1. Rotational Ambiguity in Hard-Soft Modeling Method

2. 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

3. 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.

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

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

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

17. 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

18. Spectral Profiles Matrix in Three Components System
T22T33 - T32T T32 - T T23 - T22 v1
det (T) 1 v2
T-1V =
T23T31 - T21T T33 - T31 T21 - T23 v3
T21T32 - T22T T31 - T 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)

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

20. Visualization the Area of Feasible Solutions
-Concentration Profiles Space

21. Visualization the Area of Feasible Solutions
-Spectral Profiles Space

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

23. Visualization the Area of Feasible Solutions
-Spectral Profiles Space

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