Screening for Noise Variables

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Screening for Noise Variables

• David Drain
• Lisa Trautwein
• University of Missouri-Rolla

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Terminology for robust design

• Control variables
• Deliberately fixed to obtain optimal responses
• Noise variables
• Cannot be controlled during manufacturing
• Interact with control variables to cause
  undesirable variation in responses
• Can be controlled during an experiment
• Robust process
• Performs near optimum in spite of noise variables

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Oatmeal cookie factory example

• Responses are cookie density, weight, crunch
  strength
• Control variables are oven temperature, cooking
  time and amounts of each ingredient
• Noise variables
• Humidity causes more variation in cookie density
  at low oven temperatures
• Raw material (oatmeal) aging causes crunchier
  cookies at longer cooking times

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Historical perspective

• Classical experiment design concentrated on
  discovering and estimating the effects of control
  variables
• Taguchis influence resulted in experiments to
  evaluate the effects of known or suspected noise
  variables
• Standard experiment design now incorporates some
  acknowledgement of noise variables

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Current approaches to robust design

• Resolution III fractional factorials to screen
  for control and noise variables simultaneously
• Resolution IV or V fractional factorials to
  resolve interactions from main effects and one
  another
• Include relatively few suspected noise variables
  in rather expensive experiments such as CCD or BBD

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Problems with current approaches

• Too few potential noise variables are considered
• Process upsets occur after ramp-up
• Early factorials focus on control variables
• Noise variable screening is done post hoc if
  sufficient degrees of freedom are available
• Process recipes can be expensive to change if
  noise variables are discovered too late
• Equipment changes
• Customer notification and approval

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Problem motivation

• Control variables and approximate settings are
  often known before process development even
  starts from
• Prior process generations
• Basic scientific principals
• Preventing process upsets during production
  depends on knowledge of noise variables
• So we can design robust processes
• So we can exercise control or screening in the
  presence of variation in noise variables

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Design assumption continuum

• Simultaneous screening along with possible
  control variables
• Screening with known-significant control
  variables
• Range of reasonable control variable settings is
  known
• Screening with quantified control variables
• Prior distributions for their effects are
  available

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Problem context selection

• Designed experiments
• Five approaches from conventional to unusual
• Varying numbers of suspected noise variables from
  a few to many
• Analysis of existing manufacturing data
• Plagued by the curse of dimensionality and
  happenstance correlation
• Multivariate analysis approach

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Designed experiment approach

• Algorithmic approach
• Random designs
• Conventional designs
• Fractional factorials
• Plackett-Burman
• Reduced fractional factorial
• Reduction by inspection
• D-optimal designs
• Group screening

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Algorithmic approach

• Chose a design in the interaction terms and
  induce a design in the original factors that is
  effective for evaluating interactions
• Criteria for an effective design
• Main effects and XX, XZ interactions are
  estimable
• Complete for XX interactions, XZ interactions, Z
  factors and X factors
• Be able to estimate the X, Z and XZ terms without
  aliasing to one another or other terms that might
  be significant such as XX interactions and the
  constant column
• XX and XZ interaction designs are balanced

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Completeness

• A factorial is complete if every possible
  combination of the factor levels is in the design
• For an interaction to be completely estimated all
  possible combinations of both factors need to
  appear

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Importance of completeness

• If not all treatment level combination are
  observed for the factors making up the
  interaction, then you may miss estimate the
  interaction

Y
X
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Aliasing
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Aliasing
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Aliasing
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Example for algorithmic approach
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Example for Algorithmic Approach
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Meeting criteria

• In order to meet the criteria, more runs need to
  be added
• Every attempt led to fractional factorials with
  resolution 4 or higher
• Resolution 4 only works if there is no aliasing
  between XX and XZ interactions

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Designed experiment approach

• Algorithmic approach
• Random designs
• Conventional designs
• Fractional factorials
• Plackett-Burman
• Reduced fractional factorial
• Reduction by inspection
• D-optimal designs
• Group screening

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Random designs

• Subject to the same criteria as an Algorithmic
  design
• Estimable
• Completeness
• Aliasing
• Balance
• Random designs did not meet the criteria or they
  ended up looking like fractional factorials in
  high resolution
• Why did we try this?
• Thought possibly using wrong algorithm
• Easy to do

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Random design example

• 3 noise variables and one control variable

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Random design example

• 3 noise variables and one control variable

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Random design results

• For each random design attempted the criteria
  were never met
• Therefore the best design choice is still a high
  level fractional factorial

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Designed experiment approach

• Algorithmic approach
• Random designs
• Conventional designs
• Fractional factorials
• Plackett-Burman
• Reduced fractional factorial
• Reduction by inspection
• D-optimal designs
• Group screening

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Fractional factorial designs

• Algorithmic and random designs kept leading to
  fractional factorial designs
• Resolution 4 designs carefully selected or
  resolution 5 designs meet all criteria

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27-3 fractional factorial

• This design will work dependent upon assignment
  of control and noise factors
• Let XA, Z1B, Z2C, Z3D, Z4E, Z5F, and Z6G
• Let X1A, X2B, Z1C, Z2D, Z3E, Z4 F, and Z5G

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Summary fractional factorials

• A fractional factorial design of resolution 5 or
  higher will always meet the criteria
• Aliasing must be checked for resolution four
  designs

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Plackett-Burman design

• Considered because they have very few runs
• These designs do not meet criteria because have
  significant aliasing

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Designed experiment approach

• Algorithmic approach
• Random designs
• Conventional designs
• Fractional factorials
• Plackett-Burman
• Reduced fractional factorial
• Reduction by inspection
• D-optimal designs
• Group screening

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Reduced fractional factorial designs

• Reduction by inspection
• Led to unbalanced, incomplete designs with
  aliasing
• Motivation for D-optimal designs

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D-optimal designs

• Tried D-optimal designs because it is easy to
  chose incomplete designs by inspection
• Every attempt of an unsaturated design led to
  completeness in the main effects and in the
  interactions
• Saturated designs will lead to a lack of
  completeness in the interactions
• Small, Efficient, Equireplicated Resolution V
  Fractions of 2k designs and their Applications to
  Central Composite Designs, Gary Oehlert and Pat
  Whitcomb, Oct 15, 2002

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Designed experiment approach

• Algorithmic approach
• Random designs
• Conventional designs
• Fractional factorials
• Plackett-Burman
• Reduced fractional factorial
• Reduction by inspection
• D-optimal designs
• Group screening

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Effect Sparsity

• A presumption that most of the interaction
  effects are equal to 0
• It may not be appropriate in this case since we
  are assuming the Zs are prime suspects for noise
  variables
• If effect sparsity applies then we can use group
  screening

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Group Screening

• Put factors into two groups
• Noise factors
• Control factors
• Run a factorial on the two factors
• If interaction is not significant then conclude
  that all the XZs are insignificant
• If interaction is significant
• Run a resolution five fractional factorial
• Change how the factors are grouped and run
  another factorial on these groups
• Assuming effect sparsity
• Most interactions are not important

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Analysis of manufacturing data

• Developing a new process on the basis of an
  existing process has advantages
• Millions of observations are available
• Control variables are known
• Many suspected noise variables are known
• Disadvantages
• Not a designed experiment collinearity and
  accidental tool dedication abound
• Equipment and other differences in the new
  process render extrapolation suspect

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A multivariate solution

• Admit the correlation between potential noise
  variables exist and exploit it as follows
• Find principal components in the PNVs
• Test for interactions between these PCs and
  control variables
• Significant interactions require action

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Wine Aroma Data from Minitab
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Mo, Mg, Na, K will be controlled
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Principal components
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Regression finds some noise variables
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Regression finds some noise variables
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What to do with these PCNVs?

• One option is to interpret the coefficients in
  the important components and isolate individual
  variables for further study

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Another approach

• Accept the principle components as an inherent
  characteristic of the process and dont try to
  decompose them
• Design a process robust with respect to the
  principle components
• Us multivariate statistical process control to
  monitor the principal components

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Questions?