Automatic Analysis Pipeline

Linear Regression Model - Robust

Author

Dominik Laa

Published

June 8, 2026

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	Strut Slenderness (Vol / Cell Size)	E_Break ~ Material_Hard * Lattice_Vol_Sc + I(Lattice_Vol_Sc / Cell_Size_Sc) + Structure	0.224	0.075	5.049	0.245
Epsilon_b	Rule of Mixtures + Structure + Cell Size	E_Break ~ Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc	0.190	0.151	5.144	0.470
Epsilon_b	Rule of Mixtures + Cell Size	E_Break ~ Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc	0.119	0.170	5.364	0.497
Epsilon_m	Rule of Mixtures + Cell Size	E_Max ~ Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc	0.463	0.027	0.517	0.013
Epsilon_m	Polynomial Cell Size (Quadratic)	E_Max ~ Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc + I(Cell_Size_Sc^2)	0.454	0.062	0.521	0.029
Epsilon_m	Rule of Mixtures + Structure + Cell Size	E_Max ~ Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc	0.435	0.064	0.529	0.030
Sigma_b	Rule of Mixtures: Mat × Vol	S_Break ~ Material_Hard * Lattice_Vol_Sc	0.653	0.050	4.855	0.340
Sigma_b	Strut Slenderness (Vol / Cell Size)	S_Break ~ Material_Hard * Lattice_Vol_Sc + I(Lattice_Vol_Sc / Cell_Size_Sc) + Structure	0.621	0.081	5.059	0.534
Sigma_b	Material × Cell Size	S_Break ~ Material_Hard * Lattice_Vol_Sc + Material_Hard * Cell_Size_Sc + Structure	0.618	0.068	5.086	0.441
Sigma_m	Rule of Mixtures + Cell Size	S_Max ~ Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc	0.735	0.032	3.421	0.200
Sigma_m	Rule of Mixtures: Mat × Vol	S_Max ~ Material_Hard * Lattice_Vol_Sc	0.712	0.047	3.557	0.288
Sigma_m	Strut Slenderness (Vol / Cell Size)	S_Max ~ Material_Hard * Lattice_Vol_Sc + I(Lattice_Vol_Sc / Cell_Size_Sc) + Structure	0.692	0.048	3.684	0.293

2.3 Final Models (Complete Training Data)

Formula	Model	Target	R² Training	RMSE
E_Break ~ Material_Hard * (Lattice_Vol_Sc + I(Lattice_Vol_Sc^2)) + Material_Hard * Structure	Polynomial Volume + Mat × Structure	Epsilon_b	0.672	3.287
E_Break ~ Material_Hard * Structure + Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc	Mat × Structure + Mat × Vol (Additive)	Epsilon_b	0.671	3.288
E_Break ~ Material_Hard * Lattice_Vol_Sc + Material_Hard * Structure + I(Lattice_Vol_Sc / Cell_Size_Sc)	Strut Slenderness + Mat × Structure	Epsilon_b	0.615	3.559
E_Max ~ Material_Hard * Structure + Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc	Mat × Structure + Mat × Vol (Additive)	Epsilon_m	0.671	0.404
E_Max ~ Material_Hard * (Lattice_Vol_Sc + I(Lattice_Vol_Sc^2)) + Material_Hard * Structure	Polynomial Volume + Mat × Structure	Epsilon_m	0.636	0.426
E_Max ~ Material_Hard * Lattice_Vol_Sc + Material_Hard * Cell_Size_Sc + Structure	Material × Cell Size	Epsilon_m	0.628	0.430
S_Break ~ Material_Hard * (Lattice_Vol_Sc + I(Lattice_Vol_Sc^2)) + Material_Hard * Structure	Polynomial Volume + Mat × Structure	Sigma_b	0.858	3.107
S_Break ~ Material_Hard * Structure + Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc	Mat × Structure + Mat × Vol (Additive)	Sigma_b	0.851	3.190
S_Break ~ Material_Hard * Lattice_Vol_Sc + Material_Hard * Structure + I(Lattice_Vol_Sc / Cell_Size_Sc)	Strut Slenderness + Mat × Structure	Sigma_b	0.837	3.336
S_Max ~ Material_Hard * (Lattice_Vol_Sc + I(Lattice_Vol_Sc^2)) + Material_Hard * Structure	Polynomial Volume + Mat × Structure	Sigma_m	0.873	2.372
S_Max ~ Material_Hard * Lattice_Vol_Sc + Structure * Lattice_Vol_Sc	Structure × Volume Interaction	Sigma_m	0.860	2.485
S_Max ~ Material_Hard * Structure + Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc	Mat × Structure + Mat × Vol (Additive)	Sigma_m	0.860	2.489

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	0.829	0.760
Rule of Mixtures: Mat × Vol	S_Max ~ Material_Hard * Lattice_Vol_Sc	0.796	0.723
Rule of Mixtures + Cell Size	S_Max ~ Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc	0.803	0.711

3.2.2 Target: \(\epsilon_m\)

Model	Formula	R2_Training	R2_Validation
Polynomial Cell Size (Quadratic)	E_Max ~ Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc + I(Cell_Size_Sc^2)	NA	0.623
Rule of Mixtures + Structure + Cell Size	E_Max ~ Material_Hard * Lattice_Vol_Sc + Structure + Cell_Size_Sc	0.628	0.574
Material × Cell Size	E_Max ~ Material_Hard * Lattice_Vol_Sc + Material_Hard * Cell_Size_Sc + Structure	0.628	0.573

3.2.3 Target: \(\sigma_b\)

Model	Formula	R2_Training	R2_Validation
Rule of Mixtures: Mat × Vol	S_Break ~ Material_Hard * Lattice_Vol_Sc	0.727	0.707
Rule of Mixtures + Cell Size	S_Break ~ Material_Hard * Lattice_Vol_Sc + Cell_Size_Sc	0.747	0.687
Structure–Volume (Nested)	S_Break ~ Material_Hard * Lattice_Vol_Sc + Structure:Lattice_Vol_Sc	0.775	0.686

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	0.383	0.432
Rule of Mixtures: Mat × Vol	E_Break ~ Material_Hard * Lattice_Vol_Sc	0.289	0.368
Material × Cell Size	E_Break ~ Material_Hard * Lattice_Vol_Sc + Material_Hard * Cell_Size_Sc + Structure	0.585	-0.145