Self-Modeling Curve Resolution and Constraints Hamid Abdollahi Department of Chemistry, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran

A matrix can be decomposed into the product of two significantly smaller matrices. Factorization: In many chemical studies, the measured or calculated properties of the system can be considered to be the linear sum of the term representing the fundamental effects in that system times appropriate weighing factors. D = A B + R DA B = + R

Rows and Columns Vector Spaces R = USV T = XV T = U Y T =D U SVTVT = X VTVT = U YTYT The rows of X are coordinates of rows of D in row vector space on orthonormal V bases set The rows of Y are coordinates of Columns of D in column vector space on orthonormal U bases set

Two components chromatographic system

Two components kinetic system

Two components multivariate calibration

Three components chromatographic system

Bilinearity R = XV T = C A T Bilinearity is property of a model and its definition for a data will be completely based on obeying the data from a particular model. So, when chemometricians say “a data matrix is bilinear” that means this data obey from a “bilinear model” and then we can understand the meaning of “non-bilinear data”. C = X T A=T -1 V Thus mathematically, every data is bilinear

? Is rank deficient data bilinear?

Bilinearity in Abstract Space Always in vector space of the data, all columns or rows can be produced by linear combinations of bases sets of the corresponding subspace. When the data obeys from Beer’s law, all rows and columns are linear combination of pure spectra or concentration profiles, respectively The Beer law is not fulfill when all rows and columns can not be produced by the same pure spectral or concentration profiles

Bilinearity in Abstract Space A simple first order kinetic system:

Bilinearity in Abstract Space Row (spectral) space:

Bilinearity in Abstract Space Column (concentration) space:

Non-negativity Constraint

Bilinear decomposition and non-negativity constraint Mathematically bilinear decomposition of all data matrices are possible. Generally. In r component systems, r independent vectors in row an r independent vectors in column space of the data should be defined. In most cases, chemically meaningful decomposition should produce r independent vectors which all fulfill non-negative properties of their elements. Accordingly, the question is: How we can find the bases vectors with non-negative elements?

Non-negative meaningful profiles should be in abstract spaces of the data:

Is there any exact definition for non-negative subspace? D = USV T = XV T = U Y T D ≥ 0 U Y T ≥ 0 All elements of data matrix are greater or equal to zero: The coordinates of rows and columns of D matrix in abstract space can be calculated as: Rows of X matrix are coordinate of rows of D matrix in abstract space. Rows of Y matrix are coordinate of columns of D matrix in abstract space. X V T ≥ 0 Row vectors are non-negative Column vectors are non-negative

Definition of non-negativity in abstract space: Row space: Each point in abstract space defines a spectral profile z T = [z 1 z 2 z 3 … z r ] s = z 1 v 1 + z 2 v 2 + z 3 v 3 + … z r v r z points should produce the all elements of s non- negative For each element of s, an inequality can be defined: z 11 v 11 + z 21 v 21 + z 31 v 31 + … z r1 v r1 ≥ 0 z 12 v 12 + z 22 v 22 + z 32 v 32 + … z r2 v r2 ≥ 0 z 1p v 1p + z 2p v 2p + z 3p v 3p + … z rp v rp ≥ 0 … ……

Definition of non-negativity in abstract space: z 11 v 11 + z 21 v 21 ≥ 0 z 12 v 12 + z 22 v 22 ≥ 0 z 1p v 1p + z 2p v 2p ≥ 0 … … Two components system in spectral space: z 21 ≥ (-v 11 / v 21 ) z 11 … … z 22 ≥ (-v 12 / v 22 ) z 12 z 2p ≥ (-v 1p / v 2p ) z 1p z2z2 z2z2 0 ith half-plane ith border line z 2 ≥ (-v 1i / v 2i ) z 1

Inequality borders corresponding to each element of spectral profile:

Inequality borders corresponding to each element of concentration profile:

Microscopic structure of the data: * Negative Region * Data Region * Feasible Solution Regions

Microscopic structure of the data: Normalization has not any effect on areas of different regions Normalizing to unit length in abstract space, transfer the object point on an arc of the circle with radius equal to one

Microscopic structure of the data: Normalizing to first eigenvector in abstract space, transfer the object point on a line with the first coordinates of all points equal to one

Microscopic structure of the data: Lawton-Silvester Plot is a very simple image from the data including all information which can be obtained under non-negativity constraint of bilinear decomposition

Concentration and Spectral bands: Some parts of LSP can be transferred to meaningful spectral and concentration feasible bands. Certainly the “true” solutions are here.

One of the feasible solutions of two components system under just non-negativity constraint always exist in the data. ? Discussion

? Simulate a chemical data matrix and use LSP-m file for visualizing the “data based uniqueness”

Definition of non-negativity in abstract space: z 11 v 11 + z 21 v 21 + z 31 v 31 ≥ 0 … … Three components system in spectral space: z 12 v 12 + z 22 v 22 + z 32 v 32 ≥ 0 z 1p v 1p + z 2p v 2p + z 3p v 3p ≥ 0 … z 31 ≥ –(v 11 /v 31 ) z 11 - (v 21 / v 31 ) z 21 … … … z 32 ≥ –(v 12 /v 32 ) z 12 - (v 22 / v 32 ) z 22 z 3p ≥ –(v 1p /v 3p ) z 1p - (v 2p / v 3p ) z 2p

Definition of non-negativity in abstract space: Three components system in spectral space: A plane defines the border of non-negative subspace for ith element of spectral profile z 3 ≥ –(v 1i /v 3i ) z 1 - (v 2i / v 3i ) z 2 z2z2 z1z1 z3z3 ith border plabe Generally an inequality define the non-negative half- volume in there dimensional space :

Definition of non-negativity in abstract space: Non-negativity boundaries * * * * * * * * * * * * * * * * A non-negative feasible solution

Schematic presentation of microstructure of the data: Non-negativity boundaries * * * * * * * * * * * * * * * * A non-negative feasible solution

Microstructure of 3 components chromatographic data:

Inner polyhedron Outer polyhedron Row Space

Microstructure of 3 components chromatographic data: Inner polyhedron Outer polyhedron Column Space

3D visualization of microstructure of data

Microstructure of 3 components data *Negative region *Data region *Feasible solution regions *Positive non feasible regions

Implementation of non-negativity constraint ? Discussion

Closure Constraint

c 1 + c 2 = const. D = C A d 1 = c 11 a 1 + c 12 a 2 = const. a 2 - c 11 (a 1 - a 2 ) d 2 = c 21 a 1 + c 22 a 2 = const. a 2 - c 21 (a 1 - a 2 ) d i = c i1 a 1 + c i2 a 2 = const. a 2 - c i1 (a 1 - a 2 ) Closed Two Component Systems:

d i = c i1 a 1 + c i2 a 2 = const. a 2 - c i1 (a 1 - a 2 ) (a 1 - a 2 ) a1a1 a2a2 const. a 1 c 11 (a 1 - a 2 ) d1d1 c 21 (a 1 - a 2 ) d2d2 c i1 (a 1 - a 2 ) didi Spectral Subspace of Closed Systems: D = C A In a closed two component system, the positions of all row spectra are on a line which with the origin produce two dimensional space

a1a1 c1 a1c1 a1 a2a2 (1- c 1 ) a 2 c 1 a 1 + (1- c 1 ) a 2 c2 a1c2 a1 (1- c 2 ) a 2 c 2 a 1 + (1- c 2 ) a 2 ci a1ci a1 (1- c i ) a 2 c i a 1 + (1- c i ) a 2 Convex Linear Combination and Closure: In a closed system subspace of data rows is convex

Spectral subspace reduction in a closed system is general for any component system ? Discussion

Why column mean centering in a closed system reduce the rank of the data? ?

Score plot of a closed system:

Microscopic structure of a closed system:

Closure Normalization or Constraint? Feasible solutions with limited intensities to data line are corresponding to closed concentration profiles. Closure normalization line

Implementation of closure constraint ? Discussion