rrcforest

Fit robust random cut forest model for anomaly detection

Since R2023a

Syntax

forest = rrcforest(Tbl)

forest = rrcforest(X)

forest = rrcforest(___,Name=Value)

[forest,tf] = rrcforest(___)

[forest,tf,scores] = rrcforest(___)

Description

Use the rrcforest function to fit a robust random cut forest model for outlier detection and novelty detection.

• Outlier detection (detecting anomalies in training data) — Use the output argument tf of rrcforest to identify anomalies in 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.

forest = rrcforest(Tbl) returns a RobustRandomCutForest model object for the predictor data in the table Tbl.

forest = rrcforest(X) uses the predictor data in the matrix X.

forest = rrcforest(___,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, specify ContaminationFraction=0.1 to process 10% of the training data as anomalies.

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

[forest,tf,scores] = rrcforest(___) also returns an anomaly score in the range [0,Inf) for each observation in Tbl or X. A small positive value indicates a normal observation, and a large positive value indicates an anomaly.

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: 91446×10 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.

Input Arguments

Tbl — Predictor data

table

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

X — Predictor data

numeric matrix

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: NumLearners=50,NumObservationsPerLearner=100 specifies to train a robust random cut forest model using 50 trees and 100 observations for each tree.

CategoricalPredictors — List of categorical predictors

vector of positive integers | logical vector | character matrix | string array | cell array of character vectors | "all"

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

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

For a categorical variable with more than 64 categories, the rrcforest function uses an approximate splitting method that can reduce the accuracy of the model.

Example: CategoricalPredictors="all"

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

CollusiveDisplacement — Collusive displacement calculation method

"maximal" (default) | "average"

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

The rrcforest function 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.

Example: CollusiveDisplacement="average"

Data Types: char | string

ContaminationFraction — Fraction of anomalies in training data

0 (default) | numeric scalar in the range [0,1]

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 rrcforest treats all training observations as normal observations, and sets the score threshold (ScoreThreshold property value of forest) to the maximum value of scores.

• If the ContaminationFraction value is in the range (0,1], then rrcforest 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

NumLearners — Number of robust random cut trees

100 (default) | positive integer scalar

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

Example: NumLearners=50

Data Types: single | double

NumObservationsPerLearner — Number of observations for each robust random cut tree

min(N,256) where N is the number of training observations (default) | positive integer scalar greater than or equal to 3

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 greater than or equal to 3.

Example: NumObservationsPerLearner=100

Data Types: single | double

PredictorNames — Predictor variable names

Read-only: string array of unique names | cell array of unique character vectors

This property is read-only.

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, rrcforest 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",...}.

Data Types: string | cell

StandardizeData — Flag to standardize predictor data

false or 0 (default) | true or 1

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

If you set StandardizeData=true, the rrcforest 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

UseParallel — Flag to run in parallel

false or 0 (default) | true or 1

Flag to run in parallel, specified as a numeric or logical 1 (true) or 0 (false). If you specify UseParallel=true, the rrcforest function executes for-loop iterations by using parfor. The loop runs in parallel when you have Parallel Computing Toolbox™.

Example: UseParallel=true

Data Types: logical

Output Arguments

forest — Trained robust random cut forest model

RobustRandomCutForest model object

Trained robust random cut forest model, returned as a RobustRandomCutForest model object.

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

tf — Anomaly indicators

logical column vector

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.

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

scores — Anomaly scores

numeric column vector in the range [0,Inf)

Anomaly scores, returned as a numeric column vector with values in the range [0,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 small positive value indicates a normal observation, and a large positive value indicates an anomaly.

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.

Algorithms

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

rrcforest uses observations with missing values to find splits on variables for which these observations have valid values. The function might place these observations in a branch node, not a leaf node. Then rrcforest computes the ratio (Disp(x,C)/|C|) by traversing from the branch node to the root node for each tree. The function places an observation with all missing values in the root node. Therefore, the ratio and the anomaly score become the number of training observations for each tree, which is the maximum possible anomaly score for the trained robust random cut forest model. You can specify the number of training observations for each tree by using the NumObservationsPerLearner name-value argument.

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.

Extended Capabilities

Automatic Parallel Support

Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.

To run in parallel, set the UseParallel name-value argument to true in the call to this function.

For more general information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).

Version History

Introduced in R2023a