Specialized

In this section, the following specialized designs are described:

• Definitive Screening Design
• Supersaturated Design

>>> import numpy as np
>>> 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 continuous factors that estimates all main effects independently of two-factor interactions and can detect curvature. The Jones & Nachtsheim (2013) extension adds two-level categorical factors.

>>> definitive_screening_design(
...     k,
...     n_cat=0,
...     *,
...     method="dsd",
...     min_13=False,
...     n_fake_factors=0,
... )

• k is the number of continuous factors. It may be zero when n_cat is positive.
• n_cat is the number of two-level categorical factors.
• method="dsd" de-aliases two-factor interactions involving categorical factors. method="orth" instead produces an orthogonal main-effects plan.
• min_13=True augments a small design with enough temporary factors to provide at least 13 runs.
• n_fake_factors manually adds temporary continuous factors to increase the number of runs. Their columns are not returned.

Continuous-factor columns precede categorical-factor columns. Continuous levels are coded as -1, 0, and 1, while categorical levels use PyDOE's conventional two-level coding, -1 and 1. The function always returns a floating-point NumPy array.

The historical one-argument result and row order are unchanged:

>>> 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.]])

Mixed continuous and categorical designs are generated directly:

>>> design = definitive_screening_design(3, n_cat=2)
>>> design.shape
(14, 5)
>>> np.unique(design[:, :3])
array([-1.,  0.,  1.])
>>> np.unique(design[:, 3:])
array([-1.,  1.])

The orthogonal method adds the center runs needed to make all main-effect columns orthogonal:

>>> orthogonal = definitive_screening_design(3, n_cat=2, method="orth")
>>> orthogonal.shape
(16, 5)
>>> gram = orthogonal.T @ orthogonal
>>> np.allclose(gram - np.diag(np.diag(gram)), 0)
True

Small designs can be enlarged without exposing the temporary columns:

>>> definitive_screening_design(4).shape
(9, 4)
>>> definitive_screening_design(4, min_13=True).shape
(13, 4)
>>> definitive_screening_design(4, n_fake_factors=2).shape
(13, 4)

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.]])

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.
• Jones, B., & Nachtsheim, C. J. (2013). Definitive screening designs with added two-level categorical factors. Journal of Quality Technology, 45(2), 121-129.
• 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