scoringfunctions (version 1.1)

nse: 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)

Value

Value of the Nash-Sutcliffe efficiency.

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 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:

$$S(\textbf{\textit{x}}, \textbf{\textit{y}}) \leq 1, \forall \textbf{\textit{x}}, \textbf{\textit{y}} \in \mathbb{R}^n$$

References

Gneiting T (2011) Making and evaluating point forecasts. Journal of the American Statistical Association 106(494):746--762. 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. 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))