optimizeResponse

Predictor and response values at response surface maximum or minimum of linear regression model

Since R2026a

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Syntax

optVals = optimizeResponse(mdl,"max")
optVals = optimizeResponse(mdl,"min")
optVals = optimizeResponse(___,Name=Value)

Description

optVals = optimizeResponse(mdl,"max") returns the predictor values and response value at the maximum of the response surface of the linear regression model mdl. The response surface is bounded by the minimum and maximum values of each predictor in the model.

Note

This function requires Optimization Toolbox™. Global Optimization Toolbox is also required if mdl has interaction terms with categorical variables whose values are not fixed using CategoricalValues.

example

optVals = optimizeResponse(mdl,"min") returns values at the minimum of the response surface.

example

optVals = optimizeResponse(___,Name=Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in the previous syntaxes. For example, you can specify to search for a maximum or minimum response value using fixed values for categorical predictors.

example

Examples

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This example uses:

Open Live Script

Fit a linear regression model and find the response and predictor values at maximum and minimum of the model's response surface.

Load the hald data set. Create a table containing the response variable, which is the heat evolved during hardening of a cement mixture, and the first two predictors, which are the amount of tricalcium aluminate and tricalcium silicate in the mixture.

load hald
X = ingredients(:,1:2);
tbl = array2table([heat X],VariableNames=["heat" "x1" "x2"]);

Fit a linear regression model by using fitlm.

mdl = fitlm(tbl,"heat")
mdl = 
Linear regression model:
    heat ~ 1 + x1 + x2

Estimated Coefficients:
                   Estimate       SE       tStat       pValue  
                   ________    ________    ______    __________

    (Intercept)     52.577       2.2862    22.998    5.4566e-10
    x1              1.4683       0.1213    12.105    2.6922e-07
    x2             0.66225     0.045855    14.442     5.029e-08


Number of observations: 13, Error degrees of freedom: 10
Root Mean Squared Error: 2.41
R-squared: 0.979,  Adjusted R-Squared: 0.974
F-statistic vs. constant model: 230, p-value = 4.41e-09

Find the predictor values where the heat value is a maximum, and where the heat value is a minimum.

maxResp = optimizeResponse(mdl,"max")
maxResp=1×3 table
    x1    x2     heat 
    __    __    ______

    21    71    130.43


minResp = optimizeResponse(mdl,"min")
minResp=1×3 table
    x1    x2     heat 
    __    __    ______

    1     26    71.264

Visualize the response surface and the maximum and minimum values.

% Create a grid of predictor values
x1_range = linspace(min(X(:,1)),max(X(:,1)),30);
x2_range = linspace(min(X(:,2)),max(X(:,2)),30);
[X1, X2] = meshgrid(x1_range,x2_range);
predict_tbl = table(X1(:),X2(:),VariableNames=["x1" "x2"]);

% Compute the predicted response values
Y_pred = predict(mdl,predict_tbl); 
Y_pred_grid = reshape(Y_pred,size(X1));

% Plot the response surface
figure
surf(X1, X2, Y_pred_grid)
xlabel("x1")
ylabel("x2")
zlabel("Predicted Heat")
colorbar
shading interp

% Add the maximum and minimum values to the plot
hold on
plot3(maxResp.x1,maxResp.x2,maxResp.heat,"ro", ...
MarkerSize=5,MarkerFaceColor="r")
text(maxResp.x1,maxResp.x2,maxResp.heat,"  Maximum",Color="r")
plot3(minResp.x1,minResp.x2,minResp.heat,"ro", ...
MarkerSize=5,MarkerFaceColor="b")
text(minResp.x1,minResp.x2,minResp.heat,"  Minimum",Color="b")
hold off

Figure contains an axes object. The axes object with xlabel x1, ylabel x2 contains 5 objects of type surface, line, text. One or more of the lines displays its values using only markers

The plot shows that the predicted heat has a maximum value of 130.4 for x1 = 21 and x2 = 72. The predicted heat has a minimum value of 71.3 for x1 = 1 and x2 = 26.

This example uses:

Open Live Script

Fit a linear regression model to data that includes categorical predictors, and find the maximum predicted response value and the corresponding predictor values.

Load the patients data set, which contains simulated data for 100 hospital patients.

load patients

Convert the Gender and Smoker predictors to categorical.

Gender = categorical(Gender);
Smoker = categorical(Smoker);

Create a table containing the Age, Gender, Smoker, Weight, and Diastolic predictors.

tbl = table(Age,Gender,Smoker,Weight,Diastolic);

Fit a linear regression model to the Diastolic predictor using fitlm. Specify a model that includes linear terms for Age and Weight, and an interaction term involving Gender and Smoker.

mdl = fitlm(tbl,"Diastolic ~ Age + Weight + Gender*Smoker");

Find the maximum predicted response value over the Weight value range [100 200], subject to the additional constraints that the patient is a 40-year-old nonsmoker. In this example, Global Optimization Toolbox is required because the Gender predictor is categorical and its value is not fixed. If you have Global Optimization Toolbox, you can uncomment the lines below.

% maxResp = optimizeResponse(mdl,"max",LowerBounds=[40,100], ...
%    UpperBounds=[40,200],CategoricalValues={[] categorical(false)})

If you do not have Global Optimization toolbox, you can perform separate searches for a set of predictor values when Gender=Male and Gender=Female.

maxRespMale = optimizeResponse(mdl,"max",LowerBounds=[40,100], ...
UpperBounds=[40,200],CategoricalValues={"Male" categorical(false)})
maxRespMale=1×5 table
    Age    Gender    Smoker    Weight    Diastolic
    ___    ______    ______    ______    _________

    40      Male     false      200        80.22  


maxRespFemale = optimizeResponse(mdl,"max",LowerBounds=[40,100], ...
UpperBounds=[40,200],CategoricalValues={"Female" categorical(false)})
maxRespFemale=1×5 table
    Age    Gender    Smoker    Weight    Diastolic
    ___    ______    ______    ______    _________

    40     Female    false      200       80.172

The maximum predicted response in the specified weight range is 80.22 for a 40-year-old nonsmoker male.

Input Arguments

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Linear regression model object, specified as a LinearModel object created by using fitlm or stepwiselm, or a CompactLinearModel object created by using compact.

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: optVals = optimizeResponse(mdl,"max",CategoricalValues=["nonsmoker" "female"]) finds the maximum predicted response when the values of the categorical predictors in mdl are "nonsmoker" and "female", respectively.

Fixed values for the categorical predictors, specified as a numeric vector, logical vector, string vector, categorical vector, cell array of character vectors, cell array, or table. The elements of CategoricalValues must have the same order as the categorical predictors in mdl.VariableNames. If an element of CategoricalValues is empty ([]), the function searches over all the unique values of the corresponding predictor in mdl.Variables.

Example: CategoricalValues={[] "male"}

Data Types: single | double | logical | char | string | cell | categorical | table

Optimization options, specified as an options object. If mdl has interaction terms with categorical variables whose values are not fixed using CategoricalValues, specify options for the ga solver. Otherwise, specify options for the fmincon solver. For more information about optimization options objects, see optimoptions (Optimization Toolbox).

Example: Options = myOptions

Lower bounds for the continuous predictors, specified as a numeric vector with length equal to the number of continuous predictors in mdl. Use this option to specify the lower bounds of the search range. The values in LowerBounds must follow the order of the predictors in mdl.VariableInfo. If you do not specify LowerBounds, the function searches down to the minimum value of each continuous predictor in mdl.Variables.

Example: LowerBounds=[1 -Inf 0]

Data Types: single | double

Upper bounds for the continuous predictors, specified as a numeric vector with length equal to the number of continuous predictors in mdl. Use this option to specify the upper bounds of the search range. The values in UpperBounds must follow the order of the predictors in mdl.VariableInfo. If you do not specify UpperBounds, the function searches up to the maximum value of each continuous predictor in mdl.Variables.

Example: UpperBounds=[1 Inf 0]

Data Types: single | double

Output Arguments

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Predictor and response values at the response surface minimum or maximum, returned as a table.

Version History

Introduced in R2026a

See Also

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