Rotational Ambiguity in Hard-Soft Modeling Method

Hard-Soft Modeling A approach mixing the qualities of hard-modeling and soft-modeling methods is proposed to analyze chemical data. The soft-modeling method, which obtains the pure concentration profiles and spectra of all absorbing species present in the raw measurements by using typical s constraints, a hard constraint is introduced to force some or all the concentration profiles to fulfill a known chemical model, which is refined at each iterative cycle of the optimization process. This modification of soft method drastically decreases the rotational ambiguity associated with the profiles obtained using exclusively constraints. The optional inclusion of some or all the absorbing species into the known chemical model allows the successful treatment of data matrices whose instrumental response is not exclusively due to the chemical components involved in the known process, an impossible scenario for classical hard-modeling approaches

Hard-Soft Modeling Procedure A B C X C 1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time Concentration (a.u.) A B X CSM CHM Non-linear model fitting min(CHM - CSM) CHM = f(k1, k2) All or some of the concentration profiles can be constrained. All or some of the batches can be constrained.

Hard-Soft Modeling in Two-Components system u1 + T12 u2 T21 u1 + u2 UT = [ u1 u2 ] = = [ c1 c2 ] = C c1 = u1 + T12 u2 c2 = T21 u1 + u2 v1 v2 a1 a2 1 - T12 T21 1 -T12 -T21 1 1 T-1V = = = A a1 = (v1 - T21 v2) 1 - T12 T21 1 a2 = (-T12 v1 + v2) 1 - T12 T21 1

HSM in 2 component system GridSearch2_HSM.m HSM in 2 component system

Shows the effects of introducing the hard modeling constraint on second component concentration profile ?

Concentration Profiles Matrix in Three Components System 1 1 1 UT = [ u1 u2 u3] T21 T22 T23 = [ c1 c2 c3 ] = C T31 T32 T33 c1 = u1 + T21 u2 + T31 u3 c2 = u1 + T22 u2 + T32 u3 c3 = u1 + T23 u2 + T33 u3

Spectral Profiles Matrix in Three Components System T22T33 - T32T23 T32 - T33 T23 - T22 v1 det (T) 1 v2 T-1V = T23T31 - T21T33 T33 - T31 T21 - T23 v3 T21T32 - T22T31 T31 - T32 T22 - T21 det (T) 1 a1 = ((T22T33 - T32T23) v1 + (T32 - T33 ) v2 + (T23 - T22) v3) det (T) 1 a2 = ((T23T31 – T21T33) v1 + (T33 - T31 ) v2 + (T21 - T23) v3) det (T) 1 a1 = ((T21T32 – T22T31) v1 + (T31 - T32 ) v2 + (T22 - T21) v3)

Hard-Soft Modeling for a Typical Kinetics System A B K1 ≠ k1 k2 C D A, B and D are absorbing

Visualization the Area of Feasible Solutions -Concentration Profiles Space

Visualization the Area of Feasible Solutions -Spectral Profiles Space

Introducing the hard constraint on A and B -Concentration Profiles Space

Visualization the Area of Feasible Solutions -Spectral Profiles Space

How we can use the obxerved correspondence between concentration and spectral profiles?