filloutliers

Detect and replace outliers in data

Syntax

B = filloutliers(A,fillmethod)

B = filloutliers(A,fillmethod,findmethod)

B = filloutliers(A,fillmethod,'percentiles',threshold)

B = filloutliers(A,fillmethod,movmethod,window)

B = filloutliers(___,dim)

B = filloutliers(___,Name,Value)

[B,TF,L,U,C] = filloutliers(___)

Description

B = filloutliers(A,fillmethod) finds outliers in A and replaces them according to fillmethod. For example, filloutliers(A,'previous') replaces outliers with the previous non-outlier element. By default, an outlier is a value that is more than three scaled median absolute deviations (MAD) away from the median. If A is a matrix or table, then filloutliers operates on each column separately. If A is a multidimensional array, then filloutliers operates along the first dimension whose size does not equal 1.

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

B = filloutliers(A,fillmethod,'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 = filloutliers(A,fillmethod,movmethod,window) specifies a moving method for detecting local outliers according to a window length defined by window. For example, filloutliers(A,'previous','movmean',5) identifies outliers as elements more than three local standard deviations away from the local mean within a five-element window.

B = filloutliers(___,dim) operates along dimension dim of A for any of the previous syntaxes. For example, filloutliers(A,'linear',2) operates on each row of a matrix A.

B = filloutliers(___,Name,Value) specifies additional parameters for detecting and replacing outliers using one or more name-value pair arguments. For example, filloutliers(A,'previous','SamplePoints',t) detects outliers in A relative to the corresponding elements of a time vector t.

[B,TF,L,U,C] = filloutliers(___) also returns information about the position of the outliers and thresholds computed by the detection method. TF is a logical array indicating the location of the outliers in A. The L, U, and C arguments represent the lower and upper thresholds and the center value used by the outlier detection method.

Examples

Interpolate Outliers

Create a vector of data containing an outlier, and use linear interpolation to replace the outlier. Plot the original and filled data.

A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57];
B = filloutliers(A,'linear');
plot(1:15,A,1:15,B,'o')
legend('Original Data','Interpolated Data')

Determine Outliers with Mean

Create a vector containing an outlier, and define outliers as points outside three standard deviations from the mean. Replace the outlier with the nearest element that is not an outlier, and plot the original data and the interpolated data.

A = [57 59 60 100 59 58 57 58 300 61 62 60 62 58 57];
B = filloutliers(A,'nearest','mean');
plot(1:15,A,1:15,B,'o')
legend('Original Data','Interpolated Data')

Determine Outliers with Sliding Window

Use a moving median to find local outliers within 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 away from the local median within a sliding window. Find the location of the outlier in A relative to the points in t with a window size of 5 hours. Fill the outlier with the computed threshold value using the method 'clip', and plot the original and filled data.

[B,TF,U,L,C] = filloutliers(A,'clip','movmedian',hours(5),'SamplePoints',t);
plot(t,A,t,B,'o')
legend('Original Data','Filled Data')

Display the threshold value that replaced the outlier.

L(TF)

ans = -0.8779

Matrix of Data

Fill outliers for each row of a matrix.

Create a matrix of data containing outliers along the diagonal.

A = randn(5,5) + diag(1000*ones(1,5))

A = 5×5
103 ×

    1.0005   -0.0013   -0.0013   -0.0002    0.0007
    0.0018    0.9996    0.0030   -0.0001   -0.0012
   -0.0023    0.0003    1.0007    0.0015    0.0007
    0.0009    0.0036   -0.0001    1.0014    0.0016
    0.0003    0.0028    0.0007    0.0014    1.0005

Fill outliers with zeros based on the data in each row, and display the new values.

[B,TF,lower,upper,center] = filloutliers(A,0,2);
B

B = 5×5

         0   -1.3077   -1.3499   -0.2050    0.6715
    1.8339         0    3.0349   -0.1241   -1.2075
   -2.2588    0.3426         0    1.4897    0.7172
    0.8622    3.5784   -0.0631         0    1.6302
    0.3188    2.7694    0.7147    1.4172         0

You can directly access the detected outlier values and their filled values using TF as an index vector.

[A(TF) B(TF)]

ans = 5×2
103 ×

    1.0005         0
    0.9996         0
    1.0007         0
    1.0014         0
    1.0005         0

Outlier Thresholds

Find the outlier in a vector of data, and replace it using the 'clip' method. Plot the original data, the filled data, and the thresholds and center value determined by the detection method. 'clip' replaces the outlier with the upper threshold value.

x = 1:10;
A = [60 59 49 49 58 100 61 57 48 58];
[B,TF,lower,upper,center] = filloutliers(A,'clip');
plot(x,A,x,B,'o',x,lower*ones(1,10),x,upper*ones(1,10),x,center*ones(1,10))
legend('Original Data','Filled Data','Lower Threshold','Upper Threshold','Center Value')

Input Arguments

A — Input data

vector | matrix | multidimensional array | table | timetable

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

If A is a table, then its variables must be of type double or single, or you can use the 'DataVariables' name-value pair 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 filloutliers operates only on the table elements. Row times must be unique and listed in ascending order.

Data Types: double | single | table | timetable

fillmethod — Fill method

numeric scalar | 'center' | 'clip' | 'previous' | 'next' | 'nearest' | 'linear' | 'spline' | 'pchip' | 'makima'

Fill method for replacing outliers, specified as a numeric scalar or one of the following:

Fill Method	Description
Numeric scalar	Fills with specified scalar value
'center'	Fills with the center value determined by findmethod
'clip'	Fills with the lower threshold value for elements smaller than the lower threshold determined by findmethod. Fills with the upper threshold value for elements larger than the upper threshold determined by findmethod
'previous'	Fills with the previous non-outlier value
'next'	Fills with the next non-outlier value
'nearest'	Fills with the nearest non-outlier value
'linear'	Fills using linear interpolation of neighboring, non-outlier values
'spline'	Fills using piecewise cubic spline interpolation
'pchip'	Fills using shape-preserving piecewise cubic spline interpolation
'makima'	modified Akima cubic Hermite interpolation (numeric, duration, and datetime data types only)

Data Types: double | single | char

findmethod — Method for detecting outliers

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

Method for detecting outliers, specified as one of the following:

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’s test, 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. For example, a threshold of [10 90] defines outliers as points below the 10th percentile and above the 90th percentile. The first element of threshold must be less than the second element.

movmethod — Moving method

'movmedian' | 'movmean'

Moving method for detecting outliers, specified as one of the following:

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.

Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | duration

dim — Dimension to operate along

positive integer scalar

Dimension to operate along, specified as a positive integer scalar. If no value is specified, then the default is the first array dimension whose size does not equal 1.

Consider a matrix A.

filloutliers(A,fillmethod,1) fills outliers according to the data in each column.

filloutliers(A,fillmethod,2) fills outliers according to the data in each row.

When A is a table or timetable, dim is not supported. filloutliers operates along each table or timetable variable separately.

Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: filloutliers(A,'center','mean','ThresholdFactor',4)

Data Options

'SamplePoints' — Sample points

vector

Sample points, specified as the comma-separated pair consisting of 'SamplePoints' and a vector. The sample points represent the x-axis locations of the data in A, and must be sorted and contain unique elements. Sample points do not need to be uniformly sampled. The default sample points vector is [1 2 3 ...].

Note

This name-value pair is not supported when the input data is a timetable. Timetables always 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 filloutliers(rand(1,10),'previous','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: filloutliers([1 100 3 4],'nearest','SamplePoints',[1 2.5 3 4])

Data Types: single | double | datetime | duration

'DataVariables' — Table variables to operate on

scalar | vector | cell array | function handle | table vartype subscript

Table variables to operate on, specified as the comma-separated pair consisting of 'DataVariables' and one of the options in this table. The 'DataVariables' value indicates which variables of the input table to fill. Other variables in the table not specified by 'DataVariables' pass through to the output without being operated on.

Option	Description	Examples
Variable name	A character vector or scalar string specifying a single table variable name	'Var1' "Var1"
Vector of variable names	A cell array of character vectors or string array where each element is a table variable name	{'Var1' 'Var2'} ["Var1" "Var2"]
Scalar or vector of variable indices	A scalar or vector of table variable indices	1 [1 3 5]
Logical vector	A logical vector whose elements each correspond to a table variable, where true includes the corresponding variable and false excludes it	[true false true]
Function handle	A function handle that takes a table variable as input and returns a logical scalar	@isnumeric
vartype subscript	A table subscript generated by the vartype function	vartype('numeric')

Example: filloutliers(A,'previous','DataVariables',["Var1" "Var2" "Var4"])

Outlier Detection Options

'ThresholdFactor' — Detection threshold factor

nonnegative scalar

Detection threshold factor, specified as the comma-separated pair consisting of 'ThresholdFactor' and 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'.

Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

'MaxNumOutliers' — Maximum outlier count

positive scalar

Maximum outlier count, for the 'gesd' method only, specified as the comma-separated pair consisting of 'MaxNumOutliers' and a positive scalar. The 'MaxNumOutliers' value specifies the maximum number of outliers returned by the 'gesd' method. For example, filloutliers(A,'linear','gesd','MaxNumOutliers',5) returns 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 can ensure that all outliers are detected, but at the cost of reduced computational efficiency.

Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

'OutlierLocations' — Known outlier indicator

vector | matrix | multidimensional array

Known outlier indicator, specified as the comma-separated pair consisting of 'OutlierLocations' and a logical vector, matrix, or multidimensional array of the same size as A. The known outlier indicator elements can be true to indicate an outlier in the corresponding location of A or false otherwise. Specifying 'OutlierLocations' turns off the default outlier detection method, and uses only the elements of the known outlier indicator to define outliers.

The 'OutlierLocations' name-value pair cannot be specified when findmethod is specified.

The output TF is the same as the 'OutlierLocations' value.

Data Types: logical

Output Arguments

B — Filled outlier array

vector | matrix | multidimensional array | table | timetable

Filled outlier array, returned as a vector, matrix, multidimensional array, table, or timetable. The elements of B are the same as those of A, but with all outliers replaced according to fillmethod.

Data Types: double | single | table | timetable

TF — Outlier indicator

vector | matrix | multidimensional array

Outlier indicator, returned as a vector, matrix, or multidimensional array. An element of TF is true when the corresponding element of A is an outlier and false otherwise. TF is the same size as A.

Data Types: logical

L — Lower threshold

scalar | vector | matrix | multidimensional array | table | timetable

Lower threshold used by the outlier detection method, returned as a scalar, vector, matrix, multidimensional array, table, or timetable. For example, the lower value of the default outlier detection method is three scaled MAD below the median of the input data. L has the same size as A in all dimensions except for the operating dimension where the length is 1.

Data Types: double | single | table | timetable

U — Upper threshold

scalar | vector | matrix | multidimensional array | table | timetable

Upper threshold used by the outlier detection method, returned as a scalar, vector, matrix, multidimensional array, table, or timetable. For example, the upper value of the default outlier detection method is three scaled MAD above the median of the input data. U has the same size as A in all dimensions except for the operating dimension where the length is 1.

Data Types: double | single | table | timetable

C — Center value

scalar | vector | matrix | multidimensional array | table | timetable

Center value used by the outlier detection method, returned as a scalar, vector, matrix, multidimensional array, table, or timetable. For example, the center value of the default outlier detection method is the median of the input data. C has the same size as A in all dimensions except for the operating dimension where the length is 1.

Data Types: double | single | table | timetable

More About

Median Absolute Deviation

For a random variable vector A made up of N scalar observations, the median absolute deviation (MAD) is defined as

$$\text{MAD = median}\left(|A_i-\text{median}\left(A\right)|\right)$$

for i = 1,2,...,N.

The scaled MAD is defined as c*median(abs(A-median(A))) where c=-1/(sqrt(2)*erfcinv(3/2)).

Extended Capabilities

Tall Arrays

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

Usage notes and limitations:

• The 'percentiles', 'grubs', and 'gesd' methods are not supported.

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

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

• The value of 'DataVariables' cannot be a function handle.

• Computation of filloutliers(A,fillmethod), filloutliers(A,fillmethod,'median',…) or filloutliers(A,fillmethod,'quartiles',…) along the first dimension is only supported when A is a tall column vector.

• The syntaxes filloutliers(A,'spline',…) and filloutliers(A,'makima',…) are not supported.

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.

• Only the 'center', 'clip', and numeric scalar methods for filling outliers are supported when the input data is a timetable or when the 'SamplePoints' value has type datetime or duration.

• To use the 'spline' and 'pchip' fill methods, you must enable support for variable-size arrays.

• String and character array inputs must be constant.

• The 'makima' options is not supported.