mse: Mean squared error (MSE)

In scoringfunctions: A Collection of Loss Functions for Assessing Point Forecasts

Mean squared error (MSE)

Description

The function mse computes the mean squared error when \textbf{\textit{y}} materialises and \textbf{\textit{x}} is the prediction.

Mean squared error is a realised score corresponding to the squared error scoring function serr_sf.

Usage

mse(x, y)

Arguments

x	Prediction. It can be a vector of length n (must have the same length as \textbf{\textit{y}}).
y	Realisation (true value) of process. It can be a vector of length n (must have the same length as \textbf{\textit{x}}).

Details

The mean squared error is defined by:

S(\textbf{\textit{x}}, \textbf{\textit{y}}) := (1/n) \sum_{i = 1}^{n} L(x_i, y_i)

where

\textbf{\textit{x}} = (x_1, ..., x_n)^\mathsf{T}

\textbf{\textit{y}} = (y_1, ..., y_n)^\mathsf{T}

and

L(x, y) := (x - y)^2

Domain of function:

\textbf{\textit{x}} \in \mathbb{R}^n

\textbf{\textit{y}} \in \mathbb{R}^n

Range of function:

S(\textbf{\textit{x}}, \textbf{\textit{y}}) \geq 0, \forall \textbf{\textit{x}}, \textbf{\textit{y}} \in \mathbb{R}^n

Value

Value of the mean squared error.

Note

For details on the squared error scoring function, see serr_sf.

The concept of realised (average) scores is defined by Gneiting (2011) and Fissler and Ziegel (2019).

The mean squared error is the realised (average) score corresponding to the squared error scoring function.

References

Fissler T, Ziegel JF (2019) Order-sensitivity and equivariance of scoring functions. Electronic Journal of Statistics 13(1):1166–1211. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/19-EJS1552")}.

Gneiting T (2011) Making and evaluating point forecasts. Journal of the American Statistical Association 106(494):746–762. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1198/jasa.2011.r10138")}.

Examples

# Compute the mean squared error.

set.seed(12345)

x <- 0

y <- rnorm(n = 100, mean = 0, sd = 1)

print(mse(x = x, y = y))

print(mse(x = rep(x = x, times = 100), y = y))

scoringfunctions documentation built on April 4, 2025, 12:28 a.m.