Statistics

Given the initial measured values $x_{0}$, final observed or measured values $x_{m}$ and final calculated values $x_{c}$, there are several goodness of fit statistics or skill scores which can be calculated. The definition for some of the more common ones are provided below.

Brier Skill Score

The Bier Skill Score (BSS) is given by

$BSS=1-{\frac {{\bigg \langle }{\big (}x_{m}-x_{c}{\big )}^{2}{\bigg \rangle }}{{\bigg \langle }{\big (}x_{m}-x_{0}{\big )}^{2}{\bigg \rangle }}}$

(1)

where $x_{m}$ is the measured or observed values, $x_{c}$ is the calculated values and $x_{0}$ is the initial measured values. The BSS ranges between negative infinity and one. A BSS value of 1 indicates a perfect agreement between measured and calculated values. Scores equal to or less than 0 indicates that the mean observed value is as or more accurate than the calculated values.

Nash-Sutcliffe Coefficient

$E=1-{\frac {{\bigg \langle }{\big (}x_{m}-x_{c}{\big )}^{2}{\bigg \rangle }}{{\bigg \langle }{\big (}x_{m}-{\bar {x}}{\big )}^{2}{\bigg \rangle }}}$

(2)

where where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_m} is the measured or observed values, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_c} is the calculated values and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \bar{x} = \langle x_m \rangle } . The Nash-Sutcliffe efficiency coefficient ranges from negative infinity to one. An efficiency of 1 corresponds to a perfect match between measured and calculated values. An efficiencies equal 0 or less indicates that the mean observed value is as or more accurate than the calculated values.

Root-Mean-Squared Error

The Root-Mean-Squared Error (RMSE) also referred to as Root-Mean-Squared Deviation (RMSD) is defined as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle RMSE = \sqrt{ \bigg\langle \big( x_m - x_c \big)^2 \bigg\rangle } }

(3)

where where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_m} is the measured or observed values, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_c} is the calculated values. The RMSE has the same units as the measured and calculated data.

Normalized-Root-Mean-Squared Error

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle NRMSE = \frac{\sqrt{ \bigg\langle \big( x_m - x_c \big)^2 \bigg\rangle}}{\text{Range}(x_m)} }

(4)

Mean-Absolute Error

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle MAE = \bigg\langle \big| x_m - x_c \big| \bigg\rangle }

(5)

where where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_m} is the measured or observed values, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_c} is the calculated values.

Normalized-Mean-Absolute Error

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle NMAE = \frac{MAE}{ \text{Range}(x_m) } }

(6)

where where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_m} is the measured or observed values, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_c} is the calculated values.

Correlation coefficient is defined as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R = \frac { \langle x_m x_c \rangle - \langle x_m \rangle \langle x_c \rangle }{ \sqrt{ \langle x_m^2 \rangle - \langle x_m \rangle ^2} \sqrt{ \langle x_c^2 \rangle - \langle x_c \rangle ^2} } }

(7)

where where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_m} is the measured or observed values, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_c} is the calculated values.

Bias

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B = \langle x_m \rangle - \langle x_c \rangle }

(8)

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