movvar

Moving variance

Syntax

M = movvar(A,k)

M = movvar(A,[kb kf])

M = movvar(___,w)

M = movvar(___,w,dim)

M = movvar(___,nanflag)

M = movvar(___,Name,Value)

Description

M = movvar(A,k) returns an array of local k-point variance values, where each variance is calculated over a sliding window of length k across neighboring elements of 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 variance is taken over only the elements that fill the window. M is the same size as A.

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

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

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

M = movvar(___,w) specifies a normalization factor for any of the previous syntaxes. When w = 0 (default), M is normalized by k-1 for window length k. When w = 1, M is normalized by k.

M = movvar(___,w,dim) specifies the dimension of A to operate along for any of the previous syntaxes. Always specify the weight w from the previous syntax when specifying dim. For example, if A is a matrix, then movvar(A,k,0,2) operates along the columns of A, computing the k-element sliding variance for each row. The normalization factor is the default, k-1.

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

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

Examples

Centered Moving Variance of Vector

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

A = [4 8 6 -1 -2 -3 -1 3 4 5];
M = movvar(A,3)

M = 1×10

    8.0000    4.0000   22.3333   19.0000    1.0000    1.0000    9.3333    7.0000    1.0000    0.5000

Trailing Moving Variance of Vector

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

A = [4 8 6 -1 -2 -3 -1 3 4 5];
M = movvar(A,[2 0])

M = 1×10

         0    8.0000    4.0000   22.3333   19.0000    1.0000    1.0000    9.3333    7.0000    1.0000

Specify Normalization for Moving Variance

Compute the three-point centered moving variance of a row vector and normalize each variance by the number of elements in the window.

A = [4 8 6 -1 -2 -3 -1 3 4 5];
M = movvar(A,3,1)

M = 1×10

    4.0000    2.6667   14.8889   12.6667    0.6667    0.6667    6.2222    4.6667    0.6667    0.2500

Moving Variance of Matrix

Compute the three-point centered moving variance for each row of a matrix. The window starts on the first row, slides horizontally to the end of the row, then moves to the second row, and so on. The dimension argument is two, which slides the window across the columns of A. Always specify the normalization factor when specifying the dimension.

A = [4 8 6; -1 -2 -3; -1 3 4];
M = movvar(A,3,0,2)

M = 3×3

    8.0000    4.0000    2.0000
    0.5000    1.0000    0.5000
    8.0000    7.0000    0.5000

Moving Variance 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 variance of the vector, excluding NaN values. For windows that contain any NaN value, movvar computes with the non-NaN elements.

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

M = 1×10

    8.0000    8.0000   40.5000    0.5000    1.0000    0.5000   18.0000    0.5000    1.0000    0.5000

Sample Points for Moving Variance

Compute a 3-hour centered moving variance 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 = movvar(A,k,"SamplePoints",t)

M = 1×6

    8.0000    4.0000   22.3333   19.0000    1.0000    0.5000

Return Only Full-Window Variances

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

A = [4 8 6 -1 -2 -3 -1 3 4 5];
M = movvar(A,3,"Endpoints","discard")

M = 1×8

    4.0000   22.3333   19.0000    1.0000    1.0000    9.3333    7.0000    1.0000

Input Arguments

A — Input array

vector | matrix | multidimensional array

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

Data Types: single | double | logical

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 variance includes the element in the current position plus surrounding neighbors.

For example, movvar(A,3) computes an array of local three-point variances.

[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, movvar(A,[2 1]) computes an array of local four-point variances.

w — Weight

0 (default) | 1

Weight, specified as one of these values:

• 0 — Normalize by k-1, where k is the window length. If k=1, the weight is k.

• 1 — Normalize by k.

Data Types: single | double

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 whose size does not equal 1.

Dimension dim indicates the dimension that movvar operates along, that is, the direction in which the specified window slides.

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

• movvar(A,k,0,1) computes the k-element sliding variance for each column of A and returns an m-by-n matrix.

  

• movvar(A,k,0,2) computes the k-element sliding variance 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 variance. 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 variance 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 = movvar(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 variance 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 variances

vector

Sample points for computing variances, 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 movvar(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

Variance

For a finite-length vector A made up of N scalar observations, the variance is defined as

$$V=\frac{1}{N-1}{\displaystyle \sum _{i=1}^N|A_i-\mu {|}^2}$$

where μ is the mean of A,

$$\mu =\frac{1}{N}{\displaystyle \sum _{i=1}^N{A}_i}.$$

Some definitions of variance use a normalization factor of N instead of N-1. You can specify a factor of N by setting w to 1. In either case, the mean is assumed to have the usual normalization factor 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™.

The movvar function supports GPU array input with these usage notes and limitations:

• 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 R2016a

R2023a: Specify missing value condition

Include or omit missing values in the input array when computing each variance 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 movvar function shows improved performance when computing over a matrix when there are sample points.

For example, this code computes the moving variance of a 300-by-300 matrix with corresponding sample points. The code is about 2.2x faster than in the previous release.

function timingMovvar
A = randn(300);
t = sort(rand(300,1));
tic
for k = 1:2000
    movvar(A,0.1,"SamplePoints",t);
end
toc
end

The approximate execution times are:

R2022b: 1.24 s

R2023a: 0.56

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