Fig. 1. Simulated 10-peak spectrum (solid line) and its constituent peaks (dotted lines). The peak indicated by marker points is the reference peak used to determine the effect of changes on similarity indices (λ = 4.7, σ = 0.2).

Fig. 2. The probability of H₀ that two vectors are not linearly correlated as a function of the squared correlation coefficient. Degrees of freedom: solid line–infinity, dashed line–10, dash-dot line–2. A horizontal dotted line is drawn at Pr(T ≥ t) = 0.05.

Fig. 3. The distribution of intensities of a Gaussian peak with mean 0 and standard deviation 1, calculated at 600 points between x = − 3 and x = + 3.

Fig. 4. A reference Gaussian peak (solid line, μ = 0, σ = 1), solid line, with peaks for which the correlation coefficient just fails the significance test of Eq. (5) at the 95% level (α = 0.05). Dashed line: μ = 0 and σ = 0.46, circles: with μ = 0.50 and σ = 1.

Fig. 5. Values of similarity indices as a function of changes in the parameters of a single Gaussian peak relative to a reference peak with μ = 0 and σ = 1. ○, Euclidean cosine squared, ×, correlation coefficient squared, ■, first-difference correlation coefficient squared. (A) Change in the position of the peak (μ); (B) change in the width of the peak (σ).

Fig. 6. Values of similarity indices as the parameters of a Gaussian peak with base values λ = 4.7, σ = 0.2 are changed in a 10-peak spectrum. ○, Euclidean cosine squared, ×, correlation coefficient squared, ■, first-difference correlation coefficient squared. (A) Change in the position of the peak (λ); (B) change in the width of the peak (σ).

Fig. 7. Distributions of similarity indices with 10,000 random changes in peak parameters in a 10-peak simulated spectrum. (a) Changing 2 peaks, (b) changing 5 peaks, (c) changing 10 peaks. (A) Squared Euclidean cosine; (B) squared first-difference correlation coefficient.

Fig. 8. 100 simulated and 10 template experimental spectra of Ca₂CO₃.

Fig. 9. Distributions of the similarity indices for pairwise comparisons among the simulated and template spectra of Ca₂CO₃. (A) Squared correlation coefficient, (B) squared first-difference correlation coefficient, (C) squared Euclidean cosine, (D) squared first-difference Euclidean cosine.

Definitions of similarity indices. A₁(A₂) is the vector of intensities or absorbances of spectrum 1(2), with individual element A_1,i,(A_2,i) and mean Ā₁(Ā₂)