Elbow Point Detection

Three methods for finding where a decreasing curve transitions from steep to shallow.

y = x^{-\alpha}

\alpha

1.50noise0.00

Maximum Curvature

\kappa(x) = \dfrac{|y''|}{\bigl(1+(y')^2\bigr)^{3/2}}

x^* = \arg\max_x\;\kappa(x)

Finds where the curve bends most sharply — a purely geometric criterion.

Computed via central finite differences on sampled points.

elbow atx* ≈ 3.92

Two-Segment Breakpoint

x^* = \arg\min_t\bigl[\mathrm{SSE}_1(t)+\mathrm{SSE}_2(t)\bigr]

Finds where two linear fits best partition the data into fast and slow regimes.

Grid search over candidate split positions; each segment fit by OLS.

elbow atx* ≈ 2.78

Slope Threshold

|y'(x^*)| = \theta \cdot |y'(x_0)|

\theta

0.10

First point where the slope drops below a fraction θ of the initial slope.

Adjust θ to control how much steepness drop defines the elbow.

elbow atx* ≈ 2.65