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Find elbow point using second derivative method (internal)

Source: R/elbow.finder.R
elbow.sec.deriv.Rd

Detects the "elbow" in a curve by finding where the second derivative (rate of change of the slope) is maximized. This identifies the point where the curve transitions from steep to flat, commonly used for determining optimal dimensionality in PCA or Laplacian Eigenmaps.

Usage

elbow.sec.deriv(x, smooth = TRUE, df = NULL, sort.order = c("asc", "desc"))

Arguments

x

Numeric vector of values (e.g., eigenvalues, explained variance). Must have at least 4 values for second derivative calculation.

smooth

Logical, whether to apply smoothing spline before computing second derivative. Default TRUE. Only applied if length(x) > 6.

df

Degrees of freedom for smoothing spline. If NULL (default), auto-calculated as min(length(x) * 0.7, 10). Lower values = smoother fit.

sort.order

Character, either "asc" or "desc". Specifies the expected ordering of meaningful values:

  • "asc" - values increase with importance (Laplacian Eigenmaps)

  • "desc" - values decrease with importance (PCA eigenvalues)

The algorithm works on sorted data, so this ensures correct interpretation.

Value

List with three elements:

index

Integer index of elbow point in original x vector

value

Numeric value of x at the elbow point

sec.deriv

Maximum second derivative value (diagnostic)

Details

The algorithm:

  1. Sorts values according to sort.order

  2. Optionally smooths with spline to reduce noise

  3. Normalizes to [0,1] range for scale-invariance

  4. Computes second derivative using diff(..., differences = 2)

  5. Returns index where second derivative is largest

Smoothing is recommended for noisy data but requires at least 7 points. The df parameter controls smoothing strength: lower values = smoother.