Regression Modelling in Concrete and Material Science - CHAPTER THREE The Ibearugbulem Regression Framework

CHAPTER THREE
The Ibearugbulem Regression Framework

(By Owus M. Ibearugbulem – Light for the Living)

3.1 Overview

In the previous chapter, we established that regression converts experimental data into a predictive mathematical model using matrix algebra. We also derived the general solution:

$$ [a] = ([H]^T [H])^{-1} [H]^T [Y] $$

The present chapter focuses on Ibearugbulem’s Regression Framework (IRF) — a structured, step-by-step adaptation of regression analysis tailored for engineering mixtures such as concrete, mortar, and composite materials. Ibearugbulem’s approach simplifies matrix-based regression for any number of predictors (two to five or more) and provides a transparent pathway from experimental data to validated predictive equations.

3.2 Core Philosophy of Ibearugbulem’s Approach

Traditional regression techniques such as Scheffé’s and Osadebe’s models use proportional ratios of components (e.g., cement fraction, water fraction). Ibearugbulem’s method removes this limitation by using direct experimental quantities (kg/m³, N/mm², etc.) instead of ratios. This makes the process:

• Simpler — no conversion to fractions;
• Flexible — works for partial or replacement mixes;
• Computationally neat — matrix algebra handles all relationships automatically.

It is especially suitable when available data points are few (e.g., 5–10 observations) but high accuracy is required.

3.3 The General Ibearugbulem Model

For a mixture experiment with \( k \) predictors:

$$ Y = a_0 + a_1 X_1 + a_2 X_2 + a_3 X_3 + \dots + a_k X_k \tag{3.1} $$

In matrix form:

$$ Y_j = [H][a_k] \tag{3.2} $$

where \([H]\) is the design matrix containing all predictor values plus a column of ones, and \([a_k]\) contains the regression coefficients.

3.4 Step-by-Step Computational Framework

Step 1: Form the Design Matrix

$$ [H] = \begin{bmatrix} 1 & X_{11} & X_{12} & \dots & X_{1k} \\ 1 & X_{21} & X_{22} & \dots & X_{2k} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & X_{n1} & X_{n2} & \dots & X_{nk} \end{bmatrix} $$

Step 2: Construct the Response Vector

$$ [Y] = \begin{bmatrix} Y_1 \\ Y_2 \\ \vdots \\ Y_n \end{bmatrix} $$

Step 3: Compute [CC] and [R]

$$ [CC] = [H]^T [H], \quad [R] = [H]^T [Y] $$

Step 4: Solve for [a]

$$ [a] = [CC]^{-1} [R] $$

3.5 Worked Example (Four Predictors Case)

Let’s apply this to a simplified example with four predictors corresponding to a concrete mix:

Mix	X₁ Water (kg/m³)	X₂ Cement (kg/m³)	X₃ Sand (kg/m³)	X₄ Coarse Agg. (kg/m³)	Y (MPa)
1	189	315	631	1263	3.32
2	201	366	732	1099	3.54
3	240	480	720	960	3.72
4	273	608	608	911	2.80
5	195	339	678	1187	2.69

Compute:

$$ [CC] = [H]^T [H], \quad [R] = [H]^T [Y], \quad [a] = [CC]^{-1}[R] $$

Result:

$$ Y_{mod} = 6259.19 - 2.58X_1 - 2.61X_2 - 2.61X_3 - 2.61X_4 \tag{3.7} $$

EXPLANATORY BOX – “The Detective Story”
Imagine you’re a detective trying to solve what makes concrete strong. Each ingredient — water, cement, sand, and aggregate — is a suspect. You collect evidence (lab data), line them up (the matrix [H]), and interrogate them using regression math. In the end, the coefficients tell you who really affects the strength — and by how much!

3.6 Model Validation and Goodness of Fit

Ibearugbulem’s approach uses three main statistical tools.

3.6.1 Coefficient of Determination (R²)

$$ R^2 = 1 - \frac{SSR}{TSS} \tag{3.8} $$

3.6.2 Adjusted R²

$$ R_{adj}^2 = 1 - \frac{(n - 1)}{(n - k - 1)} (1 - R^2) \tag{3.9} $$

3.6.3 Standard Error of Regression (SER)

$$ SER = \sqrt{ \frac{SSR}{n - k - 1} } \tag{3.10} $$

3.6.4 F-Test for Model Significance

$$ F = \frac{MSR}{MSE} = \frac{(RSS / k)}{(SSR / (n - k - 1))} = \frac{RSS (n - k - 1)}{SSR k} \tag{3.11} $$

$$ MSR = \frac{RSS}{k}, \quad MSE = \frac{SSR}{n - k - 1} $$

$$ F_{critical} = \frac{SS_{between}/df_{between}}{SS_{within}/df_{within}} = \frac{SS_{between}}{SS_{within}} \times \frac{df_{within}}{df_{between}} \tag{3.15} $$

$$ df_{between} = k - 1, \quad df_{within} = N - k \tag{3.16–3.17} $$

3.7 Example of Validation

Metric	Symbol	Value
Coefficient of Determination	R²	0.99
Adjusted R²	R²adj	0.961
Standard Error of Regression	SER	2.31
F-statistic	F	33.42
F-critical (87% level)	Fcrit	31.53

Since \( F > F_{crit} \), the regression model is statistically significant.

3.8 Replicability and Teaching Notes

• Excel/MATLAB: matrix and statistical computation
• Manual derivation: small datasets (5–10 points)
• Programming: a = Inverse(Transpose(H) * H) * Transpose(H) * Y

3.9 Advantages of the Ibearugbulem Framework

Feature	Description
Universality	Works for 2–5 predictor systems.
Computational Simplicity	No need for ratio transformation.
Data Efficiency	Reliable with few data points.
Interpretability	Coefficients directly represent sensitivity.
Validation Tools	Integrates R², F, and SER.
Educational Usefulness	Easy to teach and demonstrate.

EXPLANATORY BOX – “Predicting the Future Mix”
Think of Ibearugbulem’s Model like a wise elder who has tasted several pots of soup. After sampling a few, he can tell what will happen if you add more pepper or less oil. That’s what the regression model does — it learns from a few experiments and predicts the outcome of new mixtures without re-cooking them!

3.11 Summary

The Ibearugbulem Regression Framework simplifies regression in material analysis by presenting it in a clear, systematic, and replicable manner. The same logic applies for two to five predictors, providing a unified foundation for material optimisation and mix design.

Test Your Understanding (Chapter 3)

1. State the four main steps in the Ibearugbulem regression computation.
2. Write the general matrix form of the regression model.
3. Explain the meaning of \([CC] = [H]^T [H]\) and \([R] = [H]^T [Y]\).
4. Why is \(R^2\) important in validation?
5. If \(R^2 = 0.85\) and \(R_{adj}^2 = 0.84\), what does this suggest?
6. Challenge: In your own words, describe what the model is doing when it “learns” from data.