code of gillette 1983

notes Link to heading

Skim Gillette (1983) and still do not understand the matrix and spectra connection. Perhaps previous references therein should be read first.

matrices c, d Link to heading

Mixture spectra matrix $$ \mathbf{D} = \left( \begin{array}{ccc} 0.26 & 0.22 & 0.14 \newline 0.20 & 0.40 & 0.80 \newline 1.60 & 1.20 & 0.40 \newline 0.12 & 0.14 & 0.18 \end{array} \right) $$
Covariance / correlation marix $$ \mathbf{C} = \left( \begin{array}{ccc} 2.682 & 2.074 & 0.858 \newline 2.074 & 1.668 & 0.856 \newline 0.858 & 0.856 & 0.852 \end{array} \right) $$
Relation $$ \mathbf{C} = \mathbf{D}^T \mathbf{D}. $$
Code https://onecompiler.com/python/3zrz763xa

import numpy as np

D = np.array(
  [
    [0.26, 0.22, 0.14],
    [0.20, 0.40, 0.80],
    [1.60, 1.20, 0.40],
    [0.12, 0.14, 0.18]
  ]
)
print(D)

DT = D.T
print(DT)

C = DT @ D
print(C)

Result (confirmed)

[[0.26 0.22 0.14]
 [0.2  0.4  0.8 ]
 [1.6  1.2  0.4 ]
 [0.12 0.14 0.18]]
[[0.26 0.2  1.6  0.12]
 [0.22 0.4  1.2  0.14]
 [0.14 0.8  0.4  0.18]]
[[2.682 2.074 0.858]
 [2.074 1.668 0.856]
 [0.858 0.856 0.852]]

matrices c, e, l Link to heading

Using previous $\mathbf{C}$.
Relation $\mathbf{C}\mathbf{E} = \mathbf{E} \mathbf{L}$.
Result $$ \mathbf{E} = \left( \begin{array}{ccc} 0.745 & -0.400 & 0.000 \newline 0.596 & 0.039 & 0.000 \newline 0.300 & 0.916 & 0.000 \end{array} \right) $$ $$ \mathbf{L} = \left( \begin{array}{ccc} 4.688 & 0.000 & 0.000 \newline 0.000 & 0.514 & 0.000 \newline 0.000 & 0.000 & 0.000 \end{array} \right) $$
Code https://onecompiler.com/python/3zrzg7yfj

import numpy as np
from numpy import linalg as la

C = np.array(
    [
      [2.682, 2.074, 0.858],
      [2.074, 1.668, 0.856],
      [0.858, 0.856, 0.852],
      ]
  )

print("C =",)
print(C)
print()

print("CE = EL")
L, E = la.eig(C)

print("E =")
print(E.round(3))
print()

print("L =")
print(np.diag(L).round(3))
print()

print("C @ E =")
print((C @ E).round(3))
print()

print("E @ L =")
print((E @ np.diag(L)).round(3))
print()

Result (not confirmed)

C =
[[2.682 2.074 0.858]
 [2.074 1.668 0.856]
 [0.858 0.856 0.852]]

CE = EL
E =
[[-0.745 -0.535 -0.4  ]
 [-0.596  0.802  0.039]
 [-0.3   -0.267  0.916]]

L =
[[4.688 0.    0.   ]
 [0.    0.    0.   ]
 [0.    0.    0.514]]

C @ E =
[[-3.491  0.    -0.205]
 [-2.796  0.     0.02 ]
 [-1.405  0.     0.471]]

E @ L =
[[-3.491 -0.    -0.205]
 [-2.796  0.     0.02 ]
 [-1.405 -0.     0.471]]

Notice that eigenvalues are the same but not the eigenvectors.
Comparison
E Gillette (1983) NumPy
1 [0.745, 0.596, 0.300] [-0.745, -0.596, -0.300]
2 [−0.400, 0.039, 0.916] [-0.535, 0.802, -0.267]
3 [0.000, 0.000, 0.000] [-0.400, 0.039, 0.916]
Explanation: N/A.

E	Gillette (1983)	NumPy
1	`[0.745, 0.596, 0.300]`	`[-0.745, -0.596, -0.300]`
2	`[−0.400, 0.039, 0.916]`	`[-0.535, 0.802, -0.267]`
3	`[0.000, 0.000, 0.000]`	`[-0.400, 0.039, 0.916]`

matrices a, d, e Link to heading

Use previous $\mathbf{D}$ and $\mathbf{E}$ from Gillette (1983).
Code https://onecompiler.com/python/3zrznvmwg

import numpy as np

D = np.array(
  [
    [0.26, 0.22, 0.14],
    [0.20, 0.40, 0.80],
    [1.60, 1.20, 0.40],
    [0.12, 0.14, 0.18],
    ]
  )

E = np.array(
  [
    [0.745, -0.400],
    [0.596, 0.039],
    [0.300, 0.916],
    ]
  )

A = D @ E

print("D ="); print(D); print()
print("E ="); print(E); print()
print("A ="); print(A.round(3)); print()

Result (confirmed)

D =
[[0.26 0.22 0.14]
 [0.2  0.4  0.8 ]
 [1.6  1.2  0.4 ]
 [0.12 0.14 0.18]]

E =
[[ 0.745 -0.4  ]
 [ 0.596  0.039]
 [ 0.3    0.916]]

A =
[[ 0.367  0.033]
 [ 0.627  0.668]
 [ 2.027 -0.227]
 [ 0.227  0.122]]

to-do Link to heading

30-oct-2023 Matrices c, d, c, e, l, a, d, e, next last page of main ref.

refs Link to heading

Paul C. Gillette, Jerome B. Lando, Jack L. Koenig, “Factor analysis for separation of pure component spectra from mixture spectra”, Analytical Chemistry, vol 55, no 4, pp 630-633, Apr 1983, url https://doi.org/10.1021/ac00255a011. pr5ye

dicussion Link to heading

06-nov-2023 Value less that $10^{-8}$ is assumed to be $0$ is proposed by Aria.
- Results: All are confirmed 7d9d71e
- How to get the mixture spectra matrix $\mathbf{D}$ from the mixture, e.g. Figs. 7 and 8 is still still unknown.
30-oct-2023 A reference, Gillette (1983), is given. pr5ye
26-oct-2023 Google drive links is given for current available data. 762au