Page 227 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

216 Fundamentals of Ocean Renewable Energy

Correlation Coefficient
Correlation measures the direction and strength of the linear relationship
between two variables. If we have a set of n observed (O i ) and simulated (S i )
values, the linear correlation coefficient (r) (also known as Pearson’s r) can be
calculated as

n
i=1 (O i − O)(S i − S)
r = (8.42)

n
n
i=1 (O i − O) i=1 (S i − S)
For a perfect model, r = 1, whilst for a completely random prediction r = 0.
2
The square of the correlation is often used as a metric, because r indicates
the proportion of the variance in the observation that can be predicted by the
model.

Root Mean Squared Error
Root mean squared error (RMSE) is the square root of the mean of the square
of all of the error. The use of RMSE is very common, and it is considered an
excellent general-purpose error metric for numerical predictions.

n

1
RMSE = (S i − O i ) 2 (8.43)
n
i=1
where O i are the observations, S i predicted values of a variable, and n the
number of observations available for analysis. RMSE is a good measure of
accuracy, but only to compare prediction errors of different models or model
configurations for a particular variable and not between variables, as it is scale-
dependent.

Scatter Index
Scatter index (S.I.) is simply the RMSE (Eq. 8.43) divided by the mean of
the observations and multiplied by 100 to convert to a percentage error. As an
example of the worth of S.I., if an RMSE for significant wave height is 1 m, this
gives us no sense of how well the model is performing, because the mean of the
observations could, for example, be either 1 m (S.I. = 100%) or 5 m (S.I. = 20%).

Bias
Bias (also known as mean error) is the mean of the simulated values of the
selected variable minus the mean of the observed values, that is

BI = S − O (8.44)
It is an index of the average component of the error [25], with a value closer
to zero indicating a better simulation This index shows if a model is in general
overestimating or underestimating.