Method of multivariate spectral analysis

A method of determining the properties of a sample from measured spectral data collected from the sample;

• comprising:
• a) providing a two-dimensional matrix of measured spectral data A;
• b) generating a weighted spectral data matrix D by weighting matrix A;
• c) generating an initial estimate for a spectral shapes matrix S;
• d) using the initial estimate for S, factoring D into the product of two matrices C and ST, by performing a non-negativity constrained alternating least-squares analysis of D=CST, where C is a concentration intensity matrix;
• e) unweighting the matrix S generated in step d) by applying the inverse of the weighting used in step b);
• f) estimating the spectral background B;
• g) calculating P=S−B and SB=B, and redefine S=[P SB], where P is a matrix of pure spectral components without background;
• h) reweighting S and B, using the same weighting that was used in step b);
• i) solving D=[CP CB]·[P SB]T by performing alternating least squares analysis, subject to CP, CB, P≧0 and SB=B;
• j) repeating steps e) through i) until acceptable convergence is achieved;
• k) unweighting CP and S by applying the inverse of the weighting used in step b);
• l) reformatting the columns of CP as a two-dimensional image;
• and
• m) determining the properties of the sample by inspecting CP and P;
• wherein SB is the background spectral shapes matrix, CP is the concentration intensity matrix without background, and CB is the background concentration intensity matrix.

A method of determining the properties of a sample from measured spectral data collected from the sample;

• comprising
• a) providing, and then unfolding, a three-dimensional spectral data array X to form a two-dimensional matrix of measured spectral data A;
• b) generating a weighted spectral data matrix D by weighting matrix A, using Poisson statistics;
• c) generating an initial estimate for a spectral shapes matrix S by obtaining eigenvectors V of the cross product matrix DTD;
• automatically selecting the proper number of pure components q;
• compressing V to have rank=q;
• and performing a VARIMAX rotation on the reduced matrix V to produce the initial estimate for S;
• d) using the initial estimate for S, factoring D into the product of two matrices C and ST, by performing a non-negativity constrained alternating least-squares analysis of D=CST, where C is a concentration intensity matrix;
• e) unweighting S by applying the inverse of the weighting used in step b);
• f) estimating the spectral background B;
• g) reweighting S and B, using the same weighting that was used in step b);
• h) solving D=CST for C and S by performing an alternating least-squares analysis subject to the constraints C≧0 and S≧B;
• i) repeating steps e) through i) until acceptable convergence is achieved;
• j) unweighting C and S by applying the inverse of the weighting used in step b);
• k) calculating P=S−B, where P is a matrix of pure spectral components without background;
• l) reformatting the columns of C as a two-dimensional image;
• and
• m) determining the properties of the sample by inspecting C and P.

A method of determining the properties of a sample from measured spectral data collected from the sample;

• comprising
• a) providing, and then unfolding, a three-dimensional spectral data array X to form a two-dimensional matrix of measured spectral data A;
• b) generating an initial estimate for a spectral shapes matrix S by obtaining eigenvectors V of the cross product matrix ATA;
• automatically selecting the proper number of pure components q;
• compressing V to have rank=q;
• and performing a VARIMAX rotation on the reduced V to produce the initial estimate for S;
• c) using the initial estimate for S, factoring A into the product of two matrices C and ST, by performing a non-negativity constrained alternating least-squares analysis of A=CST, where C is a concentration intensity matrix;
• d) estimating the spectral background B using a robust least squares statistical method based on a linear combination of shapes based on a physical model;
• e) solving A=CST for C and S by performing an alternating least-squares analysis subject to the constraints C≧0 and S≧B;
• f) repeating steps d) through f) until acceptable convergence is achieved;
• g) subtracting B from S to generate a matrix P of pure spectral components without background;
• h) reformatting the columns of C as a two-dimensional image;
• and determining the properties of the sample by inspecting C and P.

A method of determining the properties of a sample, comprising:

• a) detecting radiation emitted by the sample;
• b) generating measured spectral data by processing the detected radiation;
• c) performing a multivariate spectral analysis of the measured spectral data, comprising:
• i) providing a two-dimensional matrix A containing the measured spectral data;
• ii) generating a weighted spectral data matrix D by weighting matrix A;
• iii) factoring D into the product of two matrices C and ST, by performing a constrained alternating least-squares analysis of D=CST, where C is a concentration intensity matrix and S is a spectral shapes matrix;
• iv) unweighting C and S by applying the inverse of the weighting used in step ii);
• and
• d) determining the properties of the sample by inspecting C and S.