# predict

Class: CompactClassificationSVM

Predict labels for support vector machine classifiers

## Syntax

• ``label = predict(SVMModel,TBL)``
• ``label = predict(SVMModel,X)``
example
• ``````[label,Score] = predict(___)``````
example

## Description

````label = predict(SVMModel,TBL)` returns a vector of predicted class labels for predictor data in the table `TBL`, based on the full or compact, trained SVM classifier `SVMModel`.```

example

````label = predict(SVMModel,X)` returns a vector of predicted class labels for predictor data in the matrix `X`, based on `SVMModel`.```

example

``````[label,Score] = predict(___)``` additionally returns class likelihood measures, i.e., either scores or posterior probabilities, using any of the previous syntaxes.```

## Input Arguments

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### `SVMModel` — SVM classifier`ClassificationSVM` classifier | `CompactClassificationSVM` classifier

SVM classifier, specified as a `ClassificationSVM` classifier or `CompactClassificationSVM` classifier returned by `fitcsvm` or `compact`, respectively.

### `TBL` — Sample datatable

Sample data, specified as a table. Each row of `TBL` corresponds to one observation, and each column corresponds to one predictor variable. Optionally, `TBL` can contain additional columns for the response variable and observation weights. `TBL` must contain all of the predictors used to train `SVMModel`. Multi-column variables and cell arrays other than cell arrays of character vectors are not allowed.

If you trained `SVMModel` using sample data contained in a `table`, then the input data for this method must also be in a table.

Data Types: `table`

### `X` — Predictor datanumeric matrix

Predictor data, specified as a numeric matrix.

Each row of `X` corresponds to one observation (also known as an instance or example), and each column corresponds to one variable (also known as a feature). The variables making up the columns of `X` must be the same as the variables that trained the `SVMModel` classifier.

The length of `Y` and the number of rows of `X` must be equal.

If you set `'Standardize',true` in `fitcsvm` to train `SVMModel`, then the software standardizes the columns of `X` using the corresponding means in `SVMModel.Mu` and standard deviations in `SVMModel.Sigma`.

Data Types: `double` | `single`

## Output Arguments

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### `label` — Predicted class labelscategorical array | character array | logical vector | vector of numeric values | cell array of character vectors

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

`label`:

For one-class learning, the elements of `label` are the one class represented in the observed class labels.

### `Score` — Predicted class scores or posterior probabilitiesnumeric column vector | numeric matrix

Predicted class scores or posterior probabilities, returned as a numeric column vector or numeric matrix.

• For one-class learning, `Score` is a column vector with the same number of rows as the training observations (`X`). The elements are the positive class scores for the corresponding observations. You cannot obtain posterior probabilities for one-class learning.

• For two-class learning, `Score` is a two-column matrix with the same number of rows as `X`.

• If you fit the optimal score-to-posterior probability transformation function using `fitPosterior` or `fitSVMPosterior`, then `Score` contains class posterior probabilities. That is, if the value of `SVMModel.ScoreTransform` is not `none`, then the elements of the first and second columns of `Score` are the negative class (`SVMModel.ClassNames{1}`) and positive class (`SVMModel.ClassNames{2}`) posterior probabilities for the corresponding observations, respectively.

• Otherwise, the elements of the first column are the negative class scores and the elements of the second column are the positive class scores for the corresponding observations.

If `SVMModel``.KernelParameters.Function` is `'linear'`, then the software estimates the classification score for the observation x using

`$f\left(x\right)=\left(x/s\right)\prime \beta +b.$`

`SVMModel` stores β, b, s in the properties `Beta`, `Bias`, and `KernelParameters``.Scale`, respectively.

## Definitions

### Classification Score

The SVM classification score for classifying observation x is the signed distance from x to the decision boundary ranging from -∞ to +∞. A positive score for a class indicates that x is predicted to be in that class, a negative score indicates otherwise.

The score for predicting x into the positive class, also the numerical, predicted response for x, $f\left(x\right)$, is the trained SVM classification function

`$f\left(x\right)=\sum _{j=1}^{n}{\alpha }_{j}{y}_{j}G\left({x}_{j},x\right)+b,$`

where $\left({\alpha }_{1},...,{\alpha }_{n},b\right)$ are the estimated SVM parameters, $G\left({x}_{j},x\right)$ is the dot product in the predictor space between x and the support vectors, and the sum includes the training set observations. The score for predicting x into the negative class is –f(x).

If G(xj,x) = xjx (the linear kernel), then the score function reduces to

`$f\left(x\right)=\left(x/s\right)\prime \beta +b.$`

s is the kernel scale and β is the vector of fitted linear coefficients.

### Posterior Probability

The probability that an observation belongs in a particular class, given the data.

For SVM, the posterior probability is a function of the score, P(s), that observation j is in class k = {-1,1}.

• For separable classes, the posterior probability is the step function

`$P\left({s}_{j}\right)=\left\{\begin{array}{l}\begin{array}{cc}0;& s<\underset{{y}_{k}=-1}{\mathrm{max}}{s}_{k}\end{array}\\ \begin{array}{cc}\pi ;& \underset{{y}_{k}=-1}{\mathrm{max}}{s}_{k}\le {s}_{j}\le \underset{{y}_{k}=+1}{\mathrm{min}}{s}_{k}\end{array}\\ \begin{array}{cc}1;& {s}_{j}>\underset{{y}_{k}=+1}{\mathrm{min}}{s}_{k}\end{array}\end{array},$`

where:

• sj is the score of observation j.

• +1 and –1 denote the positive and negative classes, respectively.

• π is the prior probability that an observation is in the positive class.

• For inseparable classes, the posterior probability is the sigmoid function

`$P\left({s}_{j}\right)=\frac{1}{1+\mathrm{exp}\left(A{s}_{j}+B\right)},$`

where the parameters A and B are the slope and intercept parameters.

### Prior Probability

The prior probability is the believed relative frequency that observations from a class occur in the population for each class.

## Examples

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### Label Test Sample Observations of SVM Classifiers

Load the `ionosphere` data set.

```load ionosphere rng(1); % For reproducibility ```

Train an SVM classifier. Specify a 15% holdout sample for testing. It is good practice to specify the class order and standardize the data.

```CVSVMModel = fitcsvm(X,Y,'Holdout',0.15,'ClassNames',{'b','g'},... 'Standardize',true); CompactSVMModel = CVSVMModel.Trained{1}; % Extract trained, compact classifier testInds = test(CVSVMModel.Partition); % Extract the test indices XTest = X(testInds,:); YTest = Y(testInds,:); ```

`CVSVMModel` is a `ClassificationPartitionedModel` classifier. It contains the property `Trained`, which is a 1-by-1 cell array holding a `CompactClassificationSVM` classifier that the software trained using the training set.

Label the test sample observations. Display the results for the first 10 observations in the test sample.

```[label,score] = predict(CompactSVMModel,XTest); table(YTest(1:10),label(1:10),score(1:10,2),'VariableNames',... {'TrueLabel','PredictedLabel','Score'}) ```
```ans = TrueLabel PredictedLabel Score _________ ______________ ________ 'b' 'b' -1.7177 'g' 'g' 2.0003 'b' 'b' -9.6841 'g' 'g' 2.5618 'b' 'b' -1.548 'g' 'g' 2.0984 'b' 'b' -2.7018 'b' 'b' -0.66291 'g' 'g' 1.6046 'g' 'g' 1.7731 ```

### Predict Labels and Posterior Probabilities of SVM Classifiers

A goal of classification is to predict labels of new observations using a trained algorithm. Many applications train algorithms on large data sets, which can use resources that are better used elsewhere. This example shows how to efficiently label new observations using an SVM classifier.

Load the ionosphere data set. Suppose that the last 10 observations become available after training the SVM classifier.

```load ionosphere n = size(X,1); % Training sample size isInds = 1:(n-10); % In-sample indices oosInds = (n-9):n; % Out-of-sample indices ```

Train an SVM classifier. It is good practice to standardize the predictors and specify the order of the classes. Conserve memory by reducing the size of the trained SVM classifier.

```SVMModel = fitcsvm(X(isInds,:),Y(isInds),'Standardize',true,... 'ClassNames',{'b','g'}); CompactSVMModel = compact(SVMModel); whos('SVMModel','CompactSVMModel') ```
``` Name Size Bytes Class Attributes CompactSVMModel 1x1 30560 classreg.learning.classif.CompactClassificationSVM SVMModel 1x1 138203 ClassificationSVM ```

The positive class is `'g'`. The `CompactClassificationSVM` classifier (`CompactSVMModel`) uses less space than the `ClassificationSVM` classifier (`SVMModel`) because the latter stores the data.

Estimate the optimal score-to-posterior-probability-transformation function.

```CompactSVMModel = fitPosterior(CompactSVMModel,... X(isInds,:),Y(isInds)) ```
```CompactSVMModel = classreg.learning.classif.CompactClassificationSVM ResponseName: 'Y' CategoricalPredictors: [] ClassNames: {'b' 'g'} ScoreTransform: '@(S)sigmoid(S,-1.968452e+00,3.121267e-01)' Alpha: [88x1 double] Bias: -0.2143 KernelParameters: [1x1 struct] Mu: [1x34 double] Sigma: [1x34 double] SupportVectors: [88x34 double] SupportVectorLabels: [88x1 double] ```

The optimal score transformation function (`CompactSVMModel.ScoreTransform`) is the sigmoid function because the classes are inseparable.

Predict the out-of-sample labels and positive class posterior probabilities. Since true labels are available, compare them with the predicted labels.

```[labels,PostProbs] = predict(CompactSVMModel,X(oosInds,:)); table(Y(oosInds),labels,PostProbs(:,2),'VariableNames',... {'TrueLabels','PredictedLabels','PosClassPosterior'}) ```
```ans = TrueLabels PredictedLabels PosClassPosterior __________ _______________ _________________ 'g' 'g' 0.98419 'g' 'g' 0.95545 'g' 'g' 0.67792 'g' 'g' 0.94448 'g' 'g' 0.98744 'g' 'g' 0.92482 'g' 'g' 0.97111 'g' 'g' 0.96986 'g' 'g' 0.97803 'g' 'g' 0.94361 ```

`PostProbs` is a 10-by-2 matrix; its first column is the negative class posterior probabilities, and second column is the positive class posterior probabilities corresponding to the new observations.

## Algorithms

• By default, the software computes optimal posterior probabilities using Platt's method [1]:

1. Performing 10-fold cross validation

2. Fitting the sigmoid function parameters to the scores returned from the cross validation

3. Estimating the posterior probabilities by entering the cross-validation scores into the fitted sigmoid function

• The software incorporates prior probabilities in the SVM objective function during training.

• For SVM, `predict` classifies observations into the class yielding the largest score (i.e., the largest posterior probability). The software accounts for misclassification costs by applying the average-cost correction before training the classifier. That is, given the class prior vector P, misclassification cost matrix C, and observation weight vector w, the software defines a new vector of observation weights (W) such that

`${W}_{j}={w}_{j}{P}_{j}\sum _{k=1}^{K}{C}_{jk}.$`

## References

[1] Platt, J. "Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods." In Advances in Large Margin Classifiers. MIT Press, 1999, pages 61–74.