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OneClassSVM

One-class support vector machine (SVM) for anomaly detection

    Description

    Use a one-class support vector machine model object OneClassSVM for outlier detection and novelty detection.

    • Outlier detection (detecting anomalies in training data) — Detect anomalies in training data by using the ocsvm function. The ocsvm function trains a OneClassSVM object and returns anomaly indicators and scores for the training data.

    • Novelty detection (detecting anomalies in new data with uncontaminated training data) — Create a OneClassSVM object by passing uncontaminated training data (data with no outliers) to ocsvm, and detect anomalies in new data by passing the object and the new data to the object function isanomaly. The isanomaly function returns anomaly indicators and scores for the new data.

    Creation

    Create a OneClassSVM object by using the ocsvm function.

    Properties

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    This property is read-only.

    Categorical predictor indices, specified as a vector of positive integers. CategoricalPredictors contains index values indicating that the corresponding predictors are categorical. The index values are between 1 and p, where p is the number of predictors used to train the model. If none of the predictors are categorical, then this property is empty ([]).

    This property is read-only.

    Fraction of anomalies in the training data, specified as a numeric scalar in the range [0,1].

    • If the ContaminationFraction value is 0, then ocsvm treats all training observations as normal observations, and sets the score threshold (ScoreThreshold property value) to the maximum anomaly score value of the training data.

    • If the ContaminationFraction value is in the range (0,1], then ocsvm determines the threshold value (ScoreThreshold property value) so that the function detects the specified fraction of training observations as anomalies.

    This property is read-only.

    Kernel scale parameter, specified as a positive scalar.

    This property is read-only.

    Regularization term strength, specified as a nonnegative scalar.

    This property is read-only.

    Predictor means, specified as a numeric vector.

    • If you specify StandardizeData=true when you train a one-class SVM model using ocsvm, the length of Mu is equal to the number of predictors.

      • The ocsvm function does not standardize columns that contain categorical variables. The elements in Mu for categorical variables contain NaN values.

      • The isanomaly function standardizes the input data by using the predictor means in Mu and standard deviations in Sigma.

    • If you set StandardizeData=false, then Mu is an empty vector ([]).

    This property is read-only.

    Number of dimensions of the expanded space, specified as a positive integer.

    This property is read-only.

    Value of the objective function that the Limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) solver minimizes to solve the one-class SVM problem, specified as a scalar.

    This property is read-only.

    Predictor variable names, specified as a cell array of character vectors. The order of the elements of PredictorNames corresponds to the order in which the predictor names appear in the training data.

    This property is read-only.

    Threshold for the anomaly score used to identify anomalies in the training data, specified as a numeric scalar in the range (–Inf,Inf).

    The software identifies observations with anomaly scores above the threshold as anomalies.

    • The ocsvm function determines the threshold value to detect the specified fraction (ContaminationFraction property) of training observations as anomalies.

    • The isanomaly object function uses the ScoreThreshold property value as the default value of the ScoreThreshold name-value argument.

    This property is read-only.

    Predictor standard deviations, specified as a numeric vector.

    • If you specify StandardizeData=true when you train a one-class SVM model using ocsvm, then the length of Sigma is equal to the number of predictors.

      • The ocsvm function does not standardize columns that contain categorical variables. The elements in Sigma for categorical variables contain NaN values.

      • The isanomaly function standardizes the input data by using the predictor means in Mu and standard deviations in Sigma.

    • If you set StandardizeData=false, then Sigma is an empty vector ([]).

    Object Functions

    isanomalyFind anomalies in data using one-class support vector machine (SVM)

    Examples

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    Detect outliers (anomalies in training data) by using the ocsvm function.

    Load the sample data set NYCHousing2015.

    load NYCHousing2015

    The data set includes 10 variables with information on the sales of properties in New York City in 2015. Display a summary of the data set.

    summary(NYCHousing2015)
    Variables:
    
        BOROUGH: 91446×1 double
    
            Values:
    
                Min          1    
                Median       3    
                Max          5    
    
        NEIGHBORHOOD: 91446×1 cell array of character vectors
    
        BUILDINGCLASSCATEGORY: 91446×1 cell array of character vectors
    
        RESIDENTIALUNITS: 91446×1 double
    
            Values:
    
                Min            0  
                Median         1  
                Max         8759  
    
        COMMERCIALUNITS: 91446×1 double
    
            Values:
    
                Min           0   
                Median        0   
                Max         612   
    
        LANDSQUAREFEET: 91446×1 double
    
            Values:
    
                Min                0
                Median          1700
                Max       2.9306e+07
    
        GROSSSQUAREFEET: 91446×1 double
    
            Values:
    
                Min                0
                Median          1056
                Max       8.9422e+06
    
        YEARBUILT: 91446×1 double
    
            Values:
    
                Min            0  
                Median      1939  
                Max         2016  
    
        SALEPRICE: 91446×1 double
    
            Values:
    
                Min                0
                Median    3.3333e+05
                Max       4.1111e+09
    
        SALEDATE: 91446×1 datetime
    
            Values:
    
                Min       01-Jan-2015
                Median    09-Jul-2015
                Max       31-Dec-2015
    

    The SALEDATE column is a datetime array, which is not supported by ocsvm. Create columns for the month and day numbers of the datetime values, and delete the SALEDATE column.

    [~,NYCHousing2015.MM,NYCHousing2015.DD] = ymd(NYCHousing2015.SALEDATE);
    NYCHousing2015.SALEDATE = [];

    Train a one-class SVM model for NYCHousing2015. Specify the fraction of anomalies in the training observations as 0.1, and specify the first variable (BOROUGH) as a categorical predictor. The first variable is a numeric array, so ocsvm assumes it is a continuous variable unless you specify the variable as a categorical variable. In addition, specify StandardizeData as true to standardize the input data, because the predictors have largely different scales.

    rng("default") % For reproducibility 
    [Mdl,tf,scores] = ocsvm(NYCHousing2015,ContaminationFraction=0.1, ...
        CategoricalPredictors=1,StandardizeData=true);

    Mdl is a OneClassSVM object. ocsvm also returns the anomaly indicators (tf) and anomaly scores (scores) for the training data NYCHousing2015.

    Plot a histogram of the score values. Create a vertical line at the score threshold corresponding to the specified fraction.

    histogram(scores)
    xline(Mdl.ScoreThreshold,"r-",["Threshold" Mdl.ScoreThreshold])

    Figure contains an axes object. The axes object contains 2 objects of type histogram, constantline.

    If you want to identify anomalies with a different contamination fraction (for example, 0.01), you can train a new one-class SVM model.

    rng("default") % For reproducibility 
    [newMdl,newtf,scores] = ocsvm(NYCHousing2015, ...
        ContaminationFraction=0.01,CategoricalPredictors=1);
    

    If you want to identify anomalies with a different score threshold value (for example, 0.65), you can pass the OneClassSVM object, the training data, and a new threshold value to the isanomaly function.

    [newtf,scores] = isanomaly(Mdl,NYCHousing2015,ScoreThreshold=0.65);
    

    Note that changing the contamination fraction or score threshold changes the anomaly indicators only, and does not affect the anomaly scores. Therefore, if you do not want to compute the anomaly scores again by using ocsvm or isanomaly, you can obtain a new anomaly indicator with the existing score values.

    Change the fraction of anomalies in the training data to 0.01.

    newContaminationFraction = 0.01;

    Find a new score threshold by using the quantile function.

    newScoreThreshold = quantile(scores,1-newContaminationFraction)
    newScoreThreshold = 0.0480
    

    Obtain a new anomaly indicator.

    newtf = scores > newScoreThreshold;

    Create a OneClassSVM object for uncontaminated training observations by using the ocsvm function. Then detect novelties (anomalies in new data) by passing the object and the new data to the object function isanomaly.

    Load the 1994 census data stored in census1994.mat. The data set consists of demographic data from the US Census Bureau to predict whether an individual makes over $50,000 per year.

    load census1994

    census1994 contains the training data set adultdata and the test data set adulttest.

    ocsvm does not use observations with missing values. Remove missing values in the data sets to reduce memory consumption and speed up training.

    adultdata = rmmissing(adultdata);
    adulttest = rmmissing(adulttest);

    Train a one-class SVM for adultdata. Assume that adultdata does not contain outliers. Specify StandardizeData as true to standardize the input data, and set KernelScale to "auto" to let the function select an appropriate kernel scale parameter using a heuristic procedure.

    rng("default") % For reproducibility
    [Mdl,~,s] = ocsvm(adultdata,StandardizeData=true,KernelScale="auto");

    Mdl is a OneClassSVM object. If you do not specify the ContaminationFraction name-value argument as a value greater than 0, then ocsvm treats all training observations as normal observations. The function sets the score threshold to the maximum score value. Display the threshold value.

    Mdl.ScoreThreshold
    ans = 0.0322
    

    Find anomalies in adulttest by using the trained one-class SVM model.

    [tf_test,s_test] = isanomaly(Mdl,adulttest);

    The isanomaly function returns the anomaly indicators tf_test and scores s_test for adulttest. By default, isanomaly identifies observations with scores above the threshold (Mdl.ScoreThreshold) as anomalies.

    Create histograms for the anomaly scores s and s_test. Create a vertical line at the threshold of the anomaly scores.

    h1 = histogram(s,NumBins=50,Normalization="probability");
    hold on
    h2 = histogram(s_test,h1.BinEdges,Normalization="probability");
    xline(Mdl.ScoreThreshold,"r-",join(["Threshold" Mdl.ScoreThreshold]))
    h1.Parent.YScale = 'log';
    h2.Parent.YScale = 'log';
    legend("Training Data","Test Data",Location="north")
    hold off

    Figure contains an axes object. The axes object contains 3 objects of type histogram, constantline. These objects represent Training Data, Test Data.

    Display the observation index of the anomalies in the test data.

    find(tf_test)
    ans =
    
      0x1 empty double column vector
    

    The anomaly score distribution of the test data is similar to that of the training data, so isanomaly does not detect any anomalies in the test data with the default threshold value. You can specify a different threshold value by using the ScoreThreshold name-value argument. For an example, see Specify Anomaly Score Threshold.

    More About

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    References

    [1] Rahimi, A., and B. Recht. “Random Features for Large-Scale Kernel Machines.” Advances in Neural Information Processing Systems. Vol. 20, 2008, pp. 1177–1184.

    [2] Le, Q., T. Sarlós, and A. Smola. “Fastfood — Approximating Kernel Expansions in Loglinear Time.” Proceedings of the 30th International Conference on Machine Learning. Vol. 28, No. 3, 2013, pp. 244–252.

    Version History

    Introduced in R2022b