Governing Equations

NIR Spectrum Simulation:
  Abs(λ) = ε_water(λ)·C_water + ε_API(λ)·C_API + noise(λ)

Key NIR Bands:
  Water O-H stretch: 1450 nm, 1940 nm
  API C-H stretch: 1680 nm (aromatic)
  Excipient: 1200 nm (cellulose)

PLS Regression (Partial Least Squares):
  Y = X·B + E
  B = (X'X)⁻¹X'Y (simplified)

Hotelling T²:
  T² = (x - x̄)'·S⁻¹·(x - x̄)
  Alert if T² > χ²(α, k)

Q Residual (Squared Prediction Error):
  Q = ||(I - P·P')·x||²
  SPE limit from calibration

Control Chart:
  μ ± 3σ limits (Western Electric rules)
  Cpk = min(USL - μ, μ - LSL) / (3σ)