mldoe module

Design classes

This section describes the two main classes of the module:

  • Two-Level Design class (TLD)

  • Mixed-Level Design class (MLD)

In both classes the design is defined by its run size (it must be a power of two) and the numbers of the columns used for its factors.

Each column numbers uniquely defines a generator, by the factors it used. Powers of two are independent factors while other numbers are combinations of independent factors. To known which factor are used in a generator, simply decompose it into powers of 2. For example \(7=3+2+1\) so the generator represented by column 7 is composed of the independent factors 1, 2 and 3, and we can write \(7=123\). All the possible generators formed by \(r\) basic factors can be summarized by the \(r \times 2^{r}-1\) matrix called the reduced design matrix. Below is the reduced design matrix for four basic factors.

\[\begin{split}\begin{array}{l|cccccccc} & \mathbf{1} & \mathbf{2} & 3 & \mathbf{4} & 5 & 6 & 7 & \mathbf{8} \\ \hline a & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ b & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 \\ c & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 \\ d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}\end{split}\]

Here is how we create a four-level factor

\[\begin{array}{rrrrcr} a & b & ab & & A \ \ \hline 0 & 0 & 0 & \rightarrow & 0 \ \ 0 & 1 & 1 & \rightarrow & 1 \ \ 1 & 0 & 1 & \rightarrow & 2 \ \ 1 & 1 & 0 & \rightarrow & 3 \end{array}\]

Matrix functions

src.mldoe.matrix.bmat(r: int, alt_coding: bool = False) → numpy.array

Create the full-interaction matrix (B) for \(r\) basic factors.

The B matrix is a \(2^r\) by \(2^r-1\) matrix with the \(2^r-1\) interactions of the columns representing the \(r\) basic factors.

Parameters

  • r (int) – number of basic factors

  • alt_coding (bool) – use (-1,+1) coding instead of (0,1) coding. Default to False (0,1).

Returns

full-interaction matrix

Return type

numpy.array

src.mldoe.matrix.gmat(r: int, alt_coding: bool = False) → numpy.array

Create the reduced interaction (G) matrix for \(r\) basic factors.

The G matrix is a \(r\) by \(2^r-1\) matrix where each column represents an interaction and each row represents a basic factor. An entry of 1 means that the \(i\)-th basic factor is used in the \(j\)-th interaction.

Parameters

  • r (int) – number of basic factors

  • alt_coding (bool) – use (-1,+1) coding instead of (0,1) coding. Default to False (0,1).

Returns

reduced interaction matrix

Return type

numpy.array

src.mldoe.matrix.rmat(r: int, alt_coding: bool = False) → numpy.array

Create the basic factor matrix (R) for \(r\) basic factors.

The R matrix is a \(2^r\) by \(r\) matrix where each column represents a basic factor and each row represents a run. The columns are ordered such that ith columns is a repetition of \(2^{(r-i)}\) zeros and \(2^{(r-i)}\) ones.

Parameters

  • r (int) – number of basic factors

  • alt_coding (bool) – use (-1,+1) coding instead of (0,1) coding. Default to False (0,1).

Returns

basic factor matrix

Return type

numpy.array

Tools functions

References

2009

Xu, H. (2009). Algorithmic construction of efficient fractional factorial designs with large run sizes. Technometrics, 51(3), 262-277.

1989

Wu, C. F. J. (1989). Construction of 2m4 n designs via a grouping scheme. The Annals of Statistics, 1880-1885.