abhishek k. shrivastava september 25 th , 2009

Abhishek K. ShrivastavaSeptember 25th, 2009

Listing Unique Fractional Factorial Designs – I

Sep. 25, 2009 Abhishek K. Shrivastava, TAMU 2

Outline1. 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 isomorphismListing 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

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


Experiments

Effect of process parameters on product qualitySource: http://www.emeraldinsight.com/fig/0680170207035.png

Miller-Urey ExperimentSource: http://www.physorg.com

• Experiments for quantifying effect of causal variables

• Experiments for testing hypothesis


Experimental Designs

1. Choose variable settings to collect data

2. Replicate runs3. Randomize run order

Collect DataA B C . . . I. . . . . . .. . . . . . .

. .

. .

. .. . . . . . .

Analyze datay = X+

Make inferences

run A B C . . . I1 0 1 0 . . . 02 0 1 1 . . . 1. . .. . .. . .20 1 0 1 . . . 1

run A B C . . . I1 0 1 0 . . . 02 0 1 1 . . . 1. . .. . .. . .20 1 0 1 . . . 121 0 1 0 . . . 022 0 1 1 . . . 1. . .. . .. . .40 1 0 1 . . . 1

run A B C . . . I9 0 1 0 . . . 015 1 1 0 . . . 0. . .. . .. . .10 1 0 1 . . . 15 0 1 1 . . . 038 0 1 1 . . . 1. . .. . .. . .22 0 1 1 . . . 1

Experimental planExperimental design


Experimental Designs

run A B C . . . I1 0 1 0 . . . 02 0 1 1 . . . 1. . .. . .. . .20 1 0 1 . . . 1

factors

a run

Levels of factor I Experimental design


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)


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??


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

2. Listing Unique FFDs


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 . . . . . .

ABCDEFG


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 . . . . . . .

ABCDEFG

ACBDFEG

A B C D E F G1 0 0 0 0 0 0 02 0 0 0 1 0 0 13 0 0 1 0 0 1 04 0 0 1 1 0 1 15 0 1 0 0 1 0 16 0 1 0 1 1 0 07 0 1 1 0 1 1 18 0 1 1 1 1 1 09 1 0 0 0 1 1 010 1 0 0 1 1 1 111 1 0 1 0 1 0 012 1 0 1 1 1 0 113 1 1 0 0 0 1 114 1 1 0 1 0 1 015 1 1 1 0 0 0 116 1 1 1 1 0 0 0

A B C D E F G1 0 0 0 0 1 1 02 0 0 0 1 1 1 13 0 0 1 0 1 0 14 0 0 1 1 1 0 05 0 1 0 0 0 1 06 0 1 0 1 0 1 17 0 1 1 0 0 0 18 0 1 1 1 0 0 09 1 0 0 0 0 0 010 1 0 0 1 0 0 111 1 0 1 0 0 1 112 1 0 1 1 0 1 013 1 1 0 0 1 0 014 1 1 0 1 1 0 115 1 1 1 0 1 1 116 1 1 1 1 1 1 0

(a) Defining words: {ABE, ACF, BDG} (b) Defining words: {ABE, ACF, CDG}


Unique designsA B C D E F G

1 0 0 0 0 1 1 02 0 0 0 1 1 1 13 0 0 1 0 1 0 14 0 0 1 1 1 0 05 0 1 0 0 0 1 06 0 1 0 1 0 1 17 0 1 1 0 0 0 18 0 1 1 1 0 0 09 1 0 0 0 0 0 010 1 0 0 1 0 0 111 1 0 1 0 0 1 112 1 0 1 1 0 1 013 1 1 0 0 1 0 014 1 1 0 1 1 0 115 1 1 1 0 1 1 116 1 1 1 1 1 1 0

A C B D F E G1 0 0 0 0 0 0 02 0 0 0 1 0 0 15 0 0 1 0 0 1 16 0 0 1 1 0 1 03 0 1 0 0 1 0 04 0 1 0 1 1 0 17 0 1 1 0 1 1 18 0 1 1 1 1 1 09 1 0 0 0 1 1 010 1 0 0 1 1 1 113 1 0 1 0 1 0 114 1 0 1 1 1 0 011 1 1 0 0 0 1 012 1 1 0 1 0 1 115 1 1 1 0 0 0 116 1 1 1 1 0 0 0

Reordered matrix, exchanged columns B↔C, E↔F, reordered

rows in (a)

(b) Defining words: {ABE, ACF, CDG}


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}

A B C D E F G1 0 0 0 0 0 0 02 0 0 0 1 0 0 13 0 0 1 0 0 1 04 0 0 1 1 0 1 15 0 1 0 0 1 0 16 0 1 0 1 1 0 07 0 1 1 0 1 1 18 0 1 1 1 1 1 09 1 0 0 0 1 1 010 1 0 0 1 1 1 111 1 0 1 0 1 0 012 1 0 1 1 1 0 113 1 1 0 0 0 1 114 1 1 0 1 0 1 015 1 1 1 0 0 0 116 1 1 1 1 0 0 0

A B C D E F G1 0 0 0 0 0 0 02 0 0 0 1 0 0 13 0 0 1 0 0 1 14 0 0 1 1 0 1 05 0 1 0 0 1 0 06 0 1 0 1 1 0 17 0 1 1 0 1 1 18 0 1 1 1 1 1 09 1 0 0 0 1 1 010 1 0 0 1 1 1 111 1 0 1 0 1 0 112 1 0 1 1 1 0 013 1 1 0 0 0 1 014 1 1 0 1 0 1 115 1 1 1 0 0 0 116 1 1 1 1 0 0 0


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.


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!


• 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

24 Full factorial

5-factor FFD

6-factor FFD

7-factor FFD

add column/ factor

add column/ factor

add column/ factor

add column/ factor …


• Consider sequential generation of 16-run designs

• Note: reducing # intermediate designs will speed up the algorithm

• How to discard isomorphs?

Listing Unique FFDs

24 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


Solving FFDI: literature reviewTwo 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)

Fastest; 2-level regular FFDs only

Legend:• Regular FFDs only• All FFDs


1. Graph models for FFDs2. Equivalence between FFDI and GI3. Solving GI

Proposed FFDI solution (in a nutshell)

…

Construct graphs from FFDs

…

Solve graph isomorphism problem

FFD class specific

3. Graphs and FFDsI. Graphs & Graph isomorphismII. 2-level regular FFDsIII. Multi-level regular FFDsIV. Non-regular FFDsV. 2-level regular split-plot FFDs


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


2-level regular FFD isomorphism (rFFDI)

• Two regular FFDs, represented by their defining contrast subgroups S1, S2 are isomorphic to 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}

S1' = {I, ACF, ABE, CDG, CBFE, ADFG, ACBDEG, BDFEG}

S1' = {I, ABE, ACF, CDG, BCEF, ADFG, ABCDEG, BDEFG} S2

B ↔ C E ↔ F

rewrite


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 = Va Vb

2. For each factor in d, add a vertex in Va

3. For each word in S, except I , add a vertex in Vb

4. For each word in S, except I , add edges between the word’s vertex (in Vb) and the factors’ vertices (in Va)


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


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


Non-regular designs as Vertex-colored graphs

• Vertex-colored graph representation

A 4-factor, 5-run design

*edges colored only for better visualization


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


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


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


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 S1, S2 are isomorphic to 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.


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

4. GI and FFDII. Solving GI: canonical labelingII. Implications to listing FFDs efficiently

…next week

abhishek k. shrivastava september 25 th , 2009

Documents

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