nse: Nash-Sutcliffe efficiency (NSE)
| nse | R Documentation |
Nash-Sutcliffe efficiency (NSE)
Description
The function nse computes the Nash-Sutcliffe efficiency when
\textbf{\textit{y}} materialises and \textbf{\textit{x}} is the
prediction.
Nash-Sutcliffe efficiency is a skill score corresponding to the squared error scoring function serr_sf. It is defined in eq. (3) in Nash and Sutcliffe (1970).
Usage
nse(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 Nash-Sutcliffe efficiency is defined by:
S_{\textnormal{skill}}(\textbf{\textit{x}}, \textbf{\textit{y}}) :=
1 - S_{\textnormal{meth}}(\textbf{\textit{x}}, \textbf{\textit{y}}) /
S_{\textnormal{ref}}(\textbf{\textit{x}}, \textbf{\textit{y}})
where
\textbf{\textit{x}} = (x_1, ..., x_n)^\mathsf{T}
\textbf{\textit{y}} = (y_1, ..., y_n)^\mathsf{T}
\textbf{1} = (1, ..., 1)^\mathsf{T}
\overline{\textbf{\textit{y}}} :=
(1/n) \textbf{1}^\mathsf{T} \textbf{\textit{y}} =
(1/n) \sum_{i = 1}^{n} y_i
L(x, y) := (x - y)^2
and the predictions of the method of interest as well as the reference method are evaluated respectively by:
S_{\textnormal{meth}}(\textbf{\textit{x}}, \textbf{\textit{y}}) :=
(1/n) \sum_{i = 1}^{n} L(x_i, y_i)
S_{\textnormal{ref}}(\textbf{\textit{x}}, \textbf{\textit{y}}) :=
(1/n) \sum_{i = 1}^{n} L(\overline{\textbf{\textit{y}}}, y_i)
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}}) \leq 1,
\forall \textbf{\textit{x}}, \textbf{\textit{y}} \in \mathbb{R}^n
Value
Value of the Nash-Sutcliffe efficiency.
Note
For details on the squared error scoring function, see serr_sf.
The concept of skill scores is defined by Gneiting (2011).
The Nash-Sutcclife efficiency is defined in eq. (3) in Nash and Sutcliffe (1970).
The Nash-Sutcclife efficiency is positevely oriented (i.e. the larger, the better).
References
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")}.
Nash JE, Sutcliffe JV (1970) River flow forecasting through conceptual models part I - A discussion of principles. Journal of Hydrology 10(3):282–290. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/0022-1694(70)90255-6")}.
Examples
# Compute the Nash-Sutcliffe efficiency.
set.seed(12345)
x <- 0
y <- rnorm(n = 100, mean = 0, sd = 1)
print(nse(x = x, y = y))
print(nse(x = rep(x = x, times = 100), y = y))
print(nse(x = mean(y), y = y))
print(nse(x = y, y = y))
