D Affine transformations

This appendix shows that affine transformations preserve the NCA effect size \(d\).

The following definitions are used:

• Let \(\Omega \subseteq \mathbb{R}^2\) be the data in the \(XY\)-plane.

• Let \(S\) be the scope defined by:

  \[ S = [X_{\min}, X_{\max}] \times [Y_{\min}, Y_{\max}] \]

• Let \(C \subseteq S\) be the ceiling zone or empty area (above the ceiling line).

The effect size in NCA is defined as:

\[ d = \frac{\text{Area}(C)}{\text{Area}(S)} \]

Let \(f: \mathbb{R}^2 \to \mathbb{R}^2\) be a general differentiable and invertible transformation:

\[ f(x, y) = (u, v) = (f_1(x), f_2(y)) \]

The Jacobian determinant \(J_f(x, y)\) indicates how an infinitesimal area around point \((x, y)\) is transformed:

\[ J_f(x, y) = \begin{vmatrix} \frac{df_1}{dx} & 0 \\ 0 & \frac{df_2}{dy} \end{vmatrix} = f_1'(x) \cdot f_2'(y) \]

The area of a transformed region \(R\) becomes:

\[ \text{Area}(f(R)) = \iint_R |J_f(x, y)| \, dx\,dy \]

So the transformed effect size becomes:

\[ d' = \frac{\iint_{{C}} |J_f(x, y)| \, dx\,dy}{\iint_{{S}} |J_f(x, y)| \, dx\,dy} \]

For the affine case:

If \(f_1(x) = ax + b\) and \(f_2(y) = cy + e\), then:

\[ f_1'(x) = a, \quad f_2'(y) = c \quad \Rightarrow \quad J_f(x, y) = ac = \text{constant} \]

So: \[ d' = \frac{ac \cdot \text{Area}({C})}{ac \cdot \text{Area}({S})} = d \]

Thus, affine transformations preserves the effect size.