Listing Unique Fractional Factorial Designs Approach

1. Listing Unique Fractional Factorial Designs – I Abhishek K. Shrivastava September 25th, 2009

2. Outline • Fractional Factorial Designs (FFD) What are experiments & designs?What are FFDs? Why is there a list? Are there many FFDs? Design isomorphismListing designs Listing unique designs – brute force gen • Listing Unique designs • Graphs & designs What are graphs?FFDs as graphs • FFDI & GI Solving GI – canonical labeling (nauty)Implications to generating design catalogs Abhishek K. Shrivastava, TAMU

3. Experiments, Designs & Fractional factorial designs (FFDs)

4. Experiments • Experiments for quantifying effect of causal variables Effect of process parameters on product quality Source: http://www.emeraldinsight.com/fig/0680170207035.png Miller-Urey Experiment Source: http://www.physorg.com • Experiments for testing hypothesis Abhishek K. Shrivastava, TAMU

5. Analyze datay = Xb+e Experimental Designs • Choose variable settings to collect data • Replicate runs • Randomize run order Collect Data Experimental plan Experimental design Make inferences Abhishek K. Shrivastava, TAMU

6. Experimental Designs factors a run Levels of factor I Experimental design Abhishek K. Shrivastava, TAMU

7. Experiments with 5 factors • Suppose each factor has 2 runs Choice of design? • Full factorial, i.e. 25 = 32 runs • Too many runs (2n) • Fractional factorial design (FFD) • Pick some subset of full factorial runs • Many fractional factorial designs exist • 25–2 design with 8 runs • Generated using defining relations D=BC and E=AB (regular FFD) Abhishek K. Shrivastava, TAMU

8. Listing FFDs • Using FFDs • Reduces experimenter’s effort • But at a cost! • Hypothetical example: 25–2 design with D=A, E=AB • Can estimate effect of A+D • Many different FFDs with different statistical capability • How do you choose an FFD?? Abhishek K. Shrivastava, TAMU

9. Design catalogs • Catalog of 16-run regular FFDs (Wu & Hamada, 2000) • Compare statistical properties to choose Issues: • Large size regular FFDs not available? • Other classes of FFDs not available Abhishek K. Shrivastava, TAMU

10. Listing Unique FFDs

11. Unique designs: 7-factor FFD example • 7 factors: • Cutting speed . . . . . . . . • Feed . . . . . . . . . . . . • Depth of cut . . . . . . . . • Hot/cold worked work piece . • Dry/wet environment . . . . • Cutting tool material . . . . . • Cutting geometry . . . . . . A B C D E F G Abhishek K. Shrivastava, TAMU

12. Unique designs: 7-factor FFD example • 7 factors: • Cutting speed . . . . . . . . • Feed . . . . . . . . . . . . • Depth of cut . . . . . . . . • Hot/cold worked work piece . • Dry/wet environment . . . . • Cutting tool material . . . . . • Cutting geometry . . . . . . . A B C D E F G A C B D F E G (a) Defining words: {ABE, ACF, BDG} (b) Defining words: {ABE, ACF, CDG} Abhishek K. Shrivastava, TAMU

13. Unique designs Reordered matrix, exchanged columns B↔C, E↔F, reordered rows in (a) (b) Defining words: {ABE, ACF, CDG} Abhishek K. Shrivastava, TAMU

14. Unique designs • Designs (a) & (b) • are isomorphic under factor relabeling & row reordering • have same statistical properties (a) Defining words: {ABE, ACF, BDG} (b) Defining words: {ABE, ACF, CDG} Abhishek K. Shrivastava, TAMU

15. 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. • FFDI problem. Computational problem of determining if two FFDs are isomorphic. Abhishek K. Shrivastava, TAMU

16. Design catalogs • No two designs should be isomorphic • Non-isomorphic catalogs • Why? • Isomorphic designs are statistically identical • Discarding isomorphs can drastically reduce catalog size • e.g., # 215–10 designs > 5 million, where # unique (i.e, non-isomorphic) designs is only 144! Abhishek K. Shrivastava, TAMU

17. add column/ factor add column/ factor add column/ factor add column/ factor … 24 Full factorial 5-factor FFD 6-factor FFD 7-factor FFD Listing Unique 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

18. 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 Listing Unique FFDs • Consider sequential generation of 16-run designs • Note: reducing # intermediate designs will speed up the algorithm • How to discard isomorphs? Abhishek K. Shrivastava, TAMU

19. Solving FFDI: literature review Two types of tests in literature • Necessary checks • faster • Word length pattern, letter pattern matrix, centered L2 discrepancy, extended word length pattern, moment projection pattern, coset pattern matrix • Necessary & Sufficient checks • slower / computationally expensive • exhaustive relabeling, Hamming distance based, minimal column base, indicator function representation based, eigenvalues of word pattern matrices (conjectured) • Legend: • Regular FFDs only • All FFDs Fastest; 2-level regular FFDs only Abhishek K. Shrivastava, TAMU

20. … … Proposed FFDI solution (in a nutshell) • Graph models for FFDs • Equivalence between FFDI and GI • Solving GI Construct graphs from FFDs Solve graph isomorphism problem FFD class specific Abhishek K. Shrivastava, TAMU

21. Graphs and FFDs • Graphs & Graph isomorphism • 2-level regular FFDs • Multi-level regular FFDs • Non-regular FFDs • 2-level regular split-plot FFDs

22. Some 2-level regular FFD terminology • Defining relations: E=AB, F=AC, G=BD • E=AB E=(A+B) mod 2 • (A+B+E) mod 2 = ABE = I (identity) • Defining words: ABE, ACF, BDG • Other words (by mod-2 sum), e.g., BCEF (= ABE+ACF) • Defining contrast subgroup – all words generated from defining words • S = {I, ABE, ACF, BDG, BCEF, ADEG, ABCDFG, CDEFG} A regular 27–3 design Abhishek K. Shrivastava, TAMU

23. 2-level regular FFD isomorphism (rFFDI) • Two regular FFDs, represented by their defining contrast subgroups S1, S2 are isomorphicto each other iff • one of S1 or S2 can be obtained from the other by some permutation of factor labels and reordering of words. • Example: two 7-factor designs, S1 = {I, ABE, ACF, BDG, BCEF, ADEG, ABCDFG, CDEFG}, S2 = {I, ABE, ACF, CDG, BCEF, ADFG, ABCDEG, BDEFG} S1 = {I, ABE, ACF, BDG, BCEF, ADEG, ABCDFG, CDEFG} B↔CE↔F S1' = {I, ACF, ABE, CDG, CBFE, ADFG, ACBDEG, BDFEG} rewrite S1' = {I, ABE, ACF, CDG, BCEF, ADFG, ABCDEG, BDEFG} S2 Abhishek K. Shrivastava, TAMU

24. 2-level regular FFDs as bipartite graphs Example: n = 7, S = {I, ABE, ACF, BDG, BCEF, ADEG, ABCDFG, CDEFG} • Start with G(V,E) = empty graph (no vertices); V = VaVb • For each factor in d, add a vertex in Va • For each word in S, except I , add a vertex in Vb • For each word in S, except I , add edges between the word’s vertex (in Vb) and the factors’ vertices (in Va) Abhishek K. Shrivastava, TAMU

25. 2-level regular FFD isomorphism problem Bipartite graph isomorphism Bipartite graph isomorphism • [Bipartite graph isomorphism problem] Given two graphs, does there exist a graph isomorphism that preserves vertex partitions. • Is GI-complete • Same computational complexity as GI • FFD to Graph conversion takes O(n|S|)steps Abhishek K. Shrivastava, TAMU

26. Multi-level designs as Multi-graphs • Multi-graph representation of a 35–2 design with defining contrast subgroup {I, ABCD2, A2B2C2D, AB2E2, A2BE, AC2DE, A2CD2E2, BC2DE2, B2CD2E} • Similar representation for mixed level designs Abhishek K. Shrivastava, TAMU

27. Non-regular designs as Vertex-colored graphs • Vertex-colored graph representation A 4-factor, 5-run design *edges colored only for better visualization Abhishek K. Shrivastava, TAMU

28. Non-regular FFD isomorphism problem Vertex colored graph isomorphism Vertex-colored graph isomorphism • [Vertex colored graph isomorphism problem] Given two graphs, does there exist a graph isomorphism that preserves vertex colors. • Is GI-complete • Same computational complexity as GI Abhishek K. Shrivastava, TAMU

29. 2-level regular split-plot FFD (FFSP) • FFDs with restricted randomization of runs • Turning part quality example • Cutting speed (A), depth of cut (B), feed (C) is not to be changed after every run • Two groups of factors • Whole plot factors: difficult to change, e.g., A, B, C in above example • Sub-plot factors: easy to change, e.g., d, e, f and g in above example • Relabeling A ↔ d not permitted anymore Abhishek K. Shrivastava, TAMU

30. Regular FFSPs • Regular fractional factorial designs with restricted randomization • Uniquely represented by defining contrast subgroup • e.g., 2(3–1)+(4–2) design with C=AB, f=de, g=Bd • Defining relations for whole plot factors have no sub-plot factors, e.g., C=AB • Defining relations for sub-plot factors have at least one sub-plot factor A 2(3–1)+(4–2) design matrix Abhishek K. Shrivastava, TAMU

31. FFSP Isomorphism • [Definition V.1] Two FFSP matrices are isomorphicto each other if one can be obtained from the other by • some relabeling of the whole-plot factor labels, sub-plot factor labels, level labels of factors and row labels. • [Proposition V.2] Two FFSPs, represented by their defining contrast subgroups S1, S2 are isomorphicto each other iff • one of S1 or S2 can be obtained from the other by some permutation of whole-plot factor labels and sub-plot factor labels, and reordering of words. Abhishek K. Shrivastava, TAMU

32. FFSPs as vertex-colored graphs • Vertex-colored graphs • Each vertex has color • Graph construction • Similar to regular FFDs • Whole-plot factors, sub-plot factors, words – all have different colors • Other variants: split-split-plot designs, non-regular split-plot designs Abhishek K. Shrivastava, TAMU

33. GI and FFDI • Solving GI: canonical labeling • Implications to listing FFDs efficiently …next week