Javier Cruz-Salgado - IIT College of Science · PDF fileJavier Cruz-Salgado ... statisticians....

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%

4

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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