OneClassSVM

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

Since R2022b

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

CategoricalPredictors — Categorical predictor indices

vector of positive integers | []

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 ([]).

ContaminationFraction — Fraction of anomalies in training data

numeric scalar in the range [0,1]

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.

KernelScale — Kernel scale parameter

positive scalar

This property is read-only.

Kernel scale parameter, specified as a positive scalar.

Lambda — Regularization term strength

nonnegative scalar

This property is read-only.

Regularization term strength, specified as a nonnegative scalar.

Mu — Predictor means

numeric vector | []

This property is read-only.

Predictor means of the training data, specified as a numeric vector.

• If you specify StandardizeData=true when you train a one-class SVM model using ocsvm:

  • 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.

  The length of Mu is equal to the number of predictors.

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

NumExpansionDimensions — Number of dimensions of expanded space

positive integer

This property is read-only.

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

ObjectiveValue — Value of objective function

scalar

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.

PredictorNames — Predictor variable names

cell array of character vectors

This property is read-only.

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

ScoreThreshold — Threshold for anomaly score

numeric scalar in the range (–Inf,Inf)

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.

Sigma — Predictor standard deviations

numeric vector | []

This property is read-only.

Predictor standard deviations of the training data, specified as a numeric vector.

• If you specify StandardizeData=true when you train a one-class SVM model using ocsvm:

  • 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.

  The length of Sigma is equal to the number of predictors.

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

Object Functions

isanomaly	Find anomalies in data using one-class support vector machine (SVM)
incrementalLearner	Convert one-class SVM model to incremental learner

Examples

Detect Outliers

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)

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. Set KernelScale to "auto" so that the software selects an appropriate kernel scale parameter using a heuristic procedure.

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

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])

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, ...
    KernelScale="auto");

If you want to identify anomalies with a different score threshold value (for example, -0.7), you can pass the OneClassSVM object, the training data, and a new threshold value to the isanomaly function.

[newtf,scores] = isanomaly(Mdl,NYCHousing2015,ScoreThreshold=-0.7);

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 = -0.3745

Obtain a new anomaly indicator.

newtf = scores > newScoreThreshold;

Detect Novelties

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. Because you specified StandardizeData=true when you trained the model, the isanomaly function standardizes the input data by using the predictor means and standard deviations of the training data stored in the Mu and Sigma properties, respectively.

[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

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

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 $$G(x_1,x_2)=\langle \phi (x_1),\phi (x_2)\rangle$$, where x_i is the ith observation (row vector) and φ(x_i) is a transformation that maps x_i 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,

$$G(x_1,x_2)=\langle \phi (x_1),\phi (x_2)\rangle \approx T(x_1)T(x_2)\text{'},$$

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

$$T(x)=m^{-1/2}\exp \left(iZx\text{'}\right)\text{'},$$

where $$Z\in {ℝ}^{m\times p}$$ is a sample drawn from $$N\left(0,{\sigma }^{-2}\right)$$ 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).

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

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.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™. (since R2023b)

Usage notes and limitations:

• The isanomaly function supports code generation.

For more information, see Introduction to Code Generation.

Version History

Introduced in R2022b

R2023b: Generate C/C++ code for prediction

You can generate C/C++ code for the isanomaly function.