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ocsvm

Fit one-class support vector machine (SVM) model for anomaly detection

    Description

    Use the ocsvm function to fit a one-class support vector machine (SVM) model for outlier detection and novelty detection.

    • Outlier detection (detecting anomalies in training data) — Use the output argument tf of ocsvm to identify anomalies in 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. Detect anomalies in new data by passing the object and the new data to the object function isanomaly.

    Mdl = ocsvm(Tbl) returns a OneClassSVM object (one-class SVM model object) for predictor data in the table Tbl.

    Mdl = ocsvm(X) uses predictor data in the matrix X.

    Mdl = ocsvm(___,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, ContaminationFraction=0.1 instructs the function to process 10% of the training data as anomalies.

    [Mdl,tf] = ocsvm(___) also returns the logical array tf, whose elements are true when an anomaly is detected in the corresponding row of Tbl or X.

    example

    [Mdl,tf,scores] = ocsvm(___) also returns an anomaly score in the range (–inf,inf) for each observation in Tbl or X. A negative score value with large magnitude indicates a normal observation, and a large positive value indicates an anomaly.

    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.

    Input Arguments

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    Predictor data, specified as a table. Each row of Tbl corresponds to one observation, and each column corresponds to one predictor variable. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed.

    To use a subset of the variables in Tbl, specify the variables by using the PredictorNames name-value argument.

    Data Types: table

    Predictor data, specified as a numeric matrix. Each row of X corresponds to one observation, and each column corresponds to one predictor variable.

    You can use the PredictorNames name-value argument to assign names to the predictor variables in X.

    Data Types: single | double

    Name-Value Arguments

    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: NumExpansionDimensions=2^15,KernelScale="auto" maps the predictor data to the 2^15 dimensional space using feature expansion with a kernel scale parameter selected by a heuristic procedure.

    Anomaly Detection Options

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

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

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

    Example: ContaminationFraction=0.1

    Data Types: single | double

    Kernel Classification Options

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    Maximum amount of allocated memory (in megabytes), specified as a positive scalar.

    If ocsvm requires more memory than the value of BlockSize to hold the transformed predictor data, then the software uses a block-wise strategy. For details about the block-wise strategy, see Algorithms.

    Example: BlockSize=1e4

    Data Types: single | double

    Kernel scale parameter, specified as "auto" or a positive scalar. The software obtains a random basis for random feature expansion by using the kernel scale parameter. For details, see Random Feature Expansion.

    If you specify "auto", then the software selects an appropriate kernel scale parameter using a heuristic procedure. This heuristic procedure uses subsampling, so estimates can vary from one call to another. Therefore, to reproduce results, set a random number seed by using rng before training.

    Example: KernelScale="auto"

    Data Types: char | string | single | double

    Regularization term strength, specified as "auto" or a nonnegative scalar.

    If you specify "auto", then the software selects an appropriate regularization parameter using a heuristic procedure.

    Example: Lambda=0.01

    Data Types: char | string | single | double

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

    If you specify "auto", then the software selects an appropriate number of dimensions using a heuristic procedure.

    Example: NumExpansionDimensions=2^15

    Data Types: char | string | single | double

    Random number stream for reproducibility of data transformation, specified as a random stream object. For details, see Random Feature Expansion.

    Use RandomStream to reproduce the random basis functions used by ocsvm to transform the predictor data to a high-dimensional space. For details, see Managing the Global Stream Using RandStream and Creating and Controlling a Random Number Stream.

    Example: RandomStream=RandStream("mlfg6331_64")

    Other Classification Options

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    List of categorical predictors, specified as one of the values in this table.

    ValueDescription
    Vector of positive integers

    Each entry in the vector is an index value indicating that the corresponding predictor is categorical. The index values are between 1 and p, where p is the number of predictors used to train the model.

    If ocsvm uses a subset of input variables as predictors, then the function indexes the predictors using only the subset. The CategoricalPredictors values do not count any variables that the function does not use.

    Logical vector

    A true entry means that the corresponding predictor is categorical. The length of the vector is p.

    Character matrixEach row of the matrix is the name of a predictor variable. The names must match the entries in PredictorNames. Pad the names with extra blanks so each row of the character matrix has the same length.
    String array or cell array of character vectorsEach element in the array is the name of a predictor variable. The names must match the entries in PredictorNames.
    "all"All predictors are categorical.

    By default, if the predictor data is in a table (Tbl), ocsvm assumes that a variable is categorical if it is a logical vector, categorical vector, character array, string array, or cell array of character vectors. If the predictor data is a matrix (X), ocsvm assumes that all predictors are continuous. To identify any other predictors as categorical predictors, specify them by using the CategoricalPredictors name-value argument.

    For the identified categorical predictors, ocsvm creates dummy variables using two different schemes, depending on whether a categorical variable is unordered or ordered. For an unordered categorical variable, ocsvm creates one dummy variable for each level of the categorical variable. For an ordered categorical variable, ocsvm creates one less dummy variable than the number of categories. For details, see Automatic Creation of Dummy Variables.

    Example: CategoricalPredictors="all"

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

    Predictor variable names, specified as a string array of unique names or cell array of unique character vectors. The functionality of PredictorNames depends on how you supply the predictor data.

    • If you supply Tbl, then you can use PredictorNames to specify which predictor variables to use. That is, ocsvm uses only the predictor variables in PredictorNames.

      • PredictorNames must be a subset of Tbl.Properties.VariableNames.

      • By default, PredictorNames contains the names of all predictor variables in Tbl.

    • If you supply X, then you can use PredictorNames to assign names to the predictor variables in X.

      • The order of the names in PredictorNames must correspond to the column order of X. That is, PredictorNames{1} is the name of X(:,1), PredictorNames{2} is the name of X(:,2), and so on. Also, size(X,2) and numel(PredictorNames) must be equal.

      • By default, PredictorNames is {'x1','x2',...}.

    Example: PredictorNames=["SepalLength" "SepalWidth" "PetalLength" "PetalWidth"]

    Data Types: string | cell

    Flag to standardize the predictor data, specified as a logical 1 (true) or 0 (false).

    If you set StandardizeData=true, the ocsvm function centers and scales each predictor variable (X or Tbl) by the corresponding column mean and standard deviation. The function does not standardize the data contained in the dummy variable columns generated for categorical predictors.

    Example: StandardizeData=true

    Data Types: logical

    Verbosity level, specified as 0 or 1. Verbose controls the display of diagnostic information at the command line.

    ValueDescription
    0ocsvm does not display diagnostic information.
    1ocsvm displays the value of the objective function, gradient magnitude, and other diagnostic information.

    Example: Verbose=1

    Data Types: single | double

    Convergence Options

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    Relative tolerance on the linear coefficients and the bias term (intercept), specified as a nonnegative scalar.

    Let Bt=[βtbt], that is, the vector of the coefficients and the bias term at optimization iteration t. If BtBt1Bt2<BetaTolerance, then optimization terminates.

    If you also specify GradientTolerance, then optimization terminates when the software satisfies either stopping criterion.

    Example: BetaTolerance=1e–6

    Data Types: single | double

    Absolute gradient tolerance, specified as a nonnegative scalar.

    Let t be the gradient vector of the objective function with respect to the coefficients and bias term at optimization iteration t. If t=max|t|<GradientTolerance, then optimization terminates.

    If you also specify BetaTolerance, then optimization terminates when the software satisfies either stopping criterion.

    Example: GradientTolerance=1e–5

    Data Types: single | double

    Maximum number of optimization iterations, specified as a positive integer.

    The default value is 1000 if the transformed data fits in memory, as specified by the BlockSize name-value argument. Otherwise, the default value is 100.

    Example: IterationLimit=500

    Data Types: single | double

    Output Arguments

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    Trained one-class SVM model, returned as a OneClassSVM object.

    You can use the object function isanomaly with Mdl to find anomalies in new data.

    Anomaly indicators, returned as a logical column vector. An element of tf is true when the observation in the corresponding row of Tbl or X is an anomaly, and false otherwise. tf has the same length as Tbl or X.

    ocsvm identifies observations with scores above the threshold (ScoreThreshold property value of Mdl) as anomalies. The function determines the threshold value to detect the specified fraction (ContaminationFraction name-value argument) of training observations as anomalies.

    Anomaly scores, returned as a numeric column vector whose values are between –Inf and Inf. scores has the same length as Tbl or X, and each element of scores contains an anomaly score for the observation in the corresponding row of Tbl or X. A negative score value with large magnitude indicates a normal observation, and a large positive value indicates an anomaly.

    More About

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    One-Class SVM

    One-class SVM, or unsupervised SVM, is an algorithm used for anomaly detection. The algorithm tries to separate data from the origin in the transformed high-dimensional predictor space. ocsvm finds the decision boundary based on the primal form of SVM with the Gaussian kernel approximation method.

    Random Feature Expansion

    Random feature expansion, such as Random Kitchen Sinks[1] or Fastfood[2], is a scheme to approximate Gaussian kernels of the kernel classification algorithm to use for big data in a computationally efficient way. Random feature expansion is more practical for big data applications that have large training sets, but can also be applied to smaller data sets that fit in memory.

    The kernel classification algorithm searches for an optimal hyperplane that separates the data into two classes after mapping features into a high-dimensional space. Nonlinear features that are not linearly separable in a low-dimensional space can be separable in the expanded high-dimensional space. All the calculations for hyperplane classification use only dot products. You can obtain a nonlinear classification model by replacing the dot product x1x2' with the nonlinear kernel function G(x1,x2)=φ(x1),φ(x2), where xi is the ith observation (row vector) and φ(xi) is a transformation that maps xi to a high-dimensional space (called the “kernel trick”). However, evaluating G(x1,x2) (Gram matrix) for each pair of observations is computationally expensive for a large data set (large n).

    The random feature expansion scheme finds a random transformation so that its dot product approximates the Gaussian kernel. That is,

    G(x1,x2)=φ(x1),φ(x2)T(x1)T(x2)',

    where T(x) maps x in p to a high-dimensional space (m). The Random Kitchen Sinks scheme uses the random transformation

    T(x)=m1/2exp(iZx')',

    where Zm×p is a sample drawn from N(0,σ2) and σ is a kernel scale. This scheme requires O(mp) computation and storage.

    The Fastfood scheme introduces another random basis V instead of Z using Hadamard matrices combined with Gaussian scaling matrices. This random basis reduces the computation cost to O(mlogp) and reduces storage to O(m).

    You can specify values for m and σ using the NumExpansionDimensions and KernelScale name-value arguments of ocsvm, respectively.

    The ocsvm function uses the Fastfood scheme for random feature expansion, and uses linear classification to train a one-class Gaussian kernel classification model.

    Algorithms

    • ocsvm considers NaN, '' (empty character vector), "" (empty string), <missing>, and <undefined> values in Tbl and NaN values in X to be missing values.

      • ocsvm removes observations with all missing values.

      • ocsvm does not use observations with some missing values. The function assigns the anomaly score of NaN and anomaly indicator of false (logical 0) to the observations.

    • ocsvm minimizes the regularized objective function using a Limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) solver with ridge (L2) regularization. If ocsvm requires more memory than the value of BlockSize to hold the transformed predictor data, then the function uses a block-wise strategy.

      • When ocsvm uses a block-wise strategy, it implements LBFGS by distributing the calculation of the loss and gradient among different parts of the data at each iteration. Also, ocsvm refines the initial estimates of the linear coefficients and the bias term by fitting the model locally to parts of the data and combining the coefficients by averaging. If you specify Verbose=1, then ocsvm displays diagnostic information for each data pass.

      • When ocsvm does not use a block-wise strategy, the initial estimates are zeros. If you specify Verbose=1, then ocsvm displays diagnostic information for each iteration.

    Alternative Functionality

    You can also use the fitcsvm function to train a one-class SVM model for anomaly detection.

    • The ocsvm function provides a simpler and preferred workflow for anomaly detection than the fitcsvm function.

      • The ocsvm function returns a OneClassSVM object, anomaly indicators, and anomaly scores. You can use the outputs to identify anomalies in training data. To find anomalies in new data, you can use the isanomaly object function of OneClassSVM. The isanomaly function returns anomaly indicators and scores for the new data.

      • The fitcsvm function supports both one-class and binary classification. If the class label variable contains only one class (for example, a vector of ones), fitcsvm trains a model for one-class classification and returns a ClassificationSVM object. To identify anomalies, you must first compute anomaly scores by using the resubPredict or predict object function of ClassificationSVM, and then identify anomalies by finding observations that have negative scores.

      • Note that a large positive anomaly score indicates an anomaly in ocsvm, whereas a negative score indicates an anomaly in predict of ClassificationSVM.

    • The ocsvm function finds the decision boundary based on the primal form of SVM, whereas the fitcsvm function finds the decision boundary based on the dual form of SVM.

    • The solver in ocsvm is computationally less expensive than the solver in fitcsvm for a large data set (large n). Unlike solvers in fitcsvm, which require computation of the n-by-n Gram matrix, the solver in ocsvm only needs to form a matrix of size n-by-m. Here, m is the number of dimensions of expanded space, which is typically much less than n for big data.

    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