designing experiments for causal networks william d heavlin advanced micro devices 2002 fall...

Designing Experiments for Causal Networks

William D Heavlin

Advanced Micro Devices

2002 Fall Technical Conference, Valley Forge, PA

October 17-18, 2002

Friday, October 18, 2002 [email protected] Fall Technical Conference, Valley Forge, PA 2

Brief Summary:

• Optimal design of experiments for

• ambitious tolerance design problem;

uses causal networks as an input data structure;

intrinsic role for blocks, interactions,

• and multivariate responses;

• skewed blocks introduced.


Experimental Design Literature

model withinblock

amongblocks

goal algorithm objectivefunction

WP:SP

Taguchi

2k-p, RSM

OAs

Factorial designs

Tukey1 df

ANOVA

Pathanalysis

LisRel

PLS

optimaldesign

computerexperiments


Outline:

1) Tolerance Design Context

2) Causal Networks

3) Problem Statement: (slide 13)

4) Split/Skew Factors

5) DiSCo model

6) Objective function & algorithm

7) Skew Factor Problem

8) Graph Partitioning

9) Summary


Context:

Process Development• targets, process fluid• focus is to work as whole

Pre-production• targets more-or-less set• factor importance, marginalities

unknown• interactions not well

characterized

Production• defect reduction, productivity

improvement, etc.• tolerance refinement


Tolerance Design:

•Big: #factors F big ~50

•Broad: #factors used=F

•Brief: # blocks

x #trials/block finite

•Local: range of levels constrained ~ ±5σf

# factors/block F1 constrained ~8

•Complete: 2nd-order interactions,

error propagation model

tolerancedesign

WP:SP

Taguchi

2k-p, RSM

OAs

Factorialdesigns

Tukey1 df

ANOVA

Pathanalysis

LisRel

PLS

optimaldesign

computerexperiments


Block self-containment:

Each block

•is a sub-experiment involving F1 factors,

•all F1 factors split within block.

•The other (F – F1 ) factors are held constant in each block.

Each block can be analyzed without regard to results from other blocks.

•A set of self-contained blocks/experimental designs constitutes an experimental strategy.

F1=4 F –F1 = 12


Causal Network Examples:

Leff

Gate OxThk

Vth

LDC

GateDopant

Poly Thk

Speed

LDD

Silicide Rs

Gate Rs

Rs s/d

Spacer

Poly DelW

RTA s/d

ScreenOxideDopant

LossSpacerOver-etch

Interfaces/d Rc

RPD Thk

CO QTimeSilicidePenetration

Gox QTime

VNI/VPIdose

N2/I2 dose

Transistorwidth

Silicide RTA

RTA2

CO Thk

LIOE*

BMDPCII

BMDThk

RPEo/e

LI Align

Polish ThkLIOE*

LI Dep Thk

LIM Exposure

IOX

I2 Damage

Udox Thk

Nitride Thk

Gate stepperSiON stripSiON dep

GM QTCDcontrol

LI btm CD

Trueactivelength

F01 F02

r01 F03

F04

F05

F06 r02

F07

F08

r03

r04

F09 r05

F10 r06

F11

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r15

R08

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F12

r11

F13 r12

R13 r14

F14 F15

F16 r16

black white

pigment

hue value

solids

solvent resin

color

viscosity

adhesion

transistor Wnt/β-catenin paint


first draft causal network:

transistor


“causal map”

•

Leff

Gate OxThk

Vth

LDC

GateDopant

Poly Thk

Speed

LDD

Silicide Rs

Gate Rs

Rs s/d

Spacer

Poly DelW

RTA s/d

ScreenOxideDopant

LossSpacerOver-etch

Interfaces/d Rc

RPD Thk


Gox QTime

VNI/VPIdose

N2/I2 dose

Transistorwidth

Silicide RTA

RTA2

CO Thk

LIOE*

BMDPCII

BMDThk

RPEo/e

LI Align

Polish ThkLIOE*

LI Dep Thk

LIM Exposure

IOX

I2 Damage

Udox Thk

Nitride Thk


GM QTCDcontrol

LI btm CD

Trueactivelength

transistor


transformation to causal map (1):

resin

solvent

solids

value

hue

pigment(s)

white

black

color

viscosity

adhesion

bla

ck

wh

ite

pig

men

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hu

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valu

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soli

ds

solv

en

t

resi

n

co

lor

vis

co

sity

ad

hesi

on

black 1 1 white 1 1 pigment 1 1 1 hue 1 value 1 solids 1 solvent 1 1 resin 1 1 color viscosity adhesion

blac

k

whi

te

pigm

ent

hue

valu

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solid

s

solv

ent

resi

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colo

r

visc

osity

adhe

sion

black 0 2 2 3 1 1 3 3 2 2 4 white 2 0 2 3 1 1 3 3 2 2 4 pigment 2 2 0 1 1 1 3 3 2 2 4 hue 3 3 1 0 2 2 4 4 1 3 5 value 1 1 1 2 0 2 4 4 1 3 5 solids 1 1 1 2 2 0 2 2 3 1 3 solvent 3 3 3 4 4 2 0 2 5 1 1 resin 3 3 3 4 4 2 2 0 5 1 1 color 2 2 2 1 1 3 5 5 0 4 6 viscosity 2 2 2 3 3 1 1 1 4 0 2 adhesion 4 4 4 5 5 3 1 1 6 2 0

dim1r dim2r black 0.262316 0.534689 white 0.332393 0.233637 pigment 0.380802 -0.346950 hue 0.781147 -0.616456 value 0.755915 0.176495 solids -0.026938 -0.014541 solvent -0.908860 0.213509 resin -0.864264 -0.398334 color 1.175607 -0.080691 viscosity -0.481124 -0.063398 adhesion -1.327874 -0.129936

XGvis

paint


transformation to causal map (2):

Causal Map properties:•Node positions are

coordinates in D-space.•Links are all about the

same length.

•Origin arbitrary.

•(With L2-distances,

orientation arbitrary.)

•Node positions (coordinates) now primary, while links must be drawn in.

- 0.5 0.0 0.5

- 1.0

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solids

solvent resin

color

viscosity

adhesion

paint


Problem Statement:

Design a series of experiments•varying F1 of F factors per block,respecting the causal network,•F moderately big.•Blocking intrinsic to the design problem.•Reasonable about interactions.

Approach:causal network “causal map”•Within-block splits, block-to-block skews•maximize dispersion among predicted responses•distance-in-space coefficients (“DiSCo”) model


Within-block (split) design:

Why:• High efficiency• Controls block-to-block

effects• Some sensitivity to

interactions• Enables block-specific,

stand-alone analysis• “block self-containment”

splitdesign

WP:SP

Taguchi

2k-p, RSM

OAs

Factorialdesigns

Tukey1 df

ANOVA

Pathanalysis

LisRel

PLS

optimaldesign

computerexperiments

Z =


among-blocks (skew) design:

Why?

• More sensitivity to interactions,

• Greater response dispersion,

• Better coverage in factor space.

Skew factors lace together the self- contained blocks. Skew factors change block-to-block.

Z =

skewdesign

WP:SP

Taguchi

2k-p, RSM

OAs

Factorialdesigns

Tukey1 df

ANOVA

Pathanalysis

LisRel

PLS

optimaldesign

computerexperiments


Paint example...

blocks X01 X02 X03 X04 X05

1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 1 -1 1 -1 -1 1 1 1 1 1 -1 1 2 -1 1 -1 -1 -1 2 -1 1 1 1 -1 2 -1 1 1 -1 1 2 -1 1 -1 1 1

block 1 skews

block 2 skews


DiSCo model:

“distance-in-space coefficients”Y = XA + ε, X = blocks, linear, and interaction terms

coefficients A:• derived from D-dim causal map• Let qy denote response in D-space• … q1, q2 two factors, same space

• Effect aqy.q1 inverse to distance:

• aqy.q1 = ±exp{–α · || qy – q1||d },• aqy.q1q2= ±aqy.q1·aqy.q2 ·exp{–β · ||q1–q2||d }

• scale parameters α, β ≈ 1...2

• distance function || • || L1 or L2; d=1 or 2; here d=2

DiSComodel

WP:SP

Taguchi

2k-p, RSM

OAs

Factorialdesigns

Tukey1 df

ANOVA

Pathanalysis

LisRel

PLS

optimaldesign

computerexperiments

Leff

Gate OxThk

Vth

LDC

GateDopant

Poly Thk

Speed

LDD

Silicide Rs

Gate Rs

Rs s/d

Spacer

Poly DelW

RTA s/d

ScreenOxideDopant

LossSpacerOver-etch

Interfaces/d Rc

RPD Thk


Gox QTime

VNI/VPIdose

N2/I2 dose

Transistorwidth

Silicide RTA

RTA2

CO Thk

LIOE*

BMDPCII

BMDThk

RPEo/e

LI Align

Polish ThkLIOE*

LI Dep Thk

LIM Exposure

IOX

I2 Damage

Udox Thk

Nitride Thk


GM QTCDcontrol

LI btm CD

Trueactivelength

linear

interactions


Paint example ...

factor1 factor2 f1:f2 dist discount Color viscosity adhesion

black 0.000000 1.000000 0.332449 0.385138 0.178442 white 0.000000 1.000000 0.406612 0.420611 0.182755 pigment(s) 0.000000 1.000000 0.432481 0.403585 0.178637 resin 0.000000 1.000000 0.000000 0.600769 0.581711 solvent 0.000000 1.000000 0.000000 0.601157 0.585261 black white 0.095543 0.826062 0.111665 0.133816 0.026939 black pigment(s) 0.791326 0.205429 0.029536 0.031931 0.006548 white pigment(s) 0.339425 0.507200 0.089192 0.086098 0.016558 black resin 2.139716 0.013851 0.000000 0.003205 0.001438 white resin 1.831375 0.025662 0.000000 0.006484 0.002728 pigment(s) resin 1.552831 0.044795 0.000000 0.010861 0.004655 black solvent 1.474810 0.052360 0.000000 0.012123 0.005468 white solvent 1.541112 0.045857 0.000000 0.011595 0.004905 pigment(s) solvent 1.977342 0.019165 0.000000 0.004650 0.002004 resin solvent 0.376340 0.471102 0.000000 0.170142 0.160388

- 0.5 0.0 0.5

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causal mapimputed coefficients

exp{–β · ||q1–q2||d } aqy.q1q2

aqy.q1


DiSCo properties:

Compatible with causal maps …

DiSCo ≈ Resolution IV•Res IV terms = #blocks – 1 + 2F

•#DiSCo terms = #blocks + (R+F )D – D – (D –1) = #blocks – 1 + 2F, when R=1, D=2

Term inclusion:•Stepwise: term “in” or “out;”

~{0,1}-binary state

•DiSCo: all terms “in,” unimportant ones far away; ~ (0,1)-analog state guided by exp{–α · || qy – q1||d }

Metrics•L2: collinearity makes causal chain•L1: 90o+ angle makes causal chain

translation

rotation


Objective function (Wynn):

objectivefunction

WP:SP

Taguchi

2k-p, RSM

OAs

Factorialdesigns

Tukey1 df

ANOVA

Pathanalysis

LisRel

PLS

optimaldesign

computerexperiments

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nd

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nIon stand

response space YB (blocks have

different centroids)

B = blocking dummy variables

Z = base design matrix, e.g. linear

P = column permutation matrix

ZU(P) = ZP with rows U appended

= [ (ZP)T | UT ]T

YB(P) = ZU(P) A (wrt linear model)

Y(B)(P) = (I–B(BTB)-1BT) YB(P)

d(Y)jk = ||yj–yk||, yj = row j of Y

c(Y)jk = exp{–d(Y)jk}

W(Y)= det(c(Y)), higher is better

W(YB(P)) x W(Y(B)(P)) max wrt P

transistor

response dispersion ...esp within blocks


Paint example: 5!=120 factor assignments

-40

-30

-20

-10

0

C(Yg(P)) C(Y(B)(P)) combined

ln(det(C(Y)))

<= 13524

<= 31254


Paint example ...

black white pigment resin solvent imputed

color imputed viscosity

X01 X03 X05 X02 X04

-1 1 1 -1 -1 0.454634 -0.690639

1 -1 1 -1 -1 0.186997 -0.864415

-1 -1 1 1 -1 -0.313643 -0.536666

1 1 1 1 -1 1.401934 1.282832 best

-1 -1 -1 1 -1 -0.941148 -1.120199

-1 1 -1 1 1 -0.529639 0.803214

-1 1 1 1 -1 0.454634 0.198898

-1 -1 1 1 1 -0.313643 0.967795

X03 X01 X02 X05 X04

1 -1 -1 1 -1 -0.558653 -0.699255

-1 1 -1 1 -1 -0.529639 -0.729028

-1 -1 1 1 -1 -0.313643 -0.536666

1 1 1 1 -1 1.401934 1.282832 worst

-1 -1 1 -1 -1 -0.313643 -1.400265

1 -1 1 -1 1 0.186997 0.007970

1 -1 1 1 -1 0.186997 0.012003

-1 -1 1 1 1 -0.313643 0.967795

-1 0 1

-1

0

1

B B

B

B

B

B

B

B

"COLOR"

"VISCOSITY"

-1 0 1

-1

0

1

W W W

W

W

W W

W

"COLOR"

"VISCOSITY"

worst:

best:

near-coincident

almost orthogonal


Traveling salesman algorithm:

travelingsalesman algorithm

WP:SP

Taguchi

2k-p, RSM

OAs

Factorialdesigns

Tukey1 df

ANOVA

Pathanalysis

LisRel

PLS

optimaldesign

computerexperiments

List 1:the F cols of genericbase design matrix Z

List 2:names of the F factors

objective function:

Wynn’s criterion in predicted response space =

W(YB(P)) x W(Y(B)(P)), permutation matrix P

algorithm:0. Each P implies different ordering Z-columns wrt factor

names.1. Solution is best P found with respect to Wynn’s criterion.2. Design F /F1 blocks (all F factors assigned once per

“cycle”).


How many skew factors F2?

•Wnt/β-catenin:

F=16 factorsR= 2 responses

•skew factors:F2=0..3 considered

base design IB(F2=3, 4 x 4)

• terms:cb=16 blocks, F=16 factors,

F(F-1)/2=120 interactions

•criteria:Wynn’s entropy“V120”=column rank wrt

model with 16+16+120 terms

“V60”=column rank wrt model with 16+16+(60 of 120) terms

–1 –1 –1 –1 –1 –1 +1 +1 –1 –1 +1 –1 –1 +1 –1 +1 –1 +1 –1 –1 +1 +1 +1 –1 –1 –1 –1 +1 –1 –1 +1 +1 –1 –1 +1 +1 –1 +1 –1 –1 –1 +1 –1 +1 +1 –1 –1 –1 +1 +1 +1 +1 +1 –1 –1 +1 +1 –1 –1 –1 +1 +1 –1 +1 +1 –1 –1 –1 +1 –1 +1 –1 +1 –1 –1 +1 –1 +1 +1 +1 –1 +1 +1 –1 +1 –1 –1 +1 –1 +1 –1 +1 +1 –1 +1 +1 +1 –1 +1 –1 +1 +1 +1 +1 –1 +1 +1 +1 +1 –1 +1 –1 +1 –1 –1 –1 –1 –1 –1 +1 –1 +1 –1 –1 +1 –1 +1 –1 –1 +1 –1 +1 –1 +1 –1 +1 +1 –1 –1 –1 +1 –1 –1 –1 +1 +1 –1 +1 –1 +1 –1 +1 –1 –1 +1 –1 –1 +1 +1 –1 –1 +1 –1 +1 +1 +1 +1 –1 –1 +1 +1 –1 –1 –1 +1 –1 +1 +1 +1 –1 –1 –1 –1 +1 +1 –1 +1 –1 –1 –1 +1 +1 +1 +1 –1 +1 –1 +1 +1 –1 –1 +1 –1 –1 +1 +1 +1 –1 +1 +1 –1 +1 +1 –1 +1 +1 +1 –1 +1 +1 +1 +1 +1 –1

IB(F2=0, 4 x 4) IB(F2=3,16 x 1)


Response dispersion:

-4 -2 0 2 4 6 8 10 12 -4

-2

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1 1

1 1 1 1

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IB(0,16X1)

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4 4

4 4 4

IB(0,4X4)

Wnt/β-catenin

IB(F2=0,c=16 x b=1) IB(F2=0,c=4 x b=4)


... with F2=3 skew factors

-4 -2 0 2 4 6 8 10 12 -4

-2

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IB(3,16X1)

-4 -2 0 2 4 6 8 10 12 -4

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1 1 1

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4 4 4

IB(3,4X4)

IB(F2=3,c=16,b=1) IB(F2=3,c=4,b=4)

Wnt/β-catenin


Efficiency results (2):

• IB(F2,c=16 x b=1) better response dispersion, very low efficiency

• IB(F2,c x b=16) column ranks increase in F2, IB(F2,4 x 4) better

... suggests using column rank to determine F2.

design V120 column rank

V60 column rank

worst factor’s efficiency

best factor’s efficiency

notes

IB( 0,16x1) 72 55 0.055 0.786 –3 factors IB( 0, 4x4) 80 69 0.250 0.252 IB( 1,16x1) 84 59 0.068 0.831 –2 factors IB( 1, 4x4) 117 76 0.245 0.369 IB( 2,16x1) 96 59 0.030 0.778 –1 factors IB( 2, 4x4) 130 83 0.248 0.359 IB( 3,16x1) 120 75 0.089 0.691 IB( 3, 4x4) 144 90 0.270 0.336 L128 Wynn 87 61 0.973 1.000 L128 V120 128 85 1.000 1.000 L128 V60 117 91 0.976 1.000 L128 V120+V60 128 89 0.974 1.000

Wnt/β-catenin


Re: Efficiency results (L128):

Most efficient design is based

on OA(n=128,k=127,q=2),

a.k.a. 2III127-120

• based on 7 generators, 4 of which are constant within 16 blocks

• 4 generators => 15 OA columns constant within blocks, hence

• 112 = 127-15 columns available to construct 2IV

16-9

• Each column gives candidates: Xj & –Xj

Select 16 of 112 columns:• L128(Wynn) maxs Wynn’s

response dispersion criterion.• L128(V120) maxs column rank

wrt to space spanned by 16x15/2=120 resolution V-interaction terms (+16 blocks +16 linear terms)

• L128(V60) maxs column rank wrt to space spanned by 60 (of 120 possible) interaction terms with shortest causal map distances (+16 blocks +16 linear terms)

• L128(V120+V60) maxs sum of latter two criteria.


Graph partitioning in DoE:

black white

pigment

hue value

solids

solvent resin

color

viscosity

adhesion

F01 F02

r01 F03

F04

F05

F06 r02

F07

F08

r03

r04

F09 r05

F10 r06

F11

r07

r15

R08

r09

r10

F12

r11

F13 r12

R13 r14

F14 F15

F16 r16

partition point

subsystem 2

subsystem 1 subsystem 1

subsystem 2

Wnt/β-catenin paint



BLACK WHITE pigment

hue value

solids

solvent RESIN

color

viscosity

adhesion

BLACK WHITE PIGMENT

hue value

solids

solvent resin

color

viscosity

adhesion

black white pigment resin solvent

X01 X03 X05 X02 X04 -1 1 1 -1 -1 1 -1 1 -1 -1

-1 -1 1 1 -1 1 1 1 1 -1

-1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 -1 -1 1 1 1

X03 X01 X02 X05 X04

1 -1 -1 1 -1 -1 1 -1 1 -1 -1 -1 1 1 -1 1 1 1 1 -1

-1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 1 1 -1

-1 -1 1 1 1

worst

best

block 1 split factors

block 1 split factors

paint


Block 1..4

Block 5..8

Block 9..12

Block 13..16


red = split factorsWnt/β-catenin


Graph partitioning and DoE:

Maximally separating subtrees•analogous to one-response-at-a-

time experiments,•do not test system as a whole, •sub-optimal wrt response

dispersion

Most blocks touch subtrees of all responses:

Objective function = dispersion among multivariate responses.

•Constraint (F1+F2 factors per block) works to assign all factors.

•Worse-case statistical efficiency splits all factors equally often.

Statistical efficiency splits unrelated factors.

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transistor


Comments, conclusions:

extensions process development, rather than

pre-production/tolerance design

summary experiment designed for a causal network;• several responses, many factors, multiple blocks;• multiple blocks = multiple experiment strategy;• each block self-contained;• skew factors: column rank rule;• DiSCo model, response-dispersion objective function

{


Selected References:

Atkinson, A.C., and Cox, D.R. (1974), “Planning Experiments for Discriminating between Models,” with discussion, Journal of the Royal Statistical Society, series B, 36, 321-348.

Bingham, D and Sitter, R (1999), “Minimum-aberration Two-level Fractional Factorial Split-plot Designs,” Technometrics, 41, 62-70.

Bisgaard, S. (1997), “Designing Experiments for Tolerancing Assembled Products,” Technometrics, 39, 142-152.

Buja, A., and Swayne, D.F. (2002), “Visualization Methodology for Multidimensional Scaling,” Journal of Classification, 19, 7-43.

Fedorov, V., and Flanagan, D. (1998), “Optimal Monitoring of Computer Networks,” in New Developments and Applications in Experimental Design, N. Flournoy, W.F. Rosenberger, and W.K. Wong, eds., IMS Lecture Notes, 34, 1-10.

Ishikawa, K. (1986), Guide to Quality Control, Tokyo: Asian Productivity Organization.

Sacks, J., Welch, W.J., Mitchell, T.J., and Wynn, H.P. (1989), “Design and Analysis of Computer Experiments,” with discussion, Statistical Science, 4, 409-435.

Shewry, M.C., and Wynn, H.P. (1987), “Maximum Entropy Sampling,” Journal of Applied Statistics, 14, 165-170.

Taguchi, G. (1986), Introduction to Quality Engineering: Designing Quality into Products and Processes, Tokyo: Asian Productivity Organization.

Tukey, J.W. (1949), “One Degree of Freedom for Non-additivity,” Biometrics, 5, 232-242.


Trademark Attribution

AMD, the AMD Arrow Logo and combinations thereof are trademarks of Advanced Micro Devices, Inc. Other product names used in this presentation are for identification purposes only and may be trademarks of their respective companies.

designing experiments for causal networks william d heavlin advanced micro devices 2002 fall...

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