crossval

Estimate loss using cross-validation

Syntax

err = crossval(criterion,X,y,'Predfun',predfun)

err = crossval(criterion,X1,...,XN,y,'Predfun',predfun)

values = crossval(fun,X)

values = crossval(fun,X1,...,XN)

___ = crossval(___,Name,Value)

Description

err = crossval(criterion,X,y,'Predfun',predfun) returns a 10-fold cross-validation error estimate for the function predfun based on the specified criterion, either 'mse' (mean squared error) or 'msc' (misclassification rate). The rows of X and y correspond to observations, and the columns of X correspond to predictor variables.

For more information, see General Cross-Validation Steps for predfun.

err = crossval(criterion,X1,...,XN,y,'Predfun',predfun) returns a 10-fold cross-validation error estimate for predfun by using the predictor variables X1 through XN and the response variable y.

values = crossval(fun,X) performs 10-fold cross-validation for the function fun, applied to the data in X. The rows of X correspond to observations, and the columns of X correspond to variables.

For more information, see General Cross-Validation Steps for fun.

values = crossval(fun,X1,...,XN) performs 10-fold cross-validation for the function fun, applied to the data in X1,...,XN. Every data set, X1 through XN, must have the same number of observations and, therefore, the same number of rows.

___ = crossval(___,Name,Value) specifies cross-validation options using one or more name-value pair arguments in addition to any of the input argument combinations and output arguments in previous syntaxes. For example, 'KFold',5 specifies to perform 5-fold cross-validation.

Examples

Compute Mean Squared Error Using Cross-Validation

Compute the mean squared error of a regression model by using 10-fold cross-validation.

Load the carsmall data set. Put the acceleration, horsepower, weight, and miles per gallon (MPG) values into the matrix data. Remove any rows that contain NaN values.

load carsmall
data = [Acceleration Horsepower Weight MPG];
data(any(isnan(data),2),:) = [];

Specify the last column of data, which corresponds to MPG, as the response variable y. Specify the other columns of data as the predictor data X. Add a column of ones to X when your regression function uses regress, as in this example.

Note: regress is useful when you simply need the coefficient estimates or residuals of a regression model. If you need to investigate a fitted regression model further, create a linear regression model object by using fitlm. For an example that uses fitlm and crossval, see Compute Mean Absolute Error Using Cross-Validation.

y = data(:,4);
X = [ones(length(y),1) data(:,1:3)];

Create the custom function regf (shown at the end of this example). This function fits a regression model to training data and then computes predicted values on a test set.

Note: If you use the live script file for this example, the regf function is already included at the end of the file. Otherwise, you need to create this function at the end of your .m file or add it as a file on the MATLAB® path.

Compute the default 10-fold cross-validation mean squared error for the regression model with predictor data X and response variable y.

rng('default') % For reproducibility
cvMSE = crossval('mse',X,y,'Predfun',@regf)

cvMSE = 17.5399

This code creates the function regf.

function yfit = regf(Xtrain,ytrain,Xtest)
b = regress(ytrain,Xtrain);
yfit = Xtest*b;
end

Compute Misclassification Error Using Logistic Regression Model and Cross-Validation

Compute the misclassification error of a logistic regression model trained on numeric and categorical predictor data by using 10-fold cross-validation.

Load the patients data set. Specify the numeric variables Diastolic and Systolic and the categorical variable Gender as predictors, and specify Smoker as the response variable.

load patients
X1 = Diastolic;
X2 = categorical(Gender);
X3 = Systolic;
y = Smoker;

Create the custom function classf (shown at the end of this example). This function fits a logistic regression model to training data and then classifies test data.

Note: If you use the live script file for this example, the classf function is already included at the end of the file. Otherwise, you need to create this function at the end of your .m file or add it as a file on the MATLAB® path.

Compute the 10-fold cross-validation misclassification error for the model with predictor data X1, X2, and X3 and response variable y. Specify 'Stratify',y to ensure that training and test sets have roughly the same proportion of smokers.

rng('default') % For reproducibility
err = crossval('mcr',X1,X2,X3,y,'Predfun',@classf,'Stratify',y)

err = 0.1100

This code creates the function classf.

function pred = classf(X1train,X2train,X3train,ytrain,X1test,X2test,X3test)
Xtrain = table(X1train,X2train,X3train,ytrain, ...
    'VariableNames',{'Diastolic','Gender','Systolic','Smoker'});
Xtest = table(X1test,X2test,X3test, ...
    'VariableNames',{'Diastolic','Gender','Systolic'});
modelspec = 'Smoker ~ Diastolic + Gender + Systolic';
mdl = fitglm(Xtrain,modelspec,'Distribution','binomial');
yfit = predict(mdl,Xtest);
pred = (yfit > 0.5);
end

Determine Number of Clusters Using Cross-Validation

For a given number of clusters, compute the cross-validated sum of squared distances between observations and their nearest cluster center. Compare the results for one through ten clusters.

Load the fisheriris data set. X is the matrix meas, which contains flower measurements for 150 different flowers.

load fisheriris
X = meas;

Create the custom function clustf (shown at the end of this example). This function performs the following steps:

1. Standardize the training data.

2. Separate the training data into k clusters.

3. Transform the test data using the training data mean and standard deviation.

4. Compute the distance from each test data point to the nearest cluster center, or centroid.

5. Compute the sum of the squares of the distances.

Note: If you use the live script file for this example, the clustf function is already included at the end of the file. Otherwise, you need to create the function at the end of your .m file or add it as a file on the MATLAB® path.

Create a for loop that specifies the number of clusters k for each iteration. For each fixed number of clusters, pass the corresponding clustf function to crossval. Because crossval performs 10-fold cross-validation by default, the software computes 10 sums of squared distances, one for each partition of training and test data. Take the sum of those values; the result is the cross-validated sum of squared distances for the given number of clusters.

rng('default') % For reproducibility
cvdist = zeros(5,1);
for k = 1:10
    fun = @(Xtrain,Xtest)clustf(Xtrain,Xtest,k);
    distances = crossval(fun,X);
    cvdist(k) = sum(distances);
end

Plot the cross-validated sum of squared distances for each number of clusters.

plot(cvdist)
xlabel('Number of Clusters')
ylabel('CV Sum of Squared Distances')

In general, when determining how many clusters to use, consider the greatest number of clusters that corresponds to a significant decrease in the cross-validated sum of squared distances. For this example, using two or three clusters seems appropriate, but using more than three clusters does not.

This code creates the function clustf.

function distances = clustf(Xtrain,Xtest,k)
[Ztrain,Zmean,Zstd] = zscore(Xtrain);
[~,C] = kmeans(Ztrain,k); % Creates k clusters
Ztest = (Xtest-Zmean)./Zstd;
d = pdist2(C,Ztest,'euclidean','Smallest',1);
distances = sum(d.^2);
end

Compute Mean Absolute Error Using Cross-Validation

Compute the mean absolute error of a regression model by using 10-fold cross-validation.

Load the carsmall data set. Specify the Acceleration and Displacement variables as predictors and the Weight variable as the response.

load carsmall
X1 = Acceleration;
X2 = Displacement;
y = Weight;

Create the custom function regf (shown at the end of this example). This function fits a regression model to training data and then computes predicted car weights on a test set. The function compares the predicted car weight values to the true values, and then computes the mean absolute error (MAE) and the MAE adjusted to the range of the test set car weights.

Note: If you use the live script file for this example, the regf function is already included at the end of the file. Otherwise, you need to create this function at the end of your .m file or add it as a file on the MATLAB® path.

By default, crossval performs 10-fold cross-validation. For each of the 10 training and test set partitions of the data in X1, X2, and y, compute the MAE and adjusted MAE values using the regf function. Find the mean MAE and mean adjusted MAE.

rng('default') % For reproducibility
values = crossval(@regf,X1,X2,y)

values = 10×2

  319.2261    0.1132
  342.3722    0.1240
  214.3735    0.0902
  174.7247    0.1128
  189.4835    0.0832
  249.4359    0.1003
  194.4210    0.0845
  348.7437    0.1700
  283.1761    0.1187
  210.7444    0.1325

mean(values)

ans = 1×2

  252.6701    0.1129

This code creates the function regf.

function errors = regf(X1train,X2train,ytrain,X1test,X2test,ytest)
tbltrain = table(X1train,X2train,ytrain, ...
    'VariableNames',{'Acceleration','Displacement','Weight'});
tbltest = table(X1test,X2test,ytest, ...
    'VariableNames',{'Acceleration','Displacement','Weight'});
mdl = fitlm(tbltrain,'Weight ~ Acceleration + Displacement');
yfit = predict(mdl,tbltest);
MAE = mean(abs(yfit-tbltest.Weight));
adjMAE = MAE/range(tbltest.Weight);
errors = [MAE adjMAE];
end

Compute Misclassification Error Using PCA and Cross-Validation

Compute the misclassification error of a classification tree by using principal component analysis (PCA) and 5-fold cross-validation.

Load the fisheriris data set. The meas matrix contains flower measurements for 150 different flowers. The species variable lists the species for each flower.

load fisheriris

Create the custom function classf (shown at the end of this example). This function fits a classification tree to training data and then classifies test data. Use PCA inside the function to reduce the number of predictors used to create the tree model.

Note: If you use the live script file for this example, the classf function is already included at the end of the file. Otherwise, you need to create this function at the end of your .m file or add it as a file on the MATLAB® path.

Create a cvpartition object for stratified 5-fold cross-validation. By default, cvpartition ensures that training and test sets have roughly the same proportions of flower species.

rng('default') % For reproducibility
cvp = cvpartition(species,'KFold',5);

Compute the 5-fold cross-validation misclassification error for the classification tree with predictor data meas and response variable species.

cvError = crossval('mcr',meas,species,'Predfun',@classf,'Partition',cvp)

cvError = 0.1067

This code creates the function classf.

function yfit = classf(Xtrain,ytrain,Xtest)

% Standardize the training predictor data. Then, find the 
% principal components for the standardized training predictor
% data.
[Ztrain,Zmean,Zstd] = zscore(Xtrain);
[coeff,scoreTrain,~,~,explained,mu] = pca(Ztrain);

% Find the lowest number of principal components that account
% for at least 95% of the variability.
n = find(cumsum(explained)>=95,1);

% Find the n principal component scores for the standardized
% training predictor data. Train a classification tree model
% using only these scores.
scoreTrain95 = scoreTrain(:,1:n);
mdl = fitctree(scoreTrain95,ytrain);

% Find the n principal component scores for the transformed
% test data. Classify the test data.
Ztest = (Xtest-Zmean)./Zstd;
scoreTest95 = (Ztest-mu)*coeff(:,1:n);
yfit = predict(mdl,scoreTest95);

end

Create Confusion Matrix Using Cross-Validation

Create a confusion matrix from the 10-fold cross-validation results of a discriminant analysis model.

Note: Use classify when training speed is a concern. Otherwise, use fitcdiscr to create a discriminant analysis model. For an example that shows the same workflow as this example, but uses fitcdiscr, see Create Confusion Matrix Using Cross-Validation Predictions.

Load the fisheriris data set. X contains flower measurements for 150 different flowers, and y lists the species for each flower. Create a variable order that specifies the order of the flower species.

load fisheriris
X = meas;
y = species;
order = unique(y)

order = 3x1 cell
    {'setosa'    }
    {'versicolor'}
    {'virginica' }

Create a function handle named func for a function that completes the following steps:

• Take in training data (Xtrain and ytrain) and test data (Xtest and ytest).

• Use the training data to create a discriminant analysis model that classifies new data (Xtest). Create this model and classify new data by using the classify function.

• Compare the true test data classes (ytest) to the predicted test data values, and create a confusion matrix of the results by using the confusionmat function. Specify the class order by using 'Order',order.

func = @(Xtrain,ytrain,Xtest,ytest)confusionmat(ytest, ...
    classify(Xtest,Xtrain,ytrain),'Order',order);

Create a cvpartition object for stratified 10-fold cross-validation. By default, cvpartition ensures that training and test sets have roughly the same proportions of flower species.

rng('default') % For reproducibility
cvp = cvpartition(y,'Kfold',10);

Compute the 10 test set confusion matrices for each partition of the predictor data X and response variable y. Each row of confMat corresponds to the confusion matrix results for one test set. Aggregate the results and create the final confusion matrix cvMat.

confMat = crossval(func,X,y,'Partition',cvp);
cvMat = reshape(sum(confMat),3,3)

cvMat = 3×3

    50     0     0
     0    48     2
     0     1    49

Plot the confusion matrix as a confusion matrix chart by using confusionchart.

confusionchart(cvMat,order)

Input Arguments

criterion — Type of error estimate

'mse' | 'mcr'

Type of error estimate, specified as either 'mse' or 'mcr'.

Value	Description
'mse'	Mean squared error (MSE) — Appropriate for regression algorithms only
'mcr'	Misclassification rate, or proportion of misclassified observations — Appropriate for classification algorithms only

X — Data set

column vector | matrix | array

Data set, specified as a column vector, matrix, or array. The rows of X correspond to observations, and the columns of X generally correspond to variables. If you pass multiple data sets X1,...,XN to crossval, then all data sets must have the same number of rows.

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

y — Response data

column vector | character array

Response data, specified as a column vector or character array. The rows of y correspond to observations, and y must have the same number of rows as the predictor data X or X1,...,XN.

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

predfun — Prediction function

function handle

Prediction function, specified as a function handle. You must create this function as an anonymous function, a function defined at the end of the .m or .mlx file containing the rest of your code, or a file on the MATLAB^® path.

This table describes the required function syntax, given the type of predictor data passed to crossval.

Value	Predictor Data	Function Syntax
@myfunction	X	function yfit = myfunction(Xtrain,ytrain,Xtest) % Calculate predicted response ... end Xtrain — Subset of the observations in X used as training predictor data. The function uses Xtrain and ytrain to construct a classification or regression model. ytrain — Subset of the responses in y used as training response data. The rows of ytrain correspond to the same observations in the rows of Xtrain. The function uses Xtrain and ytrain to construct a classification or regression model. Xtest — Subset of the observations in X used as test predictor data. The function uses Xtest and the model trained on Xtrain and ytrain to compute the predicted values yfit. yfit — Set of predicted values for observations in Xtest. The yfit values form a column vector with the same number of rows as Xtest.
@myfunction	X1,...,XN	function yfit = myfunction(X1train,...,XNtrain,ytrain,X1test,...,XNtest) % Calculate predicted response ... end X1train,...,XNtrain — Subsets of the predictor data in X1,...,XN, respectively, that are used as training predictor data. The rows of X1train,...,XNtrain correspond to the same observations. The function uses X1train,...,XNtrain and ytrain to construct a classification or regression model. ytrain — Subset of the responses in y used as training response data. The rows of ytrain correspond to the same observations in the rows of X1train,...,XNtrain. The function uses X1train,...,XNtrain and ytrain to construct a classification or regression model. X1test,...,XNtest — Subsets of the observations in X1,...,XN, respectively, that are used as test predictor data. The rows of X1test,...,XNtest correspond to the same observations. The function uses X1test,...,XNtest and the model trained on X1train,...,XNtrain and ytrain to compute the predicted values yfit. yfit — Set of predicted values for observations in X1test,...,XNtest. The yfit values form a column vector with the same number of rows as X1test,...,XNtest.

Example: @(Xtrain,ytrain,Xtest)(Xtest*regress(ytrain,Xtrain));

Data Types: function_handle

fun — Function to cross-validate

function handle

Function to cross-validate, specified as a function handle. You must create this function as an anonymous function, a function defined at the end of the .m or .mlx file containing the rest of your code, or a file on the MATLAB path.

This table describes the required function syntax, given the type of data passed to crossval.

Value	Data	Function Syntax
@myfunction	X	function value = myfunction(Xtrain,Xtest) % Calculation of value ... end Xtrain — Subset of the observations in X used as training data. The function uses Xtrain to construct a model. Xtest — Subset of the observations in X used as test data. The function uses Xtest and the model trained on Xtrain to compute value. value — Quantity or variable. In most cases, value is a numeric scalar representing a loss estimate. value can also be an array, provided that the array size is the same for each partition of training and test data. If you want to return a variable output that can change size depending on the data partition, set value to be the cell scalar {output} instead.
@myfunction	X1,...,XN	function value = myfunction(X1train,...,XNtrain,X1test,...,XNtest) % Calculation of value ... end X1train,...,XNtrain — Subsets of the data in X1,...,XN, respectively, that are used as training data. The rows of X1train,...,XNtrain correspond to the same observations. The function uses X1train,...,XNtrain to construct a model. X1test,...,XNtest — Subsets of the data in X1,...,XN, respectively, that are used as test data. The rows of X1test,...,XNtest correspond to the same observations. The function uses X1test,...,XNtest and the model trained on X1train,...,XNtrain to compute value. value — Quantity or variable. In most cases, value is a numeric scalar representing a loss estimate. value can also be an array, provided that the array size is the same for each partition of training and test data. If you want to return a variable output that can change size depending on the data partition, set value to be the cell scalar {output} instead.

Data Types: function_handle

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: crossval('mcr',meas,species,'Predfun',@classf,'KFold',5,'Stratify',species) specifies to compute the stratified 5-fold cross-validation misclassification rate for the classf function with predictor data meas and response variable species.

'Holdout' — Fraction or number of observations used for holdout validation

[] (default) | scalar value in the range (0,1) | positive integer scalar

Fraction or number of observations used for holdout validation, specified as the comma-separated pair consisting of 'Holdout' and a scalar value in the range (0,1) or a positive integer scalar.

• If the Holdout value p is a scalar in the range (0,1), then crossval randomly selects and reserves approximately p*100% of the observations as test data.

• If the Holdout value p is a positive integer scalar, then crossval randomly selects and reserves p observations as test data.

In either case, crossval then trains the model specified by either fun or predfun using the rest of the data. Finally, the function uses the test data along with the trained model to compute either values or err.

You can use only one of these four name-value pair arguments: Holdout, KFold, Leaveout, and Partition.

Example: 'Holdout',0.3

Example: 'Holdout',50

Data Types: single | double

'KFold' — Number of folds

10 (default) | positive integer scalar greater than 1

Number of folds for k-fold cross-validation, specified as the comma-separated pair consisting of 'KFold' and a positive integer scalar greater than 1.

If you specify 'KFold',k, then crossval randomly partitions the data into k sets. For each set, the function reserves the set as test data, and trains the model specified by either fun or predfun using the other k – 1 sets. crossval then uses the test data along with the trained model to compute either values or err.

You can use only one of these four name-value pair arguments: Holdout, KFold, Leaveout, and Partition.

Example: 'KFold',5

Data Types: single | double

'Leaveout' — Leave-one-out cross-validation

[] (default) | 1

Leave-one-out cross-validation, specified as the comma-separated pair consisting of 'Leaveout' and 1.

If you specify 'Leaveout',1, then for each observation, crossval reserves the observation as test data, and trains the model specified by either fun or predfun using the other observations. The function then uses the test observation along with the trained model to compute either values or err.

You can use only one of these four name-value pair arguments: Holdout, KFold, Leaveout, and Partition.

Example: 'Leaveout',1

Data Types: single | double

'MCReps' — Number of Monte Carlo repetitions

1 (default) | positive integer scalar

Number of Monte Carlo repetitions for validation, specified as the comma-separated pair consisting of 'MCReps' and a positive integer scalar. If the first input of crossval is 'mse' or 'mcr' (see criterion), then crossval returns the mean MSE or misclassification rate across all Monte Carlo repetitions. Otherwise, crossval concatenates the values from all Monte Carlo repetitions along the first dimension.

If you specify both Partition and MCReps, then the first Monte Carlo repetition uses the partition information in the cvpartition object, and the software calls the repartition object function to generate new partitions for each of the remaining repetitions.

Example: 'MCReps',5

Data Types: single | double

'Partition' — Cross-validation partition

[] (default) | cvpartition partition object

Cross-validation partition, specified as the comma-separated pair consisting of 'Partition' and a cvpartition partition object created by cvpartition. The partition object specifies the type of cross-validation and the indexing for the training and test sets.

When you use crossval, you cannot specify both Partition and Stratify. Instead, directly specify a stratified partition when you create the cvpartition partition object.

You can use only one of these four name-value pair arguments: Holdout, KFold, Leaveout, and Partition.

'Stratify' — Variable specifying groups used for stratification

column vector

Variable specifying the groups used for stratification, specified as the comma-separated pair consisting of 'Stratify' and a column vector with the same number of rows as the data X or X1,...,XN.

When you specify Stratify, both the training and test sets have roughly the same class proportions as in the Stratify vector. The software treats NaNs, empty character vectors, empty strings, <missing> values, and <undefined> values in Stratify as missing data values, and ignores the corresponding rows of the data.

A good practice is to use stratification when you use cross-validation with classification algorithms. Otherwise, some test sets might not include observations for all classes.

When you use crossval, you cannot specify both Partition and Stratify. Instead, directly specify a stratified partition when you create the cvpartition partition object.

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

'Options' — Options for running in parallel and setting random streams

structure

Options for running computations in parallel and setting random streams, specified as a structure. Create the Options structure with statset. This table lists the option fields and their values.

Field Name	Value	Default
UseParallel	Set this value to true to run computations in parallel.	false
UseSubstreams	Set this value to true to run computations in parallel in a reproducible manner. To compute reproducibly, set Streams to a type that allows substreams: 'mlfg6331_64' or 'mrg32k3a'.	false
Streams	Specify this value as a RandStream object or a cell array consisting of one such object.	If you do not specify Streams, then crossval uses the default stream.

Note

You need Parallel Computing Toolbox™ to run computations in parallel.

Example: 'Options',statset('UseParallel',true)

Data Types: struct

Output Arguments

err — Mean squared error or misclassification rate

numeric scalar

Mean squared error or misclassification rate, returned as a numeric scalar. The type of error depends on the criterion value.

values — Loss values

column vector | matrix

Loss values, returned as a column vector or matrix. Each row of values corresponds to the output of fun for one partition of training and test data.

If the output returned by fun is multidimensional, then crossval reshapes the output and fits it into one row of values. For an example, see Create Confusion Matrix Using Cross-Validation.

Tips

• A good practice is to use stratification (see Stratify) when you use cross-validation with classification algorithms. Otherwise, some test sets might not include observations for all classes.

Algorithms

General Cross-Validation Steps for predfun

When you use predfun, the crossval function typically performs 10-fold cross-validation as follows:

1. Split the observations in the predictor data X and the response variable y into 10 groups, each of which has approximately the same number of observations.

2. Use the last nine groups of observations to train a model as specified in predfun. Use the first group of observations as test data, pass the test predictor data to the trained model, and compute predicted values as specified in predfun. Compute the error specified by criterion.

3. Use the first group and the last eight groups of observations to train a model as specified in predfun. Use the second group of observations as test data, pass the test data to the trained model, and compute predicted values as specified in predfun. Compute the error specified by criterion.

4. Proceed in a similar manner until each group of observations is used as test data exactly once.

5. Return the mean error estimate as the scalar err.

General Cross-Validation Steps for fun

When you use fun, the crossval function typically performs 10-fold cross-validation as follows:

1. Split the data in X into 10 groups, each of which has approximately the same number of observations.

2. Use the last nine groups of data to train a model as specified in fun. Use the first group of data as a test set, pass the test set to the trained model, and compute some value (for example, loss) as specified in fun.

3. Use the first group and the last eight groups of data to train a model as specified in fun. Use the second group of data as a test set, pass the test set to the trained model, and compute some value as specified in fun.

4. Proceed in a similar manner until each group of data is used as a test set exactly once.

5. Return the 10 computed values as the vector values.

Alternative Functionality

Many classification and regression functions allow you to perform cross-validation directly.

• When you use fit functions such as fitcsvm, fitctree, and fitrtree, you can specify cross-validation options by using name-value pair arguments. Alternatively, you can first create models with these fit functions and then create a partitioned object by using the crossval object function. Use the kfoldLoss and kfoldPredict object functions to compute the loss and predicted values for the partitioned object. For more information, see ClassificationPartitionedModel and RegressionPartitionedModel.

• You can also specify cross-validation options when you perform lasso or elastic net regularization using lasso and lassoglm.

Extended Capabilities

Automatic Parallel Support

Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.

To run in parallel, specify the 'Options' name-value argument in the call to this function and set the 'UseParallel' field of the options structure to true using statset.

For example: 'Options',statset('UseParallel',true)

For more information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).