rmoutliers

Detect and remove outliers in data

Since R2018b

Syntax

B = rmoutliers(A)

B = rmoutliers(A,method)

B = rmoutliers(A,"percentiles",threshold)

B = rmoutliers(A,movmethod,window)

B = rmoutliers(___,dim)

B = rmoutliers(___,Name,Value)

[B,TFrm] = rmoutliers(___)

[B,TFrm,TFoutlier] = rmoutliers(___)

[B,TFrm,TFoutlier,L,U,C] = rmoutliers(___)

Description

B = rmoutliers(A) detects and removes outliers from the data in A.

• If A is a matrix, then rmoutliers detects outliers in each column of A separately and removes the entire row.

• If A is a table or timetable, then rmoutliers detects outliers in each variable of A separately and removes the entire row.

By default, an outlier is a value that is more than three scaled median absolute deviations (MAD) from the median.

B = rmoutliers(A,method) specifies a method for detecting outliers. For example, rmoutliers(A,"mean") defines an outlier as an element of A more than three standard deviations from the mean.

B = rmoutliers(A,"percentiles",threshold) defines outliers as points outside of the percentiles specified in threshold. The threshold argument is a two-element row vector containing the lower and upper percentile thresholds, such as [10 90].

B = rmoutliers(A,movmethod,window) detects local outliers using a moving window mean or median with window length window. For example, rmoutliers(A,"movmean",5) defines outliers as elements more than three local standard deviations from the local mean within a five-element window.

B = rmoutliers(___,dim) specifies the dimension of A for which to remove entries when an outlier is detected using any of the previous syntaxes. For example, rmoutliers(A,2) removes columns instead of rows for a matrix A.

B = rmoutliers(___,Name,Value) specifies additional parameters for detecting and removing outliers using one or more name-value arguments. For example, rmoutliers(A,"SamplePoints",t) detects outliers in A relative to the corresponding elements of a time vector t.

[B,TFrm] = rmoutliers(___) also returns a logical vector TFrm that indicates the rows or columns removed from A.

[B,TFrm,TFoutlier] = rmoutliers(___) also returns a logical array TFoutlier that indicates the locations of the outliers removed from A.

[B,TFrm,TFoutlier,L,U,C] = rmoutliers(___) also returns the lower threshold L, upper threshold U, and center value C used by the outlier detection method.

Examples

Remove Outliers from Vector

Create a vector containing two outliers and remove them.

A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57];
B = rmoutliers(A)

B = 1×13

    57    59    60    59    58    57    58    61    62    60    62    58    57

Use Mean Detection Method

Identify potential outliers in a timetable of data using the mean detection method, remove any outliers, and visualize the cleaned data.

Create a timetable of data, and visualize the data to detect potential outliers.

T = hours(1:15);
V = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57];
A = timetable(T',V');
plot(A.Time,A.Var1)

Remove outliers in the data, where an outlier is defined as a point more than three standard deviations from the mean.

B = rmoutliers(A,"mean")

B=14×1 timetable
    Time     Var1
    _____    ____

    1 hr      57 
    2 hr      59 
    3 hr      60 
    4 hr     100 
    5 hr      59 
    6 hr      58 
    7 hr      57 
    8 hr      58 
    10 hr     61 
    11 hr     62 
    12 hr     60 
    13 hr     62 
    14 hr     58 
    15 hr     57 

In the same graph, plot the original data and the data with the outlier removed.

hold on
plot(B.Time,B.Var1,"o-")
legend("Original Data","Cleaned Data")

Use Moving Detection Method

Use a moving median to detect and remove local outliers from a sine wave that corresponds to a time vector.

Create a vector of data containing a local outlier.

x = -2*pi:0.1:2*pi;
A = sin(x);
A(47) = 0;

Create a time vector that corresponds to the data in A.

t = datetime(2017,1,1,0,0,0) + hours(0:length(x)-1);

Define outliers as points more than three local scaled MAD from the local median within a sliding window. Find the locations of the outliers in A relative to the points in t with a window size of 5 hours, and remove them.

[B,TFrm] = rmoutliers(A,"movmedian",hours(5),"SamplePoints",t);

Plot the original data and the data with the outlier removed.

plot(t,A)
hold on
plot(t(~TFrm),B,"o-")
legend("Original Data","Cleaned Data")

Remove Columns Containing Outliers

Remove the outliers from a matrix of data, and examine the removed columns and outliers.

Create a matrix containing two outliers.

A = magic(5);
A(4,4) = 200;
A(5,5) = 300;
A

A = 5×5

    17    24     1     8    15
    23     5     7    14    16
     4     6    13    20    22
    10    12    19   200     3
    11    18    25     2   300

Remove the columns containing outliers by specifying the dimension for removal as 2. Return a logical output vector TFrm to identify which columns of A were removed, and return a logical output array TFoutlier to identify the locations of the outliers in A.

[B,TFrm,TFoutlier] = rmoutliers(A,2)

B = 5×3

    17    24     1
    23     5     7
     4     6    13
    10    12    19
    11    18    25

TFrm = 1x5 logical array

   0   0   0   1   1

TFoutlier = 5x5 logical array

   0   0   0   0   0
   0   0   0   0   0
   0   0   0   0   0
   0   0   0   1   0
   0   0   0   0   1

Find the values in the removed columns of A.

rmCol = A(:,TFrm)

rmCol = 5×2

     8    15
    14    16
    20    22
   200     3
     2   300

Find the values of the outliers in A.

rmVal = A(TFoutlier)

rmVal = 2×1

   200
   300

Specify Outlier Locations

Create a vector containing two outliers and detect their locations.

A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57];
detect = isoutlier(A)

detect = 1x15 logical array

   0   0   0   1   0   0   0   0   1   0   0   0   0   0   0

Remove the outliers. Instead of using a detection method, provide the outlier locations detected by isoutlier.

B = rmoutliers(A,"OutlierLocations",detect)

B = 1×13

    57    59    60    59    58    57    58    61    62    60    62    58    57

Visualize Outlier Thresholds

Remove an outlier from a vector of data and visualize the cleaned data.

Create a vector of data containing an outlier.

A = [60 59 49 49 58 100 61 57 48 58];

Remove the outlier using the default detection method "median".

[B,TFrm,TFoutlier,L,U,C] = rmoutliers(A);

Plot the original data, the data with outliers removed, and the thresholds and center value determined by the detection method. The center value is the median of the data, and the upper and lower thresholds are three scaled MAD above and below the median.

plot(A)
hold on
plot(find(~TFrm),B,"o-")
yline([L U C],":",["Lower Threshold","Upper Threshold","Center Value"])
legend("Original Data","Cleaned Data")

Input Arguments

A — Input data

vector | matrix | table | timetable

Input data, specified as a vector, matrix, table, or timetable.

• If A is a table, then its variables must be of type double or single, or you can use the DataVariables argument to list double or single variables explicitly. Specifying variables is useful when you are working with a table that contains variables with data types other than double or single.

• If A is a timetable, then rmoutliers operates only on the table elements. If row times are used as sample points, then they must be unique and listed in ascending order.

Data Types: double | single | table | timetable

method — Method for detecting outliers

"median" (default) | "mean" | "quartiles" | "grubbs" | "gesd"

Method for detecting outliers, specified as one of these values.

Method	Description
"median"	Outliers are defined as elements more than three scaled MAD from the median. The scaled MAD is defined as c*median(abs(A-median(A))), where c=-1/(sqrt(2)*erfcinv(3/2)).
"mean"	Outliers are defined as elements more than three standard deviations from the mean. This method is faster but less robust than "median".
"quartiles"	Outliers are defined as elements more than 1.5 interquartile ranges above the upper quartile (75 percent) or below the lower quartile (25 percent). This method is useful when the data in A is not normally distributed.
"grubbs"	Outliers are detected using Grubbs’ test for outliers, which removes one outlier per iteration based on hypothesis testing. This method assumes that the data in A is normally distributed.
"gesd"	Outliers are detected using the generalized extreme Studentized deviate test for outliers. This iterative method is similar to "grubbs" but can perform better when there are multiple outliers masking each other.

threshold — Percentile thresholds

two-element row vector

Percentile thresholds, specified as a two-element row vector whose elements are in the interval [0, 100]. The first element indicates the lower percentile threshold, and the second element indicates the upper percentile threshold. The first element of threshold must be less than the second element.

For example, a threshold of [10 90] defines outliers as points below the 10th percentile and above the 90th percentile.

movmethod — Moving method

"movmedian" | "movmean"

Moving method for detecting outliers, specified as one of these values.

Method	Description
"movmedian"	Outliers are defined as elements more than three local scaled MAD from the local median over a window length specified by window. This method is also known as a Hampel filter.
"movmean"	Outliers are defined as elements more than three local standard deviations from the local mean over a window length specified by window.

window — Window length

positive integer scalar | two-element vector of positive integers | positive duration scalar | two-element vector of positive durations

Window length, specified as a positive integer scalar, a two-element vector of positive integers, a positive duration scalar, or a two-element vector of positive durations.

When window is a positive integer scalar, the window is centered about the current element and contains window-1 neighboring elements. If window is even, then the window is centered about the current and previous elements.

When window is a two-element vector of positive integers [b f], the window contains the current element, b elements backward, and f elements forward.

When A is a timetable or SamplePoints is specified as a datetime or duration vector, window must be of type duration, and the windows are computed relative to the sample points.

dim — Dimension for removal

1 (default) | 2

Dimension for removal, specified as 1 or 2. By default, rmoutliers removes each row with a detected outlier. To remove each matrix column or table variable with a detected outlier, specify a dimension of 2.

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: rmoutliers(A,ThresholdFactor=4)

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: rmoutliers(A,"ThresholdFactor",4)

Data Options

SamplePoints — Sample points

vector | table variable name | scalar | function handle | table vartype subscript

Sample points, specified as either a vector of sample point values or one of the options in the following table when the input data is a table. The sample points represent the x-axis locations of the data, and must be sorted and contain unique elements. Sample points do not need to be uniformly sampled. The vector [1 2 3 ...] is the default.

When the input data is a table, you can specify the sample points as a table variable using one of these options.

Indexing Scheme	Examples
Variable name: A string scalar or character vector	"A" or 'A' — A variable named A
Variable index: An index number that refers to the location of a variable in the table A logical vector. Typically, this vector is the same length as the number of variables, but you can omit trailing 0 or false values	3 — The third variable from the table [false false true] — The third variable
Function handle: A function handle that takes a table variable as input and returns a logical scalar	@isnumeric — One variable containing numeric values
Variable type: A vartype subscript that selects one variable of a specified type	vartype("numeric") — One variable containing numeric values

Note

This name-value argument is not supported when the input data is a timetable. Timetables use the vector of row times as the sample points. To use different sample points, you must edit the timetable so that the row times contain the desired sample points.

Moving windows are defined relative to the sample points. For example, if t is a vector of times corresponding to the input data, then rmoutliers(rand(1,10),"movmean",3,"SamplePoints",t) has a window that represents the time interval between t(i)-1.5 and t(i)+1.5.

When the sample points vector has data type datetime or duration, then the moving window length must have type duration.

Example: rmoutliers(A,"SamplePoints",0:0.1:10)

Example: rmoutliers(T,"SamplePoints","Var1")

Data Types: single | double | datetime | duration

DataVariables — Table variables to operate on

table variable name | scalar | vector | cell array | pattern | function handle | table vartype subscript

Table variables to operate on, specified as one of the options in this table. The DataVariables value indicates which variables of the input table to examine for outliers. The data type associated with the indicated variables must be double or single.

Other variables in the table not specified by DataVariables pass through to the output without being examined for outliers.

When operating on the rows of A, rmoutliers removes any row that has outliers in the columns corresponding to the variables specified. When operating on the columns of A, rmoutliers removes the specified variables from the table.

Indexing Scheme	Examples
Variable names: A string, character vector, or cell array A pattern object	"A" or 'A' — A variable named A ["A","B"] or {'A','B'} — Two variables named A and B "Var"+digitsPattern(1) — Variables named "Var" followed by a single digit
Variable index: An index number that refers to the location of a variable in the table A vector of numbers A logical vector. Typically, this vector is the same length as the number of variables, but you can omit trailing 0 or false values	3 — The third variable from the table [2 3] — The second and third variables from the table [false false true] — The third variable
Function handle: A function handle that takes a table variable as input and returns a logical scalar	@isnumeric — All the variables containing numeric values
Variable type: A vartype subscript that selects variables of a specified type	vartype("numeric") — All the variables containing numeric values

Example: rmoutliers(T,"DataVariables",["Var1" "Var2" "Var4"])

Outlier Detection Options

ThresholdFactor — Detection threshold factor

nonnegative scalar

Detection threshold factor, specified as a nonnegative scalar.

For methods "median" and "movmedian", the detection threshold factor replaces the number of scaled MAD, which is 3 by default.

For methods "mean" and "movmean", the detection threshold factor replaces the number of standard deviations from the mean, which is 3 by default.

For methods "grubbs" and "gesd", the detection threshold factor is a scalar ranging from 0 to 1. Values close to 0 result in a smaller number of outliers, and values close to 1 result in a larger number of outliers. The default detection threshold factor is 0.05.

For the "quartiles" method, the detection threshold factor replaces the number of interquartile ranges, which is 1.5 by default.

This name-value pair is not supported when the specified method is "percentiles".

OutlierLocations — Known outlier indicator

vector | matrix

Known outlier indicator, specified as a logical vector or matrix of the same size as A. The known outlier indicator elements can be a numeric or logical 1 (true) to indicate an outlier in the corresponding location of A or 0 (false) to indicate a nonoutlier. When you specify OutlierLocations, rmoutliers does not use an outlier detection method. Instead, it uses the elements of the known outlier indicator to define outliers. The output TFoutlier contains the same logical vector or matrix.

You cannot specify the OutlierLocations name-value argument if you specify method.

Data Types: logical

MaxNumOutliers — Maximum outliers detected by GESD

positive integer scalar

Maximum outliers detected by GESD, specified as a positive integer scalar. The MaxNumOutliers value specifies the maximum number of outliers that are detected by the "gesd" method. For example, rmoutliers(A,"gesd","MaxNumOutliers",5) detects no more than five outliers.

The default value for MaxNumOutliers is the integer nearest to 10 percent of the number of elements in A. Setting a larger value for the maximum number of outliers makes it more likely that all outliers are detected but at the cost of reduced computational efficiency.

The "gesd" method assumes the nonoutlier input data is sampled from an approximate normal distribution. When the data is not sampled in this way, the number of detected outliers might exceed the MaxNumOutliers value.

MinNumOutliers — Minimum outliers required for removal

1 (default) | positive integer scalar

Minimum outliers required for removal, specified as a positive integer scalar. The MinNumOutliers value specifies the minimum number of outliers required to remove a row or column. For example, rmoutliers(A,"MinNumOutliers",3) removes a row of a matrix A when there are 3 or more outliers detected in that row.

Output Arguments

B — Data with outliers removed

vector | matrix | table | timetable

Data with outliers removed, returned as a vector, matrix, table, or timetable. The size of B depends on the number of removed rows or columns.

TFrm — Removed data indicator

vector

Removed data indicator, returned as a logical vector. Elements with a value of 1 (true) correspond to rows or columns of A that were removed. Elements with a value of 0 (false) correspond to unchanged rows or columns. The orientation and size of TFrm depend on A and the dimension of operation.

Data Types: logical

TFoutlier — Outlier indicator

vector | matrix

Outlier indicator, returned as a logical vector or matrix. Elements with a value of 1 (true) correspond to the location of outliers in A. Elements with a value of 0 (false) correspond to nonoutliers.

TFoutlier is the same size as A.

Data Types: logical

L — Lower threshold

scalar | vector | matrix | table | timetable

Since R2022b

Lower threshold used by the outlier detection method, returned as a scalar, vector, matrix, table, or timetable. For example, the lower threshold value of the default outlier detection method is three scaled MAD below the median of the input data.

If method is used for outlier detection, then L has the same size as A in all dimensions except for the operating dimension where the length is 1. If movmethod is used, then L has the same size as A.

U — Upper threshold

scalar | vector | matrix | table | timetable

Since R2022b

Upper threshold used by the outlier detection method, returned as a scalar, vector, matrix, table, or timetable. For example, the upper threshold value of the default outlier detection method is three scaled MAD above the median of the input data.

If method is used for outlier detection, then U has the same size as A in all dimensions except for the operating dimension where the length is 1. If movmethod is used, then U has the same size as A.

C — Center value

scalar | vector | matrix | table | timetable

Since R2022b

Center value used by the outlier detection method, returned as a scalar, vector, matrix, table, or timetable. For example, the center value of the default outlier detection method is the median of the input data.

If method is used for outlier detection, then C has the same size as A in all dimensions except for the operating dimension where the length is 1. If movmethod is used, then C has the same size as A.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

Usage notes and limitations:

• The "percentiles", "grubbs", and "gesd" methods are not supported.

• The "movmedian" and "movmean" methods do not support tall timetables.

• The SamplePoints and MaxNumOutliers name-value arguments are not supported.

• The value of DataVariables cannot be a function handle.

• Computation of rmoutliers(A), rmoutliers(A,"median",...), or rmoutliers(A,"quartiles",...) along the first dimension is supported only when A is a tall column vector.

• rmoutliers(A,2) is not supported for tall tables.

For more information, see Tall Arrays.

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• The "movmean" and "movmedian" methods for detecting outliers do not support timetable input data, datetime SamplePoints values, or duration SamplePoints values.

• For table input data, dim must equal 1.

• The OutlierLocation name-value argument is not supported.

• The optional output arguments TFoutlier, L, U, and C are not supported.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

• When using moving method "movmean" or "movmedian" to detect outliers, the SamplePoints name-value argument is not supported.

• The DataVariables name-value argument is not supported.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2018b

R2022b: Return outlier indicator, lower threshold, upper threshold, and center value

You can optionally return a logical outlier indicator that corresponds to the locations of outliers in the input data. You can also return the lower threshold value, upper threshold value, and center value used by the outlier detection method.

R2022b: Define outlier locations

Define the location of outliers in the input data with a known outlier indicator. You can define outlier locations, rather than using an outlier detection method, by setting the OutlierLocations name-value argument to a logical array the same size as the input data.

You cannot specify the OutlierLocations name-value argument if you specify method.

R2021b: Specify sample points as table variable

For table input data, specify the sample points as a table variable using the SamplePoints name-value argument.