Wave:GoodnessOfFit

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Description

Result chart for a GoodnessOfFit analysis with two simulated time series

GoodnessOfFit calculates various goodness of fit indicators for one or more "simulated" time series in relation to one "observed" time series.

Optionally, the analysis can be performed for individual hydrologic years.

The indicators "Volume error", "Nash-Sutcliffe efficiency", "Logarithmic Nash-Sutcliffe efficiency", "Kling-Gupta efficiency" and "Coefficient of correlation" are displayed in a radar chart. The full set of indicators is output to a result table.

Indicators

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 time series being compared 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.