# lof

## Syntax

## Description

Use the `lof`

function to create a local
outlier factor model for outlier detection and novelty detection.

Outlier detection (detecting anomalies in training data) — Use the output argument

`tf`

of`lof`

to identify anomalies in training data.Novelty detection (detecting anomalies in new data with uncontaminated training data) — Create a

`LocalOutlierFactor`

object by passing uncontaminated training data (data with no outliers) to`lof`

. Detect anomalies in new data by passing the object and the new data to the object function`isanomaly`

.

returns
a `LOFObj`

= lof(`Tbl`

)`LocalOutlierFactor`

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,
`LOFObj`

= lof(___,`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 `lof`

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

Remove nonnumeric variables from `NYCHousing2015`

. The data type of the `BOROUGH`

variable is double, but it is a categorical variable indicating the borough in which the property is located. Remove the `BOROUGH`

variable as well.

```
NYCHousing2015 = NYCHousing2015(:,vartype("numeric"));
NYCHousing2015.BOROUGH = [];
```

Train a local outlier factor model for `NYCHousing2015`

. Specify the fraction of anomalies in the training observations as 0.01.

[Mdl,tf,scores] = lof(NYCHousing2015,ContaminationFraction=0.01);

`Mdl`

is a `LocalOutlierFactor`

object. `lof`

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.

h = histogram(scores,NumBins=50); h.Parent.YScale = 'log'; xline(Mdl.ScoreThreshold,"r-",["Threshold" Mdl.ScoreThreshold])

If you want to identify anomalies with a different contamination fraction (for example, 0.05), you can train a new local outlier factor model.

[newMdl,newtf,scores] = lof(NYCHousing2015,ContaminationFraction=0.05);

Note that changing the contamination fraction 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 `lof`

, you can obtain a new anomaly indicator with the existing score values.

Change the fraction of anomalies in the training data to 0.05.

newContaminationFraction = 0.05;

Find a new score threshold by using the `quantile`

function.

newScoreThreshold = quantile(scores,1-newContaminationFraction)

newScoreThreshold = 6.7493

Obtain a new anomaly indicator.

newtf = scores > newScoreThreshold;

### Detect Novelties

Create a `LocalOutlierFactor`

object for uncontaminated training observations by using the `lof`

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`

. The predictor data must be either all continuous or all categorical to train a `LocalOutlierFactor`

object. Remove nonnumeric variables from `adultdata`

and `adulttest`

.

adultdata = adultdata(:,vartype("numeric")); adulttest = adulttest(:,vartype("numeric"));

Train a local outlier factor model for `adultdata`

. Assume that `adultdata`

does not contain outliers.

[Mdl,tf,s] = lof(adultdata);

`Mdl`

is a `LocalOutlierFactor`

object. `lof`

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 `lof`

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 = 28.6719

Find anomalies in `adulttest`

by using the trained local outlier factor 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.

h1 = histogram(s,NumBins=50,Normalization="probability"); hold on h2 = histogram(s_test,h1.BinEdges,Normalization="probability"); xline(Mdl.ScoreThreshold,"r-",join(["Threshold" Mdl.ScoreThreshold])) h1.Parent.YScale = 'log'; h2.Parent.YScale = 'log'; legend("Training Data","Test Data",Location="north") hold off

Display the observation index of the anomalies in the test data.

find(tf_test)

ans = 0x1 empty double column vector

The anomaly score distribution of the test data is similar to that of the training data, so `isanomaly`

does not detect any 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.

The predictor data must be either all continuous or all categorical. If you specify
`Tbl`

, the `lof`

function 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
`Tbl`

includes both continuous and categorical values, and you want
to identify all predictors in `Tbl`

as categorical, you must specify
`CategoricalPredictors`

as `"all"`

.

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.

The predictor data must be either all continuous or all categorical. If you specify
`X`

, the `lof`

function assumes that all
predictors are continuous. To identify all predictors in `X`

as
categorical, specify `CategoricalPredictors`

as
`"all"`

.

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: **`SearchMethod=exhaustive,Distance=minkowski`

uses the
exhaustive search algorithm with the Minkowski distance.

`BucketSize`

— Maximum data points in node

`50`

(default) | positive integer value

Maximum number of data points in the leaf node of the
*K*d-tree, specified as a positive integer value. This argument is
valid only when `SearchMethod`

is
`"kdtree"`

.

**Example: **`BucketSize=40`

**Data Types: **`single`

| `double`

`CacheSize`

— Size of Gram matrix in megabytes

`1000`

(default) | positive scalar | `"maximal"`

Size of the Gram matrix in megabytes, specified as a positive scalar or
`"maximal"`

. For the definition of the Gram matrix, see Algorithms. The
`lof`

function can use a Gram matrix when the
`Distance`

name-value argument is
`"fasteuclidean"`

.

When `CacheSize`

is `"maximal"`

,
`lof`

attempts to allocate enough memory for an entire
intermediate matrix whose size is `MX`

-by-`MX`

,
where `MX`

is the number of rows of the input data,
`X`

or `Tbl`

. `CacheSize`

does not have to be large enough for an entire intermediate matrix, but must be at
least large enough to hold an `MX`

-by-1 vector. Otherwise,
`lof`

uses the `"euclidean"`

distance.

If `Distance`

is `"fasteuclidean"`

and
`CacheSize`

is too large or `"maximal"`

,
`lof`

might attempt to allocate a Gram matrix that
exceeds the available memory. In this case, MATLAB^{®} issues an error.

**Example: **`CacheSize="maximal"`

**Data Types: **`double`

| `char`

| `string`

`CategoricalPredictors`

— Categorical predictor flag

`[]`

| `"all"`

Categorical predictor flag, specified as one of the following:

`"all"`

— All predictors are categorical. By default,`lof`

uses the Hamming distance (`"hamming"`

) for the`Distance`

name-value argument.`[]`

— No predictors are categorical, that is, all predictors are continuous (numeric). In this case, the default`Distance`

value is`"euclidean"`

.

The predictor data for `lof`

must be either all
continuous or all categorical.

If the predictor data is in a table (

`Tbl`

),`lof`

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`Tbl`

includes both continuous and categorical values, and you want to identify all predictors in`Tbl`

as categorical, you must specify`CategoricalPredictors`

as`"all"`

.If the predictor data is a matrix (

`X`

),`lof`

assumes that all predictors are continuous. To identify all predictors in`X`

as categorical, specify`CategoricalPredictors`

as`"all"`

.

`lof`

encodes categorical variables as numeric variables by
assigning a positive integer value to each category. When you use categorical
predictors, ensure that you use an appropriate distance metric
(`Distance`

).

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

`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`lof`

treats all training observations as normal observations, and sets the score threshold (`ScoreThreshold`

property value of`LOFObj`

) to the maximum value of`scores`

.If the

`ContaminationFraction`

value is in the range (`0`

,`1`

], then`lof`

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`

`Cov`

— Covariance matrix

positive definite matrix of scalar values

Covariance matrix, specified as a positive definite matrix of scalar values
representing the covariance matrix when the function computes the Mahalanobis
distance. This argument is valid only when `Distance`

is
`"mahalanobis"`

.

The default value is the covariance matrix computed from the predictor data
(`Tbl`

or `X`

) after the function excludes
rows with duplicated values and missing values.

**Data Types: **`single`

| `double`

`Distance`

— Distance metric

character vector | string scalar

Distance metric, specified as a character vector or string scalar.

If all the predictor variables are continuous (numeric) variables, then you can specify one of these distance metrics.

Value Description `"euclidean"`

Euclidean distance

`"fasteuclidean"`

Euclidean distance using an algorithm that usually saves time when the number of elements in a data point exceeds 10. See Algorithms.

`"fasteuclidean"`

applies only to the`"exhaustive"`

`SearchMethod`

.`"mahalanobis"`

Mahalanobis distance — You can specify the covariance matrix for the Mahalanobis distance by using the

`Cov`

name-value argument.`"minkowski"`

Minkowski distance — You can specify the exponent of the Minkowski distance by using the

`Exponent`

name-value argument.`"chebychev"`

Chebychev distance (maximum coordinate difference)

`"cityblock"`

City block distance

`"correlation"`

One minus the sample correlation between observations (treated as sequences of values)

`"cosine"`

One minus the cosine of the included angle between observations (treated as vectors)

`"spearman"`

One minus the sample Spearman's rank correlation between observations (treated as sequences of values)

**Note**If you specify one of these distance metrics for categorical predictors, then the software treats each categorical predictor as a numeric variable for the distance computation, with each category represented by a positive integer. The

`Distance`

value does not affect the`CategoricalPredictors`

property of the trained model.If all the predictor variables are categorical variables, then you can specify one of these distance metrics.

Value Description `"hamming"`

Hamming distance, which is the percentage of coordinates that differ

`"jaccard"`

One minus the Jaccard coefficient, which is the percentage of nonzero coordinates that differ

**Note**If you specify one of these distance metrics for continuous (numeric) predictors, then the software treats each continuous predictor as a categorical variable for the distance computation. This option does not change the

`CategoricalPredictors`

value.

The default value is `"euclidean"`

if all the predictor variables
are continuous, and `"hamming"`

if all the predictor variables are
categorical.

If you want to use the *K*d-tree algorithm
(

), then
`SearchMethod`

="kdtree"`Distance`

must be `"euclidean"`

,
`"cityblock"`

, `"minkowski"`

, or
`"chebychev"`

.

For more information on the various distance metrics, see Distance Metrics.

**Example: **`Distance="jaccard"`

**Data Types: **`char`

| `string`

`Exponent`

— Minkowski distance exponent

`2`

(default) | positive scalar value

Minkowski distance exponent, specified as a positive scalar value. This argument
is valid only when `Distance`

is
`"minkowski"`

.

**Example: **`Exponent=3`

**Data Types: **`single`

| `double`

`IncludeTies`

— Tie inclusion flag

`false`

or `0`

(default) | `true`

or `1`

Tie inclusion flag indicating whether the software includes all the neighbors
whose distance values are equal to the *k*th smallest distance,
specified as logical `0`

(`false`

) or
`1`

(`true`

). If `IncludeTies`

is `true`

, the software includes all of these neighbors. Otherwise,
the software includes exactly *k* neighbors.

**Example: **`IncludeTies=true`

**Data Types: **`logical`

`NumNeighbors`

— Number of nearest neighbors

`min(20,n-1)`

where `n`

is the
number of unique rows in predictor data (default) | positive integer value

Number of nearest neighbors in the predictor data (`Tbl`

or
`X`

) to find for computing the local outlier factor values,
specified as a positive integer value.

The default value is `min(20,n-1)`

, where `n`

is
the number of unique rows in the predictor data.

**Example: **`NumNeighbors=3`

**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,`lof`

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`

`SearchMethod`

— Nearest neighbor search method

`"kdtree"`

| `"exhaustive"`

Nearest neighbor search method, specified as `"kdtree"`

or
`"exhaustive"`

.

`"kdtree"`

— This method uses the*K*d-tree algorithm to find nearest neighbors. This option is valid when the distance metric (`Distance`

) is one of the following:`"euclidean"`

— Euclidean distance`"cityblock"`

— City block distance`"minkowski"`

— Minkowski distance`"chebychev"`

— Chebychev distance

`"exhaustive"`

— This method uses the exhaustive search algorithm to find nearest neighbors.When you compute local outlier factor values for the predictor data (

`Tbl`

or`X`

), the`lof`

function finds nearest neighbors by computing the distance values from all points in the predictor data to each point in the predictor data.When you compute local outlier factor values for new data

`Xnew`

using the`isanomaly`

function, the function finds nearest neighbors by computing the distance values from all points in the predictor data (`Tbl`

or`X`

) to each point in`Xnew`

.

The default value is `"kdtree"`

if the predictor data has 10 or
fewer columns, the data is not sparse, and the distance metric
(`Distance`

) is valid for the *K*d-tree
algorithm. Otherwise, the default value is `"exhaustive"`

.

## Output Arguments

`LOFObj`

— Trained local outlier factor model

`LocalOutlierFactor`

object

Trained local outlier factor model, returned as a `LocalOutlierFactor`

object.

You can use the object function `isanomaly`

with `LOFObj`

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 logical `1`

(`true`

) when
the observation in the corresponding row of `Tbl`

or
`X`

is an anomaly, and logical `0`

(`false`

) otherwise. `tf`

has the same length as
`Tbl`

or `X`

.

`lof`

identifies observations with
`scores`

above the threshold (`ScoreThreshold`

property value of `LOFObj`

) 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 (local outlier factor values)

numeric column vector

Anomaly scores (local outlier factor values), returned as a
numeric column vector whose values are nonnegative. `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
less than or close to 1 indicates a normal observation, and a value greater than 1 can
indicate an anomaly.

## More About

### Local Outlier Factor

The local outlier factor (LOF) algorithm detects anomalies based on the relative density of an observation with respect to the surrounding neighborhood.

The algorithm finds the *k*-nearest neighbors of an observation and computes the local reachability densities for the observation and its neighbors. The local outlier factor is the average density ratio of the observation to its neighbor. That is, the local outlier factor of observation *p* is

$$LO{F}_{k}(p)=\frac{1}{\left|{N}_{k}(p)\right|}{\displaystyle \sum _{o\in {N}_{k}(p)}\frac{lr{d}_{k}(o)}{lr{d}_{k}(p)}},$$

where

*lrd*(·) is the local reachability density of an observation._{k}*N*(_{k}*p*) represents the*k*-nearest neighbors of observation*p*. You can specify the`IncludeTies`

name-value argument as`true`

to include all the neighbors whose distance values are equal to the*k*th smallest distance, or specify`false`

to include exactly*k*neighbors. The default`IncludeTies`

value of`lof`

is`false`

for more efficient performance. Note that the algorithm in [1] uses all the neighbors.|

*N*(_{k}*p*)| is the number of observations in*N*(_{k}*p*).

For normal observations, the local outlier factor values are less than or close to 1,
indicating that the local reachability density of an observation is higher than or similar
to its neighbors. A local outlier factor value greater than 1 can indicate an anomaly. The
`ContaminationFraction`

argument of `lof`

and the `ScoreThreshold`

argument of `isanomaly`

control the threshold for the local outlier
factor values.

The algorithm measures the density based on the reachability distance. The reachability distance of observation *p* with respect to observation *o* is defined as

$${\tilde{d}}_{k}(p,o)=\mathrm{max}({d}_{k}(o),d(p,o)),$$

where

*d*(_{k}*o*) is the*k*th smallest distance among the distances from observation*o*to its neighbors.*d*(*p*,*o*) is the distance between observation*p*and observation*o*.

The algorithm uses the reachability distance to reduce the statistical fluctuations of *d*(*p*,*o*) for the observations close to observation *o*.

The local reachability density of observation *p* is the reciprocal of the average reachability distance from observation *p* to its neighbors.

$$lr{d}_{k}(p)=1/\frac{{\displaystyle \sum _{o\in {N}_{k}(p)}{\tilde{d}}_{k}(p,o)}}{\left|{N}_{k}(p)\right|}.$$

The density value can be infinity if the number of duplicates is greater than the number of
neighbors (*k*). Therefore, if the training data contains duplicates, the
`lof`

and `isanomaly`

functions use the weighted
local outlier factor (WLOF) algorithm. This algorithm computes the weighted local outlier
factors using the weighted local reachability density (*wlrd*).

$$WLO{F}_{k}(p)=\frac{1}{{\displaystyle \sum _{o\in {N}_{k}(p)}w(o)}}{\displaystyle \sum _{o\in {N}_{k}(p)}\frac{wlr{d}_{k}(o)}{wlr{d}_{k}(p)}},$$

where

$$wlr{d}_{k}(p)=1/\frac{{\displaystyle \sum _{o\in {N}_{k}(p)}w(o){\tilde{d}}_{k}(p,o)}}{{\displaystyle \sum _{o\in {N}_{k}(p)}w(o)}},$$

and *w*(*o*) is the number of duplicates for observation *o* in the
training data. After computing the weight values, the algorithm treats each set of
duplicates as one observation.

### Distance Metrics

A distance metric is a function that defines a distance between two observations. `lof`

supports various distance metrics for continuous variables and categorical variables.

Given an *mx*-by-*n* data matrix *X*, which is treated as *mx* (1-by-*n*) row vectors *x _{1}*,

*x*, ...,

_{2}*x*, and an

_{mx}*my*-by-

*n*data matrix

*Y*, which is treated as

*my*(1-by-

*n*) row vectors

*y*,

_{1}*y*, ...,

_{2}*y*, the various distances between the vector

_{my}*x*and

_{s}*y*are defined as follows:

_{t}Distance metrics for continuous (numeric) variables

Euclidean distance

$${d}_{st}^{2}=({x}_{s}-{y}_{t})({x}_{s}-{y}_{t}{)}^{\prime}.$$

The Euclidean distance is a special case of the Minkowski distance, where

*p*= 2.Specify Euclidean distance by setting the

`Distance`

parameter to`'euclidean'`

.Fast Euclidean distance is the same as Euclidean distance, but uses an algorithm that usually saves time when the number of variables in an observation

*n*exceeds 10. See Algorithms.Mahalanobis distance

$${d}_{st}^{2}=({x}_{s}-{y}_{t}){C}^{-1}({x}_{s}-{y}_{t}{)}^{\prime},$$

where

*C*is the covariance matrix.Specify Mahalanobis distance by setting the

`Distance`

parameter to`'mahalanobis'`

.City block distance

$${d}_{st}={\displaystyle \sum _{j=1}^{n}\left|{x}_{sj}-{y}_{tj}\right|}.$$

The city block distance is a special case of the Minkowski distance, where

*p*= 1.Specify city block distance by setting the

`Distance`

parameter to`'cityblock'`

.Minkowski distance

$${d}_{st}=\sqrt[p]{{\displaystyle \sum _{j=1}^{n}{\left|{x}_{sj}-{y}_{tj}\right|}^{p}}}.$$

For the special case of

*p*= 1, the Minkowski distance gives the city block distance. For the special case of*p*= 2, the Minkowski distance gives the Euclidean distance. For the special case of*p*= ∞, the Minkowski distance gives the Chebychev distance.Specify Minkowski distance by setting the

`Distance`

parameter to`'minkowski'`

.Chebychev distance

$${d}_{st}={\mathrm{max}}_{j}\left\{\left|{x}_{sj}-{y}_{tj}\right|\right\}.$$

The Chebychev distance is a special case of the Minkowski distance, where

*p*= ∞.Specify Chebychev distance by setting the

`Distance`

parameter to`'chebychev'`

.Cosine distance

$${d}_{st}=\left(1-\frac{{x}_{s}{{y}^{\prime}}_{t}}{\sqrt{\left({x}_{s}{{x}^{\prime}}_{s}\right)\left({y}_{t}{{y}^{\prime}}_{t}\right)}}\right).$$

Specify cosine distance by setting the

`Distance`

parameter to`'cosine'`

.Correlation distance

$${d}_{st}=1-\frac{\left({x}_{s}-{\overline{x}}_{s}\right){\left({y}_{t}-{\overline{y}}_{t}\right)}^{\prime}}{\sqrt{\left({x}_{s}-{\overline{x}}_{s}\right){\left({x}_{s}-{\overline{x}}_{s}\right)}^{\prime}}\sqrt{\left({y}_{t}-{\overline{y}}_{t}\right){\left({y}_{t}-{\overline{y}}_{t}\right)}^{\prime}}},$$

where

$${\overline{x}}_{s}=\frac{1}{n}{\displaystyle \sum _{j}{x}_{sj}}$$

and

$${\overline{y}}_{t}=\frac{1}{n}{\displaystyle \sum _{j}{y}_{tj}}.$$

Specify correlation distance by setting the

`Distance`

parameter to`'correlation'`

.Spearman distance is one minus the sample Spearman's rank correlation between observations (treated as sequences of values):

$${d}_{st}=1-\frac{\left({r}_{s}-{\overline{r}}_{s}\right){\left({r}_{t}-{\overline{r}}_{t}\right)}^{\prime}}{\sqrt{\left({r}_{s}-{\overline{r}}_{s}\right){\left({r}_{s}-{\overline{r}}_{s}\right)}^{\prime}}\sqrt{\left({r}_{t}-{\overline{r}}_{t}\right){\left({r}_{t}-{\overline{r}}_{t}\right)}^{\prime}}},$$

where

*r*is the rank of_{sj}*x*taken over_{sj}*x*_{1j},*x*_{2j}, ...*x*, as computed by_{mx,j}`tiedrank`

.*r*is the rank of_{tj}*y*taken over_{tj}*y*_{1j},*y*_{2j}, ...*y*, as computed by_{my,j}`tiedrank`

.*r*and_{s}*r*are the coordinate-wise rank vectors of_{t}*x*and_{s}*y*, that is,_{t}*r*= (_{s}*r*_{s}_{1},*r*_{s}_{2}, ...*r*) and_{sn}*r*= (_{t}*r*_{t1},*r*_{t2}, ...*r*)._{tn}$${\overline{r}}_{s}=\frac{1}{n}{\displaystyle \sum _{j}{r}_{sj}}=\frac{\left(n+1\right)}{2}$$.

$${\overline{r}}_{t}=\frac{1}{n}{\displaystyle \sum _{j}{r}_{tj}}=\frac{\left(n+1\right)}{2}$$.

Specify Spearman distance by setting the

`Distance`

parameter to`'spearman'`

.

Distance metrics for categorical variables

Hamming distance is the percentage of coordinates that differ:

$${d}_{st}=(\#({x}_{sj}\ne {y}_{tj})/n).$$

Specify Hamming distance by setting the

`Distance`

parameter to`'hamming'`

.Jaccard distance is one minus the Jaccard coefficient, which is the percentage of nonzero coordinates that differ:

$${d}_{st}=\frac{\#\left[\left({x}_{sj}\ne {y}_{tj}\right)\cap \left(\left({x}_{sj}\ne 0\right)\cup \left({y}_{tj}\ne 0\right)\right)\right]}{\#\left[\left({x}_{sj}\ne 0\right)\cup \left({y}_{tj}\ne 0\right)\right]}.$$

Specify Jaccard distance by setting the

`Distance`

parameter to`'jaccard'`

.

## Algorithms

### Missing Values

`lof`

considers `NaN`

, `''`

(empty character vector), `""`

(empty string), `<missing>`

, and `<undefined>`

values in `Tbl`

and `NaN`

values in `X`

to be missing values.

`lof`

does not use observations with missing values.`lof`

assigns the anomaly score of`NaN`

and anomaly indicator of`false`

(logical 0) to observations with missing values.

### Fast Euclidean Distance Algorithm

The `"fasteuclidean"`

`Distance`

calculates Euclidean distances using extra memory to save
computational time. This algorithm is named "Euclidean Distance Matrix Trick" in Albanie
[2] and elsewhere. Internal
testing shows that this algorithm saves time when the number of predictors exceeds 10. The
`"fasteuclidean"`

distance does not support sparse data.

To find the matrix D of distances between all the points
*x _{i}* and

*x*, where each

_{j}*x*has

_{i}*n*variables, the algorithm computes distance using the final line in the following equations:

$$\begin{array}{c}{D}_{i,j}^{2}=\Vert {x}_{i}-{x}_{j}{\Vert}^{2}\\ ={(}^{{x}_{i}}({x}_{i}-{x}_{j})\\ =\Vert {x}_{i}{\Vert}^{2}-2{x}_{i}^{T}{x}_{j}+\Vert {x}_{j}{\Vert}^{2}.\end{array}$$

The matrix $${x}_{i}^{T}{x}_{j}$$ in the last line of the equations is called the *Gram
matrix*. Computing the set of squared distances is faster, but slightly less
numerically stable, when you compute and use the Gram matrix instead of computing the
squared distances by squaring and summing. For more information, see Albanie [2].

To store the Gram matrix, the software uses a cache with the default size of
`1e3`

megabytes. You can set the cache size using the
`CacheSize`

name-value argument. If the value of
`CacheSize`

is too large or `"maximal"`

,
`lof`

might try to allocate a Gram matrix that exceeds the
available memory. In this case, MATLAB issues an error.

## References

[1] Breunig, Markus M., et al. “LOF: Identifying Density-Based Local Outliers.” *Proceedings of the 2000 ACM SIGMOD International Conference on Management of Data*, 2000, pp. 93–104.

[2] Albanie, Samuel.
*Euclidean Distance Matrix Trick.* June, 2019. Available at https://www.robots.ox.ac.uk/%7Ealbanie/notes/Euclidean_distance_trick.pdf.

## Version History

**Introduced in R2022b**

### R2023b: `"fasteuclidean"`

distance support

The `lof`

function supports the `"fasteuclidean"`

`Distance`

algorithm. This algorithm usually computes distances faster
than the default `"euclidean"`

algorithm when the number of variables in a
data point exceeds 10. The algorithm uses extra memory to store an intermediate Gram matrix
(see Algorithms). Set the size of this
memory allocation using the `CacheSize`

name-value argument.

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