msre: Mean squared relative error (MSRE)

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

Mean squared relative error (MSRE)

Description

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

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

Usage

msre(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 relative 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)/x)^{2}

Domain of function:

\textbf{\textit{x}} > \textbf{0}

\textbf{\textit{y}} > \textbf{0}

where

\textbf{0} = (0, ..., 0)^\mathsf{T}

is the zero vector of length n and the symbol > indicates pairwise inequality.

Range of function:

S(\textbf{\textit{x}}, \textbf{\textit{y}}) \geq 0, \forall \textbf{\textit{x}}, \textbf{\textit{y}} > \textbf{0}

Value

Value of the mean squared relative error.

Note

For details on the squared relative error scoring function, see srelerr_sf.

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

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

set.seed(12345)

x <- 0.5

y <- rlnorm(n = 100, mean = 0, sdlog = 1)

print(msre(x = x, y = y))

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

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