Method for factor analysis of GC/MS data
The method of the present invention provides a fast, robust, and automated multivariate statistical analysis of gas chromatography/mass spectroscopy (GC/MS) data sets. The method can involve systematic elimination of undesired, saturated peak masses to yield data that follow a linear, additive model. The cleaned data can then be subjected to a combination of PCA and orthogonal factor rotation followed by refinement with MCR-ALS to yield highly interpretable results.
1. A method for factor analysis of GC/MS data, comprising:
a) measuring a mass-charge (m/z) spectrum of a sample with a gas chromatograph/mass spectrometer (GC/MS), whereby the spectrum is represented an m×n matrix of GC/MS data D, wherein m is a number of mass spectra and n is a number of eluted chromatographic peaks;
b) factoring the data matrix D into the product of an m×p matrix of orthogonal scores vectors T and an n×p matrix of orthonormal loading vectors P using Principal Component Analysis (PCA) to provide a PCA solution D=TP T , wherein p is a pseudorank;
c) rotating the loading matrix P, according to P =PR, to provide a rotated loading matrix P of chromatographic factors, wherein R is an orthogonal rotation matrix; and
d) rotating the scores matrix T, according to T =TR, to provide a rotated scores matrix T of mass spectral factors and a rotated data matrix, according to D= T P T .
2. The method of claim 1 , further comprising the step of selective ion depletion of peaks in the m/z spectrum for one or more dominant species in the sample prior to step a).
3. The method of claim 1 , wherein step b) comprises nonlinear iterative partial least squares, eigenanalysis, or singular value decomposition.
4. The method of claim 1 , wherein the pseudorank p in step b) represents a number of chemically meaningful species in the sample.
5. The method of claim 1 , wherein the rotation matrix R in step b) is determined using a varimax procedure.
6. The method of claim 1 , further comprising scaling the data matrix D prior to step b) and rescaling the rotated data matrix after step d).
7. The method of claim 6 , wherein the data matrix D is scaled according to {tilde over (D)}=HD where the diagonal matrix H is a diagonal matrix whose diagonal elements are d m −1/2 and wherein the m×n data matrix D is oriented as mass spectral domain by chromatographic domain with mean m/z spectrum d m given by
d
_
m
=
1
n
D
1
n
.
8. The method of claim 1 , further comprising refining the PCA solution, according to D= T P T =MC T , to provide an m×p matrix M of refined mass spectral factors and an n×p matrix C of refined chromatographic factors.
9. The method of claim 8 , wherein the refining comprises refining the rotated loading matrix P .
10. The method of claim 9 , wherein the refining comprises constrained least-squares, according to
P
_
←
min
P
|
T
D
-
P
_
T
_
T
F
.
11. The method of claim 9 , further comprising refining the rotated scores matrix T .
12. The method of claim 11 , wherein the refining comprises constrained alternating least squares, wherein given a refined P , a refined T is computed, subject to constraints, according to
T
_
←
min
T
|
P
D
-
P
_
T
_
T
F
.
13. The method of claim 12 , wherein the refining comprises non-negative matrix factorization.