Documentation

# moment

Central moment

## Syntax

``m = moment(X,order)``
``m = moment(X,order,'all')``
``m = moment(X,order,dim)``
``m = moment(X,order,vecdim)``

## Description

example

````m = moment(X,order)` returns the central moment of `X` for the order specified by `order`. If `X` is a vector, then `moment(X,order)` returns a scalar value that is the k-order central moment of the elements in `X`.If `X` is a matrix, then `moment(X,order)` returns a row vector containing the k-order central moment of each column in `X`.If `X` is a multidimensional array, then `moment(X,order)` operates along the first nonsingleton dimension of `X`. ```

example

````m = moment(X,order,'all')` returns the central moment of the specified order for all elements of `X`.```

example

````m = moment(X,order,dim)` takes the central moment along the operating dimension `dim` of `X`.```

example

````m = moment(X,order,vecdim)` returns the central moment over the dimensions specified in the vector `vecdim`. For example, if `X` is a 2-by-3-by-4 array, then `moment(X,1,[1 2])` returns a 1-by-1-by-4 array. Each element of the output array is the first-order central moment of the elements on the corresponding page of `X`.```

## Examples

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Set the random seed for reproducibility of the results.

`rng('default')`

Generate a matrix with 6 rows and 5 columns.

`X = randn(6,5)`
```X = 6×5 0.5377 -0.4336 0.7254 1.4090 0.4889 1.8339 0.3426 -0.0631 1.4172 1.0347 -2.2588 3.5784 0.7147 0.6715 0.7269 0.8622 2.7694 -0.2050 -1.2075 -0.3034 0.3188 -1.3499 -0.1241 0.7172 0.2939 -1.3077 3.0349 1.4897 1.6302 -0.7873 ```

Find the third-order central moment of `X`.

`m = moment(X,3)`
```m = 1×5 -1.1143 -0.9973 0.1234 -1.1023 -0.1045 ```

`m` is a row vector containing the third-order central moment of each column in `X`.

Find the central moment along different dimensions for a multidimensional array.

Set the random seed for reproducibility of the results.

`rng('default') `

Create a 4-by-3-by-2 array of random numbers.

`X = randn([4,3,2])`
```X = X(:,:,1) = 0.5377 0.3188 3.5784 1.8339 -1.3077 2.7694 -2.2588 -0.4336 -1.3499 0.8622 0.3426 3.0349 X(:,:,2) = 0.7254 -0.1241 0.6715 -0.0631 1.4897 -1.2075 0.7147 1.4090 0.7172 -0.2050 1.4172 1.6302 ```

Find the fourth-order central moment of `X` along the default dimension.

`m1 = moment(X,4)`
```m1 = m1(:,:,1) = 11.4427 0.3553 33.6733 m1(:,:,2) = 0.0360 0.4902 2.3821 ```

By default, `moment` operates along the first dimension of `X` whose size does not equal 1. In this case, this dimension is the first dimension of `X`. Therefore, `m1` is a 1-by-3-by-2 array.

Find the fourth-order central moment of `X` along the second dimension.

`m2 = moment(X,4,2)`
```m2 = m2(:,:,1) = 7.3476 13.8702 0.4625 2.7741 m2(:,:,2) = 0.0341 2.2389 0.0171 0.6766 ```

`m2` is a 4-by-1-by-2 array.

Find the fourth-order central moment of `X` along the third dimension.

`m3 = moment(X,4,3)`
```m3 = 4×3 0.0001 0.0024 4.4627 0.8093 3.8273 15.6340 4.8866 0.7205 1.1412 0.0811 0.0833 0.2433 ```

`m3` is a 4-by-3 matrix.

Find the central moment over multiple dimensions by using the `'all'` and `vecdim` input arguments.

Set the random seed for reproducibility of the results.

`rng('default')`

Create a 4-by-3-by-2 array of random numbers.

`X = randn([4 3 2])`
```X = X(:,:,1) = 0.5377 0.3188 3.5784 1.8339 -1.3077 2.7694 -2.2588 -0.4336 -1.3499 0.8622 0.3426 3.0349 X(:,:,2) = 0.7254 -0.1241 0.6715 -0.0631 1.4897 -1.2075 0.7147 1.4090 0.7172 -0.2050 1.4172 1.6302 ```

Find the third-order central moment of `X`.

`mall = moment(X,3,'all')`
```mall = 0.2431 ```

`mall` is the third-order central moment of the entire input data set `X`.

Find the third-order moment of each page of `X` by specifying the first and second dimensions.

`mpage = moment(X,3,[1 2])`
```mpage = mpage(:,:,1) = 0.6002 mpage(:,:,2) = -0.3475 ```

For example, `mpage(1,1,2)` is the third-order central moment of the elements in `X(:,:,2)`.

Find the third-order moment of the elements in each `X(i,:,:)` slice by specifying the second and third dimensions.

`mrow = moment(X,3,[2 3])`
```mrow = 4×1 2.7552 0.0443 -0.7585 0.5340 ```

For example, `mrow(1)` is the third-order central moment of the elements in `X(1,:,:)`.

## Input Arguments

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Input data that represents a sample from a population, specified as a vector, matrix, or multidimensional array.

• If `X` is a vector, then `moment(X,order)` returns a scalar value that is the k-order central moment of the elements in `X`.

• If `X` is a matrix, then `moment(X,order)` returns a row vector containing the k-order central moment of each column in `X`.

• If `X` is a multidimensional array, then `moment(X,order)` operates along the first nonsingleton dimension of `X`.

To specify the operating dimension when `X` is a matrix or an array, use the `dim` input argument.

Data Types: `single` | `double`

Order of the central moment, specified as a positive integer.

Data Types: `single` | `double`

Dimension along which to operate, specified as a positive integer. If you do not specify a value for `dim`, then the default is the first nonsingleton dimension of `X`.

Consider the third-order central moment of a matrix `X`:

• If `dim` is equal to 1, then `moment(X,3,1)` returns a row vector that contains the third-order central moment of each column in `X`.

• If `dim` is equal to 2, then `moment(X,3,2)` returns a column vector that contains the third-order central moment of each row in `X`.

If `dim` is greater than `ndims(X)` or if `size(X,dim)` is 1, then `moment` returns an array of zeros the same size as `X`.

Data Types: `single` | `double`

Vector of dimensions, specified as a positive integer vector. Each element of `vecdim` represents a dimension of the input array `X`. The output `m` has length 1 in the specified operating dimensions. The other dimension lengths are the same for `X` and `m`.

For example, if `X` is a 2-by-3-by-3 array, then `moment(X,1,[1 2])` returns a 1-by-1-by-3 array. Each element of the output array is the first-order central moment of the elements on the corresponding page of `X`.

Data Types: `single` | `double`

## Output Arguments

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Central moments, returned as a scalar, vector, matrix, or multidimensional array.

## Algorithms

The central moment of order k for a distribution is defined as

`${m}_{k}=E{\left(x-\mu \right)}^{k},$`

where µ is the mean of x, and E(t) represents the expected value of the quantity t. The `moment` function computes a sample version of this population value.

`${m}_{k}=\frac{1}{n}\sum _{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{k}.$`

Note that the first-order central moment is zero, and the second-order central moment is the variance computed using a divisor of n rather than n – 1, where n is the length of the vector `x` or the number of rows in the matrix `X`.