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

2. Sep. 25, 2009Abhishek K. Shrivastava, TAMU2 Outline 1.Fractional Factorial Designs (FFD) What are experiments & designs? What are FFDs? Why is there a list? Are there many FFDs? 2.Listing Unique designs Design isomorphism Listing designs Listing unique designs – brute force gen 3.Graphs & designs What are graphs? FFDs as graphs 4.FFDI & GI Solving GI – canonical labeling (nauty) Implications to generating design catalogs

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

4. Sep. 25, 2009Abhishek K. Shrivastava, TAMU4 Experiments 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 quantifying effect of causal variables Experiments for testing hypothesis

5. Sep. 25, 2009Abhishek K. Shrivastava, TAMU5 Experimental Designs 1. Choose variable settings to collect data 2. Replicate runs 3. Randomize run order Collect Data ABC...I........................... Analyze data y = X  +  Make inferences run ABC...I 1 010...0 2 011...1......... 20 101...1 run ABC...I 1 010...0 2 011...1......... 20 101...1 21 010...0 22 011...1......... 40 101...1 run ABC...I 9 010...0 15 110...0......... 10 101...1 5 011...0 38 011...1......... 22 011...1 Experimental planExperimental design

6. Sep. 25, 2009Abhishek K. Shrivastava, TAMU6 Experimental Designs run ABC...I 1 010...0 2 011...1......... 20 101...1 factors a run Levels of factor I Experimental design

7. Sep. 25, 2009Abhishek K. Shrivastava, TAMU7 Experiments with 5 factors Suppose each factor has 2 runs Choice of design? Full factorial, i.e. 2 5 = 32 runs – Too many runs (2 n ) Fractional factorial design (FFD) – Pick some subset of full factorial runs – Many fractional factorial designs exist – 2 5–2 design with 8 runs Generated using defining relations D=BC and E=AB (regular FFD)

8. Sep. 25, 2009Abhishek K. Shrivastava, TAMU8 Listing FFDs Using FFDs – Reduces experimenter’s effort – But at a cost! Hypothetical example: 2 5–2 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??

9. Sep. 25, 2009Abhishek K. Shrivastava, TAMU9 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

10. 2.Listing Unique FFDs

11. Sep. 25, 2009Abhishek K. Shrivastava, TAMU11 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...... ABCDEFGABCDEFG

12. Sep. 25, 2009Abhishek K. Shrivastava, TAMU12 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....... ABCDEFGABCDEFG ACBDFEGACBDFEG ABCDEFG 10000000 20001001 30010010 40011011 50100101 60101100 70110111 80111110 91000110 101001111 111010100 121011101 131100011 141101010 151110001 161111000 ABCDEFG 10000110 20001111 30010101 40011100 50100010 60101011 70110001 80111000 91000000 101001001 111010011 121011010 131100100 141101101 151110111 161111110 (a) Defining words: { ABE, ACF, BDG } (b) Defining words: { ABE, ACF, CDG }

13. Sep. 25, 2009Abhishek K. Shrivastava, TAMU13 Unique designs ABCDEFG 10000110 20001111 30010101 40011100 50100010 60101011 70110001 80111000 91000000 101001001 111010011 121011010 131100100 141101101 151110111 161111110 ACBDFEG 10000000 20001001 50010011 60011010 30100100 40101101 70110111 80111110 91000110 101001111 131010101 141011100 111100010 121101011 151110001 161111000 Reordered matrix, exchanged columns B ↔ C, E ↔ F, reordered rows in (a) (b) Defining words: { ABE, ACF, CDG }

14. Sep. 25, 2009Abhishek K. Shrivastava, TAMU14 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 } ABCDEFG 10000000 20001001 30010010 40011011 50100101 60101100 70110111 80111110 91000110 101001111 111010100 121011101 131100011 141101010 151110001 161111000 ABCDEFG 10000000 20001001 30010011 40011010 50100100 60101101 70110111 80111110 91000110 101001111 111010101 121011100 131100010 141101011 151110001 161111000

15. Sep. 25, 2009Abhishek K. Shrivastava, TAMU15 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. FFDI problem. Computational problem of determining if two FFDs are isomorphic.

16. Sep. 25, 2009Abhishek K. Shrivastava, TAMU16 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., # 2 15–10 designs > 5 million, where # unique (i.e, non-isomorphic) designs is only 144!

17. Sep. 25, 2009Abhishek K. Shrivastava, TAMU17 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 Listing Unique FFDs 2 4 Full factorial 5-factor FFD 6-factor FFD 7-factor FFD add column/ factor …

18. Sep. 25, 2009Abhishek K. Shrivastava, TAMU18 Consider sequential generation of 16-run designs Note: reducing # intermediate designs will speed up the algorithm How to discard isomorphs? Listing Unique FFDs 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

19. Sep. 25, 2009Abhishek K. Shrivastava, TAMU19 Solving FFDI: literature review Two types of tests in literature Necessary checks – faster – Word length pattern, letter pattern matrix, centered L 2 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) Fastest; 2-level regular FFDs only Legend: Regular FFDs only All FFDs

20. Sep. 25, 2009Abhishek K. Shrivastava, TAMU20 1. Graph models for FFDs 2. Equivalence between FFDI and GI 3. Solving GI Proposed FFDI solution (in a nutshell) … Construct graphs from FFDs … Solve graph isomorphism problem FFD class specific

21. 3. Graphs and FFDs I. Graphs & Graph isomorphism Graphs & Graph isomorphism II. 2-level regular FFDs III. Multi-level regular FFDs IV. Non-regular FFDs V. 2-level regular split-plot FFDs 3. Graphs and FFDs I. Graphs & Graph isomorphism Graphs & Graph isomorphism II. 2-level regular FFDs III. Multi-level regular FFDs IV. Non-regular FFDs V. 2-level regular split-plot FFDs

22. Sep. 25, 2009Abhishek K. Shrivastava, TAMU22 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 2 7–3 design

23. Sep. 25, 2009Abhishek K. Shrivastava, TAMU23 2-level regular FFD isomorphism (rFFDI) Two regular FFDs, represented by their defining contrast subgroups S 1, S 2 are isomorphic to each other iff – one of S 1 or S 2 can be obtained from the other by some permutation of factor labels and reordering of words. Example: two 7-factor designs, S 1 = { I, ABE, ACF, BDG, BCEF, ADEG, ABCDFG, CDEFG }, S 2 = { I, ABE, ACF, CDG, BCEF, ADFG, ABCDEG, BDEFG } S 1 = { I, ABE, ACF, BDG, BCEF, ADEG, ABCDFG, CDEFG } S 1 ' = { I, ACF, ABE, CDG, CBFE, ADFG, ACBDEG, BDFEG } S 1 ' = { I, ABE, ACF, CDG, BCEF, ADFG, ABCDEG, BDEFG }  S 2 B ↔ C E ↔ F rewrite

24. Sep. 25, 2009Abhishek K. Shrivastava, TAMU24 2-level regular FFDs as bipartite graphs Example: n = 7, S = {I, ABE, ACF, BDG, BCEF, ADEG, ABCDFG, CDEFG} 1. Start with G(V,E) = empty graph (no vertices); V = V a  V b 2. For each factor in d, add a vertex in V a 3. For each word in S, except I, add a vertex in V b 4. For each word in S, except I, add edges between the word’s vertex (in V b ) and the factors’ vertices (in V a )

25. Sep. 25, 2009Abhishek K. Shrivastava, TAMU25 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 2-level regular FFD isomorphism problem Bipartite graph isomorphism

26. Sep. 25, 2009Abhishek K. Shrivastava, TAMU26 Multi-level designs as Multi-graphs Multi-graph representation of a 3 5–2 design with defining contrast subgroup {I, ABCD 2, A 2 B 2 C 2 D, AB 2 E 2, A 2 BE, AC 2 DE, A 2 CD 2 E 2, BC 2 DE 2, B 2 CD 2 E} Similar representation for mixed level designs

27. Sep. 25, 2009Abhishek K. Shrivastava, TAMU27 Non-regular designs as Vertex-colored graphs Vertex-colored graph representation A 4-factor, 5-run design *edges colored only for better visualization

28. Sep. 25, 2009Abhishek K. Shrivastava, TAMU28 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 Non-regular FFD isomorphism problem Vertex colored graph isomorphism

29. Sep. 25, 2009Abhishek K. Shrivastava, TAMU29 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

30. Sep. 25, 2009Abhishek K. Shrivastava, TAMU30 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

31. Sep. 25, 2009Abhishek K. Shrivastava, TAMU31 FFSP Isomorphism [ Definition V.1 ] Two FFSP matrices are isomorphic to 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 S 1, S 2 are isomorphic to each other iff – one of S 1 or S 2 can be obtained from the other by some permutation of whole-plot factor labels and sub-plot factor labels, and reordering of words.

32. Sep. 25, 2009Abhishek K. Shrivastava, TAMU32 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

33. 4. GI and FFDI I. Solving GI: canonical labeling II. Implications to listing FFDs efficiently … next week 4. GI and FFDI I. Solving GI: canonical labeling II. Implications to listing FFDs efficiently … next week