code of gillette 1983

notes

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

matrices c, d

• 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

• 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.

matrices a, d, e

• 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

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

refs

• 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

• 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