serrlog_rs: Realised squared error log score

In scoringfunctions: Loss Functions for Assessing Point Forecasts

Realised squared error log score

Description

The function serrlog_rs computes the realised squared error log score when \textbf{\textit{y}} materialises and \textbf{\textit{x}} is the prediction.

Realised squared error log score is a realised score corresponding to the squared error log scoring function serrlog_sf.

Usage

serrlog_rs(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 realised squared error log score 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) := (\log(x) - \log(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 realised squared error log score.

Note

For details on the squared error log scoring function, see serrlog_sf.

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

The realised squared error log score is the realised (average) score corresponding to the squared error log 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")}.

See Also

serrlog_sf, meanlog_if

Examples

# Compute the realised squared error log score.

set.seed(12345)

x <- 0.5

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

print(serrlog_rs(x = x, y = y))

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

scoringfunctions documentation built on Aug. 30, 2026, 5:07 p.m.