mspe: Mean squared percentage error (MSPE)

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

Mean squared percentage error (MSPE)

Description

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

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

Usage

mspe(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 percentage 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)/y)^{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 percentage error.

Note

For details on the squared percentage error scoring function, see sperr_sf.

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

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

set.seed(12345)

x <- 0.5

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

print(mspe(x = x, y = y))

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

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