rmse

Root-mean-square error between arrays

Since R2022b

Syntax

E = rmse(F,A)

E = rmse(F,A,"all")

E = rmse(F,A,dim)

E = rmse(F,A,vecdim)

E = rmse(___,nanflag)

E = rmse(___,Weight=W)

Description

E = rmse(F,A) returns the root-mean-square error (RMSE) between the forecast (predicted) array 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 RMSE for each column.

• If F and A are multidimensional arrays, then E contains the RMSE 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.

E = rmse(F,A,"all") returns the RMSE of all elements in F and A.

E = rmse(F,A,dim) operates along dimension dim. For example, if F and A are matrices, then rmse(F,A,2) operates on the elements in each row and returns a column vector containing the RMSE of each row.

E = rmse(F,A,vecdim) operates along the dimensions specified in the vector vecdim. For example, if F and A are matrices, then rmse(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.

E = rmse(___,nanflag) specifies whether to include or omit NaN values in F and A for any of the previous syntaxes. For example, rmse(F,A,"omitnan") ignores NaN values when computing the RMSE. By default, rmse includes NaN values.

E = rmse(___,Weight=W) specifies a weighting scheme W and returns the weighted RMSE. If W is a vector, its length must equal the length of the operating dimension. If W is a matrix or multidimensional array, it must have the same dimensions as F, A, or F-A. You cannot specify a weighting scheme if you specify vecdim or "all".

Examples

RMSE of Two Forecasts

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 RMSE between each forecast and the actual data.

E1 = rmse(F1,A)

E1 = 0.8165

E2 = rmse(F2,A)

E2 = 2.3805

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

F = [F1 F2]

F = 3×2

     1     2
    10     5
     9    10

E = rmse(F,A)

E = 1×2

    0.8165    2.3805

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

RMSE of Matrix Rows

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

E = rmse(F,A,2)

E = 3×1

    4.2426
    2.0000
    1.4142

RMSE of Array Pages

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 RMSE 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 = rmse(F,A,[1 2])

E = 
E(:,:,1) =

    3.2787


E(:,:,2) =

    5.2678

The first page of E contains the RMSE between the first page of F and the matrix A. The second page of E contains the RMSE between the second page of F and the matrix A.

RMSE Excluding Missing Values

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 RMSE between the forecast and the actual data, ignoring NaN values. For columns that contain all NaN values in F or A, the RMSE is NaN.

E = rmse(F,A,"omitnan")

E = 1×3

    1.4142    4.2426       NaN

Specify RMSE Weight Vector

Create a forecast column vector and an actual column vector.

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

Compute the RMSE between the forecast and actual data according to a weighting scheme specified by W.

W = [0.5; 0.25; 0.25];
E = rmse(F,A,Weight=W)

E = 1.7321

Input Arguments

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-n matrix and A is a 1-by-n row vector. For more information, see Compatible Array Sizes for Basic Operations.

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-n matrix and A is a 1-by-n row vector. For more information, see Compatible Array Sizes for Basic Operations.

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:

• rmse(F,A,1) computes the RMSE 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.

• rmse(F,A,2) computes the RMSE 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. rmse(F,A,[1 2]) computes the RMSE 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 RMSE. 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 RMSE. If all elements in the operating dimension are NaN in F, A, or W, then the corresponding element in E is NaN. "omitmissing" and "omitnan" have the same behavior.

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. If W is a matrix or multidimensional array, it must have the same dimensions as F, A, or F-A.

You cannot specify this argument if you specify vecdim or "all".

Data Types: single | double

More About

Root-Mean-Square Error

For a forecast array F and actual array A made up of n scalar observations, the root-mean-square error is defined as

$$E=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{\left|A_i-F_i\right|}^2}}$$

with the summation performed along the specified dimension.

Weighted Root-Mean-Square Error

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

$$E_W=\sqrt{\frac{{\displaystyle \sum _{i=1}^n{W}_i{\left|A_i-F_i\right|}^2}}{{\displaystyle \sum _{i=1}^n{W}_i}}}$$

with the summation performed along the specified dimension.

Extended Capabilities

Tall Arrays

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

This function fully supports tall arrays. For more 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

R2023a: Specify missing value condition

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

R2023a: Improved performance with small group size

The rmse function shows improved performance when computing over a real vector when the operating dimension is not specified. The function determines the default operating dimension more quickly in R2023a than in R2022b.

For example, this code computes the root-mean-square error along the default vector dimension. The code is about 2x faster than in the previous release.

function timingRmse
F = rand(10,1);
A = rand(10,1);
for i = 1:8e5
    rmse(F,A);
end
end

The approximate execution times are:

R2022b: 4.12 s

R2023a: 2.07 s

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

timeit(@timingRmse)

R2023a: Code generation support

Generate C or C++ code for the rmse function.