15.3 Computer-aided optimal design generation

Computer-aided optimal design generation revolutionizes experimental planning. By harnessing algorithms and software, researchers can create designs that maximize efficiency and minimize costs. This approach automates the complex process of balancing various factors to achieve the most informative experiments.

These tools employ sophisticated mathematical techniques to optimize design criteria. From exchange algorithms to genetic algorithms, they explore vast design spaces to find the best configurations. Software packages make these powerful methods accessible to researchers across disciplines.

Algorithmic Design Generation

Exchange Algorithms for Optimal Design

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• Exchange algorithms are a class of algorithms used for generating optimal experimental designs
• Involve exchanging points between a candidate set and a design set to improve the design's optimality criterion
• Coordinate-exchange algorithm is a specific type of exchange algorithm
  • Exchanges individual coordinates of design points instead of entire points
  • Can be more efficient than exchanging entire points, especially for high-dimensional designs
• Fedorov algorithm is another well-known exchange algorithm
  • Starts with an initial design and iteratively exchanges points to improve the optimality criterion
  • Continues until no further improvements can be made or a maximum number of iterations is reached
• DETMAX (determinant maximization) algorithm is a variant of the Fedorov algorithm
  • Focuses on maximizing the determinant of the information matrix, which is related to D-optimality
  • Often used when D-optimality is the desired criterion for the experimental design

Optimization Criteria and Algorithms

• Algorithmic design generation often involves optimizing a specific criterion, such as D-optimality or A-optimality
• D-optimality aims to maximize the determinant of the information matrix, which minimizes the generalized variance of the parameter estimates
  • Commonly used criterion due to its desirable statistical properties and computational tractability
• A-optimality focuses on minimizing the average variance of the parameter estimates
  • Can be more computationally challenging than D-optimality but may be preferred in certain situations
• Multiplicative algorithm is an optimization algorithm that can be used for generating optimal designs
  • Updates the design weights multiplicatively based on the directional derivative of the optimality criterion
  • Can be faster than exchange algorithms for some problems but may be more sensitive to the initial design
• Simulated annealing is a probabilistic optimization algorithm inspired by the annealing process in metallurgy
  • Allows for occasional acceptance of worse designs to escape local optima
  • Can be effective for complex design spaces with many local optima but may be slower than other algorithms
• Genetic algorithms are inspired by biological evolution and use operators like selection, crossover, and mutation
  • Maintain a population of designs that evolve over generations based on their fitness (optimality criterion)
  • Can be effective for complex, high-dimensional design spaces but may require careful tuning of parameters

Design Software Packages

• Several software packages are available for generating optimal experimental designs using various algorithms and criteria
• Common packages include:
  • JMP: Offers a user-friendly interface for generating optimal designs and analyzing experimental data
  • SAS ADX: Provides a comprehensive set of tools for adaptive and optimal design of experiments
  • MATLAB: Allows users to implement custom algorithms and criteria using the optimization and statistics toolboxes
  • R packages (AlgDesign, OptimalDesign): Enable researchers to generate and evaluate optimal designs using a variety of algorithms and criteria
• These packages often provide functions for generating designs based on specific models (linear, nonlinear, mixture) and optimality criteria (D, A, I, G)
• Some packages also offer graphical user interfaces (GUIs) for designing experiments and visualizing design properties (power, estimation accuracy)
• Using design software can greatly simplify the process of generating optimal designs, especially for complex models and high-dimensional design spaces