Chemometrics — IUPAC Division I / V analytical chemistry at the multivariate-statistics grain. Foundations: principal-components analysis (PCA; Pearson 1901, Hotelling 1933) decomposes a centered data matrix X ∈ ℝ^{n×p} via its…
chemometrics
PCA via SVD: centred X = UΣVᵀ, Eckart-Young best rank-k in Frobenius
Principal-components analysis via the singular-value decomposition (Pearson 1901, Hotelling 1933; SVD foundations Eckart-Young 1936, Mirsky…
Beer-Lambert c = A/(εl): univariate calibration / multivariate OLS extension
Beer-Lambert law A = εcl (Beer 1852; Lambert 1760): optical absorbance is linear in analyte concentration c, path-length l, and…
Mahalanobis d² = (x−μ)ᵀΣ⁻¹(x−μ): covariance-respecting distance metric
Mahalanobis distance (Mahalanobis 1936): the quadratic-form distance between a point x and a distribution with mean μ and covariance Σ,…
SVD σ₁ of [[3],[4]] = 5 (Pythagorean-Frobenius witness)
Sympy-exact witness of the SVD/Frobenius identity on a rank-1 matrix. X = [[3], [4]] has XᵀX = [25], so σ₁ = √25 = 5 exactly — the…
OLS β = (XᵀX)⁻¹Xᵀy on X=[[1,1],[1,2],[1,3]], y=[1,2,3] ⇒ β=[0,1]
Sympy-exact witness of the OLS normal-equations solution on a perfectly-linear 3-point calibration. Design matrix X (column 1 = intercept,…
Mahalanobis on Σ=diag(1,4), μ=0, x=(2,2): d² = 5
Sympy-exact witness of the Mahalanobis quadratic-form metric on a diagonal 2×2 covariance. Setup: Σ = diag(1, 4) (feature-2 has 4× the…