of Chemical Reaction Systems Julien Billeter, Michael Amrhein, - - PowerPoint PPT Presentation

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ALS Scheme using Extent-based Constraints for the Analysis

• f Chemical Reaction Systems

Julien Billeter, Michael Amrhein, Dominique Bonvin

Laboratoire d’Automatique Ecole Polytechnique Fédérale de Lausanne Switzerland

XVI Chemometrics in Analytical Chemistry June 7, 2016, Bracelona

Outline

• Introduction and Motivation
• Typical ALS algorithm
• Use of implicit calibration in ALS
• Use of extents in ALS

– A brief introduction to Extents – Constraints based on Extents – An initialization based on conc. submatrices and local rank information – ALS algorithm with Extents and implicit calibration

• Simulated case study
• Conclusion and Perspectives

2

Introduction and Motivation

Introduction

ALS algorithm leads to a solution (C, E) for the factorization of L-

• dim. spectroscopic data A of S species at K times, so that A = C E.

Motivation

• Working in a d-dim. space with d ≤ S (C  extents X)
• Constraints in X are numerous and stronger than in C
• More constraints in the time direction (on X) means fewer

constraints in the wavelength direction (on E).

Scope of this work

Absorbance data measured under batch and fed-batch conditions

3

ALS algorithm with a posteriori constraints

4

ˆ

i i +

= E C A ( ) ˆ

i ≤

h E Normalize ˆ

i

E ˆ ˆ

i i +

= A C E ( ) ˆ

i ≤

g C 1 ← + i i ˆ C ALS with a posteriori constraints  Estimates at points  and  are not least-squares estimates! Problems of convergence

 

Soft-modeling (PCA, local rank…)

ALS algorithm with constrained optimization

5

min s.t. ( )

i

F i i i

− ≤

E

C E A h E Normalize ˆ

i

E 1 ← + i i ˆ C min s.t. ( )

i

F i i i

− ≤

C

C C A g E ALS constrained

• ptimization

 Estimates at points  and  are least-squares estimates!

 

ALS algorithm with implicit calibration

• Solve the problem of finding C and E as a combined

constrained optimization problem where only C is adjusted and E is estimated by implicit calibration (E = C+A)

• Typical constraints
• g(C): nonnegativity, monotonicity, unimodality, closure
• h(E): nonnegativity
• Constraints and normalization of E are required,

as well as rank C = rank E = S !

6

min s.t. ( ) , ( ) Normalize

F +

= − ≤ ≤

C

C C C A A g h E E E E

Concept of Extents

Homogeneous reaction systems with inlets

• Material balance in terms of numbers of moles N (K × S)
• S numbers of moles N → d = R + p ≤ S extents X
• Reconstruction equation

7

T

( ) ( ) ( ), ( )

in in

t t t = + = r q n n N C n  ( ) ( ), ( ) ( ) ( ), ( )

r R i r in n in p

t t t t = = = = x r q x x x  

T T i r n n in t

= + + N 1 X C n X N

( )

T T

, with ] [ ;

n i t n r in +

  = = − =   T 1 n X T N C X X N

Constraints on Extents based on prior knowledge

• x(0) = 0d (initial conditions of X)
• X ³ 0K×d and N(X) ³ 0K×S (nonnegative)
• Xin monotonically increasing,

xin,j(t) concave (convex) if qin,j monotonically decreasing (increasing)

• Xr monotonically increasing (for irreversible reactions)

xr,i(t) concave (convex) if ri(t) monotonically decreasing (increasing)

• Initial and Terminal equality constraints on N(X) are enforced

n0 = nk(0) and n(x(tend)) = nk(xk(tend)), sub k indicates a known value

• Path equality constraints on X can be enforced

xi(t) = xi,k(t) (e.g. an extent is known a priori to be zero)

8

Convex Concave convex, then concave

Constraints on Extents based on measurements

1. Estimate numerically the 1st and 2nd time derivatives of X, i.e. and 2. Design convex/concave constraints based on the sign of 3. If step 2 failed, design monotonicity constraints based on the sign of

9

X  X  X  X 

X 

Monotonically increasing

Upper limit Lower limit

X 

Convex Concave

Upper limit Lower limit

X

this approach could also be applied to concentration profiles to detect regions where monotonicity and/or unimodality constraints apply. Remark: time time time

Initialization with Concentration

submatrices and local rank information

Assumption: The initial and final concentrations

• f Sa ³ d species are known for any experiment

The (S − Sa) remaining conc. are reconstructed via the extents E is estimated via N = ½(S + S mod 2) experiments N0 and X0 are computed from the estimate of E,

10

c: calibration, a: available species, f: final conditions

   

2

, (

)

v c

N L × A

1 1 ( ) ( ) T , , ( ) ( ) T , , , , ( ) ( ) T , ,

[ ; ] [ ; ] [ ; ˆ ˆ ˆ ˆ ]

a

a f a j j a f a c a c c v c v N N a f a c

+

+ +

      = → = → =       →   → →

T T T

E A A n n n n n X n X E N N N N   ↓

2 ( )

a

N S × 2 ( ) N d × 2 ( ) N S × ( ) K S × ( ) K d ×

   

ALS algorithm

with Extents and implicit calibration

• A = CE  Av := VA = NE , with V the volume
• Solve the constrained optimization where X is adjusted

and E is estimated by implicit calibration (E = N(X)+Av).

• Typical constraints
• f(X): nonnegativity, monotonicity, convexity/concavity, path constraints
• g(N(X)): nonnegativity, initial and final equality constraints
• No constraints on E are required!

11

min ( ) s.t. ( ) ( ) , ( ( ))

v F v +

= − ≤ ≤

X

A E A f g X X X X E N N N

t w

• 0.03

2

• 0.02
• 0.01

1.5 1

d dwA

0.01 0.8 1 0.02 0.6 0.03 0.4 0.5 0.2

w t

2 0.2 1.5 1 0.4

A

0.8 0.6 1 0.6 0.8 0.4 0.5 0.2

Simulated case study Difference absorbance spectra

12

→ → A B C

2 combined experiments:

• Experiment 1 (only A initially present)
• Experiment 2 (only B initially present)

Pretreatment: 1st derivative in the wavelength direction Noise: 1% uniformly distributed

Simulated case study Constraints applied

13

• Regular ALS does not work as E cannot be constrained positively
• ALS based on X with implicit calibration resolves both the

rotational and intensity ambiguities with the following constraints:

• Initialization X0 from conc. submatrices and local rank information
• Constraints on Experiment 1
• Initial and terminal n’s imposed
• x1 and x2 monotonically increasing
• x1 concave, x2 convex then concave

Remarks: No constraint or normalization on E is required! Constraints X0 ≥ 0, N(X) ≥ 0 are not even necessary!

• Constraints on Experiment 2
• Initial and terminal n’s imposed
• x1(t) = 0, ∀t (path constraint)
• x2 concave

Simulated case study

ALS based on X with implicit calibration

14

t w R

• 1

2

• 0.5

1.5 1 ×10-5 0.5 0.8 1 1 0.6 1.5 0.4 0.5 0.2

Residuals (ssq 1.6×10-7)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

t

1 2 3 4 5 6 7 8 9 10

N

×10-3 N(ssq 7.5×10-8) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

t

1 2 3 4 5 6 7 8 9 10

X

×10-3 X (ssq 4.6×10-4)

A B C x1 x2

— true

• ALS-estimated

Exp 1 Exp 2 Exp 1 Exp 2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

w

• 0.04
• 0.03
• 0.02
• 0.01

0.01 0.02 0.03 0.04

E

E (ssq 1.4×10-5)

A B C

Conclusion

ALS with extents and implicit calibration

• Optimization in a reduced space
• S⋅K decision variables in C versus d⋅K in X, with d ≤ S
• Better handling of the constraints
• Simpler constraints formulation
• Large number of constraints based on prior knowledge
• Stronger constraints (concavity/convexity vs unimodality)
• No constraints on E
• Use of data pre-treatment along wavelength direction

(e.g. 1st derivative correction…)

15

Perspectives

ALS with extents and implicit calibration

• Analysis of rank-deficient data
• Subtraction of the initial and inlet contributions

A  H = Xr (NE)  ALS on Xr and (NE) with rank R < S

• Use of hard constraints in terms of extents
• Each extent of reaction represents the effect of a single

reaction independently of all the others. The use of hard constraints in terms of extents should allow a constant diagnosis of each postulated kinetic step.

16

Final word

17

References

• Bhatt N., Amrhein M., Bonvin D., Incremental identification of reaction and mass-

transfer kinetics using the concept of extents, Ind. Eng. Chem. Res. 50 (2011) 12960

• Billeter J., Srinivasan S., Bonvin D., Extent-based kinetic identification using

spectroscopic measurements and multivariate calibration, Anal. Chim. Acta. 767 (2013) 21

• Rodrigues D., Srinivasan S., Billeter J., Bonvin D., Variant and invariant states

for chemical reaction systems, Comp. Chem. Eng. 73 (2015) 23

• S. Srinivasan, D. Kumar, J. Billeter, S. Narasimhan, D. Bonvin, DYCOPS (2016)

Thank you for your attention