Process-Oriented Basis Representations (POBREP) for Multivariate SPC: Tracing Errors to their Source

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Process-Oriented Basis Representations (POBREP)
for Multivariate SPC Tracing Errors to their
Source

• Russell R. Barton
• Penn State, Smeal College of Business
• Acknowledgments J. McCool, D. Gonzalez-Barreto,
  E. Foster

Many products consist of multiple similar
measurements, such as temperatures, thickness or
registration errors at multiple locations. Under
such conditions, it is possible to produce
process diagnostics analyzing the multivariate
process quality vector using a process-oriented
basis. Many potential production problems have
characteristic signatures that can be detected in
the multivariate quality vector.

• Multivariate Repeated Measurements - Motivation
• A Process-Oriented Approach
• Math Details
• Example
• If time POBREP for Multivariate Capability

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POBREP for Capturing Process Knowledge
Chip Capacitor Manufacturing
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POBREP for Capturing Process Knowledge
Screen Printing of Silver Squares Registration
Errors Problematic
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POBREP for Capturing Process Knowledge
Measuring Registration Error
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A littl math notation for multivariate quality
vector
Define the set of n measured deviations from
nominal to be a multivariate quality vector x.
In this example suppose horizontal and vertical
registration errors measured only for the pads at
each corner of the sheet x1 is horizontal error
at upper left pad, x2 is vertical error at upper
left pad, x3 is horizontal error at upper right
pad, etc.
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Notation for multivariate quality vector
Measuring Registration Error
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A process-oriented approach

• The situation a set of 8 misregistration numbers
  is hard to interpret
• SPC using Hotellings T 2 or principal components
  is complicated and not intuitive
• Ideally, the link to specific causes would be
  clear

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Process-specific causes of misregister and
characteristic signatures
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Mathematical notation POBREP vector
Suppose have n different characteristic
signatures for n different process causes, say
a1, a2, ... , an. If the process-oriented
basis vectors a1, a2, ... , an are linearly
independent (not linear combinations of each
other) then they provide an alternative basis
for representing the x data as a linear
combination of the signatures x z1a1
z2a2 ... znan.
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Mathematical notation POBREP vector
x z1a1 z2a2 ... znan. The z
(z1, z2, ..., zn)' found by computing a matrix
inverse and then doing a simple linear
calculation z A-1x A is the matrix
consisting of the column vectors a1, a2, ... ,
an. These are the characteristic signatures. We
call this basis a1, a2, ... , an a
process-oriented basis. Thus POBREP z is a
process-oriented basis representation of the
original data vector, x.
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Screen printing example
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Screen printing example
standard basis
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1
process-oriented
basis
uniform
differential
diagonal
uniform errors
rotation
stretch/shrink
stretch/shrink
stretch/shrink
1
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Screen printing example
Inverse easy use Excel or Octave (free MATLAB)
???
z A-1x
A-1
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Excel instructions http//judgeg.myweb.port.ac.uk
/SAME/Xmatinv.pdf
Step 1 Highlight the block of cells for the
inverse (for example if you are inverting a 3x3
matrix this should also be 3x3). In my
illustration the matrix is in cells A1C3 and the
inverse is going to go in cells A6C8. Step 2
In the top left hand cell of the new block (A6)
type the following MINVERSE( Step 3 Then use
the mouse to paste over the cells where the
matrix to invert is situated (i.e. A1C3) Step 4
Enter the close bracket symbol ) Step 5 Press
the following keys together Ctrl Shift Enter
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Screen printing example
Inverse easy use Excel or Octave (free MATLAB)
z A-1x
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Screen Printing Example
Standard Representationx (0, 1, 2, -1, 0, -1,
-2, 1)'
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The calculation
z (z1, z2, ..., zn) Lets show the
calculation of z1. We will need the standard
representation, x, and the first row of A-1 x
(0, 1, 2, -1, 0, -1, -2, 1) First row of A-1
(0, 1/4, 0, 1/4, 0, 1/4, 0, 1/4) So the
calculation is z1 00 ¼1 02 ¼-1
00 ¼-1 0-2 ¼1 0 That means we
observe NO HORIZONTAL SHIFT. KEY once A-1 is
constructed, this calculation is easy in Excel.
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Screen Printing Example
Standard Representation of x (0, 1, 2, -1, 0,
-1, -2, 1)'
POBREP Representation of x (0, 1, 2, -1, 0,
-1, -2, 1) is z (0, 0, 1, 0, 1, 0, 0, 0)
uniform
differential
diagonal
uniform errors
rotation
stretch/shrink
stretch/shrink
stretch/shrink
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The power of POBREP
x z1a1 z2a2 ... znan. Using the
process-oriented basis representation z, of the
original vector x, diagnosis is
possible Potential causes are associated with
patterns (ai) having positive or negative
coefficients (zi) that are large in magnitude.
These patterns are linked with one or more
specific causes. Further, if the ai are scaled
so that the maximum magnitude is 1, the zi value
indicates the worst error magnitude introduced by
this cause. (our example one unit of error from
rotation, one from horizontal stretch).
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POBREP for Data Reduction

• In many interesting cases, would like to keep
  full set of measurements (e.g. all
  misregistration errors) but have a relatively
  small set of signatures giving both data
  reduction and cause connection.
• POPBREP facilitates this instead of computing
  A-1, solve
• x Az
• by least squares

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POBREP for Data Reduction
Fine Pitch Component with CCD .020? and 208
leads
Q in this case is ((z1), (z2), , (z208))
PRACTICAL??
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POBREP for Data Reduction
a1
a2
a3
a4
Four Basis Elements for Fine Pitch Component
Example
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Summary POBREP Diagnosis Methodology
Hypothesized or Observed Process Deviations
Process
...
Observed Error Patterns (x)
Process Oriented Basis Matrix A A a1
a2 .....an
1 0 1 0 1 0 1 0
a1
x Az ? - solving the linear system (via
least squares if A is not
full rank) will provide a representation of the
error vector in the basis matrix
space zi are coefficients for the ai
.........
z1
z2
z4
z3
zn
Potential process causes are associated with
patterns having large zi coefficients
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POBREP vector for SPC
Since causes are associated with signatures (ai)
having positive or negative coefficients (zi)
that are large in magnitude, multivariate SPC
with POBREP can be more informative than
univariate SPC. Univariate SPC monitor for
special cause variation, then separately
investigate to find cause. Multivariate SPC
POBREP z coefficients give the cause!
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POBREP for Diagnosis-based SPC Charts for z
Coefficients
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Process-Oriented Basis Representations (POBREP)
for Multivariate SPC Tracing Errors to their
Source
Conclusions

• Multivariate SPC usual methods (principal
  components, Hotellings T 2) difficult to
  interpret
• A Process-Oriented Multivariate Vector, z
• interpretable
• practical (can be calculated easily and with
  adequate precision) in many cases
• can induce independence between components

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POBREP and Multivariate Capability

• POBREP can address similar issues in multivariate
  capability.
• A brief overview

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Three widely accepted univariate indices

• Cp (USL-LSL)/6s

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Process Capability and Multivariate Capability
Indices
(Taam et. al (1993), Shahriari et. al (1995),
Chen (1994),Wierda (1992))

• Taam et al. Assumed elliptical specifications
• Shahriari et al. Presented three numbers that
  describe multivariate capability
• Chen A general approach allowing rectangular or
  elliptical specifications and non-normal
  distributions
• Wierda Direct computation of percentage
  conforming approach

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Multivariate capability index literature review
summary

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Wierda (1993) approach to the multivariate index

• Multivariate index proposed that uses
  p-dimensional rectangular specification area.
• Minimum expected or potentially attainable
  proportion of non-conformance items approach.
• Original proportion conforming definition of
  capability indices is explicitly preserved
• ? probability of producing a good part

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Wierda (1993) multivariate indexdetails

• Compute ? when quality variables independent

• Compute ? when quality variables dependent
• (? known)
• np is MVN density
• ? is covariance matrix
• L and U are vectors of
• specifications

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Wierda multivariate capability index graphical
aid

• ? is a bivariate reliability capability
  measure
• ? gives multivariate proportion conforming
  Integrate over bivariate normal density for the
  dependent case
• Independent case ? ?1?2

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Current Limitations in Multivariate Capability

• Estimating ?x is difficult when there are many
  quality variables.
• Interpretation is difficult when one number
  represents the joint affect of many variables.

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Multivariate Process-oriented Capability Method
Example

• z A-1x (Eight zs per part)
• Z z1 z 2 z 100.
• Using Z and the specification limits, capability
  can be computed
• Often, covariance matrix ?z will have zero
  non-diagonal elementsindependent causes

 
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Multivariate Process-oriented Capability Method
Example

• If the values 1 and 1 are in each column at
  least once, the full affect of basis elements is
  estimated
• x rectangular specifications LSL lt x lt USL may
  make less sense than specifications on z
  components because of cause connection and
  scaling to match maximum x-deviation for a
  specific cause.

 
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POBREP multivariate capability index graphical
aid

• Using z values instead of x likely to yield
  independence
• Independent case ? ?1?2

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Multivariate Capability Indices using
Process-Oriented Basis Representations
Conclusions

• Multivariate Capability Indices - difficult to
  interpret
• A Process-Oriented Multivariate Capability
  Vector
• interpretable
• practical (can be calculated with adequate
  precision) in many cases
• can induce independence between components