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
oflof
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) tolof
. Detect anomalies in new data by passing the object and the new data to the object functionisanomaly
.
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)
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
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
Kd-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 theDistance
name-value argument.[]
— No predictors are categorical, that is, all predictors are continuous (numeric). In this case, the defaultDistance
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. IfTbl
includes both continuous and categorical values, and you want to identify all predictors inTbl
as categorical, you must specifyCategoricalPredictors
as"all"
.If the predictor data is a matrix (
X
),lof
assumes that all predictors are continuous. To identify all predictors inX
as categorical, specifyCategoricalPredictors
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), thenlof
treats all training observations as normal observations, and sets the score threshold (ScoreThreshold
property value ofLOFObj
) to the maximum value ofscores
.If the
ContaminationFraction
value is in the range (0
,1
], thenlof
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 theCategoricalPredictors
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 Kd-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 kth 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 usePredictorNames
to specify which predictor variables to use. That is,lof
uses only the predictor variables inPredictorNames
.PredictorNames
must be a subset ofTbl.Properties.VariableNames
.By default,
PredictorNames
contains the names of all predictor variables inTbl
.
If you supply
X
, then you can usePredictorNames
to assign names to the predictor variables inX
.The order of the names in
PredictorNames
must correspond to the column order ofX
. That is,PredictorNames{1}
is the name ofX(:,1)
,PredictorNames{2}
is the name ofX(:,2)
, and so on. Also,size(X,2)
andnumel(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 Kd-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
orX
), thelof
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 theisanomaly
function, the function finds nearest neighbors by computing the distance values from all points in the predictor data (Tbl
orX
) to each point inXnew
.
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 Kd-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
where
lrdk(·) is the local reachability density of an observation.
Nk(p) represents the k-nearest neighbors of observation p. You can specify the
IncludeTies
name-value argument astrue
to include all the neighbors whose distance values are equal to the kth smallest distance, or specifyfalse
to include exactly k neighbors. The defaultIncludeTies
value oflof
isfalse
for more efficient performance. Note that the algorithm in [1] uses all the neighbors.|Nk(p)| is the number of observations in Nk(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
where
dk(o) is the kth 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.
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).
where
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 x1, x2, ..., xmx, and an my-by-n data matrix Y, which is treated as my (1-by-n) row vectors y1, y2, ...,ymy, the various distances between the vector xs and yt are defined as follows:
Distance metrics for continuous (numeric) variables
Euclidean distance
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
where C is the covariance matrix.
Specify Mahalanobis distance by setting the
Distance
parameter to'mahalanobis'
.City block distance
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
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
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
Specify cosine distance by setting the
Distance
parameter to'cosine'
.Correlation distance
where
and
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):
where
Specify Spearman distance by setting the
Distance
parameter to'spearman'
.
Distance metrics for categorical variables
Hamming distance is the percentage of coordinates that differ:
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:
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 ofNaN
and anomaly indicator offalse
(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 xi and xj, where each xi has n variables, the algorithm computes distance using the final line in the following equations:
The matrix 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 R2022bR2023b: "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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