mse {scoringfunctions} | R Documentation |
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 |
y |
Realisation (true value) of process. It can be a vector of length
|
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. doi:10.1214/19-EJS1552.
Gneiting T (2011) Making and evaluating point forecasts. Journal of the American Statistical Association 106(494):746–762. 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))