Mean absolute percentage error between arrays

*Since R2022b*

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## Syntax

`E = mape(F,A)`

`E = mape(F,A,"all")`

`E = mape(F,A,dim)`

`E = mape(F,A,vecdim)`

`E = mape(___,nanflag)`

`E = mape(___,zeroflag)`

`E = mape(___,Weights=W)`

## Description

example

returns the mean absolute percentage error (MAPE) between the forecast (predicted) array `E`

= mape(F,A)`F`

and the actual (observed) array `A`

.

`F`

and`A`

must either be the same size or have sizes that are compatible.If

`F`

and`A`

are vectors of the same size, then`E`

is a scalar.If

`F-A`

is a matrix, then`E`

is a row vector containing the MAPE for each column.If

`F`

and`A`

are multidimensional arrays, then`E`

contains the MAPE computed along the first array dimension of size greater than 1, with elements treated as vectors. The size of`E`

in this dimension is 1, while the sizes of all other dimensions are the same as in`F-A`

.

returns the MAPE of all elements in `E`

= mape(F,A,"all")`F`

and `A`

.

example

operates along dimension `E`

= mape(F,A,dim)`dim`

. For example, if `F`

and `A`

are matrices, then `mape(F,A,2)`

operates on the elements in each row and returns a column vector containing the MAPE of each row.

example

operates along the dimensions specified in the vector `E`

= mape(F,A,vecdim)`vecdim`

. For example, if `F`

and `A`

are matrices, then `mape(F,A,[1 2])`

operates on all the elements in `F`

and `A`

because every element of a matrix is contained in the array slice defined by dimensions 1 and 2.

example

specifies whether to include or omit `E`

= mape(___,nanflag)`NaN`

values in `F`

and `A`

for any of the previous syntaxes. For example, `mape(F,A,"omitnan")`

ignores `NaN`

values when computing the MAPE. By default, `mape`

includes `NaN`

values.

example

specifies whether to include or omit zero values in `E`

= mape(___,zeroflag)`A`

. For example, `mape(F,A,"includezero")`

includes the zeros in the calculation, while `mape(F,A,"omitzero")`

ignores them.

example

specifies a weighting scheme `E`

= mape(___,Weights=W)`W`

and returns the weighted MAPE.

## Examples

collapse all

### MAPE of Two Forecasts

Open Live Script

Create two column vectors of forecast (predicted) data and one column vector of actual (observed) data.

F1 = [1; 10; 9];F2 = [2; 5; 10];A = [1; 9; 10];

Compute the MAPE between each forecast and the actual data.

E1 = mape(F1,A)

E1 = 7.0370

E2 = mape(F2,A)

E2 = 48.1481

Alternatively, create a matrix containing both forecasts and compute the MAPE between each forecast and the actual data in one command.

F = [F1 F2]

`F = `*3×2* 1 2 10 5 9 10

E = mape(F,A)

`E = `*1×2* 7.0370 48.1481

The first element of `E`

is the MAPE between the first forecast column and the actual data. The second element of `E`

is the MAPE between the second forecast column and the actual data.

### MAPE of Matrix Rows

Open Live Script

Create a matrix of forecast data and a matrix of actual data.

F = [17 19; 1 6; 16 15];A = [17 25; 3 4; 16 13];

Compute the MAPE between the forecast and the actual data across each row by specifying the operating dimension as 2. The smallest MAPE corresponds to the MAPE between the third rows of the forecast data and actual data.

E = mape(F,A,2)

`E = `*3×1* 12.0000 58.3333 7.6923

### MAPE of Array Pages

Open Live Script

Create a 3-D array with pages containing forecast data and a matrix of actual data.

F(:,:,1) = [2 4; -2 1];F(:,:,2) = [4 4; 8 -3];A = [6 7; 1 4];

Compute the MAPE between the predicted data in each page of the forecast array and the actual data matrix by specifying a vector of operating dimensions 1 and 2.

E = mape(F,A,[1 2])

E = E(:,:,1) = 121.1310E(:,:,2) = 237.7976

The first page of `E`

contains the MAPE between the first page of `F`

and the matrix `A`

. The second page of `E`

contains the MAPE between the second page of `F`

and the matrix `A`

.

### MAPE Excluding Missing Values

Open Live Script

Create a matrix of forecast data and a matrix of actual data containing `NaN`

values.

F = [17 19 3; 6 16 NaN];A = [17 25 NaN; 4 16 NaN];

Compute the MAPE between the forecast and the actual data, ignoring `NaN`

values. For columns that contain all `NaN`

values in `F`

or `A`

, the MAPE is `NaN`

.

`E = mape(F,A,"omitnan")`

`E = `*1×3* 25 12 NaN

### MAPE Excluding Zeros

Open Live Script

Create a row vector of forecast data and a row vector of actual data containing a zero.

F = [1 6 10 5];A = [2 6 0 3];

Compute the MAPE between the forecast and the actual data.

E = mape(F,A)

E = Inf

Because zero values in `A`

are included in the MAPE calculation by default, the result is `Inf`

. Ignore the zero in the actual data by specifying `"omitzero"`

. Now, the function computes the MAPE for only the first, second, and fourth columns of the input data.

`Eomit = mape(F,A,"omitzero")`

Eomit = 38.8889

### Specify MAPE Weight Vector

Open Live Script

Create a forecast column vector and an actual column vector.

F = [2; 10; 13];A = [1; 9; 10];

Compute the MAPE between the forecast and actual data according to a weighting scheme specified by `W`

.

W = [0.5; 0.25; 0.25];E = mape(F,A,Weights=W)

E = 60.2778

## Input Arguments

collapse all

`F`

— Forecast array

vector | matrix | multidimensional array

Forecast or predicted array, specified as a vector, matrix, or multidimensional array.

Inputs `F`

and `A`

must either be the same size or have sizes that are compatible. For example, `F`

is an * m*-by-

*matrix and*

`n`

`A`

is a 1-by-*row vector. For more information, see Compatible Array Sizes for Basic Operations.*

`n`

**Data Types: **`single`

| `double`

**Complex Number Support: **Yes

`A`

— Actual array

vector | matrix | multidimensional array

Actual or observed array, specified as a vector, matrix, or multidimensional array.

Inputs `F`

and `A`

must either be the same size or have sizes that are compatible. For example, `F`

is an * m*-by-

*matrix and*

`n`

`A`

is a 1-by-*row vector. For more information, see Compatible Array Sizes for Basic Operations.*

`n`

**Data Types: **`single`

| `double`

**Complex Number Support: **Yes

`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.

The size of `E`

in the operating dimension is 1. All other dimensions of `E`

have the same size as the result of `F-A`

.

For example, consider four forecasts in a 3-by-4 matrix, `F`

, and actual data in a 3-by-1 column vector, `A`

:

`mape(F,A,1)`

computes the MAPE of the elements in each column and returns a 1-by-4 row vector.The size of

`E`

in the operating dimension is 1. The difference of`F`

and`A`

is a 3-by-4 matrix. The size of`E`

in the nonoperating dimension is the same as the second dimension of`F-A`

, which is 4. The overall size of`E`

becomes 1-by-4.`mape(F,A,2)`

computes the MAPE of the elements in each row and returns a 3-by-1 column vector.The size of

`E`

in the operating dimension is 1. The difference of`F`

and`A`

is a 3-by-4 matrix. The size of`E`

in the nonoperating dimension is the same as the first dimension of`F-A`

, which is 3. The overall size of`E`

becomes 3-by-1.

`vecdim`

— Vector of dimensions to operate along

vector of positive integers

Vector of dimensions to operate along, specified as a vector of positive integers. Each element represents a dimension of the input arrays. The size of `E`

in the operating dimensions is 1. All other dimensions of `E`

have the same size as the result of `F-A`

.

For example, consider forecasts in a 2-by-3-by-3 array, `F`

, and actual data in a 1-by-3 row vector, `A`

. `mape(F,A,[1 2])`

computes the MAPE over each page of `F`

and returns a 1-by-1-by-3 array. The size of `E`

in the operating dimensions is 1. The difference of `F`

and `A`

is a 2-by-3-by-3 array. The size of `E`

in the nonoperating dimension is the same as the third dimension of `F-A`

, which is 3.

`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 the input arrays when computing the MAPE. If any element in the operating dimension is`NaN`

, then the corresponding element in`E`

is`NaN`

.`"includemissing"`

and`"includenan"`

have the same behavior.`"omitmissing"`

or`"omitnan"`

— Ignore`NaN`

values in the input arrays when computing the MAPE. If all elements in the operating dimension are`NaN`

in`F`

,`A`

, or`W`

, then the corresponding elements in`E`

are`NaN`

.`"omitmissing"`

and`"omitnan"`

have the same behavior.

`zeroflag`

— Zero condition

`"includezero"`

(default) | `"omitzero"`

Zero condition, specified as one of these values:

`"includezero"`

— Include zeros in`A`

when computing the MAPE. If`A`

contains one or more zeros, then`E`

is`Inf`

.`"omitzero"`

— Ignore zeros in`A`

when computing the MAPE. Small nonzero values of`A`

that would result in a MAPE of`Inf`

are also ignored. If all elements of`A`

are ignored, then`E`

is`NaN`

.

`W`

— Weighting scheme

vector | matrix | multidimensional array

Weighting scheme, specified as a vector, matrix, or multidimensional array. The elements of `W`

must be nonnegative.

If `W`

is a vector, it must have the same length as the operating dimension or must have the same size as `F-A`

. If `W`

is a matrix or multidimensional array, it must have the same size as `F`

, `A`

, or `F-A`

.

You cannot specify this argument if you specify `vecdim`

or `"all"`

.

**Data Types: **`double`

| `single`

## More About

collapse all

### Mean Absolute Percentage Error

For a forecast array *F* and actual array *A* made up of *n* scalar observations, the mean absolute percentage error is defined as

$$E=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\left|\frac{{A}_{i}-{F}_{i}}{{A}_{i}}\right|}\times 100$$

with the summation performed along the specified dimension. When `F`

or `A`

is complex, `mape`

computes the mean absolute percentage error using the complex magnitude of `(F-A)./A`

.

### Weighted Mean Absolute Percentage Error

For a forecast array *F* and actual array *A* made up of *n* scalar observations and weighting scheme *W*, the weighted mean absolute percentage error is defined as

$${E}_{W}=\frac{{\displaystyle \sum _{i=1}^{n}{W}_{i}\left|\frac{{A}_{i}-{F}_{i}}{{A}_{i}}\right|}\times 100}{{\displaystyle \sum _{i=1}^{n}{W}_{i}}}$$

with the summation performed along the specified dimension.

## Tips

Zeros or small nonzero values in the actual data

`A`

might indicate that MAPE is not the appropriate metric to measure error for`F`

and`A`

.

## Extended Capabilities

### Tall Arrays

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

This function fully supports tall arrays. Formore information, see Tall Arrays.

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

### GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. 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 R2022b**

expand all

### R2023a: Specify missing value condition

Include or omit missing values in the input arrays when computing the MAPE by using the `"includemissing"`

or `"omitmissing"`

options. These options have the same behavior as the `"includenan"`

and `"omitnan"`

options, respectively.

### R2023a: Code generation support

Generate C or C++ code for the `mape`

function.

## See Also

rmse | mean | abs

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