1. Címlap Ernő Keszei Eötvös Loránd University Budapest, HUNGARY http://keszei.chem.elte.hu/ Efficient model-free deconvolution of measured femtosecond kinetic data using a genetic algorithm

2. Outline Genetic algorithms: a ”historical” intro A few words about femtochemical data and convolution A brief summary of deconvolution methods Genetic algorithms: how they work in general Implementation of a genetic algorithm for deconvolution Examples of the performance: on a simulated data set on an experimental data set Conclusions and perspectives

3. no

4. idézet2 And God said, Behold, I have given you every herb bearing seed, which is upon the face of all the earth, and every tree, in which is the fruit of a tree yielding seed; to you it shall be for meat. So God created man in his own image, in the image of God created he him; male and female created he them. And God blessed them, and God said unto them, Be fruitful, and multiply, and replenish the earth, and subdue it: and have dominion over the fish of the sea, and over the fowl of the air, and over every living thing that moveth upon the earth. (Genezis 1.27-1.29, authorized King James version)

5. idézet2 So God created man in his own image, in the image of God created he him; Be fruitful, and multiply, and replenish the earth, (Genezis 1.27-1.29, authorized King James version)

6. genalg C. Darwin: On the Origin of Species, John Murray, London, 1859 J. H. Holland. Adaptation in Natural and Artificial Systems, The University of Michigan Press, Michigan, 1975............................................. 2008...... ????...............................................

7. Femtochemistry Aim: time-resolved data on elementary reactions Time-resolution needed : 10 –11 -10 –14 seconds 10 –15 seconds = 1 femtosecond problem: electronically accessible time resolution not less than 10 –9 s (nanosecond) Ahmed Zewail (1987) first time-resolved results on an elementary reaction (Nobel-prize 1999) femtochemistry 10-10000 fs

8. CPM lézererősítő Nd:YAG lézer Ar - ion lézer detektor D2OD2O minta Kísérleti berendezés CPM laser amplifier pumping laser driving laser detector D2OD2O pump probe reference delay line Femtosecond pump-probe measurement sample 0.3 μm = 1 fs

9. Lézerfotolízis A– B – CA– B – CA + BC ground state excited state higher excited state Potential energy A – BC distance Femtosecond pump-probe measurement

10. határozatlansági reláció Let f (t) and F (  ) be each others Fourier-transforms in time and frequency domain: Let us define their ”widths” as their second moments: N being the 2-norm: If f is differentiable and, then Consequences of the uncertainty relation Visible range: Δ t ~ 100 fs Δ ω ~ 5 nm

11. Matematikai leírás Detected signal can be written as a convolution: Maths of the detected femtosecond signal pump (I g )probe (I m ) time (n is the number of exciting photons) instrument response function

12. Torzítás a kinetikában Distortion of the signal due to convolution time kinetic signal

13. instrument response function kinetic signal time Distortion of the signal due to convolution

14. measured signal kinetic signal time Distortion of the signal due to convolution instrument response function

15.  = Needed: reconstruction of the undistorted object from the image object  spread = image It can be found as the solution of the integral equation i = o  s dt ' or more explicitly o bject s pread i mage Reformulation using image processing terms Problem: there exists an infinite number of solutions

16. Dekonvolúciós eljárások iterative parameter estimation of the convolved model a known model function is needed computationally extensive (convolution at each iteration) estimated parameters are correlated with IRF parameters simple algorithms short computation time examples: Van Cittert iteration inverse filtering complicated algorithms long computation time easily adapted as ”ad hoc ” methods to a given problem Linear methodsNonlinear methods Most widely used: reconvolution Model-free deconvolution methods Methods of deconvolution

17. Fourier-transform of a continuous function: Discrete Fourier-transform: F ourier-transzform áció Fourier-transformation time, t frequency, ω amplitude

18. The undistorted object o can be computed (in principle) by a simple inverse Fourier-transformation: Convolution in frequency space: I (  ) = S (  ) · O (  ) Deconvolution in frequency space: O (  ) = S ()S () I ()I () Inverz szűrés Inverse filtering ”filtering” ”inverse filtering”

19. Deconvolution by inverse filtering amplitude channel Amplitude spectrum of the filtered deconvolved signal In addition to inverse filtering, a smoothing filter is also used to damp high frequencies in order to filter out noise deconvolved

20. amplitude undistorted signal Deconvolution by inverse filtering channel Amplitude spectrum of the filtered deconvolved signal deconvolved In addition to inverse filtering, a smoothing filter is also used to damp high frequencies in order to filter out noise

21. Iterációs módszerek Iteration methods o (i +1) = o (i) (x) + [i(x) – s(x)  o (i) (x)] is a suitable function to ensure convergence If is a constant: linear iterative deconvolution If is a function of x : nonlinear iterative deconvolution is called the relaxation function

22. Bayes: 4. lépés Deconvolution by (Bayesian) iteration step4.4. deconvolved image amplitude channel

23. Bayes: 16. lépés Deconvolution by (Bayesian) iteration step16. deconvolved image amplitude channel

24. Bayes: 128. lépés Deconvolution by (Bayesian) iteration step128. deconvolved image amplitude channel

25. Bayes: 512. lépés Deconvolution by (Bayesian) iteration step512. deconvolved image amplitude channel

26. Bayes: 1883. lépés Deconvolution by (Bayesian) iteration step1883. deconvolved undistorted signal amplitude channel

27. genetikus algoritmusok Genetic algorithms (”eugenics”) create an initial population measure the fitness of each individual select individuals to reproduce (parents) let parents mate (crossover) perform mutation on each offspring select individuals of the new generation repeat production of new generations (evolution) until you find an individual with the expected features result: individual(s) with optimal features production of a new generation

28. Creation of the initial population („genesis”) The initial population should be made via inversion of the above distortion effects convolution makeswiden the signal temporally, diminish its amplitude, shallow its rise and descent, smooth out steplike jumps

29. Creation of the initial population („genesis”) From the experiment, the image i (and the spread s ) is known

30. Creation of the initial population („genesis”) To reconstruct the object o : compress the image temporally, From the experiment, the image i (and the spread s ) is known

31. Creation of the initial population („genesis”) increase its amplitude, To reconstruct the object o : compress the image temporally, From the experiment, the image i (and the spread s ) is known

32. Creation of the initial population („genesis”) increase the steepness of its rise and decay, increase its amplitude, To reconstruct the object o : compress the image temporally, From the experiment, the image i (and the spread s ) is known

33. Creation of the initial population („genesis”) restitute the stepwise jump by ”cutting” the first few data To reconstruct the object o : From the experiment, the image i (and the spread s ) is known increase the steepness of its rise and decay, increase its amplitude, compress the image temporally,

34. Creation of the initial population („genesis”) random factors are used in all the operations for the compression ratio, amplitude increase, steepness increase of the rise and decay location of the initial cut The resulting initial population is made of different ”individuals”:

35. Reproduction of the population (”evolution”) 1. computation of the suitability (fitness) of individuals to be a proper object function: large fitness = small difference between reconvolved individual and image (measured by the sum of squared differences) 2. selection of 2 parents with a probability proportional to their fitness 3. crossover of selected parents results in a would-be offspring (simple average or fitness-weighted average of parents) 4. mutation of the would-be offspring, to get an individual of the new generation 5. after sufficient new individuals, select the new generation (”elitism”: if the most fit parent(s) are also selected) To get another new generation, repetition of 1-5. is performed, until a satisfactory deconvolved will be found. Stopping: MSE error, Durbin-Watson statistics, No. of generations

36. Balancing creation and evolution a carefully generated initial population is usually quite close to a suitable deconvolved – a fairly good estimate of the object To get the right initial population, well-chosen parameters (compression, amplitude increase, steepness enhancement, initial cut) are needed – but random parameter variation is also necessary ! during reproduction of the population, randomness is also important (selection of parents, mutation), but mutation is a key element determining the quality of solution ! - too large mutations lead to noisy deconvolved data set - too small mutations result in a wavy deconvolved data set a „smooth” correction in a larger interval avoids both noisy and wavy behavior (actual implementation: correction by adding a random Gaussian)

37. Applied genetic algorithm in technical terms Data structure: a chromosome is the deconvolved data set (coded genes are floating point numbers - ∞ alleles) Individuals: single-chromosome haploid gene-sequence; no phenotype Fitness: a scaled inverse of the sum of squared differences between the image and the reconvolved individual Parent selection: fitness-proportional probability, roulette-wheel (natural selection, not breeding) Crossover: arithmetic; non-weighted average or fitness-weighted average of 2 parents Mutation: changes neighbouring genes in a given interval by adding a smooth random function Selection of the new generation: one-parent elitism offsprings make the new generation, except for the fittest parent

38. eredmény ek1 Deconvolution of synthetic data

39. eredmény ek1 Deconvolution of synthetic data

40. eredmény ek1 Deconvolution of synthetic data

41. eredmények2 Deconvolution of synthetic data

42. eredmények2 Deconvolution of synthetic data

43. eredmény3 Deconvolution of experimental data fluorescence of adenosine monophosphate in water upconversion detection excited at 267 nm observed at 310 nm Bányász & Gustavsson

44. eredmény4 Deconvolution of experimental data

45. eredmény4 Deconvolution of experimental data

46. Conclusions Genetic algorithms are suitable deconvolution methods They can be well adapted to deconvolve femtochemical data (or transient responses in general) Deconvolved data sets do not contain neither enhanced noise nor extra low-frequency oscillations The entire frequency range of the undistorted signal can be reconstructed The method performs excellently on experimental data There are good perspectives to develop a largely automated version with an easy-to-use Graphical User Interface Moral: 1. it is worth reading even the oldest literature 2. both creation and evolution have their place in science

47. Acknowledgement Ákos Bányász & Thomas Gustavsson CNRS Saclay (experimental data) Péter Pataki, grad. student in mathematics Eötvös Loránd University Budapest (parts of the Matlab code) € € € €............ Hungarian National Research Fund (OTKA) Balaton / TéT bilateral exchange program (France-Hungary) R & D Ulrafast Lasers Kft. (Róbert Szipőcs)

48. vége

49. eredmény3 Smoothing effect – synthetic data

50. eredmény4 Smoothing effect – synthetic data

51. eredmény3 Effect of mutations

52. MSE: 0.06 DW: 0.07 2 generations

53. MSE: 0.001 DW: 1.93 2000 generations