mixtureDOE

Design of experiments (DOE) for mixture experiments

Since R2024b

Description

A mixtureDOE object contains a design for a mixture experiment, and additional information about the design, model, and algorithm used to generate the design. The elements of a run in a mixture design sum to one. Use the properties of a mixtureDOE object to investigate the relative effects of a set of factors on a response variable.

Creation

Syntax

dmix = mixtureDOE(n)

dmix = mixtureDOE(bounds)

dmix = mixtureDOE(___,A,b)

dmix = mixtureDOE(___,Name=Value)

Description

dmix = mixtureDOE(n) generates a mixture design for n factors and returns the design information in a mixtureDOE object dmix.

example

dmix = mixtureDOE(bounds) specifies the number of factors and the bounds for the design runs. The elements of bounds must be between 0 and 1.

example

dmix = mixtureDOE(___,A,b) specifies linear constraints for the design runs using any of the input argument combinations in the previous syntaxes. The linear constraints are given by the formula A * x ≤ b, for a design run x.

example

dmix = mixtureDOE(___,Name=Value) specifies additional options using one or more name-value arguments. For example, you can specify the experiment model and the algorithm used to generate the design runs.

example

Input Arguments

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`n` — Number of factors
positive integer

Number of factors in the design, specified as a positive integer.

Data Types: single | double

`bounds` — Factor bounds
2-by-n matrix

Factor bounds, specified as a 2-by-n matrix, where n is the number of factors in the design. Each column of bounds corresponds to a factor. The first row of bounds contains the lower bounds for the factors, and the second row contains the upper bounds.

The sum of the lower bounds for a mixture design must be less than or equal to one. The sum of the upper bounds must be greater than or equal to one.

This argument sets the Bounds property.

Example: [0.1 0.1 0; 0.5 0.7 0.7]

Data Types: single | double

`A` — Linear inequality constraints
m-by-n numeric matrix

Linear inequality constraints, specified as an m-by-n matrix, where n is the number of factors in the design and m is the number of constraints. You must also specify b when you specify A.

When you specify A and b, mixtureDOE uses the following constraints to calculate the design runs:

A(i,1)*Factor1+A(i,2)*Factor2+...+A(i,n)*FactorN≤b(i)

You can reverse the direction of the sign in the inequality by multiplying the elements of A and b by –1.

This argument sets the LinearConstraints property.

Example: A=[0.3 5 100; 0.1 7 90; 0.25 2 180]

Data Types: single | double

`b` — Linear inequality constraints
numeric vector with m elements

Linear inequality constraints, specified as a numeric vector with m elements, where m is the number of constraints. You must also specify A when you specify b.

When you specify b and A, mixtureDOE uses the following constraints to calculate the design runs:

A(i,1)*Factor1+A(i,2)*Factor2+...+A(i,n)*FactorN≤b(i)

You can reverse the direction of the sign in the inequality by multiplying the elements of A and b by –1.

This argument sets the LinearConstraints property.

Example: b=[1 5 2]

Data Types: single | double

Name-Value Arguments

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Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: mixtureDOE(3,A,b,Degree=1,FactorNames=["drug1","drug2","drug3"]) specifies the factor names and degree for a design with three factors and linear constraints.

`Degree` — Degree of design
positive integer

Degree of the design, specified as a positive integer. Together with the Method name-value argument, Degree controls the number and location of the design runs. The possible values for Degree depend on the value of Method.

`Method`	Values for `Degree`	Number of Design Runs
`"simplex-centroid"`	`1` to `n`	$\sum_{i = 1}^{d} (\begin{matrix} n \\ i \end{matrix})$
`"simplex-lattice"`	Integers greater than `0`	$(\begin{matrix} n + d - 1 \\ d \end{matrix})$
`"extreme-vertices"`	`0` to `n – 2`	`mixtureDOE` uses the shape of the design space to iteratively calculate design runs. The degree of the design specifies the number of iterations in the calculation. For more information, see `Method`.

In the above table, n is the number of factors in the design and d is the degree of the design. The default value for Degree is 0 when Method is "extreme-vertices", and n otherwise.

This argument sets the Degree property.

Example: Degree=5

Data Types: single | double

`FactorNames` — Factor names
string vector | cell array of character vectors

Factor names, specified as a string vector or a cell array of character vectors.

If you specify ModelSpecification as a character vector or string scalar formula in Wilkinson Notation, then FactorNames must contain only valid variable names.

Example: FactorNames=["compound1","compound2"]

Data Types: char | string | cell

`Method` — Algorithm for generating design
`"simplex-centroid"` | `"simplex-lattice"` | `"extreme-vertices"`

Algorithm for generating the mixture design, specified as one of the following values.

Value Description

Value	Description
`"simplex-centroid"`	This method generates design runs that contain elements equal to 0, 1, or a fraction. The fraction elements of a run are equal to each other, have numerators equal to 1, and have denominators less than or equal to the number of factors in the model. For example, a design with three factors includes the runs `(1,0,0)`, `(1/2,1/2,0)`, and `(1/3,1/3,1/3)`.
`"simplex-lattice"`	This method generates design runs that contain elements equal to 0, 1, or a fraction. The fraction elements of a run have denominators less than or equal to the number of factors in the model. For example, a design with three factors includes the runs `(1,0,0)`, `(1/2,1/2,0)`, and `(2/3,1/3,0)`.
`"extreme-vertices"`	This method uses the shape of the design space to generate runs for the design. The design space is a polygon whose elements satisfy the bounds and linear constraints for the mixture design. The design includes the runs located at the corners and center of the design space. `mixtureDOE` adds runs to the design depending on its degree. Degree 0 — No additional runs are added. Degree d — `mixtureDOE` generates additional runs by first taking all possible groupings of two runs in the design and then calculating the average of the groupings. The function adds the new runs to the design before taking the averages from all possible groupings of three. This process repeats up to groups of d+1 runs. You can specify the degree d of the design by using the `Degree` name-value argument.

"simplex-centroid"

This method generates design runs that contain elements equal to 0, 1, or a fraction. The fraction elements of a run are equal to each other, have numerators equal to 1, and have denominators less than or equal to the number of factors in the model.

For example, a design with three factors includes the runs (1,0,0), (1/2,1/2,0), and (1/3,1/3,1/3).

"simplex-lattice"

This method generates design runs that contain elements equal to 0, 1, or a fraction. The fraction elements of a run have denominators less than or equal to the number of factors in the model.

For example, a design with three factors includes the runs (1,0,0), (1/2,1/2,0), and (2/3,1/3,0).

"extreme-vertices"

This method uses the shape of the design space to generate runs for the design. The design space is a polygon whose elements satisfy the bounds and linear constraints for the mixture design.

The design includes the runs located at the corners and center of the design space. mixtureDOE adds runs to the design depending on its degree.

Degree 0 — No additional runs are added.
Degree d — mixtureDOE generates additional runs by first taking all possible groupings of two runs in the design and then calculating the average of the groupings. The function adds the new runs to the design before taking the averages from all possible groupings of three. This process repeats up to groups of d+1 runs.

You can specify the degree d of the design by using the Degree name-value argument.

When you pass the bounds input argument to mixtureDOE, the supported values for Method depend on the type of bounds you specify.

When you specify only lower bounds by setting all upper bounds equal to 1, Method can be any value listed in the table above, and the default value is "simplex-centroid".
When you specify only upper bounds by setting all lower bounds equal to 0, the value for Method depends on whether the design has an implied lower bound. The design has an implied lower bound if the sum of all the upper bounds minus the smallest upper bound is less than 1.
- No implied lower bound — Method can be any value listed in the table above, and the default value is "simplex-centroid".
- Implied lower bound — Method must be "extreme-vertices".

The default value for Method is "simplex-centroid" when you specify n, and "extreme-vertices" when you specify A and b.

This argument sets the Method property.

Example: Method="simplex-lattice"

Data Types: char | string

`ModelSpecification` — Experiment model
string scalar | character vector | t-by-p terms matrix

Experiment model, specified as one of the following values.

A character vector or string scalar with the model name.

Value	Model Description
`"linear"` (Default)	The model contains an intercept and linear term for each factor.
`"constant"`	The model contains only a constant (intercept) term.
`"interactions"`	The model contains an intercept, a linear term for each factor, and all products of pairs of distinct factors (no squared terms).
`"purequadratic"`	The model contains an intercept term, and linear and squared terms for each factor.
`"quadratic"`	The model contains an intercept term, linear and squared terms for each factor, and all products of pairs of distinct factors.
`"scheffe-linear"`	The model contains a linear term for each factor and does not include an intercept term.
`"scheffe-quad"`	The model is given by the formula: $\sum_{i = 1}^{n} b_{i} x_{i} + \sum_{i = 1}^{n} \sum_{j < i}^{n - 1} b_{i j} x_{i} x_{j}$
`"scheffe-special-cubic"`	The model is given by the formula: $\sum_{i = 1}^{n} b_{i} x_{i} + \sum_{i = 1}^{n} \sum_{j < i}^{n - 1} b_{i j} x_{i} x_{j} + \sum_{i = 1}^{n} \sum_{j < i}^{n - 1} \sum_{k < j}^{n - 2} b_{i j k} x_{i} x_{j} x_{k}$
`"polyijk"`	The model is a polynomial with all terms up to degree `i` in the first factor, degree `j` in the second factor, and so on. Specify the maximum degree for each factor by using numerals 0 though 9. The model contains interaction terms, but the degree of each interaction term does not exceed the maximum value of the specified degrees. For example, `"poly13"` has an intercept and x₁, x₂, x₂², x₂³, x₁x₂, and x₁x₂² terms, where x₁ and x₂ are the first and second factors, respectively.

In the above table, each x_i corresponds to the ith factor in the mixture design, and b_i, b_ij, b_ijk, and d_ij are coefficients for the model terms.

A character vector or string scalar formula in Wilkinson notation. The factor names in the formula must be valid variable names specified by FactorNames.
A t-by-n terms matrix, where t is the number of terms and n is the number of factors in the design. A terms matrix is convenient when the number of factors is large and you want to generate the terms programmatically. For more information about terms matrices, see Terms Matrix.

ModelSpecification does not include the response variable and does not affect the location of the design runs.

This argument sets the ModelSpecification property.

Example: ModelSpecification="Factor1+Factor2*Factor3"

Example: ModelSpecification="scheffe-special-cubic"

Data Types: single | double | char | string

Properties

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`Degree` — Degree of design
Read-only: positive integer

This property is read-only.

Degree of the design, specified as a positive integer.

Data Types: single | double

`Design` — Generated design runs
Read-only: table

This property is read-only.

Generated design runs, represented as a table. Each column of Design corresponds to a factor in the design, and each row corresponds to a run.

`Bounds` — Factor bounds
Read-only: 2-by-n numeric matrix

This property is read-only.

Factor bounds, specified as a 2-by-n numeric matrix.

Data Types: double

`LinearConstraints` — Linear constraints
Read-only: numeric matrix

This property is read-only.

Linear constraints, specified as a numeric matrix. When you specify A and b, LinearConstraints is [A,b].

Data Types: single | double

`Method` — Algorithm for generating design
Read-only: `"simplex-centroid"` | `"simplex-lattice"` | `"extreme-vertices"`

This property is read-only.

Algorithm for generating the design, specified as "simplex-centroid", "simplex-lattice", or "extreme-vertices".

Data Types: string

`ModelSpecification` — Experiment model
Read-only: formula in Wilkinson notation

This property is read-only.

Experiment model, represented as a formula in Wilkinson notation. ModelSpecification indicates the model you want to fit with the specified design. ModelSpecification does not include the response variable.

Data Types: string

Object Functions

fitlm Fit linear regression model using design runs

Examples

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Generate Mixture Design

Open Live Script

Generate a mixture design for three factors.

dmix = mixtureDOE(3)

dmix = 
  mixtureDOE with properties:

                Design: [7×3 table]
    ModelSpecification: "Factor1 + Factor2 + Factor3"
                Bounds: [2×3 double]
                Degree: 3
                Method: "simplex-centroid"
     LinearConstraints: []

dmix is a mixtureDOE object that contains information about the generated mixture design. The output displays the degree of the design, the method used to generate the design runs, and the size of the tables describing the design and factors. The output also displays the model for the design, although the model does not affect how the software generates design runs.

Display the design table.

dmix.Design

ans=7×3 table
    Factor1    Factor2    Factor3
    _______    _______    _______

          1          0          0
          0          1          0
          0          0          1
        0.5        0.5          0
        0.5          0        0.5
          0        0.5        0.5
    0.33333    0.33333    0.33333

The design table displays the values for the seven runs in the mixture design. The elements of each run sum to one.

Specify Factor Bounds and Fit Linear Model

Open Live Script

Generate a mixture design and specify bounds for the design runs.

dmix = mixtureDOE([0.01 0.5 0.3; 0.1 0.6 0.7])

dmix = 
  mixtureDOE with properties:

                Design: [5×3 table]
    ModelSpecification: "Factor1 + Factor2 + Factor3"
                Bounds: [2×3 double]
                Degree: 0
                Method: "extreme-vertices"
     LinearConstraints: []

dmix is a mixtureDOE object that contains information about the generated mixture design. Because factor bounds are specified for the design, the function uses the "extreme-vertices" method to generate the design runs.

Generate some response data for the design runs.

rng(0,"twister") %  For reproducibility
runs = dmix.Design;
h = height(runs);
response = 2*runs.Factor1+3*runs.Factor2+runs.Factor3+0.01*randn(h,1);

Fit a linear model using the fitlm function. Specify the design runs in dmix as the predictor data and response as the response data.

mdl = fitlm(dmix,response)

mdl = 
Linear regression model:
    y ~ Factor1 + Factor2 + Factor3

Estimated Coefficients:
               Estimate       SE       tStat      pValue  
               ________    ________    ______    _________

    Factor1     2.0104      0.18533    10.848    0.0083916
    Factor2     2.9234     0.071299    41.002    0.0005943
    Factor3     1.1118     0.086745    12.817    0.0060326


Number of observations: 5, Error degrees of freedom: 2
Root Mean Squared Error: 0.0156

mdl is a LinearModel object that contains the results of fitting a linear model to the data. The model display includes the model formula, estimated coefficients, and model summary statistics.

Specify Linear Constraints

Open Live Script

Specify bounds and linear constraints for the design runs in a mixture experiment. The sum of the lower bounds for the design runs must be less than or equal to 1.

bounds = [0.1 0.1 0; 0.5 0.7 0.7];
A = [-0.85 -0.9 -1; 0.85 0.9 1; -0.7 0 -1];
b = [-0.9; 0.95; -0.4];

Generate design runs that satisfy the bounds and linear constraints.

dmix = mixtureDOE(bounds,A,b)

dmix = 
  mixtureDOE with properties:

                Design: [7×3 table]
    ModelSpecification: "Factor1 + Factor2 + Factor3"
                Bounds: [2×3 double]
                Degree: 0
                Method: "extreme-vertices"
     LinearConstraints: [3×4 double]

dmix is a mixtureDOE object that contains information about the generated mixture design. The design runs satisfy the following system of inequalities:

-0.85*Factor1-0.9*Factor2-Factor3 ≤ -0.9

0.85*Factor1+0.9*Factor2+Factor3 ≤ 0.95

-0.7*Factor1-Factor3 ≤ -0.4

Inspect the LinearConstraints property of dmix.

dmix.LinearConstraints

ans = 3×4

   -0.8500   -0.9000   -1.0000   -0.9000
    0.8500    0.9000    1.0000    0.9500
   -0.7000         0   -1.0000   -0.4000

The first three columns of the matrix contain the left side of the system of inequalities. The last column contains the right side.

Specify Factor Names, Method, and Degree of Design

Open Live Script

Generate a mixture design for an experiment that measures the effectiveness of different amounts of three ingredients in a drug. Specify the factor names, method for generating the design runs, and degree for the design.

dmix = mixtureDOE(3,FactorNames=["ingredient1","ingredient2","ingredient3"],Degree=10,Method="simplex-lattice")

dmix = 
  mixtureDOE with properties:

                Design: [66×3 table]
    ModelSpecification: "ingredient1 + ingredient2 + ingredient3"
                Bounds: [2×3 double]
                Degree: 10
                Method: "simplex-lattice"
     LinearConstraints: []

dmix is a mixtureDOE object that contains information about the generated mixture design. The output shows that the function generates the design runs using the simplex-lattice method, and the degree of the design is 10. Together with the method, the degree controls how many design runs are generated.

Display the design table.

dmix.Design

ans=66×3 table
    ingredient1    ingredient2    ingredient3
    ___________    ___________    ___________

          1              0              0    
        0.9            0.1              0    
        0.9              0            0.1    
        0.8            0.2              0    
        0.8            0.1            0.1    
        0.8              0            0.2    
        0.7            0.3              0    
        0.7            0.2            0.1    
        0.7            0.1            0.2    
        0.7              0            0.3    
        0.6            0.4              0    
        0.6            0.3            0.1    
        0.6            0.2            0.2    
        0.6            0.1            0.3    
        0.6              0            0.4    
        0.5            0.5              0    
      ⋮

The design table displays the amount of each ingredient to be tested for the 66 design runs.

More About

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Terms Matrix

A terms matrix T is a t-by-n matrix specifying the terms in a model, where t is the number of terms, and n is the number of factors in the design. The value of T(i,j) is the exponent of variable j in term i.

For example, suppose that a design includes three factors x1, x2, and x3. Each row of T represents one term:

[0 0 0] — Constant term or intercept
[0 1 0] — x2; equivalently, x1^0 * x2^1 * x3^0
[1 0 1] — x1*x3
[2 0 0] — x1^2
[0 1 2] — x2*(x3^2)

References

[1] Cornell, J. A. Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data. United Kingdom: Wiley, 1990.

Version History

Introduced in R2024b

mixtureDOE

Description

Creation

Syntax

Description

Input Arguments

n — Number of factors positive integer

bounds — Factor bounds 2-by-n matrix

A — Linear inequality constraints m-by-n numeric matrix

b — Linear inequality constraints numeric vector with m elements

Name-Value Arguments

Degree — Degree of design positive integer

FactorNames — Factor names string vector | cell array of character vectors

Method — Algorithm for generating design "simplex-centroid" | "simplex-lattice" | "extreme-vertices"

ModelSpecification — Experiment model string scalar | character vector | t-by-p terms matrix

Properties

Degree — Degree of design Read-only: positive integer

Design — Generated design runs Read-only: table

Bounds — Factor bounds Read-only: 2-by-n numeric matrix

LinearConstraints — Linear constraints Read-only: numeric matrix

Method — Algorithm for generating design Read-only: "simplex-centroid" | "simplex-lattice" | "extreme-vertices"

ModelSpecification — Experiment model Read-only: formula in Wilkinson notation

Object Functions

Examples

Generate Mixture Design

Specify Factor Bounds and Fit Linear Model

Specify Linear Constraints

Specify Factor Names, Method, and Degree of Design

More About

Terms Matrix

References

Version History

See Also

`n` — Number of factors
positive integer

`bounds` — Factor bounds
2-by-n matrix

`A` — Linear inequality constraints
m-by-n numeric matrix

`b` — Linear inequality constraints
numeric vector with m elements

`Degree` — Degree of design
positive integer

`FactorNames` — Factor names
string vector | cell array of character vectors

`Method` — Algorithm for generating design
`"simplex-centroid"` | `"simplex-lattice"` | `"extreme-vertices"`

`ModelSpecification` — Experiment model
string scalar | character vector | t-by-p terms matrix

`Degree` — Degree of design
Read-only: positive integer

`Design` — Generated design runs
Read-only: table

`Bounds` — Factor bounds
Read-only: 2-by-n numeric matrix

`LinearConstraints` — Linear constraints
Read-only: numeric matrix

`Method` — Algorithm for generating design
Read-only: `"simplex-centroid"` | `"simplex-lattice"` | `"extreme-vertices"`

`ModelSpecification` — Experiment model
Read-only: formula in Wilkinson notation