1. Abhishek K. Shrivastava October 2 nd, 2009 Listing Unique Fractional Factorial Designs – II

2. Oct. 2, 2009Abhishek K. Shrivastava, TAMU2 Outline 1.Fractional Factorial Designs (FFD) 2.Listing Unique designs 3.Graphs & designs 4.FFDI & GI Solving GI – canonical labeling (nauty) Implications to generating design catalogs RECAP Sequential generation of design catalogs Design isomorphism problem Designs as graphs

3. Oct. 2, 2009Abhishek K. Shrivastava, TAMU3 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 Sequential generation of FFDs 2 4 Full factorial 5-factor FFD 6-factor FFD 7-factor FFD add column/ factor …

4. Oct. 2, 2009Abhishek K. Shrivastava, TAMU4 Consider sequential generation of 16-run designs Note: reducing # intermediate designs will speed up the algorithm How to discard isomorphs? Sequential generation of FFD catalogs 2 4 design Non-isomorphic 5-factor designs Non-isomorphic 6-factor designs... Non-isomorphic 7-factor designs 7-factor designs from 6-factor designs discard isomorphs Intermediate step

5. Oct. 2, 2009Abhishek K. Shrivastava, TAMU5 FFD Isomorphism (FFDI) Definition. Two FFD matrices are isomorphic to each other if one can be obtained from the other by – some relabeling of the factor labels, level labels of factors and row labels. ABCDEFG 10000111 20001110 30010101 40011100 50100010 60101011 70110000 80111001 91000001 101001000 111010011 121011010 131100100 141101101 151110110 161111111 ABCDEFG 10000111 20001110 30010100 40011101 50100011 60101010 70110000 80111001 91000001 101001000 111010010 121011011 131100101 141101100 151110110 161111111 B ↔ CE ↔ FB ↔ CE ↔ F

6. Oct. 2, 2009Abhishek K. Shrivastava, TAMU6 1. Graph models (bipartite, vertex-colored) for FFDs (2/multi/mixed-level, regular/non-regular, split-plot) 2. Equivalence between FFDI and GI 3. Solving GI Proposed FFDI solution (in a nutshell) … Construct graphs from FFDs … Solve graph isomorphism problem

7. Oct. 2, 20097Abhishek K. Shrivastava, TAMU 4. GI and FFDI I. Solving GI: canonical labeling II. Implications to listing FFDs efficiently 4. GI and FFDI I. Solving GI: canonical labeling II. Implications to listing FFDs efficiently

8. Oct. 2, 20098Abhishek K. Shrivastava, TAMU Graph Isomorphism Problem 1. Direct comparison of two graphs – search for an f between G 1 & G 2 – Good if many isomorphisms between two graphs 2. Canonical labeling – Compute a signature C(  ) defined s.t. C(G 1 )=C(G 2 ) iff G 1 & G 2 isomorphic – Good if many graphs to compare GI Problem. Given two graphs G 1, G 2 does there exist a bijective function f:V(G 1 )  V(G 2 ) that preserves vertex adjacencies? A B D C E F H G f( A ) = H f( B ) = G f( C ) = F f( D ) = E

9. Oct. 2, 20099Abhishek K. Shrivastava, TAMU Canonical labeling in nauty (McKay 1981) nauty Input 1.graph G Output Canonically labeled graph C(G) Automorphisms

10. Oct. 2, 200910Abhishek K. Shrivastava, TAMU Canonical labeling by example degree d(v,U), v  V, U  V – # edges between v and U Partition with 3 cells

11. 11 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)?

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13. Oct. 2, 200913Abhishek K. Shrivastava, TAMU Using the {C,F} split may result in a different candidate for canonically labeled graph C(G) is one among these alternatives 1 2 3 4 5 6 123456123456 Canonical labeling by example 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}

14. Oct. 2, 200914Abhishek K. Shrivastava, TAMU 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 1 2 3 4 5 6 123456123456 Concatenate columns [000001, 001010, 010100, 001011, 010101, 100110] Concatenate rows b(G)=000001001010010100001011010101100110 2

15. 15 Summary of Canonical labeling algorithm Use degree to form partitions Split non-singleton ( search tree ) Use degree to refine partitions Find automorphisms and C(G) from discrete partitions 000001001010010100001011010101100110 2 Vertex invariant Many vertex invariants exist

16. Oct. 2, 200916Abhishek K. Shrivastava, TAMU Sequential generation of 16-run designs We know how to discard isomorphs Note: reducing # intermediate designs will speed up the algorithm Sequential generation of FFD catalogs 2 4 design Non-isomorphic 5-factor designs Non-isomorphic 6-factor designs... Non-isomorphic 7-factor designs 7-factor designs from 6-factor designs discard isomorphs Intermediate step

17. Oct. 2, 200917Abhishek K. Shrivastava, TAMU Implications of Graph approach to Seq. Gen. Note: reducing # intermediate designs will speed up the algorithm 1. Canonical labeling – # expensive computations for comparing m designs = m 2. Using automorphisms to reduce # intermediate designs

18. Oct. 2, 200918Abhishek K. Shrivastava, TAMU 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

19. Oct. 2, 200919Abhishek K. Shrivastava, TAMU Example contd. 7-factor designs from the 6-factor design – Add defining words ADG or BDG or CDG or ABCG, etc… – Consider graphs obtained by using defining words 1. BDG  S 1 = { I, ABE, ACF, BDG, BCEF, ADEG, ABCDFG, CDEFG } 2. CDG  S 2 = { I, ABE, ACF, CDG, BCEF, ADFG, ABCDEG, BDEFG } d1d1 d2d2

20. Oct. 2, 200920Abhishek K. Shrivastava, TAMU Example contd. + + = = B  C, E  F (automorphism) B  C, E  F (isomorphism)

21. Oct. 2, 200921Abhishek K. Shrivastava, TAMU Reducing Intermediate designs with Autom. Non-isomorphic 2 n–p designs... Non-isomorphic 2 (n+1) – (p+1) designs Intermediate 2 (n+1) – (p+1) designs Construct 2 (n+1) – (p+1) designs by adding defining words to d Find non-isomorphic defining words for d Find automorphisms of d a 2 n–p design d discard isomorphs by Graph-based check repeat for each d...

22. Oct. 2, 200922Abhishek K. Shrivastava, TAMU Results: Computational efficiency We compare three methods for 2-level regular FFDs: 1. EigVal – Lin & Sitter (2008)’s algorithm 2.GBAnoR – Same as EigVal, except using our new isomorphism check 3.GBA – GBAnoR with our design reduction method added

23. Oct. 2, 200923Abhishek K. Shrivastava, TAMU Results: Cumulative CPU times (in secs.) GBAnoR vs. EigVal – reduction over 90% in most cases GBA vs. EigVal – reduction over 97% in most cases n–k 256 run (R ≥ 5) n–kn–k 512 run (R ≥ 5) EigVal GBAnoRGBAEigVal GBAnoR GBA 9–10.0460.015010–10.2030.0310.015 10–20.3120.0310.01511–22.1090.1090.046 11–31.9060.0930.03112–320.1550.4990.156 12–46.6090.2180.09313–4126.3412.3280.765 13–517.6710.5150.26514–5750.34411.3275.453 14–631.5300.9530.54615–65,119.4558.59238.484 15–741.4671.4060.92116–7(30 hours)328.462269.687 16–842.8581.7031.18717–8(12 days)1,927.9201,771.250 17–943.0611.8901.265

24. Oct. 2, 200924Abhishek K. Shrivastava, TAMU Results: Number of intermediate designs EigVal & GBAnoR give same # designs (no additional reduction) GBA further reduces intermediate designs by 30–70% in most cases n–kn–k 256 run (R ≥ 5) n–kn–k 512 run (R ≥ 5) EigVal GBAnoRGBAEigVal GBAnoR GBA 9–1162 10–1381 10–2227 6811–2703 166 11–3409 14612–32,063 496 12–4480 20613–44,739 1,497 13–5453 26714–511,077 5,731 14–6205 13715–625,913 18,444 15–751 4216–7*60,54552,917 16–822217–8*132,909128,292

25. Oct. 2, 200925Abhishek K. Shrivastava, TAMU 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)

26. Oct. 2, 200926Abhishek K. Shrivastava, TAMU Results for 2-level regular split-plot FFDs Catalogs of non-isomorphic minimum aberration FFSPs Run-sizeResolution Largest n 1 +n 2 CPU Time (mins) 32320*13.1 6431316.1 12841312.2 25651711.4 51251810.6 102461624.5 h 20487233.4 h 40968247.6 h New catalogs *Up to 10 factor 32-run designs appear in Bingham and Sitter (2001)

27. Oct. 2, 200927Abhishek K. Shrivastava, TAMU 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

28. Oct. 2, 200928Abhishek K. Shrivastava, TAMU A Related Problem: Complicated Engg. Designs Schematic of phone quality testing system EQ 5 1 2 … EQ 1 1 … EQ 2 1 … EQ 3 1 … EQ 4 1 … EQ 6 1 … EQ n 1 … 1234 5678 89 … 1234567 … Fixtures Buses Equipments Colored graph representation of a test configuration

29. Oct. 2, 200929Abhishek K. Shrivastava, TAMU Thank you ! Thank you !