text-align:center;"> 0.651 </td>
   <td style="text-align:center;"> 0.428 </td>
   <td style="text-align:center;"> 0.816 </td>
   <td style="text-align:center;"> -0.556 </td>
   <td style="text-align:center;"> 0.584 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> M05 </td>
   <td style="text-align:center;"> 0.649 </td>
   <td style="text-align:center;"> 0.861 </td>
   <td style="text-align:center;"> 0.424 </td>
   <td style="text-align:center;"> 0.628 </td>
   <td style="text-align:center;"> 0.518 </td>
   <td style="text-align:center;"> 0.830 </td>
   <td style="text-align:center;"> -0.822 </td>
   <td style="text-align:center;"> 0.578 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> M06 </td>
   <td style="text-align:center;"> 0.757 </td>
   <td style="text-align:center;"> 0.830 </td>
   <td style="text-align:center;"> 0.249 </td>
   <td style="text-align:center;"> 0.576 </td>
   <td style="text-align:center;"> 0.699 </td>
   <td style="text-align:center;"> 0.775 </td>
   <td style="text-align:center;"> -0.420 </td>
   <td style="text-align:center;"> 0.442 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> M07 </td>
   <td style="text-align:center;"> 0.657 </td>
   <td style="text-align:center;"> 0.857 </td>
   <td style="text-align:center;"> 0.622 </td>
   <td style="text-align:center;"> 0.651 </td>
   <td style="text-align:center;"> 0.523 </td>
   <td style="text-align:center;"> 0.830 </td>
   <td style="text-align:center;"> -0.283 </td>
   <td style="text-align:center;"> 0.600 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> M08 </td>
   <td style="text-align:center;"> 0.535 </td>
   <td style="text-align:center;"> 0.843 </td>
   <td style="text-align:center;"> 0.574 </td>
   <td style="text-align:center;"> 0.618 </td>
   <td style="text-align:center;"> 0.360 </td>
   <td style="text-align:center;"> 0.814 </td>
   <td style="text-align:center;"> -0.615 </td>
   <td style="text-align:center;"> 0.575 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> M09 </td>
   <td style="text-align:center;"> 0.582 </td>
   <td style="text-align:center;"> 0.839 </td>
   <td style="text-align:center;"> 0.623 </td>
   <td style="text-align:center;"> 0.651 </td>
   <td style="text-align:center;"> 0.428 </td>
   <td style="text-align:center;"> 0.816 </td>
   <td style="text-align:center;"> -0.556 </td>
   <td style="text-align:center;"> 0.584 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> M10 </td>
   <td style="text-align:center;"> -0.658 </td>
   <td style="text-align:center;"> 0.464 </td>
   <td style="text-align:center;"> -0.213 </td>
   <td style="text-align:center;"> 0.389 </td>
   <td style="text-align:center;"> -0.876 </td>
   <td style="text-align:center;"> 0.499 </td>
   <td style="text-align:center;"> -2.527 </td>
   <td style="text-align:center;"> 0.474 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> M11 </td>
   <td style="text-align:center;"> -0.517 </td>
   <td style="text-align:center;"> 0.624 </td>
   <td style="text-align:center;"> -0.038 </td>
   <td style="text-align:center;"> 0.536 </td>
   <td style="text-align:center;"> -0.667 </td>
   <td style="text-align:center;"> 0.709 </td>
   <td style="text-align:center;"> -1.526 </td>
   <td style="text-align:center;"> 0.683 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> M12 </td>
   <td style="text-align:center;"> 0.664 </td>
   <td style="text-align:center;"> 0.852 </td>
   <td style="text-align:center;"> 0.506 </td>
   <td style="text-align:center;"> 0.583 </td>
   <td style="text-align:center;"> 0.540 </td>
   <td style="text-align:center;"> 0.834 </td>
   <td style="text-align:center;"> -0.620 </td>
   <td style="text-align:center;"> 0.613 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> M13 </td>
   <td style="text-align:center;"> 0.615 </td>
   <td style="text-align:center;"> 0.855 </td>
   <td style="text-align:center;"> 0.400 </td>
   <td style="text-align:center;"> 0.609 </td>
   <td style="text-align:center;"> 0.458 </td>
   <td style="text-align:center;"> 0.837 </td>
   <td style="text-align:center;"> -1.203 </td>
   <td style="text-align:center;"> 0.633 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> M15 </td>
   <td style="text-align:center;"> 0.614 </td>
   <td style="text-align:center;"> 0.897 </td>
   <td style="text-align:center;"> 0.447 </td>
   <td style="text-align:center;"> 0.734 </td>
   <td style="text-align:center;"> 0.448 </td>
   <td style="text-align:center;"> 0.885 </td>
   <td style="text-align:center;"> -1.348 </td>
   <td style="text-align:center;"> 0.749 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> M16 </td>
   <td style="text-align:center;"> 0.614 </td>
   <td style="text-align:center;"> 0.861 </td>
   <td style="text-align:center;"> 0.487 </td>
   <td style="text-align:center;"> 0.698 </td>
   <td style="text-align:center;"> 0.484 </td>
   <td style="text-align:center;"> 0.851 </td>
   <td style="text-align:center;"> -0.845 </td>
   <td style="text-align:center;"> 0.683 </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> M17 </td>
   <td style="text-align:center;"> 0.639 </td>
   <td style="text-align:center;"> 0.873 </td>
   <td style="text-align:center;"> 0.336 </td>
   <td style="text-align:center;"> 0.662 </td>
   <td style="text-align:center;"> 0.561 </td>
   <td style="text-align:center;"> 0.858 </td>
   <td style="text-align:center;"> -0.382 </td>
   <td style="text-align:center;"> 0.683 </td>
  </tr>
</tbody>
</table>

:::