Factorial Designs

In this sectionthe following kinds of factorial designs will be described:

• General Full-Factorial
• 2-Level Full-Factorial
• 2-Level Fractional-Factorial
• Plackett-Burman
• Generalized Subset Design

>>> from pyDOE3 import fullfract, ff2n, fracfact, pbdesign, gsd

General Full-Factorial (fullfact)¶

This kind of design offers full flexibility as to the number of discrete levels for each factor in the design. Its usage is simple:

>>> fullfact(levels) # (1)!
>>> fullfact([2,3])
array([[ 0. 0.],
       [ 1. 0.],
       [ 0. 1.],
       [ 1. 1.],
       [ 0. 2.],
       [ 1. 2.]])

1. levels is array of integers.

As can be seen in the outputthe design matrix has as many columns as items in the input array.

2-Level Full Factorial (ff2n)¶

This function is a convenience wrapper to fullfact that forces all the factors to have two levels eachyou simple tell it how many factors to create a design for.

>>> ff2n(3)
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.]])

2-Level Fractional-Factorial (fracfact)¶

This function requires a little more knowledge of how the confounding will be allowed (this means that some factor effects get muddled with other interaction effectsso it's harder to distinguish between them).

Let's assume that we just can't afford (for whatever reason) the number of runs in a full-factorial design. We can systematically decide on a fraction of the full-factorial by allowing some of the factor main effects to be confounded with other factor interaction effects. This is done by defining an alias structure that definessymbolicallythese interactions. These alias structures are written like \(C = AB\) or \(I = ABC\)or \(AB = CD\)etc. These define how one column is related to the others.

For examplethe alias \(C = AB\) or \(I = ABC\) indicate that there are three factors (\(A\)\(B\)and \(C\)) and that the main effect of factor \(C\) is confounded with the interaction effect of the product \(AB\)and by extension\(A\) is confounded with \(BC\) and \(B\) is confounded with \(AC\). A full- factorial design with these three factors results in a design matrix with 8 runsbut we will assume that we can only afford 4 of those runs. To create this fractional designwe need a matrix with three columnsone for \(A\)\(B\)and \(C\)only now where the levels in the \(C\) column is created by the product of the \(A\) and \(B\) columns.

The input to fracfact is a generator string of symbolic characters (lowercase or uppercasebut not both) separated by spaceslike::

>>> gen = "a b ab"

This design would result in a 3-column matrixwhere the third column is implicitly defined as "c = ab". This means that the factor in the third column is confounded with the interaction of the factors in the first two columns. The design ends up looking like this;

>>> fracfact("a b ab")
array([[-1.-1. 1.],
       [ 1.-1.-1.],
       [-1. 1.-1.],
       [ 1. 1. 1.]])

Fractional factorial designs are usually specified using the notation \(2^{(k-p)}\)where \(k\) is the number of columns and \(p\) is the number of effects that are confounded. In terms of resolution levelhigher is "better". The above design would be considered a \(2^{(3-1)}\) fractional factorial designa 1/2-fraction designor a Resolution III design (since the smallest alias \(I=ABC\) has three terms on the right-hand side). Another common design is a Resolution III\(2^{(7-4)}\) fractional factorial and would be created using the following string generator.

>>> fracfact("a b ab c ac bc abc")
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.-1.-1.-1.-1.],
       [-1.-1. 1. 1.-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 sophisticated generator strings can be created using the "+" and "-" operators. The "-" operator swaps the levels of that column like this:

>>> fracfact("a b -ab")
array([[-1.-1.-1.],
       [ 1.-1. 1.],
       [-1. 1. 1.],
       [ 1. 1.-1.]])

In order to reduce confoundingwe can utilize the fold function:

>>> m = fracfact("a b ab")
>>> fold(m)
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.]])

Applying the fold to all columns in the design breaks the alias chains between every main factor and two-factor interactions. This means that we can then estimate all the main effects clear of any two-factor interactions. Typicallywhen all columns are foldedthis "upgrades" the resolution of the design.

By defaultfold applies the level swapping to all columnsbut we can fold specific columns (first column = 0)if desiredby supplying an array to the keyword columns:

>>> fold(m, columns=[2])
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.]])

Another way to reduce confounding it to scan several (or all) available fractional designs and pick the one that has less confounding. The function fracfact_opt performs just that. For a \(2^{k-p}\) fractional factorial the function scans all generators that create at most \(2^{k-p}\) experimentsand pick the one that has confounding on interactions of order as high as possible:

>>> design, alias_map, alias_cost = fracfact_opt(6, 2)
>>> design
"a b c d bcd acd"
>>> print("\n".join(alias_map))
a = bef = cdf = abcde
b = aef = cde = abcdf
c = adf = bde = abcef
d = acf = bce = abdef
e = abf = bcd = acdef
f = abe = acd = bcdef
af = be = cd = abcdef
ab = ef = acde = bcdf
ac = df = abde = bcef
ad = cf = abce = bdef
ae = bf = abcd = cdef
bc = de = abdf = acef
bd = ce = abcf = adef
abc = ade = bdf = cef
abd = ace = bcf = def
abef = acdf = bcde

You can generate the human-readable alias_map of any design with the function fracfact_aliasing:

>>> print("\n".join(fracfact_aliasing(fracfact("a b ab"))[0]))
a = bc
b = ac
c = ab
abc

2-Level Fractional-Factorial specified by resolution (fracfact_by_res)¶

This function constructs a minimal design at given resolution. It does so by constructing a generator string with a minimal number of base factors and passes it to fracfact. This approach favors convenience over fine-grained control over which factors that are confounded.

To construct a 6-factorresolution III-designfractfact_by_res is used like this;

>>> fracfact_by_res(6, 3)
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.],
       [-1.-1. 1. 1.-1.-1.],
       [ 1.-1. 1.-1. 1.-1.],
       [-1. 1. 1.-1.-1. 1.],
       [ 1. 1. 1. 1. 1. 1.]])

Plackett-Burman (pbdesign)¶

Another way to generate fractional-factorial designs is through the use of Plackett-Burman designs. These designs are unique in that the number of trial conditions (rows) expands by multiples of four (e.g. 4812etc.). The max number of columns allowed before a design increases the number of rows is always one less than the next higher multiple of four.

For exampleI can use up to 3 factors in a design with 4 rows:

>>> pbdesign(3)
array([[-1.-1. 1.],
       [ 1.-1.-1.],
       [-1. 1.-1.],
       [ 1. 1. 1.]])

But if I want to do 4 factorsthe design needs to increase the number of rows up to the next multiple of four (8 in this case):

>>> pbdesign(4)
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.],
       [-1. 1.-1. 1.],
       [ 1. 1. 1. 1.]])

Thusan 8-run Plackett-Burman design can handle up to (8 - 1) = 7 factors.

As a side noteIt just so happens that the Plackett-Burman and \(2^{(7-4)}\) fractional factorial design are identical:

>>> np.all(pbdesign(7)==fracfact("a b ab c ac bc abc"))
True

Generalized Subset Design (gsd)¶

GSD is a generalization of traditional fractional factorial designs to problems where factors can have more than two levels.

In many application problemsfactors can have categorical or quantitative factors on more than two levels. Previous reduced designs have not been able to deal with such types of problems. Full multi-level factorial designs can handle such problems but are however not economical regarding the number of experiments.

The GSD provide balanced designs in multi-level experiments with the number of experiments reduced by a user-specified reduction factor. Complementary reduced designs are also provided analogous to fold-over in traditional fractional factorial designs.

An example with three factors using threefour and six levels respectively reduced with a factor 4:

>>> gsd([3, 4, 6], 4)
array([[000],
       [004],
       [011],
       [015],
       [022],
       [033],
       [101],
       [105],
       [112],
       [123],
       [130],
       [134],
       [202],
       [213],
       [220],
       [224],
       [231],
       [235]])

More Information¶

If the user needs more information about appropriate designsplease consult the following articles on Wikipedia:

• Factorial designs
• Plackett-Burman designs

There is also a wealth of information on the NIST website about the various design matrices that can be created as well as detailed information about designing/setting-up/running experiments in general.