IsolationForest

Isolation forest for anomaly detection

Since R2021b

Description

Use an isolation forest (ensemble of isolation trees) model object IsolationForest for outlier detection and novelty detection.

• Outlier detection (detecting anomalies in training data) — Detect anomalies in training data by using the iforest function. The iforest function builds an IsolationForest object and returns anomaly indicators and scores for the training data.

• Novelty detection (detecting anomalies in new data with uncontaminated training data) — Create an IsolationForest object by passing uncontaminated training data (data with no outliers) to iforest, 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 an IsolationForest object by using the iforest 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 iforest 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 iforest determines the threshold value (ScoreThreshold property value) so that the function detects the specified fraction of training observations as anomalies.

NumLearners — Number of isolation trees

positive integer scalar

This property is read-only.

Number of isolation trees, specified as a positive integer scalar.

NumObservationsPerLearner — Number of observations for each isolation tree

positive integer scalar

This property is read-only.

Number of observations to draw from the training data without replacement for each isolation tree, specified as a positive integer 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 [0,1]

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 [0,1].

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

• The iforest 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.

Object Functions

isanomaly

Find anomalies in data using isolation forest

Examples

Detect Outliers

Detect outliers (anomalies in training data) by using the iforest 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: 91446x1 double

        Values:

            Min          1    
            Median       3    
            Max          5    

    NEIGHBORHOOD: 91446x1 cell array of character vectors

    BUILDINGCLASSCATEGORY: 91446x1 cell array of character vectors

    RESIDENTIALUNITS: 91446x1 double

        Values:

            Min            0  
            Median         1  
            Max         8759  

    COMMERCIALUNITS: 91446x1 double

        Values:

            Min           0   
            Median        0   
            Max         612   

    LANDSQUAREFEET: 91446x1 double

        Values:

            Min                0
            Median          1700
            Max       2.9306e+07

    GROSSSQUAREFEET: 91446x1 double

        Values:

            Min                0
            Median          1056
            Max       8.9422e+06

    YEARBUILT: 91446x1 double

        Values:

            Min            0  
            Median      1939  
            Max         2016  

    SALEPRICE: 91446x1 double

        Values:

            Min                0
            Median    3.3333e+05
            Max       4.1111e+09

    SALEDATE: 91446x1 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 iforest. 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 = [];

The columns BOROUGH, NEIGHBORHOOD, and BUILDINGCLASSCATEGORY contain categorical predictors. Display the number of categories for the categorical predictors.

length(unique(NYCHousing2015.BOROUGH))

ans = 5

length(unique(NYCHousing2015.NEIGHBORHOOD))

ans = 254

length(unique(NYCHousing2015.BUILDINGCLASSCATEGORY))

ans = 48

For a categorical variable with more than 64 categories, the iforest function uses an approximate splitting method that can reduce the accuracy of the isolation forest model. Remove the NEIGHBORHOOD column, which contains a categorical variable with 254 categories.

NYCHousing2015.NEIGHBORHOOD = [];

Train an isolation forest 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 iforest assumes it is a continuous variable unless you specify the variable as a categorical variable.

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

Mdl is an IsolationForest object. iforest 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 isolation forest model.

rng("default") % For reproducibility 
[newMdl,newtf,scores] = iforest(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 IsolationForest 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 iforest 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 = 0.7045

Obtain a new anomaly indicator.

newtf = scores > newScoreThreshold;

Detect Novelties

Create an IsolationForest object for uncontaminated training observations by using the iforest 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.

Train an isolation forest model for adultdata. Assume that adultdata does not contain outliers.

rng("default") % For reproducibility
[Mdl,tf,s] = iforest(adultdata);

Mdl is an IsolationForest object. iforest also returns the anomaly indicators tf and anomaly scores s for the training data adultdata. If you do not specify the ContaminationFraction name-value argument as a value greater than 0, then iforest treats all training observations as normal observations, meaning all the values in tf are logical 0 (false). The function sets the score threshold to the maximum score value. Display the threshold value.

Mdl.ScoreThreshold

ans = 0.8600

Find anomalies in adulttest by using the trained isolation forest 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.

histogram(s,Normalization="probability")
hold on
histogram(s_test,Normalization="probability")
xline(Mdl.ScoreThreshold,"r-",join(["Threshold" Mdl.ScoreThreshold]))
legend("Training Data","Test Data",Location="northwest")
hold off

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

find(tf_test)

ans = 15655

The anomaly score distribution of the test data is similar to that of the training data, so isanomaly detects a small number of 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

Isolation Forest

The isolation forest algorithm [1] detects anomalies by isolating anomalies from normal points using an ensemble of isolation trees.

The iforest function creates an isolation forest model (ensemble of isolation trees) for training observations and detects outliers (anomalies in the training data). Each isolation tree is trained for a subset of training observations as follows:

1. iforest draws samples without replacement from the training observations for each tree.

2. iforest grows a tree by choosing a split variable and split position uniformly at random. The function continues until every sample reaches a separate leaf node for each tree.

This algorithm assumes the data has only a few anomalies and they are different from normal points. Therefore, an anomaly reaches a separate leaf node closer to the root node and has a shorter path length (the distance from the root node to the leaf node) than normal points. The iforest function identifies outliers using anomaly scores that are defined based on the average path lengths over all isolation trees.

The isanomaly function uses a trained isolation forest model to detect anomalies in the data. For novelty detection (detecting anomalies in new data with uncontaminated training data), you can train an isolation forest model with uncontaminated training data (data with no outliers) and use it to detect anomalies in new data. For each observation of the new data, the function finds the corresponding leaf node in each tree, finds the average path length to reach a leaf node from the root node in the trained isolation forest model, and returns an anomaly indicator and score.

For more details, see Anomaly Detection with Isolation Forest.

Anomaly Scores

The isolation forest algorithm computes the anomaly score s(x) of an observation x by normalizing the path length h(x):

$$s(x)=2^{-\frac{E[h(x)]}{c(n)}},$$

where E[h(x)] is the average path length over all isolation trees in the isolation forest, and c(n) is the average path length of unsuccessful searches in a binary search tree of n observations.

• The score approaches 1 as E[h(x)] approaches 0. Therefore, a score value close to 1 indicates an anomaly.

• The score approaches 0 as E[h(x)] approaches n – 1. Also, the score approaches 0.5 when E[h(x)] approaches c(n). Therefore, a score value smaller than 0.5 and close to 0 indicates a normal point.

Tips

• You can use interpretability features, such as lime, shapley, partialDependence, and plotPartialDependence, to interpret how predictors contribute to anomaly scores. Define a custom function that returns anomaly scores, and then pass the custom function to the interpretability functions. For an example, see Specify Model Using Function Handle.

References

[1] Liu, F. T., K. M. Ting, and Z. Zhou. "Isolation Forest," 2008 Eighth IEEE International Conference on Data Mining. Pisa, Italy, 2008, pp. 413-422.

Extended Capabilities

C/C++ Code Generation

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

Usage notes and limitations:

• The isanomaly function supports code generation.

For more information, see Introduction to Code Generation.

Version History

Introduced in R2021b