Root Mean Square Error (RMSE)
The Root Mean Square Error (RMSE) (also called the root mean square deviation, RMSD) is a frequently used measure of the difference between values predicted by a model and the values actually observed from the environment that is being modelled. These individual differences are also called residuals, and the RMSE serves to aggregate them into a single measure of predictive power.
The Root Mean Square Error (RMSE) (also called the root mean square deviation, RMSD) is a frequently used measure of the difference between values predicted by a model and the values actually observed from the environment that is being modelled. These individual differences are also called residuals, and the RMSE serves to aggregate them into a single measure of predictive power.
The RMSE of a
model prediction with respect to the estimated variable Xmodel is defined as the square root of the mean squared
error:
where Xobs
is observed values and Xmodel
is modelled values at time/place i.
The RMSE
values can be used to distinguish model performance in a calibration period
with that of a validation period as well as to compare the individual model
performance to that of other predictive models.
Normalized Root Mean Square Error (NRMSE)
Non-dimensional
forms of the RMSE are useful because often one wants to compare RMSE with
different units. There are two approaches: normalize the RMSE to the range of
the observed data, or normalize to the mean of the observed data.
(the latter one
is also called CV,RMSE for the resemblance with calculating the
coefficient of variance).
Pearson Correlation Coefficient (r)
Correlation –
often measured as a correlation coefficient – indicates the strength and
direction of a linear relationship between two variables (for example model
output and observed values). A number of different coefficients are used for
different situations. The best known is the Pearson product-moment correlation
coefficient (also called Pearson correlation coefficient or the sample
correlation coefficient), which is obtained by dividing the covariance of the
two variables by the product of their standard deviations. If we have a series
n observations and n model values, then the Pearson product-moment correlation
coefficient can be used to estimate the correlation between model and
observations.
The correlation
is +1 in the case of a perfect increasing linear relationship, and -1 in case
of a decreasing linear relationship, and the values in between indicates the
degree of linear relationship between for example model and observations. A
correlation coefficient of 0 means the there is no linear relationship between
the variables.
The square of
the Pearson correlation coefficient (r2),
known as the coefficient of determination, describes how much of the variance
between the two variables is described by the linear fit.
Nash-Sutcliffe Coefficient
(E)
The
Nash-Sutcliffe model efficiency coefficient (E) is commonly used to assess the
predictive power of hydrological discharge models. However, it can also be used
to quantitatively describe the accuracy of model outputs for other things than
discharge (such as nutrient loadings, temperature, concentrations etc.). It is defined
as:
where Xobs
is observed values and Xmodel
is modelled values at time/place i.
Nash-Sutcliffe efficiencies can range from -¥ to
1. An efficiency of 1 (E = 1) corresponds to a perfect match between model and
observations. An efficiency of 0 indicates that the model predictions are as
accurate as the mean of the observed data, whereas an efficiency less than zero
(-¥ < E < 0) occurs when the observed mean
is a better predictor than the model.
Essentially, the closer the model efficiency
is to 1, the more accurate the model is.
