Rotational Ambiguity in Hard-Soft Modeling Method

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

• All or some of the concentration profiles can be constrained.

• All or some of the batches can be constrained.

A B C X

C C0 1 2 3 4 5 6 7 8 9 10

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cent

ratio

n (a

.u.)

A

B

C

X

0 1 2 3 4 5 6 7 8 9 100

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ratio

n (a

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A B C XA

B

C

X

CSM CHM

Non-linear model fitting

min(CHM - CSM)CHM = f(k1, k2)

Hard-Soft Modeling Procedure

UT = 1 T12

T21 1 u1 + T12 u2

T21 u1 + u2

=[ u1 u2 ] [ c1 c2 ] = C

v1

v2

a1

a21 - T12 T21

1 -T12

-T21 11

T-1V = = = A

c1 = u1 + T12 u2

c2 = T21 u1 + u2

Hard-Soft Modeling in Two-Components system

=

a1 = (v1 - T21 v2) 1 - T12 T21

1

a2 = (-T12 v1 + v2) 1 - T12 T21

1

GridSearch2_HSM.m

HSM in 2 component system

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

UT = [ u1 u2 u3] = [ c1 c2 c3 ] = C

1 1 1

T21 T22 T23

T31 T32 T33

c1 = u1 + T21 u2 + T31 u3

c2 = u1 + T22 u2 + T32 u3

c3 = u1 + T23 u2 + T33 u3

Concentration Profiles Matrix in Three Components System

Spectral Profiles Matrix in Three Components System

T22T33 - T32T23 T32 - T33 T23 - T22

T23T31 - T21T33 T33 - T31 T21 - T23

T21T32 - T22T31 T31 - T32 T22 - T21

v1

T-1V =det (T)1

v3

v2

a1 = ((T22T33 - T32T23) v1 + (T32 - T33 ) v2 + (T23 - T22) v3)

a2 = ((T23T31 – T21T33) v1 + (T33 - T31 ) v2 + (T21 - T23) v3)

a1 = ((T21T32 – T22T31) v1 + (T31 - T32 ) v2 + (T22 - T21) v3)

det (T)1

det (T)1

det (T)1

Hard-Soft Modeling for a Typical Kinetics System

A BC D

k1

k2

K1 ≠ k1

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?