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Javier Cruz-Salgado Visitor Student
Department of Applied Mathematics Il l inois
Institute of Technology
Joint work with professor Lulu
Kang and William A. Brenneman
Mesh-Free Seminar
October 9, 2013
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Introduction
Mixture Models
Differences Between Mixture Models
Slack Variable Model and Filler Component
Choice of Slack Variable
Examples
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
x1
25%
x2
25%
x3
50%
x1
37.5%
x2
25%
x3
37.5%
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
x1
25%
x2
25%
x3
50%
x1
37.5%
x2
25%
x3
37.5%
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Single-component constraints
Multiple-component constraints
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Scheffé Model
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Scheffé Model
Lineal:
Quadratic:
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Kronecker Model (Draper and Pukelsheim 1998)
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Kronecker Model (Draper and Pukelsheim 1998)
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Kronecker Model
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Slack Variable
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Slack Variable
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Slack Variable
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
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Incorporate the mixture
constraint in the
simplest way.
Is not possible to interpret
the effects.
Variable selection of the
effects is restricted.
Some statistical software
does not recognize
this model
Is recommended for its
better numerical
stability.
Shares all the
disadvantages of the
Scheffe model.
The design and the
analysis can be
rendered into the
classic non-constrain
experiments.
It is very popular among
practitioners,
especially
formulators.
It is not so favored by
academic
statisticians.
Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
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Cornell (2000) Does it matter which component is chosen as the slack variable?
Khuri (2005) Model equivalence.
Piepel and Landmesser (2009) made a very complete review on SV-model.
Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
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• Factorial Design
• Fractional
Factorial Design
• Central
Composite
• Box Behnken
• Statistical
significance tests
of estimated
model
coefficients
• Variable
selection
methods
• Consider the
relationship
between SV and
S model in order
to avoid
misleading
results and
conclusions.
Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
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• Water in
shampoo
• Flour in bakery.
• Sugar in drug
tablets an pills
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
There are cases when it is not clear which component should be the Slack Variable
The experiments have been carried out using the constrained mixture
Variable selection is more accurate for models that have less collinearity
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Conditional Number Variance Inflation Factor
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Mean Variance Inflation Factor
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Average Squared Correlation Criterion
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Average Squared Correlation Criterion
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Average Squared Correlation Criterion
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Average Squared Correlation Criterion
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Example 1: Procter and Gamble Company. There are 8 non fillers
components and water. 65-run I-optimal design was used for the
complete quadratic SV-model.
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Example 1: Procter and Gamble Company. There are 8 active
components and water. 65-run I-optimal design was used for the
complete quadratic SV-model.
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Example 1: Procter and Gamble Company. There are 8 active
components and water. 65-run I-optimal design was used for the
complete quadratic SV-model.
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Linear Transformation
Scale [-1, 1] Lower-
Pseudocomponents
Modified L-
Pseudocomponents Upper-
Pseudocomponents
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Linear Transformation
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Comparison with other Models
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Example 2: Piepel (2009) used an artificial example on mixtures of
two drugs x1 and x2, an enhancer x3, and a filler x4.
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Average Squared
Correlation
Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Example 2: Piepel (2009) used an artificial example on mixtures of
two drugs x1 and x2, an enhancer x3, and a filler x4.
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Example 3: Prescott (2002) described an experiment to study the
effects of different mixtures.
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Average Squared
Correlation
Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Example 3: Prescott (2002) described an experiment to study the
effects of different mixtures.
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Example 4: Cornell (2000) described an experiment with four
components.
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Average Squared
Correlation
Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Example 4: Cornell (2000) described an experiment with four
components.
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Example 5: Cornell (2003) showed a study involving three
components.
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Average Squared
Correlation
Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
Comparison with other Models
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Introduction – Mixture Models – Differences Between Mixture Models – Slack Variable Model and Filler Component – Choice of Slack
Variable – Examples
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