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Specialized

In this section, the following specialized designs are described:

Note

All available designs can be accessed after a simple import statement: 

>>> from pydoe import definitive_screening_design, supersaturated_design

Definitive Screening Design (definitive_screening_design)

A definitive screening design (Jones & Nachtsheim, 2011) is a three-level design for k factors requiring only \(2k + 1\) runs that estimates all main effects independent of two-factor interactions and of each other, estimates all quadratic effects, and keeps two-factor interactions clear of main effects.

>>> definitive_screening_design(k)  # (1)!
  1. k — number of factors. k - 1 must be an odd prime (e.g. k = 4, 6, 8, 12, 14, 18, 20, 24, ...).
>>> definitive_screening_design(4)
array([[ 0.,  0.,  0.,  0.],
       [ 0.,  1.,  1.,  1.],
       [-1.,  0.,  1., -1.],
       [-1., -1.,  0.,  1.],
       [-1.,  1., -1.,  0.],
       [ 0., -1., -1., -1.],
       [ 1.,  0., -1.,  1.],
       [ 1.,  1.,  0., -1.],
       [ 1., -1.,  1.,  0.]])

Note

The design is built from a Paley conference matrix \(C\) of order \(k\) as \(D = [0; C; -C]\), stacking the all-zero center run on top of \(C\) and its fold-over \(-C\).

Supersaturated Design (supersaturated_design)

A supersaturated design has more two-level factors than runs (\(k > n\)). It cannot estimate all main effects simultaneously, but is useful for screening when only a small fraction of factors are expected to be active. supersaturated_design performs a random search to minimize \(E(s^2)\), the average squared off-diagonal element of \(X^T X\).

>>> supersaturated_design(n_factors, n_runs, iterations=1000, seed=None)  # (1)!
  1. n_factors — number of two-level factors \(k\) (must exceed n_runs). n_runs — number of runs \(n\) (≥ 2). iterations — number of random candidates to evaluate. seed — for reproducibility.
>>> supersaturated_design(6, 4, iterations=200, seed=0)
array([[-1.,  1.,  1., -1., -1.,  1.],
       [ 1.,  1., -1.,  1.,  1.,  1.],
       [ 1.,  1.,  1., -1.,  1., -1.],
       [ 1.,  1.,  1.,  1., -1., -1.]])

Note

Smaller \(E(s^2)\) values indicate lower average correlation between factor columns, allowing cleaner estimation of the active effects under effect sparsity.

More Information

For further reading, see:

  • Jones, B., & Nachtsheim, C. J. (2011). A class of three-level designs for definitive screening in the presence of second-order effects. Journal of Quality Technology, 43(1), 1-15.
  • Xiao, L., Lin, D. K. J., & Bai, F. (2012). Constructing definitive screening designs using conference matrices. Journal of Quality Technology, 44(1), 2-8.
  • Lin, D. K. J. (1993). A new class of supersaturated designs. Technometrics, 35(1), 28-31.
  • NIST Handbook Section 5.3.3.4 — Supersaturated Designs