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**Class: **ClassificationLinear

Predict labels for linear classification models

`Label = predict(Mdl,X)`

`Label = predict(Mdl,X,Name,Value)`

```
[Label,Score]
= predict(___)
```

returns
predicted class labels with additional options specified by one or
more `Label`

= predict(`Mdl`

,`X`

,`Name,Value`

)`Name,Value`

pair arguments. For example, you
can specify that columns in the predictor data correspond to observations.

`[`

also returns classification scores for
both classes using any of the previous syntaxes. `Label`

,`Score`

]
= predict(___)`Score`

contains
classification scores for each regularization strength in `Mdl`

.

`Mdl`

— Binary, linear classification model`ClassificationLinear`

model objectBinary, linear classification model, specified as a `ClassificationLinear`

model object.
You can create a `ClassificationLinear`

model object
using `fitclinear`

.

`X`

— Predictor datafull matrix | sparse matrix

Predictor data, specified as an *n*-by-*p* full or sparse matrix. This orientation of `X`

indicates that rows correspond to individual observations, and columns correspond to individual predictor variables.

If you orient your predictor matrix so that observations correspond to columns and specify `'ObservationsIn','columns'`

, then you might experience a significant reduction in computation time.

**Data Types: **`single`

| `double`

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`

.

`'ObservationsIn'`

— Predictor data observation dimension`'rows'`

(default) | `'columns'`

Predictor data observation dimension, specified as the comma-separated
pair consisting of `'ObservationsIn'`

and `'columns'`

or `'rows'`

.

If you orient your predictor matrix so that observations correspond
to columns and specify `'ObservationsIn','columns'`

,
then you might experience a significant reduction in optimization-execution
time.

`Label`

— Predicted class labelscategorical array | character array | logical matrix | numeric matrix | cell array of character vectors

Predicted class labels, returned as a categorical or character array, logical or numeric matrix, or cell array of character vectors.

In most cases, `Label`

is an *n*-by-*L*
array of the same data type as the observed class labels (`Y`

) used to
train `Mdl`

. (The software treats string arrays as cell arrays of character
vectors.)
*n* is the number of observations in `X`

and
*L* is the number of regularization strengths in
`Mdl.Lambda`

. That is,
`Label(`

is the predicted class label for observation * i*,

`j`

`i`

`Mdl.Lambda(``j`

)

.If `Y`

is a character array and *L* >
1, then `Label`

is a cell array of class labels.

`Score`

— Classification scoresnumeric array

Classification
scores, returned as a *n*-by-2-by-*L* numeric
array. *n* is the number of observations in `X`

and *L* is
the number of regularization strengths in `Mdl.Lambda`

. `Score(`

is
the score for classifying observation * i*,

`k`

`j`

`i`

`k`

`Mdl.Lambda(``j`

)

. `Mdl.ClassNames`

stores
the order of the classes.If `Mdl.Learner`

is `'logistic'`

,
then classification scores are posterior probabilities.

Load the NLP data set.

`load nlpdata`

`X`

is a sparse matrix of predictor data, and `Y`

is a categorical vector of class labels. There are more than two classes in the data.

The models should identify whether the word counts in a web page are from the Statistics and Machine Learning Toolbox™ documentation. So, identify the labels that correspond to the Statistics and Machine Learning Toolbox™ documentation web pages.

`Ystats = Y == 'stats';`

Train a binary, linear classification model using the entire data set, which can identify whether the word counts in a documentation web page are from the Statistics and Machine Learning Toolbox™ documentation.

```
rng(1); % For reproducibility
Mdl = fitclinear(X,Ystats);
```

`Mdl`

is a `ClassificationLinear`

model.

Predict the training-sample, or resubstitution, labels.

label = predict(Mdl,X);

Because there is one regularization strength in `Mdl`

, `label`

is column vectors with lengths equal to the number of observations.

Construct a confusion matrix.

ConfusionTrain = confusionchart(Ystats,label);

The model misclassifies only one `'stats'`

documentation page as being outside of the Statistics and Machine Learning Toolbox documentation.

Load the NLP data set and preprocess it as in Predict Training-Sample Labels. Transpose the predictor data matrix.

load nlpdata Ystats = Y == 'stats'; X = X';

Train a binary, linear classification model that can identify whether the word counts in a documentation web page are from the Statistics and Machine Learning Toolbox™ documentation. Specify to hold out 30% of the observations. Optimize the objective function using SpaRSA.

rng(1) % For reproducibility CVMdl = fitclinear(X,Ystats,'Solver','sparsa','Holdout',0.30,... 'ObservationsIn','columns'); Mdl = CVMdl.Trained{1};

`CVMdl`

is a `ClassificationPartitionedLinear`

model. It contains the property `Trained`

, which is a 1-by-1 cell array holding a `ClassificationLinear`

model that the software trained using the training set.

Extract the training and test data from the partition definition.

trainIdx = training(CVMdl.Partition); testIdx = test(CVMdl.Partition);

Predict the training- and test-sample labels.

labelTrain = predict(Mdl,X(:,trainIdx),'ObservationsIn','columns'); labelTest = predict(Mdl,X(:,testIdx),'ObservationsIn','columns');

Because there is one regularization strength in `Mdl`

, `labelTrain`

and `labelTest`

are column vectors with lengths equal to the number of training and test observations, respectively.

Construct a confusion matrix for the training data.

ConfusionTrain = confusionchart(Ystats(trainIdx),labelTrain);

The model misclassifies only three documentation pages as being outside of Statistics and Machine Learning Toolbox documentation.

Construct a confusion matrix for the test data.

ConfusionTest = confusionchart(Ystats(testIdx),labelTest);

The model misclassifies three documentation pages as being outside the Statistics and Machine Learning Toolbox, and two pages as being inside.

Estimate test-sample, posterior class probabilities, and determine the quality of the model by plotting a ROC curve. Linear classification models return posterior probabilities for logistic regression learners only.

Load the NLP data set and preprocess it as in Predict Test-Sample Labels.

load nlpdata Ystats = Y == 'stats'; X = X';

Randomly partition the data into training and test sets by specifying a 30% holdout sample. Identify the test-set indices.

```
cvp = cvpartition(Ystats,'Holdout',0.30);
idxTest = test(cvp);
```

Train a binary linear classification model. Fit logistic regression learners using SpaRSA. To hold out the test set, specify the partitioned model.

CVMdl = fitclinear(X,Ystats,'ObservationsIn','columns','CVPartition',cvp,... 'Learner','logistic','Solver','sparsa'); Mdl = CVMdl.Trained{1};

`Mdl`

is a `ClassificationLinear`

model trained using the training set specified in the partition `cvp`

only.

Predict the test-sample posterior class probabilities.

[~,posterior] = predict(Mdl,X(:,idxTest),'ObservationsIn','columns');

Because there is one regularization strength in `Mdl`

, `posterior`

is a matrix with 2 columns and rows equal to the number of test-set observations. Column *i* contains posterior probabilities of `Mdl.ClassNames(i)`

given a particular observation.

Obtain false and true positive rates, and estimate the AUC. Specify that the second class is the positive class.

[fpr,tpr,~,auc] = perfcurve(Ystats(idxTest),posterior(:,2),Mdl.ClassNames(2)); auc

auc = 0.9985

The AUC is `1`

, which indicates a model that predicts well.

Plot an ROC curve.

figure; plot(fpr,tpr) h = gca; h.XLim(1) = -0.1; h.YLim(2) = 1.1; xlabel('False positive rate') ylabel('True positive rate') title('ROC Curve')

The ROC curve and AUC indicate that the model classifies the test-sample observations almost perfectly.

To determine a good lasso-penalty strength for a linear classification model that uses a logistic regression learner, compare test-sample values of the AUC.

Load the NLP data set. Preprocess the data as in Predict Test-Sample Labels.

load nlpdata Ystats = Y == 'stats'; X = X';

Create a data partition that specifies to holdout 10% of the observations. Extract test-sample indices.

rng(10); % For reproducibility Partition = cvpartition(Ystats,'Holdout',0.10); testIdx = test(Partition); XTest = X(:,testIdx); n = sum(testIdx)

n = 3157

YTest = Ystats(testIdx);

There are 3157 observations in the test sample.

Create a set of 11 logarithmically-spaced regularization strengths from $$1{0}^{-6}$$ through $$1{0}^{-0.5}$$.

Lambda = logspace(-6,-0.5,11);

Train binary, linear classification models that use each of the regularization strengths. Optimize the objective function using SpaRSA. Lower the tolerance on the gradient of the objective function to `1e-8`

.

CVMdl = fitclinear(X,Ystats,'ObservationsIn','columns',... 'CVPartition',Partition,'Learner','logistic','Solver','sparsa',... 'Regularization','lasso','Lambda',Lambda,'GradientTolerance',1e-8)

CVMdl = classreg.learning.partition.ClassificationPartitionedLinear CrossValidatedModel: 'Linear' ResponseName: 'Y' NumObservations: 31572 KFold: 1 Partition: [1x1 cvpartition] ClassNames: [0 1] ScoreTransform: 'none' Properties, Methods

Extract the trained linear classification model.

Mdl1 = CVMdl.Trained{1}

Mdl1 = ClassificationLinear ResponseName: 'Y' ClassNames: [0 1] ScoreTransform: 'logit' Beta: [34023x11 double] Bias: [1x11 double] Lambda: [1x11 double] Learner: 'logistic' Properties, Methods

`Mdl`

is a `ClassificationLinear`

model object. Because `Lambda`

is a sequence of regularization strengths, you can think of `Mdl`

as 11 models, one for each regularization strength in `Lambda`

.

Estimate the test-sample predicted labels and posterior class probabilities.

[label,posterior] = predict(Mdl1,XTest,'ObservationsIn','columns'); Mdl1.ClassNames; posterior(3,1,5)

ans = 1.0000

`label`

is a 3157-by-11 matrix of predicted labels. Each column corresponds to the predicted labels of the model trained using the corresponding regularization strength. `posterior`

is a 3157-by-2-by-11 matrix of posterior class probabilities. Columns correspond to classes and pages correspond to regularization strengths. For example, `posterior(3,1,5)`

indicates that the posterior probability that the first class (label `0`

) is assigned to observation 3 by the model that uses `Lambda(5)`

as a regularization strength is 1.0000.

For each model, compute the AUC. Designate the second class as the positive class.

auc = 1:numel(Lambda); % Preallocation for j = 1:numel(Lambda) [~,~,~,auc(j)] = perfcurve(YTest,posterior(:,2,j),Mdl1.ClassNames(2)); end

Higher values of `Lambda`

lead to predictor variable sparsity, which is a good quality of a classifier. For each regularization strength, train a linear classification model using the entire data set and the same options as when you trained the model. Determine the number of nonzero coefficients per model.

Mdl = fitclinear(X,Ystats,'ObservationsIn','columns',... 'Learner','logistic','Solver','sparsa','Regularization','lasso',... 'Lambda',Lambda,'GradientTolerance',1e-8); numNZCoeff = sum(Mdl.Beta~=0);

In the same figure, plot the test-sample error rates and frequency of nonzero coefficients for each regularization strength. Plot all variables on the log scale.

figure; [h,hL1,hL2] = plotyy(log10(Lambda),log10(auc),... log10(Lambda),log10(numNZCoeff + 1)); hL1.Marker = 'o'; hL2.Marker = 'o'; ylabel(h(1),'log_{10} AUC') ylabel(h(2),'log_{10} nonzero-coefficient frequency') xlabel('log_{10} Lambda') title('Test-Sample Statistics') hold off

Choose the index of the regularization strength that balances predictor variable sparsity and high AUC. In this case, a value between $$1{0}^{-2}$$ to $$1{0}^{-1}$$ should suffice.

idxFinal = 9;

Select the model from `Mdl`

with the chosen regularization strength.

MdlFinal = selectModels(Mdl,idxFinal);

`MdlFinal`

is a `ClassificationLinear`

model containing one regularization strength. To estimate labels for new observations, pass `MdlFinal`

and the new data to `predict`

.

For linear classification models, the raw *classification
score* for classifying the observation *x*, a row vector,
into the positive class is defined by

$${f}_{j}(x)=x{\beta}_{j}+{b}_{j}.$$

For the model with regularization strength *j*, $${\beta}_{j}$$ is the estimated column vector of coefficients (the model property
`Beta(:,j)`

) and $${b}_{j}$$ is the estimated, scalar bias (the model property
`Bias(j)`

).

The raw classification score for classifying *x* into
the negative class is –*f*(*x*).
The software classifies observations into the class that yields the
positive score.

If the linear classification model consists of logistic regression learners, then the
software applies the `'logit'`

score transformation to the raw
classification scores (see `ScoreTransform`

).

Calculate with arrays that have more rows than fit in memory.

This function fully supports tall arrays. For more information, see Tall Arrays (MATLAB).

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

Use

`saveCompactModel`

,`loadCompactModel`

, and`codegen`

to generate code for the`predict`

function. Save a trained model by using`saveCompactModel`

. Define an entry-point function that loads the saved model by using`loadCompactModel`

and calls the`predict`

function. Then use`codegen`

to generate code for the entry-point function.This table contains notes about the arguments of

`predict`

. Arguments not included in this table are fully supported.Argument Notes and Limitations `Mdl`

For the usage notes and limitations of the model object, see Code Generation of the

`ClassificationLinear`

object.`X`

Must be a single-precision or double-precision matrix and can be variable-size.

If you specify

`'ObservationsIn','rows'`

(default), then the number of columns in`X`

must be`numel(Mdl.PredictorNames)`

. Rows and columns must correspond to observations and predictors, respectively.If you specify

`'ObservationsIn','columns'`

, then the number of rows in`X`

must be`numel(Mdl.PredictorNames)`

. Rows and columns must correspond to predictors and observations, respectively.

Name-value pair arguments Names in name-value pair arguments must be compile-time constants.

The value for the

`'ObservationsIn'`

name-value pair argument must be a compile-time constant. For example, to use the`'ObservationsIn','columns'`

name-value pair argument in the generated code, include`{coder.Constant('ObservationsIn'),coder.Constant('columns')}`

in the`-args`

value of`codegen`

.

For more information, see Introduction to Code Generation.

`ClassificationLinear`

| `confusionchart`

| `fitclinear`

| `loss`

| `perfcurve`

| `testcholdout`

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