scoringfunctions (version 1.1)

mse: 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)

Value

Value of the mean squared error.

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:

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

References

Fissler T, Ziegel JF (2019) Order-sensitivity and equivariance of scoring functions. Electronic Journal of Statistics 13(1):1166--1211. 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. 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))