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+\epsilon)-ln(Q_m^t+\epsilon)\right)^2}{\sum_{t=1}^T\left(ln(Q_o^t)-\overline{ln(Q_o+\epsilon)}\right)^2} }[/math]

with

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

Qot: observed value at time t

Qmt: simulated value at time t

ε: small constant set to 1% of the average value of Qo as recommended by Pushpalatha et al. (2012).

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 of the time series are discarded.