Wave:GoodnessOfFit

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Beschreibung

GoodnessOfFit calculates various goodness of fit parameters for two time series:

Volume error

[math]\displaystyle{ m = 100 \cdot \frac{\sum_{t=1}^T{(Q_m^t - Q_o^t)}}{\sum_{t=1}^T{Q_o^t}} }[/math]

with:

m: Volume error [%]

Qot: observed value at time t

Qmt: simulated value at time t

Sum of squared errors

[math]\displaystyle{ F^2 = \sum_{t=1}^T{\left(Q_o^t - Q_m^t\right)^2} }[/math]

with

F²: Sum of squared errors

Qot: observed value at time t

Qmt: simulated value at time t

Correlation coefficient / coefficient of determination

[math]\displaystyle{ r = \frac{s_{o,m}}{s_o \cdot s_m} }[/math]

with

[math]\displaystyle{ s_{o,m} = \frac{1}{n - 1} \sum_{t=1}^T{(Q_o^t - \overline{Q_o}) \cdot (Q_m^t - \overline{Q_m})} }[/math]

[math]\displaystyle{ s_o = \sqrt{\frac{1}{n - 1} \sum_{t=1}^T{(Q_o^t - \overline{Q_o})^2}} }[/math]

[math]\displaystyle{ s_m = \sqrt{\frac{1}{n - 1} \sum_{t=1}^T{(Q_m^t - \overline{Q_m})^2}} }[/math]

with

r: correlation coefficient (-1 ≤ r ≤ 1)

r²: coefficient of determination (0 ≤ r² ≤ 1)

so,m: covariance

so: standard deviation of observed values

sm: standard deviation of simulated values

Qot: observed value at time t

Qmt: simulated value at time t

n: Number of values

Qo: observed average

Qm: simulated average

Rating

Coefficient of determination	Rating
< 0.2	unsatisfactory
0.2 - 0.4	satisfactory
0.4 - 0.6	good
0.6 - 0.8	very good
> 0.8	excellent

Nash-Sutcliffe efficiency

[math]\displaystyle{ E = 1-\frac{\sum_{t=1}^T\left(Q_o^t-Q_m^t\right)^2}{\sum_{t=1}^T\left(Q_o^t-\overline{Q_o}\right)^2} }[/math]

with

E: Nash-Sutcliffe efficiency [-]

Qo: observed average

Qot: observed value at time t

Qmt: simulated value at time t

Nash-Sutcliffe efficiencies can range from -∞ to 1. An efficiency of 1 (E=1) corresponds to a perfect match of modeled discharge to the observed data. An efficiency of 0 (E=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.

— Wikipedia

Logarithmic Nash-Sutcliffe efficiency

[math]\displaystyle{ E,ln = 1-\frac{\sum_{t=1}^T\left(ln(Q_o^t)-ln(Q_m^t)\right)^2}{\sum_{t=1}^T\left(ln(Q_o^t)-ln(\overline{Q_o})\right)^2} }[/math]

with

E,ln: Logarithmic Nash-Sutcliffe efficiency [-]

Qo: observed average

Qot: observed value at time t

Qmt: simulated value at time t

Da in die Modelleffizienz [(Nash-Sutcliffe Effizenz)] der quadratische Fehler zwischen den simulierten und gemessenen Werten eingeht, werden Abweichungen hoher Werte (z. B. Hochwasserabflüsse) gegenüber geringen Werten (z. B. Niedrigwasserabflüsse) überbewertet. Daher wird häufig die logarithmierte Modelleffizienz berechnet. Das Gütemaß Reff,ln ist besser zur Bewertung der Modellierung von geringeren Werten (z. B. Niedrigwasserabflüssen) geeignet.

— BWK

Kling-Gupta efficiency

[math]\displaystyle{ \text{KGE} = 1 - \sqrt{ (r - 1)^2 + (\beta - 1)^2 + (\gamma - 1)^2 } }[/math]

with

r: correlation coefficient

β: bias ratio

γ: variability ratio

-∞ ≤ KGE ≤ 1. Larger is better.

Reference: Gupta et al. 2009

Hydrologic deviation

[math]\displaystyle{ D = 200 \cdot \frac{\sum_{t=1}^T{|Q_m^t - Q_o^t| \cdot Q_o^t}}{n \cdot {Q_{o,max}}^2} }[/math]

mit

D: Hydrologic deviation [-]

Qot: observed value at time t

Qmt: simulated value at time t

n: Number of values

Qo,max: observed maximum

[Die hydrologische Deviation] kann verstanden werden als gewogene mittlere Abweichung, angegeben in Prozent des jeweiligen Spitzenabflusses. Bei völliger Übereinstimmung der beiden Kurven würde sich somit die Deviation zu Null ergeben; bei Vorhandensein der gemessenen Kurve (in Dreiecksform) und Nichtvorhandensein der gerechneten Kurve (alle Ordinaten gleich Null) ergäbe sich eine Deviation von 50,0 — um nur zwei Extremfälle zu nennen.

— Schultz (1968), S.53

Rating

Deviation	Rating
0 - 3	very good
3 - 10	good
10 - 18	usable

— Schultz (1968)

Notes

The two time series are cleaned before conducting the analysis, i.e. they are cut to each other's extents and all non-coincident nodes and nodes that have a NaN value in one the time series are discarded.