mixtureDOE

Design of experiments (DOE) for mixture experiments

Since R2024b

Description

A mixtureDOE object contains a design for a mixture experiment. The elements of a point 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. The object properties include information about the design, model, and algorithm used to generate the design.

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.

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

dmix = mixtureDOE(___,A,b) specifies linear constraints for the design points 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 point x.

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 points.

Input Arguments

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.

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 points:

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.

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 points:

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.

Example: b=[1 5 2]

Data Types: single | double

Name-Value Arguments

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 points. The possible values for Degree depend on the value of Method.

Method	Values for Degree	Number of Design Points
"simplex-centroid"	1 to n	$${\displaystyle \sum _{i=1}^d\left(\begin{array}{c}n\\ i\end{array}\right)}$$
"simplex-lattice"	Integers greater than 0	$$\left(\begin{array}{c}n+d-1\\ d\end{array}\right)$$
"extreme-vertices"	0 to n – 2	mixtureDOE uses the shape of the design space to iteratively calculate design points. 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.

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.

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
"simplex-centroid"	This method generates design points that contain elements equal to 0, 1, or a fraction. The fraction elements of a point 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 points (1,0,0), (1/2,1/2,0), and (1/3,1/3,1/3).
"simplex-lattice"	This method generates design points that contain elements equal to 0, 1, or a fraction. The fraction elements of a point have denominators less than or equal to the number of factors in the model. For example, a design with three factors includes the points (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 points 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 points at the corners and center of the design space. mixtureDOE adds points to the design depending on its degree. Degree 0 — No additional points are added. Degree d — mixtureDOE generates additional points by first taking all possible groupings of two points in the design and then calculating the average of the groupings. The function adds the new points to the design before taking the averages from all possible groupings of three. This process repeats up to groups of d+1 points. 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.

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"	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, 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: $${\displaystyle \sum _{i=1}^n{b}_i{x}_i}+{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j<i}^{n-1}{b}_{ij}{x}_i{x}_j}}$$
  "scheffe-special-cubic"	The model is given by the formula: $${\displaystyle \sum _{i=1}^n{b}_i{x}_i}+{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j<i}^{n-1}{b}_{ij}{x}_i{x}_j}}+{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j<i}^{n-1}{\displaystyle \sum _{k<j}^{n-2}{b}_{ijk}{x}_i{x}_j{x}_k}}}$$
  "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, x1*x2, and x1*x22 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 factor 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 points.

Example: ModelSpecification="Factor1+Factor2*Factor3"

Example: ModelSpecification="scheffe-special-cubic"

Data Types: single | double | char | string

Properties

Degree — Degree of design

positive integer

This property is read-only.

Degree of the design, specified as a positive integer. This property is set by the Degree name-value argument when you create the mixtureDOE object.

Data Types: single | double

Design — Generated design points

table

This property is read-only.

Generated design points, specified as a table. Each column of Design corresponds to a factor in the design, and each row corresponds to a point.

Data Types: table

Bounds — Factor bounds

2-by-n numeric matrix

This property is read-only.

Factor bounds, specified as a 2-by-n numeric matrix. This property is set by the bounds input argument when you create the mixtureDOE object.

Data Types: double

LinearConstraints — Linear constraints

numeric matrix

This property is read-only.

Linear constraints, specified as a numeric matrix.

This property is set by the A and b input arguments when you create the mixtureDOE object. When you specify A and b, LinearConstraints is [A,b].

Data Types: single | double

Method — Algorithm for generating design

"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". This property is set by the Method name-value argument when you create the mixtureDOE object.

Data Types: string

ModelSpecification — Experiment model

formula in Wilkinson notation

This property is read-only.

Experiment model, specified 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.

This property is set by the ModelSpecification name-value argument when you create the mixtureDOE object.

Data Types: string

Object Functions

fitlm

Fit linear regression model using design points

Examples

Generate Mixture Design

Generate a mixture design for three factors.

dmix = mixtureDOE(3)

dmix = 
  mixtureDOE with properties:

                Design: [7x3 table]
    ModelSpecification: "Factor1 + Factor2 + Factor3"
                Bounds: [2x3 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 points, 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 points.

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 points in the mixture design. The elements of each point sum to one.

Specify Factor Bounds and Fit Linear Model

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

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

dmix = 
  mixtureDOE with properties:

                Design: [5x3 table]
    ModelSpecification: "Factor1 + Factor2 + Factor3"
                Bounds: [2x3 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 points.

Generate some response data for the design points.

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

Fit a linear model using the fitlm function. Specify the design points 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

Specify bounds and linear constraints for the design points in a mixture experiment. The sum of the lower bounds for the design points 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 points that satisfy the bounds and linear constraints.

dmix = mixtureDOE(bounds,A,b)

dmix = 
  mixtureDOE with properties:

                Design: [7x3 table]
    ModelSpecification: "Factor1 + Factor2 + Factor3"
                Bounds: [2x3 double]
                Degree: 0
                Method: "extreme-vertices"
     LinearConstraints: [3x4 double]

dmix is a mixtureDOE object that contains information about the generated mixture design. The design points 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

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 points, and degree for the design.

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

dmix = 
  mixtureDOE with properties:

                Design: [66x3 table]
    ModelSpecification: "ingredient1 + ingredient2 + ingredient3"
                Bounds: [2x3 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 points using the simplex-lattice method, and the degree of the design is 10. Together with the method, the degree controls how many design points 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 points.

More About

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)

Wilkinson Notation

Wilkinson notation describes the terms in a model. The notation relates to the terms included in the model, not to the multipliers (coefficients) of those terms.

Wilkinson notation uses these symbols:

• + means include the next variable.

• – means do not include the next variable.

• : defines an interaction, which is a product of the terms.

• * defines an interaction and all lower order terms.

• ^ raises the predictor to a power, exactly as in * repeated, so ^ includes lower order terms as well.

• () groups the terms.

This table shows typical examples of Wilkinson notation.

Wilkinson Notation	Terms in Standard Notation
1	Constant (intercept) term
x1^k, where k is a positive integer	x1, x12, ..., x1k
x1 + x2	x1, x2
x1*x2	x1, x2, x1*x2
x1:x2	x1*x2 only
–x2	Do not include x2
x1*x2 + x3	x1, x2, x3, x1*x2
x1 + x2 + x3 + x1:x2	x1, x2, x3, x1*x2
x1*x2*x3 – x1:x2:x3	x1, x2, x3, x1*x2, x1*x3, x2*x3
x1*(x2 + x3)	x1, x2, x3, x1*x2, x1*x3

For more details, see Wilkinson Notation.

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