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The American University in Cairo Interdisciplinary Engineering Programs ENGR 592 Dr. Lotfi Gaafar. Mixture Designs Simplex Lattice Simplex Centroid . Presented by Ahmed Hassan Sayed. Introduction. - PowerPoint PPT PresentationTRANSCRIPT
Mixture Designs
Presented byAhmed Hassan Sayed
The American University in Cairo
Interdisciplinary Engineering Programs
ENGR 592
Dr. Lotfi GaafarMixture DesignsSimplex Lattice Simplex
Centroid
Mixture Designs
Introduction• In many cases, products are made by blending more than one
ingredient together. • Usually the manufacturer of each of these products is
interested in one or more properties of the final product, which depends on the proportions of the ingredients used
• Examples: – Cake formulations (by blending baking powder, shortening, four,
sugar, and water), property of interest is the fluffiness of cake.– Construction Concrete (made by mixing sand, water, and
cement), property of interest is the compressive strength
Mixture DesignsMixture Designs: When to
use?• Experiments in which the response depends on
factors that represents proportions of a blend.
• The fact that the proportions must sum up to a constant (usually 1 or 100%) makes this type of experiments a class on its own.
Mixture DesignsMixture designs:
Representation• From previous discussion:
• Geometrically, it’s represented by a q-1 dimensional simplex (q is no. of components)
• All experimental points lies on, or inside the simplex region:– 2-component: a straight line.– 3-component: a triangle.– 4-component: tetrahedron.
0.1...211
q
q
ii xxxx
Mixture DesignsMixture designs:
Representation• Points on the vertices of the simplex region are called
pure or single component mixture.• Triangular coordinates (3 component system):
– Points (1,0,0), (0,1,0), and (0,0,1) corresponds to the pure blends of A, B, and C.
Mixture DesignsMixture designs:
Representation• The response can be represented by the surface above
the triangle in 3D or as a contour plot, where each contour line represents a specific response.
Mixture DesignsSteps in Planning a Mixture
Experiment1. Objective and problem statement.2. Select the mixture component, and the levels they will assume
with (proportions).3. Identify the property of interest or the response variable.4. Decide what's the appropriate model to fir your data, and choose
a design that will be enough to both fit the model and allows for statistical test of the model.
5. Perform the experiment based on the design points.6. Analyze the data (Ex: ANOVA)7. Draw conclusions from the analysis and state your
recommendations.
Mixture DesignsObjectives & Underlying
Assumptions• The objective of a mixture experiments can be
one of the following:
– Empirically predict the response of interest for certain components proportions.
– Obtain some "measure of influence" of single components and with other components as well on the response of interest.
Mixture DesignsObjectives & Underlying
Assumptions• The assumptions made for this type of
experiments:– Errors and normally independently distributed with a
mean of zero and a constant variance.– The true response surface is continuous over the
entire region.– The response is assumed to be only dependent on
the ingredients proportions and not on the amount of the mixture itself.
Mixture DesignsFit Models: Canonical
Polynomials• Consider fitting a 1st order polynomial to a 3 –
component mixture
• Due to the fact that all proportions must sum up to one, this can be rewritten as (Multiply intercept by the components sum):
• Where:
332211 xxxy o
3*32
*21
*1 xxxy
1*1 o 2
*2 o 3
*3 o
Mixture Designs
Simplex Lattice Designs• Points are evenly distributed allover the
simplex range. (a lattice)• For a model of degree m, the proportions
assumed by each components are:
• For a q component mixture, the lattice is referred to by the notation {q, m}
1,...,2,1,0mm
xi
Mixture DesignsSimplex Lattice : {3,2}
Example• For q= 3, m = 2. The proportions assumed are:
• Using all possible Combinations:
1,21,0ix
)2/1,2/1,0(),2/1,0,2/1(),0,2/1,2/1(),1,0,0(),0,1,0(),0,0,1(),,( 321 xxx
Mixture DesignsSimplex Lattice : {3,2}
Example• Points on the vertices represent pure
mixtures, points on the edges represent binary blends. )0,5.0,5.0(
)1,0,0()5.0,5.0,0(
)0,0,1(
)0,5.0,5.0(
)1,0,0(
)5.0,5.0,0(
)5.0,0,5.0(
)0,1,0(
Mixture Designs
Simplex Centroid Designs• Consists of all possible mixtures that have
equally weighted proportions of one to q components.
• q permutations of (1,0,0..,0), (q Choose 2) permutations of (1/2,1/2,0,0,…,0) till we reach the overall centroid (1/q, 1/q, …, 1/q)ز
Mixture Designs
)3/1,3/1,3/1(
)1,0,0()5.0,5.0,0(
)0,0,1(
)0,5.0,5.0(
)1,0,0(
)5.0,5.0,0(
)5.0,0,5.0(
)0,1,0(
)3/1,3/1,3/1(
Simplex Centroid : 3 component Example
• For q = 3, the simplex space will contain the following points:
(1,0,0), (0,1,0), (0,0,1), (.5,.5,0), (.5,0,.5), (0,.5,.5), (1/3, 1/3, 1/3).
Mixture Designs
Mixture Designs: ANOVA• Augmented points are usually added to give
more degrees of freedom for error lack of fit, and model significance analysis.
• Simplex Lattice: – Overall simplex center, if not present in the design.– A 50-50 combination between the simplex center and
its vertices.• Simplex Centroid:
– The 50-50 combination between the simplex center and its vertices.
Mixture Designs
Mixture Designs: ANOVA• Augmented points as it appears on both the
simplex lattice and the simplex centroid designs.)3/1,3/1,3/1(
)1,0,0()5.0,5.0,0(
)0,0,1(
)0,5.0,5.0(
)1,0,0(
)5.0,5.0,0(
)5.0,0,5.0(
)0,1,0(
)3/2,6/1,6/1()6/1,3/2,6/1(
(6/1,6/1,3/2)
(3/1,3/1,3/1)
Mixture Designs
Mixture Designs: ANOVA• Pure Error and Lack of Fit Test:
– Replicated Runs should exist to enable it– If the residuals variability >> the pure error
variability, then it can be concluded that there are differences in the blends that your model cannot explain, and hence there's lack of fit.
Mixture Designs
Mixture Designs: ANOVA• Sequential Build up of Model:
– If one is not sure about the degree of the model, start with simple linear model and go up.
– Model significance is statistically tested each time a higher order is fitted.
– test continues until there's no significant improvement in the model fit.
Mixture Designs
Mixture Designs: ANOVA• Sequential Build up of Model: Example
ANOVA; Var.:DV (mixt4.sta)
3 Factor mixture design; Mixture total=1., 14 Runs
Sequential fit of models of increasing complexity
SSdfMSSSdfMS R-sqr
ModelEffectEffectEffectErrorErrorErrorFpR-sqrAdj.
Linear44.755222.37846.872114.26115.25160.02510.48840.3954
Quadratic30.558310.18616.31482.03934.99490.03070.8220.7107
Special Cubic0.71910.71915.59672.22790.32250.58780.82980.6839
Cubic8.22932.7437.36741.84171.48930.34520.91960.7387
Total Adjusted91.627137.048
Mixture DesignsSimplex Lattice & Simplex
Centroid: Comparison• Based on a 3 component, 10 runs
experiment.
– {3,3} simplex lattice design.– 3 component simplex centroid design with 3
augmented points
Mixture DesignsSimplex Lattice & Simplex
Centroid: Comparison• {3,3} simplex lattice (left), 3 component augmented
simplex centroid (right))0,0,1(
)3/2,0,3/1(
)3/1,0,3/2()0,3/1,3/2(
)0,3/2,3/1(
)3/2,3/1,0()3/1,3/2,0( )1,0,0()0,1,0(
)3/1,3/1,3/1(
)3/1,3/1,3/1(
)1,0,0()5.0,5.0,0(
)5.0,0,5.0(
)0,5.0,5.0(
)1,0,0(
)5.0,5.0,0(
)5.0,0,5.0(
)0,1,0(
)3/1,3/1,3/1(
)6/1,6/1,3/2(
)3/2,6/1,6/1()6/1,3/2,6/1(
Mixture DesignsSimplex Lattice & Simplex
Centroid: Comparison• Fitted Model:
– Simplex lattice:• Supports a cubic model fit:
• The last three terms allows studying changes in response shape of orders higher than quadratic for binary blends.
)()()( 323223313113212112
221123322331132112332211
xxxxxxxxxxxxxxxxxxxxxxxxy
Mixture DesignsSimplex Lattice & Simplex
Centroid: Comparison• Fitted Model:
– Simplex Centroid:• Supports fitting a special quadratic model:
• This model helps to detect curvature of the response surface in the interior of the triangle, which cannot be done by the cubic model of the simplex lattice design
232112333
221122332
211123
322331132112332211
xxxxxxxxx
xxxxxxxxxy
Mixture DesignsSimplex Lattice & Simplex
Centroid: Comparison• Information Distribution throughout the
experimental Region:– Simplex Centroid: More uniform distribution in
the interior of the triangle.– Simplex Lattice: More information about
response surface behavior for binary blends.
Mixture DesignsSimplex Lattice & Simplex
Centroid: Comparison• Model Sequential Buildup:
– Augment Simplex Centroid is more powerful, where power refers to the rejection of zero lack of fit when the true surface is more complicated than it can be described by terms in the fitted model.
Mixture Designs
Important Remarks• Lower Bounds:
– New space is scaled.– Use of pseudo-components.
Mixture Designs
Important Remarks• Upper and Lower Bounds:
– Simplex space is no longer a triangle.– Cannot use standard simplex lattice or
centroid designs to predict the response.
Mixture Designs
Important Remarks• Mixture-amount Experiments
– If the third assumption of mixture experiments is violated, that is, the mixture depends on both the components proportions and the mixture amount, the experiment becomes Mixture-amount experiments.
Mixture Designs
References
Cornell, J.A., Experiments With Mixtures. John Wiley & Sons Inc. 1990Experimental Designs by statease.pdf (Statease software help file)http://www.statsoft.com/textbook/stexdes.htmlhttp://www.6sigma.us/handbook/pri/section5/pri54.htm
Mixture Designs
Thank You!
Questions?