# RobustRandomCutForest

## 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 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 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 |

## 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)

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 `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:

`rrcforest`

draws samples without replacement from the training observations for each tree.`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*.

$$\text{CoDisp}(x,Z)={E}_{T}\left[\underset{x\in C\subseteq Z}{\mathrm{max}}\frac{1}{\left|C\right|}{\displaystyle \sum _{y\in Z-C}\left(f\left(y,Z,T\right)-f\left(y,Z-C,{T}^{\u2033}\right)\right)}\right],$$

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

, the
software computes the average of the ratios for each tree, and uses the averaged values to
compute the collusive displacement value.`CollusiveDisplacement`

="average"

## 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**

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