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Ph. D. ThesisPh. D. Thesis 2. Theory  Fundamentals of the Multivariate Data Analysis 2. Theory Fundamentals of the Multivariate Data Analysis 2.8. Too Much Information Deteriorates Calibration2.8. Too Much Information Deteriorates Calibration 2.8.6. Variable Selection by Simulated Annealing2.8.6. Variable Selection by Simulated Annealing
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Ph. D. Thesis
  Abstract
  Table of Contents
  1. Introduction
  2. Theory Fundamentals of the Multivariate Data Analysis
    2.1. Overview of the Multivariate Quantitative Data Analysis
    2.2. Experimental Design
    2.3. Data Preprocessing
    2.4. Data Splitting and Validation
    2.5. Calibration of Linear Relationships
    2.6. Calibration of Nonlinear Relationships
    2.7. Neural Networks Universal Calibration Tools
    2.8. Too Much Information Deteriorates Calibration
      2.8.1. Overfitting, Underfitting and Model Complexity
      2.8.2. Neural Networks and the Complexity Problem
      2.8.3. Brute Force Variable Selection
      2.8.4. Variable Selection by Stepwise Algorithms
      2.8.5. Variable Selection by Genetic Algorithms
      2.8.6. Variable Selection by Simulated Annealing
      2.8.7. Variable Compression by Principal Component Analysis
      2.8.8. Topology Optimization by Pruning Algorithms
      2.8.9. Topology Optimization by Genetic Algorithms
      2.8.10. Topology Optimization by Growing Neural Network Algorithms
    2.9. Measures of Error and Validation
  3. Theory Quantification of the Refrigerants R22 and R134a: Part I
  4. Experiments, Setups and Data Sets
  5. Results Kinetic Measurements
  6. Results Multivariate Calibrations
  7. Results Genetic Algorithm Framework
  8. Results Growing Neural Network Framework
  9. Results All Data Sets
  10. Results Various Aspects of the Frameworks and Measurements
  11. Summary and Outlook
  12. References
  13. Acknowledgements
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2.8.6.   Variable Selection by Simulated Annealing

Simulated Annealing (SA) is a method that simulates the thermodynamic process in which metal is heated to its melting temperature and then slowly cooled to its crystal configuration of lowest energy. The system is in thermal equilibrium when the probability of a certain state is governed by a Boltzmann distribution:

 
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with E as energy, T as temperature and k as Boltzmann's constant. Kirkpatrick et al. [133] applied SA to an optimization problem. During the minimization of a multivariate function, a candidate solution is generated by randomly perturbing the current configuration and the energy (similar to the fitness of GA) is calculated. If the new energy is lower than the current, the displacement is accepted. If the energy is higher, the displacement is accepted with a probability given by the Boltzmann distribution . These uphill steps allow the algorithm to escape from local minima. The probability of accepting an uphill step is a function of the change of the energy and of the temperature, which is gradually lowered during the search process. Due to the similar approach with random steps in the search process, SA has several times been compared with GA for the selection of variables whereby SA achieved comparable or slightly worse results and consequently will not be used in this study [91]-[96].

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