movmad

Moving median absolute deviation

Syntax

M = movmad(A,k)

M = movmad(A,[kb kf])

M = movmad(___,dim)

M = movmad(___,nanflag)

M = movmad(___,Name,Value)

Description

M = movmad(A,k) returns an array of local k-point median absolute deviations (MADs), where each MAD is calculated over a sliding window of length k across neighboring elements of A. M is the same size as A.

When k is odd, the window is centered about the element in the current position. When k is even, the window is centered about the current and previous elements. The window size is automatically truncated at the endpoints when there are not enough elements to fill the window. When the window is truncated, the MAD is taken over only the elements that fill the window.

• If A is a vector, then movmad operates along the length of the vector A.

• If A is a multidimensional array, then movmad operates along the first dimension of A whose size does not equal 1.

M = movmad(A,[kb kf]) computes the MAD with a window of length kb+kf+1 that includes the element in the current position, kb elements backward, and kf elements forward.

M = movmad(___,dim) specifies the dimension of A to operate along for any of the previous syntaxes. For example, movmad(A,k,2) for a matrix A operates across the columns of A, computing the k-element sliding MAD for each row.

M = movmad(___,nanflag) specifies whether to include or omit NaN values in A. For example, movmad(A,k,"omitnan") ignores NaN values when computing each MAD. By default, movmad includes NaN values.

M = movmad(___,Name,Value) specifies additional parameters for the moving MAD using one or more name-value arguments. For example, if x is a vector of time values, then movmad(A,k,"SamplePoints",x) computes the moving MAD of A relative to the times in x.

Examples

Centered Moving MAD of Vector

Compute the three-point centered moving MAD of a row vector. When there are fewer than three elements in the window at the endpoints, compute over the elements that are available.

A = [1 2 4 -1 -2 -3 -1 3 2 1];
M = movmad(A,3)

M = 1×10

    0.5000    1.0000    2.0000    1.0000    1.0000    1.0000    2.0000    1.0000    1.0000    0.5000

Trailing Moving MAD of Vector

Compute the three-point trailing moving MAD of a row vector. When there are fewer than three elements in the window at the endpoints, compute over the elements that are available.

A = [1 2 1 -1 -2 -3 -1 3 4 1];
M = movmad(A,[2 0])

M = 1×10

         0    0.5000         0    1.0000    1.0000    1.0000    1.0000    2.0000    1.0000    1.0000

Moving MAD of Matrix

Compute the 3-point centered moving MAD for each row of a matrix. The dimension argument is 2, which slides the window across the columns of A. The window starts on the first row, slides horizontally to the end of the row, then moves to the second row, and so on.

A = [1 2 1; -1 -2 -3; -1 3 4]

A = 3×3

     1     2     1
    -1    -2    -3
    -1     3     4

M = movmad(A,3,2)

M = 3×3

    0.5000         0    0.5000
    0.5000    1.0000    0.5000
    2.0000    1.0000    0.5000

Moving MAD Excluding Missing Values

Create a row vector containing NaN values.

A = [4 8 NaN -1 -2 -3 NaN 3 4 5];

Compute the three-point centered moving MAD of the vector, excluding NaN values. For windows that contain any NaN value, movmad computes with the non-NaN elements.

M = movmad(A,3,"omitnan")

M = 1×10

    2.0000    2.0000    4.5000    0.5000    1.0000    0.5000    3.0000    0.5000    1.0000    0.5000

Sample Points for Moving MAD

Compute a 3-hour centered moving MAD of the data in A according to the time vector t.

A = [4 8 6 -1 -2 -3];
k = hours(3);
t = datetime(2016,1,1,0,0,0) + hours(0:5)

t = 1x6 datetime
   01-Jan-2016 00:00:00   01-Jan-2016 01:00:00   01-Jan-2016 02:00:00   01-Jan-2016 03:00:00   01-Jan-2016 04:00:00   01-Jan-2016 05:00:00

M = movmad(A,k,"SamplePoints",t)

M = 1×6

    2.0000    2.0000    2.0000    1.0000    1.0000    0.5000

Return Only Full-Window MADs

Compute the three-point centered moving MAD of a row vector, but discard any calculation that uses fewer than three points from the output. In other words, return only the MADs computed from a full three-element window, discarding endpoint calculations.

A = [1 2 1 -1 -2 -3 -1 3 4 1];
M = movmad(A,3,"Endpoints","discard")

M = 1×8

     0     1     1     1     1     2     1     1

Input Arguments

A — Input array

vector | matrix | multidimensional array

Input array, specified as a vector, matrix, or multidimensional array.

k — Window length

numeric or duration scalar

Window length, specified as a numeric or duration scalar. When k is a positive integer scalar, the centered MAD includes the element in the current position plus surrounding neighbors.

For example, movmad(A,3) computes an array of local three-point MAD values.

[kb kf] — Directional window length

numeric or duration row vector containing two elements

Directional window length, specified as a numeric or duration row vector containing two elements. When kb and kf are positive integer scalars, the calculation is over kb+kf+1 elements. The calculation includes the element in the current position, kb elements before the current position, and kf elements after the current position.

For example, movmad(A,[2 1]) computes an array of local four-point MAD values.

dim — Dimension to operate along

positive integer scalar

Dimension to operate along, specified as a positive integer scalar. If you do not specify the dimension, then the default is the first array dimension of size greater than 1.

Consider an m-by-n input matrix, A:

• movmad(A,k,1) computes the k-element sliding MADs for each column of A and returns an m-by-n matrix.

  

• movmad(A,k,2) computes the k-element sliding MADs for each row of A and returns an m-by-n matrix.

  

nanflag — Missing value condition

"includemissing" (default) | "includenan" | "omitmissing" | "omitnan"

Missing value condition, specified as one of these values:

• "includemissing" or "includenan" — Include NaN values in A when computing each MAD. If any element in the window is NaN, then the corresponding element in M is NaN. "includemissing" and "includenan" have the same behavior.

• "omitmissing" or "omitnan" — Ignore NaN values in A, and compute each MAD over fewer points. If all elements in the window are NaN, then the corresponding element in M is NaN. "omitmissing" and "omitnan" have the same behavior.

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.

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

Example: M = movmad(A,k,"Endpoints","fill")

Endpoints — Method to treat windows near endpoints

"shrink" (default) | "discard" | "fill" | numeric or logical scalar

Method to treat windows near endpoints, specified as one of these options:

Value	Description
"shrink"	Shrink the window size near the endpoints of the input to include only existing elements.
"discard"	Do not output any MAD values when the window does not completely overlap with existing elements.
"fill"	Replace nonexisting elements with NaN.
numeric or logical scalar	Replace nonexisting elements with the specified numeric or logical value.

SamplePoints — Sample points for computing MADs

vector

Sample points for computing MADs, specified as a vector. The sample points represent the locations of the data in A. Sample points do not need to be uniformly sampled. By default, the sample points vector is [1 2 3 ... ].

Moving windows are defined relative to the sample points, which must be sorted and contain unique elements. For example, if t is a vector of times corresponding to the input data, then movmad(rand(1,10),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.

If the sample points are nonuniformly spaced and Endpoints is specified, then its value must be "shrink".

More About

Median Absolute Deviation

For a finite-length 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.

Extended Capabilities

Tall Arrays

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

This function supports tall arrays with the limitations:

• The SamplePoints name-value argument is not supported.

For more information, see Tall Arrays.

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

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:

• The maximum window length is 31 for gpuArray data.

• The SamplePoints name-value argument is not supported.

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

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced in R2017a

R2023a: Specify missing value condition

Include or omit missing values in the input array when computing each MAD by using the "includemissing" or "omitmissing" options. These options have the same behavior as the "includenan" and "omitnan" options, respectively.

R2023a: Improved performance when computing over matrix with sample points

The movmad function shows improved performance when computing over a matrix when there are sample points.

For example, this code computes the moving median absolute deviation of a 300-by-300 matrix with corresponding sample points. The code is about 1.3x faster than in the previous release.

function timingMovmad
A = randn(300);
t = sort(rand(300,1));
tic
for k = 1:250
    movmad(A,0.1,"SamplePoints",t);
end
toc
end

The approximate execution times are:

R2022b: 1.06 s

R2023a: 0.80

The code was timed on a Windows^® 10, Intel^® Xeon^® CPU E5-1650 v4 @ 3.60 GHz test system by calling the timingMovmad function.