# rrcforest

## Syntax

## 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`

.

returns a `forest`

= rrcforest(`Tbl`

)`RobustRandomCutForest`

model object for the predictor data in the table
`Tbl`

.

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
`forest`

= rrcforest(___,`Name=Value`

)

to process 10% of the
training data as anomalies.`ContaminationFraction`

=0.1

## 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.

## 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 If |

Logical vector | A |

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

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:

`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"

## 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**

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