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…
Savitzky-Golay framework: least-squares polynomial filter as convolution kernel (Savitzky & Golay 1964)
The Savitzky-Golay (1964) digital smoothing filter — the canonical chemometric convolution kernel derived by fitting a low-order polynomial…
Beer-Lambert mixture framework: A = L·Σᵢ εᵢ cᵢ (additive absorbance for independent chromophores)
The multi-component Beer-Lambert law — the additive extension of the single-species A = ε·c·L law to a mixture of non-interacting…
PCA trace framework: Σᵢ λᵢ = tr(C) (sum of eigenvalues equals total variance)
The PCA trace-invariance theorem — a direct linear-algebra corollary underpinning the 'percent-variance-explained' ladder ubiquitous in…
Savitzky-Golay 5-pt quadratic: centre 17/35, wings 12/35, tips -3/35; Σc = 1; mirror-symmetric
Sympy-exact witness of the canonical 5-point quadratic Savitzky-Golay smoothing kernel. Setup: construct c = sp.Matrix([-3, 12, 17, 12,…
Beer-Lambert 2-species: A = L·(ε₁c₁ + ε₂c₂); A(2L)/A(L) = 2; unit ε ⇒ A/L = c₁+c₂
Sympy-exact witness of the multi-species Beer-Lambert additivity and its two canonical regime fingerprints. Setup: A = L·(ε₁·c₁ + ε₂·c₂)…
PCA 2×2 trace: λ₁+λ₂ = a+c = tr(C); λ₁·λ₂ = a·c−b² = det(C)
Sympy-exact witness of the 2×2 PCA trace/determinant anchor — the smallest analytically-solvable case of Σλᵢ = tr(C) that still exhibits…
PLS regression (Wold 1975)
S Wold 1975 PLS-1/PLS-2: latent-variables maximizing covariance Cov(X u, Y v); robust under multicollinearity; standard in spectroscopy +…
PCA (Pearson 1901 / Hotelling 1933)
Pearson 1901 + Hotelling 1933: orthogonal transformation diagonalizes covariance matrix; PC1 = direction of maximum variance; loadings +…
MCR-ALS (Tauler 1995)
R Tauler 1995 multivariate-curve-resolution: D = C S^T + E; alternating-least-squares with non-negativity + closure constraints; resolves…
DoE (Fisher 1935 + Box-Behnken)
R A Fisher 1935 + Box-Wilson 1951 + Box-Behnken 1960: full + fractional + central-composite + Box-Behnken designs; orthogonality +…
Savitzky-Golay filter (1964)
Savitzky-Golay 1964: polynomial-fit smoothing + differentiation in moving window; preserves higher-frequency content vs. moving-average;…
Calibration transfer
Feudale 2002 + Bouveresse 1996: transfer of calibration model across instruments via slope-bias correction or…
Kennard-Stone (1969)
R Kennard-L Stone 1969 maximum-distance training-set selection; modern uniform-coverage + AL active-learning calibration.
SVR (Vapnik 1995)
V Vapnik 1995 support-vector-regression; modern kernel-PLS + Gaussian-process regression spectroscopy ML calibration.
Ridge regression (Hoerl-Kennard 1970)
A Hoerl-R Kennard 1970 ridge-regression β=(X'X+λI)^-1 X'y; modern LASSO + elastic-net + Bayesian-LASSO chemometric variable-selection.
OPLS (Trygg 2002)
J Trygg-S Wold 2002 orthogonal-PLS; modern Y-uncorrelated systematic variation removal in metabolomic / sensor / NIR.
PLS-DA (Barker-Rayens 2003)
M Barker-W Rayens 2003 PLS-discriminant-analysis; modern variable-importance VIP + permutation-tests multivariate classification.
ANN (Rumelhart 1986)
Rumelhart-Hinton-Williams 1986 backprop; modern deep-CNN spectral / chromatographic / image classification chemometrics.