Global analysis peak fitting for chemical spectroscopy data

1. A method for analyzing a chemical test sample, the method comprising:

(i) applying a thermal or photo-induced aging process to a chemical test sample, thereby producing an aged chemical test sample, wherein the aged chemical test sample is the polymer comprising nitrile or polytetrafluoroethylene;

(ii) irradiating the aged chemical test sample;

(iii) obtaining an image array of the aged chemical test sample after irradiation;

(iv) employing a computer-implemented method to analyze the image array, wherein the computer-implemented method comprises:

(a) receiving, with a processor, the image array of the aged chemical test sample after irradiation, wherein the array comprises d n number of emission images for each n th energy level, and wherein each image comprises dimensions of a by b, and wherein each of d n , n, a, and b is, independently, an integer of 1 or more;

(b) generating, with a processor, a data matrix D based on the image array, wherein D has a dimension of m by n, and wherein m is a times b;

(c) performing, with a processor, global analysis of D, thereby generating a matrix A comprising linear terms and a matrix S comprising non-linear terms, wherein D=AS T , A has a dimension of m by q, S has a dimension of n by q, and q is an integer of 1 or more that represents a number of factors,

wherein generating the matrix A and the matrix S comprises initially non-linearly solving the non-linear terms in S and then linearly solving separately for the linear terms in A, and

wherein each of q factors is characterized by a mathematical function that describes a spectral feature of a representation of the matrix D, and

wherein q is less than n, and

wherein the one or more factors are selected from the group consisting of a Gaussian function, a Lorentzian function, a pseudo-Voigt function, an asymmetric peak function, a shaped step function, and an offset function; and

(d) converting the matrix A and the matrix S into one or more chemical spectra comprising q number of spectral features that characterizes the aged chemical test sample, thereby identifying one or more chemical bonds, atoms, or charges present in the aged chemical test sample; and

(v) analyzing the one or more factors present in the aged chemical test sample, as compared to a native chemical sample, thereby characterizing an age of the aged chemical test sample.

2. The method of claim 1 , wherein step (c) comprises a non-negativity constraint.

3. The method of claim 1 , wherein step (c) comprises:

(c4) solving for an estimated matrix Ŝ by non-linearly solving D=AST, wherein the estimated matrix Ŝ is an estimate of S;

(c5) solving for an estimated matrix  using the estimated matrix Ŝ as S in the following equation D=AS T , wherein the estimated matrix  is an estimate of A;

(c6) computing a convergence metric; and

(c7) iterating steps (c4) to (c6) until the convergence metric meets a convergence criterion.

4. The method of claim 3 , further comprising:

(c4′) constructing an initial matrix Ŝ initial composed of expected non-linear terms, wherein (c4′) is conducted prior to step (c4).

5. The method of claim 4 , wherein step (c4) comprises solving for an estimated matrix Ŝ by non-linearly solving D=AS T by beginning from the initial matrix Ŝ initial .

6. The method of claim 1 , wherein step (c) comprises:

(c8) constructing an initial matrix Ŝ initial from expected non-linear terms;

(c9) solving for an estimated matrix Ŝ by non-linearly solving D=AS T by beginning from the initial matrix Ŝ initial , wherein the estimated matrix Ŝ is an estimate of S;

(c10) solving for an estimated matrix  using the estimated matrix Ŝ as S in the following equation D=AS T , wherein the estimated matrix  is an estimate of A;

(c11) computing a convergence metric; and

(c12) iterating steps (c9) to (c11) until the convergence metric meets a convergence criterion.

7. The method of claim 1 , wherein step (c) comprises:

(c13) constructing an initial matrix Ŝ initial from expected non-linear terms;

(c14) solving for an estimated matrix  using the initial matrix Ŝ initial as S in the following equation D=AS T , wherein the estimated matrix  is an estimate of A;

(c15) computing a convergence metric;

(c16) if not converged, solving for an estimated matrix Ŝ by non-linearly solving D=AS T , wherein the estimated matrix Ŝ is an estimate of S,

(c17) iterating steps (c14) to (c16) until the convergence metric meets a convergence criterion.

8. The method of claim 1 , wherein step (c) comprises applying a least squares minimization criterion to generate the matrix A and the matrix S.

9. The method of claim 1 , further comprising:

(c′) compressing the data matrix D to provide a score matrix T and a loading matrix P, wherein D=TP T and (c′) is conducted prior to (c).

10. The method of claim 9 , wherein step (c) comprises performing, with a processor, global analysis of compressed D, thereby generating a matrix A comprising linear terms and a matrix S comprising non-linear terms, wherein D=TP T =AS T .

11. The method of claim 1 , wherein one or more factors comprise a plurality of pseudo-Voigt functions.

12. The method of claim 1 , wherein the image array comprises one or more absorption images or spectra of the aged chemical test sample.

13. The method of claim 12 , wherein the absorption images or spectra comprise one or more X-ray absorption spectra, near edge X-ray absorption fine structure spectra, fluorescence spectra, infrared spectra, photon emission spectra, or photon absorption spectra.

14. A method for analyzing a chemical test sample, the method comprising:

(i) applying a thermal or photo-induced aging process to a chemical test sample, thereby producing an aged chemical test sample, wherein the aged chemical test sample is the polymer comprising nitrile or polytetrafluoroethylene;

(ii) irradiating the aged chemical test sample;

(iii) obtaining an image array of the aged chemical test sample after irradiation;

(iv) employing a computer-implemented method to analyze the image array, wherein the computer-implemented method comprises:

(a) receiving, with a processor, the image array of the aged chemical test sample after irradiation, wherein the array comprises d n number of emission images for each n th energy level, and wherein each image comprises dimensions of a by b, and wherein each of d n , n, a, and b is, independently, an integer of 1 or more;

(b) generating, with a processor, a data matrix D based on the image array, wherein D has a dimension of m by n, and wherein m is a times b;

(c) compressing the data matrix D to provide a score matrix T and a loading matrix P, wherein D=TP T ;

(d) performing, with a processor, global analysis of D, thereby generating a matrix A comprising linear terms and a matrix S comprising non-linear terms, wherein D=TP T =AS T , A has a dimension of m by q, S has a dimension of n by q, and q is an integer of 1 or more that represents a number of factors,

wherein generating the matrix A and the matrix S comprises initially non-linearly solving the non-linear terms in S and then linearly solving separately for the linear terms in A, and

wherein each of q factors is characterized by a mathematical function that describes a spectral feature of a representation of the matrix D, and

wherein q is less than n, and

wherein the one or more factors are selected from the group consisting of a Gaussian function, a Lorentzian function, a pseudo-Voigt function, an asymmetric peak function, a shaped step function, and an offset function; and

(e) converting the matrix A and the matrix S into one or more chemical spectra comprising q number of spectral features that characterizes the aged chemical test sample, thereby identifying one or more chemical bonds, atoms, or charges present in the aged chemical test sample; and

(v) analyzing the one or more factors present in the aged chemical test sample, as compared to a native chemical sample, thereby characterizing an age of the aged chemical test sample.

15. The method of claim 14 , wherein step (c) comprises an eigenanalysis of D=TP T .

16. The method of claim 14 , wherein step (c) comprises solving for the loading matrix P by analysis of D=TP T .

17. The method of claim 14 , wherein step (c) comprises:

(c1) solving for the loading matrix P by analysis of D=TP T ; and

(c2) solving for the score matrix T by using the loading matrix P.

18. The method of claim 17 , wherein step (d) comprises:

(d1) performing, with a processor, global analysis of D, thereby generating a first matrix A 1 comprising linear terms and a first matrix S 1 comprising non-linear terms;

(d2) computing a first residual matrix R 1 that characterizes error between D and generated matrix A 1 and matrix S 1 ;

(d3) iterating steps (d1) and (d2) until convergence to provide matrix A and matrix S for step (e).

19. The method of claim 17 , wherein step (d) comprises:

(d4) solving for an estimated matrix Ŝ by non-linearly solving D=AS T , wherein the estimated matrix Ŝ is an estimate of S;

(d5) solving for an estimated matrix  using the estimated matrix Ŝ as S, the matrix P, and the matrix T in the following equation AS T =TP T , wherein the estimated matrix  is an estimate of A;

(d6) computing a convergence metric; and

(d7) iterating steps (d4) to (d6) until the convergence metric meets a convergence criterion.

20. The method of claim 19 , wherein step (d) comprises:

(d12) computing the matrix A by using the estimator matrix à in the following equation A=TÃ, wherein T=DP and (d12) is conducted after (d11).

21. The method of claim 17 , wherein step (d) comprises:

(d8) solving for an estimated matrix Ŝ by non-linearly solving D=AS T , wherein the estimated matrix Ŝ is an estimate of S;

(d9) solving for an estimator matrix à using the estimated matrix Ŝ as S and the matrix P in the following equation P T =ÃS T , wherein the estimator matrix à is an estimator of A such that A=TÃ;

(d10) computing a convergence metric; and

(d11) iterating steps (d8) to (d10) until the convergence metric meets a convergence criterion.

22. The method of claim 17 , wherein step (d) comprises applying a least squares minimization criterion to generate the matrix A and the matrix S.

23. The method of claim 22 , wherein the absorption images or spectra comprise one or more X-ray absorption spectra, near edge X-ray absorption fine structure spectra, fluorescence spectra, infrared spectra, photon emission spectra, or photon absorption spectra.

24. The method of claim 14 , wherein D is an m×n matrix, T is an m×p matrix, and P T is an p×n matrix; each of m, n, and p is an integer of 1 or more; and p is less than both m and n.

25. The method of claim 14 , wherein step (d) comprises a non-negativity constraint.

26. The method of claim 14 , wherein the image array comprises one or more absorption images or spectra of the aged chemical test sample.