# movstd

Moving standard deviation

## Description

example

M = movstd(A,k) returns an array of local k-point standard deviation values. Each standard deviation 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 standard deviation is taken over only the elements that fill the window. M is the same size as A.

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

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

example

M = movstd(A,[kb kf]) computes the standard deviation with a window of length kb+kf+1. The calculation includes the element in the current position, kb elements backward, and kf elements forward.

example

M = movstd(___,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.

example

M = movstd(___,w,dim) returns the array of sliding standard deviations along dimension dim for any of the previous syntaxes. Always specify the weight w from the previous syntax when specifying dim. For example, movstd(A,k,0,2) operates along the columns of a matrix A, computing the k-element sliding standard deviation for each row. The normalization factor is the default, k-1.

example

M = movstd(___,nanflag) specifies whether to include or omit NaN values from the calculation for any of the previous syntaxes. movstd(A,k,'includenan') includes all NaN values in the calculation while movstd(A,k,'omitnan') ignores them and computes the standard deviation over fewer points.

example

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

## Examples

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

A = [4 8 6 -1 -2 -3 -1 3 4 5];
M = movstd(A,3)
M = 1×10

2.8284    2.0000    4.7258    4.3589    1.0000    1.0000    3.0551    2.6458    1.0000    0.7071

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

A = [4 8 6 -1 -2 -3 -1 3 4 5];
M = movstd(A,[2 0])
M = 1×10

0    2.8284    2.0000    4.7258    4.3589    1.0000    1.0000    3.0551    2.6458    1.0000

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

A = [4 8 6 -1 -2 -3 -1 3 4 5];
M = movstd(A,3,1)
M = 1×10

2.0000    1.6330    3.8586    3.5590    0.8165    0.8165    2.4944    2.1602    0.8165    0.5000

Compute the three-point centered moving standard deviation 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 = movstd(A,3,0,2)
M = 3×3

2.8284    2.0000    1.4142
0.7071    1.0000    0.7071
2.8284    2.6458    0.7071

Compute the three-point centered moving standard deviation of a row vector containing two NaN elements.

A = [4 8 NaN -1 -2 -3 NaN 3 4 5];
M = movstd(A,3)
M = 1×10

2.8284       NaN       NaN       NaN    1.0000       NaN       NaN       NaN    1.0000    0.7071

Recalculate the standard deviation, but omit the NaN values. When movstd discards NaN elements, it takes the standard deviation over the remaining elements in the window.

M = movstd(A,3,'omitnan')
M = 1×10

2.8284    2.8284    6.3640    0.7071    1.0000    0.7071    4.2426    0.7071    1.0000    0.7071

Compute a 3-hour centered moving standard deviation 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
Columns 1 through 3

01-Jan-2016 00:00:00   01-Jan-2016 01:00:00   01-Jan-2016 02:00:00

Columns 4 through 6

01-Jan-2016 03:00:00   01-Jan-2016 04:00:00   01-Jan-2016 05:00:00

M = movstd(A,k,'SamplePoints',t)
M = 1×6

2.8284    2.0000    4.7258    4.3589    1.0000    0.7071

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

A = [4 8 6 -1 -2 -3 -1 3 4 5];
M = 1×8

2.0000    4.7258    4.3589    1.0000    1.0000    3.0551    2.6458    1.0000

## Input Arguments

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

Data Types: single | double | logical

Window length, specified as a numeric or duration scalar. When k is a positive integer scalar, the centered standard deviation includes the element in the current position plus surrounding neighbors. For example, a three-point standard deviation defined by a window of length three results in the following calculation for a vector A:

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

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, a four-point standard deviation defined by the directional window [2 1] results in the following calculation for a vector A:

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

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

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.

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

Consider a two-dimensional input array, A.

• If dim = 1, then movstd(A,k,0,1) starts with the first column and slides vertically over each row. The standard deviation is taken over k elements at a time. Then it moves to the second column and repeats the computation. This process continues until all columns are exhausted. The argument 0 specifies the default weight, which is required when specifying dim.

• If dim = 2, then movstd(A,k,0,2) starts with the first row and slides horizontally across each column. The standard deviation is taken over k elements at a time. Then it moves to the second row and repeats the computation. This process continues until all rows are exhausted. The argument 0 specifies the default weight, which is required when specifying dim.

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

NaN condition, specified as one of these values:

• 'includenan' — Include NaN values from the input when computing the standard deviation, resulting in the output NaN.

• 'omitnan' — Ignore all NaN values in the input. If a window contains only NaN values, then movstd returns NaN.

### 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: M = movstd(A,k,'Endpoints','fill')

Method to treat leading and trailing windows, specified as the comma-separated pair consisting of 'Endpoints' and one of the following:

'Endpoints' ValueDescription
'shrink'Shrink the window size near the endpoints of the input to include only existing elements.
'discard'Do not output any standard deviations when the window does not completely overlap with existing elements.
'fill'Substitute nonexisting elements with NaN.
numeric or logical scalarSubstitute nonexisting elements with a specified numeric or logical value.

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

Sample points for computing standard deviations, specified as the comma-separated pair consisting of 'SamplePoints' and a vector. The sample points represent the location 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 movstd(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 the 'Endpoints' name-value pair is specified, then its value must be 'shrink'.

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

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### Standard Deviation

For a random variable vector A made up of N scalar observations, the standard deviation is defined as

$\begin{array}{l}S={\sqrt{\frac{1}{N-1}\sum _{i=1}^{N}|{A}_{i}-\mu {|}^{2}}}^{},\hfill \\ \hfill \end{array}$

where μ is the mean of A:

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

The standard deviation is the square root of the variance. Some definitions of standard deviation use a normalization factor of N instead of N-1, which you can specify by setting w to 1.