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R: The Pearson product-moment correlation coefficient, also known as r, R, or Pearson's r, is a measure of the strength and direction of the linear relationship between two variables that is defined as the covariance
of the
variables divided by the product of their standard deviations. This is the best known and most commonly used type of correlation coefficient; when the term "correlation coefficient" is used without further qualification, it
usually
refers to the Pearson product-moment correlation coefficient.
$$\text{R} = \dfrac{ \sum_{i=1}^{n}(o_i - \overline{o})(m_i - \overline{m}) }{ \sqrt{ \sum_{i=1}^{n}(o_i - \overline{o_i})^2} \sqrt{ \sum_{i=1}^{n}(m_i - \overline{m_i})^2} }$$
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R Spearman: Spearman's rank correlation coefficient or Spearman's rho, named after Charles Spearman and often denoted by the Greek letter ρ, is a nonparametric measure of rank correlation (statistical dependence
between the
rankings of two variables). It assesses how well the relationship between two variables can be described using a monotonic function.
$$\rho = 1 - \dfrac{ 6 \sum_{i=1}^{n} (m_i - o_i)^2 }{ n (n^2-1) }$$
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NMB (Normalized Mean Bias): NMB captures the average deviations between two datasets. On this website it is reported in units of percent. Values near 0 are the best, negative values indicate underestimation and
positive values
indicate overestimation.
$$\text{NMB} = \dfrac{\sum_{i=1}^{n} \left( m_i - o_i \right)}{\sum_{i=1}^{n} o_i} $$
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MNMB (Modified Normalized Mean Bias): MNMB is a normalisation based on the mean of the observed and forecast value. It ranges between −2 and 2 and when multiplied by 100 %, it can be interpreted as a percentage bias.
$$\text{MNMB} = \dfrac{2}{n} \sum_{i=1}^{n}( \dfrac{m_i - o_i}{m_i + o_i} )$$
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FGE (Fractionnal Gross Error): FGE is a measure of model error, ranging between 0 and 2 and behaves symmetrically with respect to under- and overestimation, without over emphasizing outliers.
$$\text{FGE} = \dfrac{2}{n} \sum_{i=1}^{n} \left| \dfrac{m_i - o_i}{m_i+o_i} \right|$$
Overall Evaluation only
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R-Space (Spatial Correlation) focuses on the spatial dimension by removing the time dimension. In order to do so, a time averaging is first computed for each time series. Then, a standard Pearson correlation is computed from the time averaged observation and model datasets.
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R-Time (Time Correlation) focuses on the time dimension by removing the space dimension. In order to do so, a standard Pearson correlation is computed for the timeseries (in the chosen time resolution, e.g daily or monthly) at every station, and the median of the timeseries is then computed.