# iforest

## Syntax

## 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`

.

returns an `forest`

= iforest(`Tbl`

)`IsolationForest`

object for 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,
`forest`

= iforest(___,`Name=Value`

)

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

=0.1

## 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. Display a summary of the data set.

summary(NYCHousing2015)

NYCHousing2015: 91446x10 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 `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 train a new 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 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 `iforest`

or `isanomaly`

, you can obtain a new anomaly indicator 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 = 0.7045

Obtain a new anomaly indicator.

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 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 `iforest`

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 = 0.8600

Find anomalies in `adulttest`

by using the trained isolation forest model.

[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 = 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

`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 an isolation forest using 50 isolation trees and 100 observations for each isolation
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`

), `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 model.

**Example: **`CategoricalPredictors="all"`

**Data Types: **`single`

| `double`

| `logical`

| `char`

| `string`

| `cell`

`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`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`

`NumLearners`

— Number of isolation trees

100 (default) | positive integer scalar

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`

`NumObservationsPerLearner`

— Number of observations for each isolation 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 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`

`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,`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",...}`

.

**Data Types: **`string`

| `cell`

`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 `iforest`

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 isolation forest model

`IsolationForest`

object

Trained isolation forest model, returned as an `IsolationForest`

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`

.

`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.

`scores`

— Anomaly scores

numeric column vector in the range `[0,1]`

Anomaly scores, returned as a numeric column vector whose values are in the
range `[0,1]`

. `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.

## 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 creates an isolation forest model (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 as follows:

`iforest`

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

grows a tree by choosing a split variable and split position uniformly at random. The function continues until every sample reaches a separate leaf node for each tree.

This algorithm assumes the data has only a few anomalies and they are different from
normal points. Therefore, an anomaly reaches 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 `iforest`

function identifies outliers using anomaly scores that are defined
based on the average path lengths over all isolation trees.

The `isanomaly`

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

## Tips

After training a model, you can generate C/C++ code that finds anomalies for new data. Generating C/C++ code requires MATLAB

^{®}Coder™. For details, see Code Generation of the`isanomaly`

function and Introduction to Code Generation.

## 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`

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
`iforest`

computes anomaly scores by using the
distance from the root node to the branch node. The function places an observation
with all missing values in the root node, so the score value becomes 1.

## 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.

## 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 R2021b**

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