RobustRandomCutForest

Robust random cut forest model for anomaly detection

Since R2023a

Description

Use a robust random cut forest model object RobustRandomCutForest for outlier detection and novelty detection.

• Outlier detection (detecting anomalies in training data) — Detect anomalies in training data by using the rrcforest function. The rrcforest function returns a RobustRandomCutForest model object, anomaly indicators, and scores for the training data.

• Novelty detection (detecting anomalies in new data with uncontaminated training data) — Create a RobustRandomCutForest model object by passing uncontaminated training data (data with no outliers) to rrcforest. 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 RobustRandomCutForest model object by using the rrcforest 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 ([]).

CollusiveDisplacement — Collusive displacement calculation method

'maximal' | 'average'

This property is read-only.

Collusive displacement calculation method, specified as 'maximal' or 'average'.

The software finds the maximum change ('maximal') or the average change ('average') in model complexity for each tree, and computes the collusive displacement (anomaly score) for each observation. For details, see Anomaly Scores.

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 rrcforest 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 rrcforest determines the threshold value (ScoreThreshold property value) so that the function detects the specified fraction of training observations as anomalies.

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 robust random cut forest model using rrcforest:

  • The rrcforest 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 ([]).

NumLearners — Number of robust random cut trees

positive integer scalar

This property is read-only.

Number of robust random cut trees (trees in the robust random cut forest model), specified as a positive integer scalar.

NumObservationsPerLearner — Number of observations for each robust random cut tree

positive integer scalar

This property is read-only.

Number of observations to draw from the training data without replacement for each robust random cut tree (tree in the robust random cut forest model), 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,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 [0,Inf).

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

• The rrcforest 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 robust random cut forest model using rrcforest:

  • The rrcforest 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 robust random cut forest
incrementalLearner	Convert robust random cut forest model to incremental learner

Examples

Detect Outliers

Detect outliers (anomalies in training data) by using the rrcforest 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)

NYCHousing2015: 91446x10 table

Variables:

    BOROUGH: double
    NEIGHBORHOOD: cell array of character vectors
    BUILDINGCLASSCATEGORY: cell array of character vectors
    RESIDENTIALUNITS: double
    COMMERCIALUNITS: double
    LANDSQUAREFEET: double
    GROSSSQUAREFEET: double
    YEARBUILT: double
    SALEPRICE: double
    SALEDATE: datetime

Statistics for applicable variables:

                             NumMissing          Min              Median               Max               Mean               Std      

    BOROUGH                      0                     1                  3                  5             2.8431            1.3343  
    NEIGHBORHOOD                 0                                                                                                   
    BUILDINGCLASSCATEGORY        0                                                                                                   
    RESIDENTIALUNITS             0                     0                  1               8759             2.1789           32.2738  
    COMMERCIALUNITS              0                     0                  0                612             0.2201            3.2991  
    LANDSQUAREFEET               0                     0               1700           29305534         2.8752e+03        1.0118e+05  
    GROSSSQUAREFEET              0                     0               1056            8942176         4.6598e+03        4.3098e+04  
    YEARBUILT                    0                     0               1939               2016         1.7951e+03          526.9998  
    SALEPRICE                    0                     0             333333         4.1111e+09         1.2364e+06        2.0130e+07  
    SALEDATE                     0           01-Jan-2015        09-Jul-2015        31-Dec-2015        07-Jul-2015        2470:47:17  

The SALEDATE column is a datetime array, which is not supported by rrcforest. Create columns for the month and day numbers of the datetime values, and then 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 rrcforest function uses an approximate splitting method that can reduce the accuracy of the robust random cut forest model. Remove the NEIGHBORHOOD column, which contains a categorical variable with 254 categories.

NYCHousing2015.NEIGHBORHOOD = [];

Train a robust random cut 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 rrcforest assumes it is a continuous variable unless you specify the variable as a categorical variable.

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

Mdl is a RobustRandomCutForest model object. rrcforest 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 robust random cut forest model.

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

If you want to identify anomalies with a different score threshold value (for example, 65), you can pass the RobustRandomCutForest model object, the training data, and a new threshold value to the isanomaly function.

[newtf,scores] = isanomaly(Mdl,NYCHousing2015,ScoreThreshold=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 rrcforest or isanomaly, you can obtain a new anomaly indicator using 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 = 
63.2642

Obtain a new anomaly indicator.

newtf = scores > newScoreThreshold;

Detect Novelties

Create a RobustRandomCutForest model object for uncontaminated training observations by using the rrcforest 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 contains 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.

Assume that adultdata does not contain outliers. Train a robust random cut forest model for adultdata. Specify StandardizeData as true to standardize the input data.

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

Mdl is a RobustRandomCutForest model object. rrcforest 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 rrcforest 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 = 
86.5315

Find anomalies in adulttest by using the trained robust random cut forest 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.

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 = 
3541

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.

Zoom in to see the anomaly and the observations near the threshold.

xlim([50 92])
ylim([0 0.001])

You can specify a different threshold value by using the ScoreThreshold name-value argument. For an example, see Specify Anomaly Score Threshold.

More About

Robust Random Cut Forest

The robust random cut forest algorithm [1] classifies a point as a normal point or an anomaly based on the change in model complexity introduced by the point. Similar to the Isolation Forest algorithm, the robust random cut forest algorithm builds an ensemble of trees. The two algorithms differ in how they choose a split variable in the trees and how they define anomaly scores.

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

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

2. rrcforest grows a tree by choosing a split variable in proportion to the ranges of variables, and choosing the split position uniformly at random. The function continues until every sample reaches a separate leaf node for each tree.

Using the range information in to choose a split variable makes the algorithm robust to irrelevant variables.

Anomalies are easy to describe, but make describing the remainder of the data more difficult. Therefore, adding an anomaly to a model increases the model complexity of a forest model [1]. The rrcforest function identifies outliers using anomaly scores that are defined based on the change in model complexity.

The isanomaly function uses a trained robust random cut forest model to detect anomalies in the data. For novelty detection (detecting anomalies in new data with uncontaminated training data), you can train a robust random cut 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, computes the change in model complexity introduced by the leaf nodes, and returns an anomaly indicator and score.

Anomaly Scores

The robust random cut forest algorithm uses collusive displacement as an anomaly score. The collusive displacement of a point x indicates the contribution of x to the model complexity of a forest model. A small positive anomaly score value indicates a normal observation, and a large positive value indicates an anomaly.

As defined in [1], the model complexity |M(T)| of a tree T is the sum of path lengths (the distance from the root node to the leaf nodes) over all points in the training data Z.

$$\left|\text{M}(T)\right|={\displaystyle \sum _{y\in Z}f\left(y,Z,T\right)},$$

where f(y,Z,T) is the depth of y in tree T. The displacement of x is defined to indicate the expected changes in the model complexity introduced by x.

$$\text{Disp}(x,Z)={\displaystyle \sum _{T,y\in Z-\left\{x\right\}}P\left(T\right)\left(f\left(y,Z,T\right)-f\left(y,Z-\left\{x\right\},T^{\prime }\right)\right)},$$

where T' is a tree over Z – {x}. Disp(x,Z) is the expected number of points in the sibling node of the leaf node containing x. This definition is not robust to duplicates or near-duplicates, and can cause outlier masking. To avoid outlier masking, the robust random cut forest algorithm uses the collusive displacement CoDisp, where a set C includes x and the colluders of x.

$$\text{CoDisp}(x,Z)=E_T\left[\underset{x\in C\subseteq Z}{\max }\frac{1}{\left|C\right|}{\displaystyle \sum _{y\in Z-C}\left(f\left(y,Z,T\right)-f\left(y,Z-C,T^{″}\right)\right)}\right],$$

where T" is a tree over Z – C, and |C| is the number of points in the subtree of T for C.

The default value for the CollusiveDisplacement name-value argument of rrcforest is "maximal". For each tree, by default, the software finds the set C that maximizes the ratio Disp(x,C)/|C| by traversing from the leaf node of x to the root node, as described in [2]. If you specify CollusiveDisplacement="average", the software computes the average of the ratios for each tree, and uses the averaged values to compute the collusive displacement value.

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] Guha, Sudipto, N. Mishra, G. Roy, and O. Schrijvers. "Robust Random Cut Forest Based Anomaly Detection on Streams," Proceedings of The 33rd International Conference on Machine Learning 48 (June 2016): 2712–21.

[2] Bartos, Matthew D., A. Mullapudi, and S. C. Troutman. "rrcf: Implementation of the Robust Random Cut Forest Algorithm for Anomaly Detection on Streams." Journal of Open Source Software 4, no. 35 (2019): 1336.

Version History

Introduced in R2023a