Small orthogonal arrays of strength 3

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Small orthogonal arrays of strength 3

• Eric D. Schoen
• TNO Science and Industry,
• Eindhoven University of Technology

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Definition

• Orthogonal array (OA) of strength t
• Arrangement of N rows and n columns
• Column i has si different symbols
• Each set of t columns has each t-tuple of symbols
  occurring equally often

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Motivation

• Strength 2 main effects aliased with 2-factor
  interactions.
• Strength 3 main effects orthogonal to 2-factor
  interactions 2fi correlated.
• Use strength 3 if main effects and 2fi obstruct
  each others detection in arrays with lesser
  strength.

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Organization

• Computer-algebraic methods for the construction
  of designs of experiments
• Dutch Technology Foundation STW
• Discrete mathematics Arjeh Cohen (PhD advisor)
  and Man Nguyen (PhD student defence of thesis
  September 22, 2005)
• Andries Brouwer
• Applied statistics Eric Schoen

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Construction

• Explicit construction some rule to add columns
  to a basic array of t columns
• Helps to formulate generalizations
• Helps to conduct statistical analysis of the
  results.
• Example see poster of Man Nguyen
• JSPI construction paper with N ? 64

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Enumeration

• Explicit construction does not give all
  non-isomorphic arrays given N, n, and the si
• Enumeration algorithm to generate all ni
  lexicographically minimal arrays.
• No subsequent isomorphism testing needed.
• All OA (N ? 100 t 3) are being enumerated.

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Selection

• Availability of various ni arrays
• Arrays differ in
• Degrees of freedom for 2fi
• Estimation capacity
• Blocking potential
• Enumeration and selection paper (submitted)

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Applications

• Wood construction study OA(48 4x3x24 3)
• 19 ni arrays
• 14 with all 2fi estimable
• 2 with optimum value of det(XX)
• Development of dairy product OA(64 44 x 26 3)
• Unique array
• Blocking orthogonal to main effects
• Application paper (non-scientific in Dutch)

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Open problems

• Classification of all ni arrays obtained
• Finding construction methods for arrays of
  particular interest
• Finding the relation of the 2-level parts with
  Hadamard matrices
• Finding an efficient and general blocking
  algorithm.

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Follow-up project

• High-Throughput screening
• 100 ? N ? 1000
• Large values of n
• Motivation miniature vessels for chemical
  reactions
• We are looking for PhD candidates

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Details available from the author
e.d.schoen_at_tue.nl