Ales Gottvald

Institute of Scientific Instruments, Academy of Sciences of the CR,
Kralovopolska 147, CZ-612 64 Brno, Czech Republic

Abstract: Meta-Evolutionary optimization brings some new qualities into stochastic optimization methods: enforced global convergence, reduced sensitivity to initial parameter estimates, detailed uncertainty analysis, etc. This robust methodology may be advantageously used for quantifying signal parameters in Magnetic Resonance Spectroscopy (MRS) under extreme conditions. Inhomogeneity, noise, phase, truncation, apodization and some other spectral artifacts are involved in this approach. An important prior information, namely inhomogeneity functions, are carefully identified using experimental data. Additional prior information, both deterministic and probabilistic, may be imposed as well. By contrast to conventional DFT-based quantifications, this Evolutionary MRS is (much) more immune to various artifacts, while providing detailed accuracy analysis. These features may be very useful for both In Vivo and In Vitro MRS. Conventional DFT-based quantifications may be used as the first guess, which may be substantially improved in many non-ideal regimes.

Recently, we have witnessed an intensive development of stochastic non-linear optimization methods inspired by some biological, thermodynamic or genetic ideas. Typical methods within this class are Evolution Strategies [1, 5], Simulated Annealing [2, 3] and Genetic Algorithms [3, 4]. These methods, though being usually treated as completely independent, also share many similarities [1-5, 14, 15, 17]. Some critical features of these methods may be substantially improved by introducing a concept of Meta-optimization [15]. Following our previous MENDEL's paper [18], a rationale underlying this generalized view on the stochastic optimizations may be outlined as follows:

During a long history of biological evolution, nature "tested" various optimization strategies. Systems ("species") performing some ineffective optimization strategies vanished because of their limited adaptibility to their environment. Thus, if a biological system has been created as a result of a long evolutionary process, this process would be inevitably an optimized optimization strategy in its final stage. The evolution is a meta-optimizing strategy, which adapts not only a system to be optimized, but also the optimization process itself.

Meta-optimizing algorithms should optimize not only some degrees of freedom, but also some auxiliary parameters governing the optimization itself. Having natural examples of evolutionary optimizations available, one can learn not only some optimizing mechanisms, but also their meta-optimization settings (mortality, variability, etc.). It was discovered that the best evolutionary adaptability is reached only within some narrow "evolutionary window". Consequently, a surprising interpretation of some biological facts (e.g., mortality and variability of species) is possible via purely mathematical features of some Meta-Evolutionary algorithms. Note also that conventional Evolution Strategies or Genetic Algorithms mimic actually an old Lamarck's theory of heredity (a direct transfer of "optimizing experience" via genotypes). A modern view by Darwin and Mendel is better simulated by the concept of Meta-Evolutionary optimization. These ideas was discovered independently by several authors (Rechenberg [6], Gottvald [18, 15]). For more discussion and particular Meta-Evolutionary algorithms, the reader is referred to our previous papers [17-19]. Some archetypes of meta-optimizing (meta-genetic) algorithms may also be traced in [23, 24, 25].

Meta-Evolutionary algorithms generate a class of optimization trajectories, which implies some useful features [15, 17, 19]: (i) enforced global convergence; (ii) lower sensitivity to initial configurations; (iii) lower sensitivity to auxiliary optimization parameters; (iv) detailed uncertainty analysis; (v) unified implementation of several global optimization techniques (multistart, relaxation, tabu-search, etc.) [9, 17-19].

In the present paper, a Meta-Evolutionary optimization will be applied to quantifying signal parameters in biomedical Magnetic Resonance Spectroscopy (MRS) [10, 11, 17, 20]. By contrast to conventional Fourier-based quantifications, this Evolutionary MRS is less susceptible to many artifacts. Consequently, it shows much promise in biomedical MRS under various extreme conditions (inhomogeneity, noise, truncations, etc.).

[1] Schwefel, P. H.: "Numerical Optimization of Computer Models". Wiley, Chichester, 1981
[2] Aarts E. - Korst J.: "Simulated Annealing and Boltzmann Machines". Wiley, Chichester, 1989
[3] Scales J. A. - Smith L. M. - Fischer T. L.: "Global Optimization Methods for Multimodal Inversion Problems". J. Comput. Phys. 103 (1992), pp. 258 - 268
[4] Hoffmeister F. - Bck T.: "Genetic Algorithms and Evolution Strategies: Similarities and Diferences". Tech. Report No. SYS-1/92, Systems Analysis Res. Group, Univ. of Dortmund, Feb. 1992
[5] Fogel D. B.: "Evolutionary Computation. Toward a New Philosophy of Machine Intelligence." IEEE Press, New York, 1995
[6] Rechenberg I.: "Evolution Strategy". In: Computational Intelligence: Imitating Life (Zurada J. M., Marks R. J., Robinson Ch. J., Ed's), IEEE Press, New York, 1994
[7] Tarantola A.: "Inverse Problems Theory (Methods for Data Fitting and Moment Parameter Estimation)". Elsevier, Amsterdam, 1987
[8] Sabatier P. C. (ed.): "Inverse Problems: An Interdisciplinary Study". Academic Press, London, 1987
[9] Dixon L. C. W.: "Global Optima Without Convexity". In: Greenberg H. J. (ed.): Design and Implementation of Optimization Software". Sijnoff & Noordhof, Alphen aan den Rijn, 1978
[10] de Beer R. - van Ormondt D.: "Analysis of NMR Data Using Time Domain Fitting Procedures". NMR - Basic Concepts and Progress, Springer-Verlag, Berlin, 1992, pp. 202 - 248
[11] Haselgrove J. C. et al.: "Analysis of In Vivo NMR Spectra". Reviews of Magn. Reson. in Medicine 2, 2, 1987, pp. 167 - 222
[12] Kotyk et al.: "Comparison of Fourier and Bayesian Analysis of NMR Signals". Part I. - J. Magn. Reson. 98, 1992, pp. 483 - 500; Part II. - J. Magn Reson. A 116, 1995, pp. 1 - 9
[13] Johnson G. et al.: "Multiple-Window Spectrum Estimation Applied to in Vivo NMR Spectroscopy". J. Magn. Reson. B 110, 1996, pp. 138 - 149
[14] Gottvald A.: "Comparative Analysis of Optimization Methods for Magnetostatics". IEEE Trans. Magn. 24 (1988), 1, pp. 411 - 414
[15] Gottvald A.: "Inverse Problems, Non-linear Phenomena and Global Optimization Methods". In: "Non-Linear Phenomena in Electromagnetic Fields" (T. Furuhashi - Y. Uchikawa, Eds.), Elsevier, Amsterdam, 1992, pp. 181 - 184
[16] Gottvald A.: "Inverse Problems of ECT: Part 1 - Modelling, Part 2 - Meta-Optimization & Statistical Regularization". In: Proc. of the 2nd Japanese - Czech - Slovak Joint Seminar on Applied Electromagnetics in Materials, Kyoto (Japan), 1994, pp. 15 - 22
[17] Gottvald A.: "Inverse and Optimization Methodologies in MRS: Part 1: Evolution Strategies; Part 2: Meta-Optimization; Part 3: Applications to Bloch and Maxwell Systems; Part 4: Applications to In Vivo Spectra Quantifications". In: Proc. of the 3rd Japanese-Czech-Slovak Joint Seminar on Applied Electromagnetics, Prague, July 5 - 7, 1995, pp. 5 - 20
[18] Gottvald A.: "Meta-Evolutionary Optimization". In: Proc. of the MENDEL'95 Conference, Brno (Czech Rep.), Sept. 26 - 28, 1995
[19] Gottvald A.: "A Survey of Inverse Problems, Meta-Evolutionary Optimization and Bayesian Statistics: ". Int. J. of Applied Electromagnetics and Mechanics, 1996 (submitted)
[20] Gottvald A.: "MRS Beyond Inhomogeneity and Noise Limits". In: Proc. of the Biosignal '96 Conf., Brno (CR), 1996
[21] Malczyk R. - Gottvald A.: "Modelling Inhomogeneity Phenomena in MRS". In: Proc. of the Biosignal '96 Conf., Brno (CR), 1996
[22] Kucharova H. - Gottvald A.: "Identifying Inhomogeneity Functions in MRS". In: Proc. of the Biosignal '96 Conf., Brno, 1996
[23] Mercer R. E.: "Adaptive Search Using a Reproductive Meta-plan". Unpublished master's thesis, Univ. of Alberta, Edmonton, 1977
[24] Grefenstette J. J.: "Optimization of Control Parameters for Genetic Algorithms". IEEE Transactions on Systems, Man, and Cybernetics 16 (1986), 1, pp. 122 - 128
[25] Grefenstette J. J.: "Incorporating Problem Specific Knowledge into Genetic Algorithms". In: Davis L., (ed.), "Genetic Algorithms and Simulated Annealing", Morgan Kaufmann Publishers, Los Altos, CA, 1987, pp. 42 - 60

Back to: