# IsolationForest

Isolation forest for anomaly detection

## 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 `iforest`

.

## 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 between 0 and 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 of `PredictorNames`

corresponds to the order in which the predictor names appear in the training data.

**Data Types: **`cell`

`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 between 0 and 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. Print 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 retrain an 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 does not change 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 identifier 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 = 0.6597

Obtain a new anomaly identifier.

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 indicator `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, that is, the values in `tf`

are all logical 0 (`false`

). The function sets the score threshold to the maximum score value. Display the threshold value.

Mdl.ScoreThreshold

ans = 0.8600

Use the trained isolation forest model to find anomalies in `adulttest`

.

[tf_test,s_test] = isanomaly(Mdl,adulttest);

The `isanomaly`

function returns 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 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 builds an isolation forest (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, sampled without
replacements. `iforest`

grows an isolation tree by choosing a split
variable and split position at random until every observation in a subset lands in a
separate leaf node. Anomalies are few and different; therefore, an anomaly lands in 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 function identifies outliers using
anomaly scores defined based on the
average path lengths over all isolation trees.

The `isanomaly`

function uses a trained isolation forest to detect
anomalies in data. For novelty detection (detecting anomalies in new data with
uncontaminated training data), you can train an isolation forest 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 average path length to reach a leaf node
from the root node in the trained isolation forest, and returns an anomaly identifier 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.

## 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.

## See Also

**Introduced in R2021b**

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