ALS Scheme using Extent-based Constraints for the Analysis § of 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 Normalize ˆ ˆ ( i ≤ ) E h E 0 i ALS ˆ ˆ + = A ˆ ˆ C E + = C E C A with a posteriori i i 0 i i constraints Soft-modeling (PCA, local rank…) ˆ i ← i + 1 ( i ≤ ) g C 0 Estimates at points and are not least-squares estimates! Problems of convergence 4
ALS algorithm with constrained optimization Normalize ˆ E i ALS min − min − ˆ A C E A C E C i i i i F F constrained E C 0 i i s.t. ( ) ≤ s.t. ( ) ≤ h E 0 g C 0 i optimization i i ← i + 1 Estimates at points and are least-squares estimates! 5
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 ) min − A C E F C + s.t. = E C A ( ) ≤ , g C 0 ( ) ≤ h E 0 Normalize E • Typical constraints o g ( C ): nonnegativity, monotonicity, unimodality, closure o h ( E ): nonnegativity • Constraints and normalization of E are required, as well as rank C = rank E = S ! 6
Concept of Extents Homogeneous reaction systems with inlets Material balance in terms of numbers of moles N ( K × S ) • T ( ) t ( ) t ( ), ( ) t = + 0 = n N r C q n n in in 0 S numbers of moles N → d = R + p ≤ S extents X • ( ) T T + N = , = − with = [ ; ] X X X 1 n T T N C r in n t 0 i n ( ) t = ( ), t ( ) 0 = x r x 0 r r R ( ) t = ( ), t ( ) 0 = x q x 0 i n in in p Reconstruction equation • N T T = + + X N X C 1 n r in i n n t 0 7
Constraints on Extents based on prior knowledge convex, x (0) = 0 d (initial conditions of X ) • then concave X ³ 0 K × d and N ( X ) ³ 0 K × S (nonnegative) • Concave Convex X in monotonically increasing , • x in,j ( t ) concave (convex) if q in,j monotonically decreasing (increasing) X r monotonically increasing (for irreversible reactions ) • x r,i ( t ) concave (convex) if r i ( t ) monotonically decreasing (increasing) Initial and Terminal equality constraints on N ( X ) are enforced • n 0 = n k (0) and n ( x ( t end )) = n k ( x k ( t end )), sub k indicates a known value Path equality constraints on X can be enforced • x i ( t ) = x i,k ( t ) (e.g. an extent is known a priori to be zero) 8
Constraints on Extents based on measurements Estimate numerically the 1 st and 2 nd time derivatives of X , i.e. and 1. X X 2. Design convex/concave constraints based on the sign of X 3. If step 2 failed, design monotonicity constraints based on the sign of X X X X Upper limit Upper limit Lower limit Lower limit 0 0 time time time Monotonically Concave Convex increasing Remark : this approach could also be applied to concentration profiles to detect regions where monotonicity and/or unimodality constraints apply. 9
Initialization with Concentration submatrices and local rank information Assumption : The initial and final concentrations of S a ³ d species are known for any experiment The ( S − S a ) remaining conc. are reconstructed via the extents E is estimated via N = ½( S + S mod 2) experiments N 0 and X 0 are computed from the estimate of E , , ( 2 N × L ) ( ) 1 ( ) 1 T [ ; ] A n n v c 0 , a f a , ↓ + ˆ ˆ ˆ ˆ T T T ( ) j ( ) T j N N N + N + = [ ; ] → → → = → = → a n n X E A A E X 0 , a f a , c a , c c c v c , 0 v 0 ( 2 N × S ) ( 2 N × d ) ( 2 N × S ) ( K × S ) ( K × d ) a ( N ) ( N ) T [ ; ] n n 0 , a f a , c : calibration, a : available species, f : final conditions 10
ALS algorithm with Extents and implicit calibration • A = CE A v := VA = N E , with V the volume • Solve the constrained optimization where X is adjusted and E is estimated by implicit calibration ( E = N ( X ) + A v ). N min − ( ) A X E v F X N + s.t. = ( ) E X A v ( ) ≤ , f X 0 N ( ( )) ≤ g X 0 • Typical constraints o f ( X ): nonnegativity, monotonicity, convexity/concavity, path constraints o g ( N ( X )): nonnegativity, initial and final equality constraints • No constraints on E are required! 11
Simulated case study Difference absorbance spectra A → B → C 2 combined experiments: Experiment 1 (only A initially present) • Experiment 2 (only B initially present) • 0.8 0.03 0.02 0.6 0.01 dw A 0.4 0 A d -0.01 0.2 -0.02 0 -0.03 2 2 1.5 1 1.5 1 Pretreatment: 0.8 0.8 1 1 0.6 1 st derivative in the 0.6 t 0.4 0.4 0.5 0.5 t 0.2 0.2 w w 0 0 0 0 wavelength direction Noise: 1% uniformly distributed 12
Simulated case study Constraints applied 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: o Initialization X 0 from conc. submatrices and local rank information o Constraints on Experiment 2 o Constraints on Experiment 1 o Initial and terminal n ’s imposed o Initial and terminal n ’s imposed o x 1 and x 2 monotonically increasing o x 1 ( t ) = 0, ∀ t (path constraint) o x 1 concave, x 2 convex then concave o x 2 concave Remarks: No constraint or normalization on E is required! Constraints X 0 ≥ 0 , N ( X ) ≥ 0 are not even necessary! 13
Simulated case study ALS based on X with implicit calibration × 10 -3 N (ssq 7.5 × 10 -8 ) E (ssq 1.4 × 10 -5 ) 0.04 10 A A 9 0.03 B B 8 0.02 7 C C 0.01 6 5 N E 0 4 -0.01 3 -0.02 Exp 2 Exp 1 2 1 -0.03 0 -0.04 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t w × 10 -3 X (ssq 4.6 × 10 -4 ) R esiduals (ssq 1.6 × 10 -7 ) 10 x 1 × 10 -5 9 x 2 8 1.5 7 1 6 0.5 R X 5 0 4 -0.5 3 -1 2 Exp 1 Exp 2 2 — true 1 1.5 1 0.8 1 0.6 0 • ALS-estimated 0.4 0.5 t 0.2 w 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0 t 14
Conclusion ALS with extents and implicit calibration • Optimization in a reduced space o S ⋅ K decision variables in C versus d ⋅ K in X , with d ≤ S • Better handling of the constraints o Simpler constraints formulation o Large number of constraints based on prior knowledge o Stronger constraints (concavity/convexity vs unimodality) • No constraints on E o Use of data pre-treatment along wavelength direction (e.g. 1 st derivative correction…) 15
Perspectives ALS with extents and implicit calibration • Analysis of rank-deficient data o Subtraction of the initial and inlet contributions A H = X r (NE) ALS on X r and ( NE ) with rank R < S • Use of hard constraints in terms of extents o 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 Thank you for your attention References Bhatt N., Amrhein M., Bonvin D., Incremental identification of reaction and mass- o 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 o spectroscopic measurements and multivariate calibration, Anal. Chim. Acta. 767 ( 2013 ) 21 Rodrigues D., Srinivasan S., Billeter J., Bonvin D., Variant and invariant states o for chemical reaction systems, Comp. Chem. Eng. 73 ( 2015 ) 23 S. Srinivasan, D. Kumar, J. Billeter, S. Narasimhan, D. Bonvin, DYCOPS ( 2016 ) o 17
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