# iforest

Fit isolation forest for anomaly detection

## Description

Use the iforest function to fit an isolation forest model for outlier detection and novelty detection.

• Outlier detection (detecting anomalies in training data) — Use the output argument tf of iforest to identify anomalies in 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. Detect anomalies in new data by passing the object and the new data to the object function isanomaly.

example

forest = iforest(Tbl) returns an IsolationForest object for predictor data in the table Tbl.

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

forest = iforest(___,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, ContaminationFraction=0.1 instructs the function to process 10% of the training data as anomalies.

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

example

[forest,tf,scores] = iforest(___) also returns an anomaly score in the range [0,1] for each observation in Tbl or X. A score value close to 0 indicates a normal observation, and a value close to 1 indicates an anomaly.

## Examples

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Detect outliers (anomalies in training data) by using the iforest function.

Load the sample data set 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;

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.

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 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.

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.

## Input Arguments

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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

Predictor data, specified as a numeric matrix. Each row of X corresponds to one observation, and each column corresponds to one predictor variable.

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 an isolation forest using 50 isolation trees and 100 observations for each isolation tree.

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

ValueDescription
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 iforest 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 matrixEach 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 vectorsEach 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 in a table (Tbl), iforest 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), iforest 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 iforest function uses an approximate splitting method that can reduce the accuracy of the isolation forest model.

Example: CategoricalPredictors='all'

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

Fraction of anomalies in the training data, specified as a numeric scalar between 0 and 1.

• If the ContaminationFraction value is 0 (default), then iforest 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 iforest 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

Number of isolation trees, specified as a positive integer scalar.

The average path lengths used by the isolation forest algorithm to compute anomaly scores usually converge well before growing 100 isolation trees for both normal points and anomalies [1].

Example: NumLearners=50

Data Types: single | double

Number of observations to draw from the training data without replacement for each isolation tree, specified as a positive integer scalar greater than or equal to 3.

The isolation forest algorithm performs well with a small NumObservationsPerLearner value, because using a small sample size helps to detect dense anomalies and anomalies close to normal points. However, you need to experiment with the sample size if N is small. For an example, see Examine NumObservationsPerLearner for Small Data.

Example: NumObservationsPerLearner=100

Data Types: single | double

Predictor variable names, specified as a string array of unique names or cell array of unique character vectors. The functionality of PredictorNames depends on the way you supply the predictor data.

• If you supply Tbl, then you can use PredictorNames to choose which predictor variables to use. That is, iforest 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',...}.

Example: PredictorNames=["SepalLength" "SepalWidth" "PetalLength" "PetalWidth"]

Data Types: string | cell

Flag to run in parallel, specified as true or false. If you specify UseParallel=true, the iforest function executes for-loop iterations in parallel by using parfor. This option requires Parallel Computing Toolbox™.

Example: UseParallel=true

Data Types: logical

## Output Arguments

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Trained isolation forest model, returned as an IsolationForest object.

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

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.

iforest 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.

Anomaly scores, returned as a numeric column vector whose values are between 0 and 1. 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 score value close to 0 indicates a normal observation, and a value close to 1 indicates an anomaly.

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### 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):

$s\left(x\right)={2}^{-\frac{E\left[h\left(x\right)\right]}{c\left(n\right)}},$

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.

## Algorithms

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

• iforest does not use observations with all missing values and assigns the anomaly score of 1 to the observations.

• iforest uses observations with some missing values to find splits on variables for which these observations have valid values.

## 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.