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Listing Unique Fractional Factorial Designs – II. Abhishek K. Shrivastava October 2 nd , 2009. Outline. Fractional Factorial Designs (FFD). RECAP Sequential generation of design catalogs Design isomorphism problem Designs as graphs. Listing Unique designs. Graphs & designs. FFDI & GI.

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abhishek k shrivastava october 2 nd 2009

Listing Unique Fractional Factorial Designs – II

Abhishek K. Shrivastava

October 2nd, 2009

outline
Outline
  • Fractional Factorial Designs (FFD)

RECAP

  • Sequential generation of design catalogs
  • Design isomorphism problem
  • Designs as graphs
  • Listing Unique designs
  • Graphs & designs
  • FFDI & GI

Solving GI – canonical labeling (nauty)Implications to generating design catalogs

Abhishek K. Shrivastava, TAMU

sequential generation of ffds

add column/ factor

add column/ factor

add column/ factor

add column/ factor

24 Full factorial

5-factor FFD

6-factor FFD

7-factor FFD

Sequential generation of FFDs
  • Consider 16-run designs – sequential generation
  • How do you pick these columns?? FFD class
    • Regular FFD: defining relation E=AB, F=AC, G=BD
    • Orthogonal arrays: added column keeps orthogonal array property
  • All possible choices of columns gives the catalog

Abhishek K. Shrivastava, TAMU

sequential generation of ffd catalogs

discard isomorphs

7-factor designs from 6-factor designs

...

24 design

Non-isomorphic 5-factor designs

Non-isomorphic 6-factor designs

Non-isomorphic 7-factor designs

Intermediate step

Sequential generation of FFD catalogs
  • Consider sequential generation of 16-run designs
  • Note: reducing # intermediate designs will speed up the algorithm
  • How to discard isomorphs?

Abhishek K. Shrivastava, TAMU

ffd isomorphism ffdi
FFD Isomorphism (FFDI)
  • Definition. Two FFD matrices are isomorphicto each other if one can be obtained from the other by
    • some relabeling of the factor labels, level labels of factors and row labels.

B↔C

E↔F

Abhishek K. Shrivastava, TAMU

proposed ffdi solution in a nutshell

Proposed FFDI solution (in a nutshell)
  • Graph models (bipartite, vertex-colored) for FFDs (2/multi/mixed-level, regular/non-regular, split-plot)
  • Equivalence between FFDI and GI
  • Solving GI

Construct graphs from FFDs

Solve graph isomorphism problem

Abhishek K. Shrivastava, TAMU

slide7
GI and FFDI
    • Solving GI: canonical labeling
    • Implications to listing FFDs efficiently

Abhishek K. Shrivastava, TAMU

graph isomorphism problem

f(A) = H

f(B) = G

f(C) = F

f(D) = E

A

D

E

H

B

C

F

G

Graph Isomorphism Problem
  • Direct comparison of two graphs – search for an f between G1 & G2
    • Good if many isomorphisms between two graphs
  • Canonical labeling – Compute a signature C() defined s.t. C(G1)=C(G2) iff G1 & G2 isomorphic
    • Good if many graphs to compare

GI Problem. Given two graphs G1, G2does there exist a bijective function f:V(G1) V(G2) that preserves vertex adjacencies?

Abhishek K. Shrivastava, TAMU

canonical labeling in nauty mckay 1981
Canonical labeling in nauty(McKay 1981)

nauty

Input

  • graph G

Output

  • Canonically labeled graph C(G)
  • Automorphisms

Abhishek K. Shrivastava, TAMU

canonical labeling by example

Partitionwith 3 cells

Canonical labeling by example
  • degreed(v,U), vV, UV
    • # edges between v and U

Abhishek K. Shrivastava, TAMU

slide11
No further refinement of partition possible using ‘degree’
  • Exchanging labels between vertices in different cells gives non-isomorphic graphs
    • Try relabeling AE({A,F} is an edge, {E,F} is not!)
    • What about AB (same cell)?
canonical labeling by example1

1 2 3 4 5 6

1

2

3

4

5

6

Canonical labeling by example
  • Using the {C,F} split may result in a different candidate for canonically labeled graph
  • C(G) is one among these alternatives

Canonically labeled graph C(G) with {E,A,B,C,F,D}(or {E,B,A,F,C,D}) {1,2,3,4,5,6}

Abhishek K. Shrivastava, TAMU

finding c g among alternatives

1 2 3 4 5 6

1

2

3

4

5

6

Finding C(G) among alternatives
  • Pick the smallest ‘candidate’ based on some total ordering
    • E.g., a binary number b(G)
  • nauty uses this and some other rules to quickly find the canonical graph

Concatenate columns

[000001, 001010, 010100, 001011, 010101, 100110]

Concatenate rows b(G)=0000010010100101000010110101011001102

Abhishek K. Shrivastava, TAMU

summary of canonical labeling algorithm

Split non-singleton (search tree)

Use degree to refine partitions

0000010010100101000010110101011001102

0000010010100101000010110101011001102

Summary of Canonical labeling algorithm

Use degree to form partitions

Vertex invariant

Many vertex invariants exist

Find automorphisms and C(G) from discrete partitions

sequential generation of ffd catalogs1

discard isomorphs

7-factor designs from 6-factor designs

...

24 design

Non-isomorphic 5-factor designs

Non-isomorphic 6-factor designs

Non-isomorphic 7-factor designs

Intermediate step

Sequential generation of FFD catalogs
  • Sequential generation of 16-run designs
  • We know how to discard isomorphs
  • Note: reducing # intermediate designs will speed up the algorithm

Abhishek K. Shrivastava, TAMU

implications of graph approach to seq gen
Implications of Graph approach to Seq. Gen.
  • Note: reducing # intermediate designs will speed up the algorithm
  • Canonical labeling
    • # expensive computations for comparing m designs = m
  • Using automorphisms to reduce # intermediate designs

Abhishek K. Shrivastava, TAMU

automorphisms intermediate designs
Automorphisms & Intermediate designs
  • Example: n=6, S = {I, ABE, ACF, BCEF}
  • 6-factor 2-level regular fractional factorial design
    • BC, EF is an automorphism

Abhishek K. Shrivastava, TAMU

example contd
Example contd.
  • 7-factor designs from the 6-factor design
    • Add defining words ADG or BDGor CDGorABCG, etc…
    • Consider graphs obtained by using defining words
      • BDG S1 = {I, ABE, ACF, BDG, BCEF, ADEG, ABCDFG, CDEFG}
      • CDGS2 = {I, ABE, ACF, CDG, BCEF, ADFG, ABCDEG, BDEFG}

d1

d2

Abhishek K. Shrivastava, TAMU

example contd1
Example contd.

+

=

BC, EF (isomorphism)

BC, EF (isomorphism)

BC, EF (automorphism)

+

=

Abhishek K. Shrivastava, TAMU

reducing intermediate designs with autom

...

Non-isomorphic 2n–pdesigns

a 2n–p design d

Find automorphisms of d

repeat for each d

Find non-isomorphic defining words for d

Construct 2(n+1) – (p+1) designs by adding defining words to d

Intermediate 2(n+1) – (p+1) designs

discard isomorphs by Graph-based check

Non-isomorphic 2(n+1) – (p+1)designs

...

Reducing Intermediate designs with Autom.

Abhishek K. Shrivastava, TAMU

results computational efficiency
Results: Computational efficiency

We compare three methods for 2-level regular FFDs:

  • EigVal – Lin & Sitter (2008)’s algorithm
  • GBAnoR – Same as EigVal, except using our new isomorphism check
  • GBA – GBAnoR with our design reduction method added

Abhishek K. Shrivastava, TAMU

results cumulative cpu times in secs
Results: Cumulative CPU times (in secs.)
  • GBAnoR vs. EigVal – reduction over 90%in most cases
  • GBA vs. EigVal – reduction over 97%in most cases

Abhishek K. Shrivastava, TAMU

results number of intermediate designs
Results: Number of intermediate designs
  • EigVal & GBAnoR give same # designs (no additional reduction)
  • GBA further reduces intermediate designs by 30–70% in most cases

Abhishek K. Shrivastava, TAMU

results design catalogs
Results: design catalogs
  • Generated new catalogs of 1024 (R ≥ 6), all 2048-run (R ≥ 7) & 4096-run (R ≥ 8) designs

[1967] Draper & Mitchell

[1993] Chen & Wu (64-run)

[2008] Lin & Sitter (512-run)

[2009] Shrivastava & Ding (4096-run)

Abhishek K. Shrivastava, TAMU

results for 2 level regular split plot ffds
Results for 2-level regular split-plot FFDs
  • Catalogs of non-isomorphic minimum aberration FFSPs

New catalogs

*Up to 10 factor 32-run designs appear in Bingham and Sitter (2001)

Abhishek K. Shrivastava, TAMU

summary contributions
Summary & Contributions
  • Generic framework for generating catalogs of non-isomorphic FFDs
    • New, efficient isomorphism check
    • Fast design generation algorithm
    • Extensible to different classes of FFDs by constructing graph representations
  • New catalogs of designs up to 4096 runs, much more than existing in current literature

Abhishek K. Shrivastava, TAMU

a related problem complicated engg designs

1

2

3

4

5

6

7

8

EQ6

Fixtures

EQn

1

2

3

4

5

6

7

8

9

1

1

Buses

EQ2

EQ1

EQ5

Equipments

EQ3

1

1

1

EQ4

1

2

1

Schematic of phone quality testing system

A Related Problem: Complicated Engg. Designs

Colored graph representation of a test configuration

Abhishek K. Shrivastava, TAMU

slide29

Thank you!

Abhishek K. Shrivastava, TAMU