# Wave:GoodnessOfFit

## 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

$\displaystyle{ m = 100 \cdot \frac{\sum_{t=1}^T{(Q_m^t - Q_o^t)}}{\sum_{t=1}^T{Q_o^t}} }$

with:

m: Volume error [%]
Qot: observed value at time t
Qmt: simulated value at time t

### Sum of squared errors

$\displaystyle{ F^2 = \sum_{t=1}^T{\left(Q_o^t - Q_m^t\right)^2} }$

with

F²: Sum of squared errors
Qot: observed value at time t
Qmt: simulated value at time t

### Correlation coefficient / coefficient of determination

$\displaystyle{ r = \frac{s_{o,m}}{s_o \cdot s_m} }$

with

$\displaystyle{ s_{o,m} = \frac{1}{n - 1} \sum_{t=1}^T{(Q_o^t - \overline{Q_o}) \cdot (Q_m^t - \overline{Q_m})} }$
$\displaystyle{ s_o = \sqrt{\frac{1}{n - 1} \sum_{t=1}^T{(Q_o^t - \overline{Q_o})^2}} }$
$\displaystyle{ s_m = \sqrt{\frac{1}{n - 1} \sum_{t=1}^T{(Q_m^t - \overline{Q_m})^2}} }$

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

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

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[2]

### Logarithmic Nash-Sutcliffe efficiency

$\displaystyle{ E,ln = 1-\frac{\sum_{t=1}^T\left(ln(Q_o^t+\epsilon)-ln(Q_m^t+\epsilon)\right)^2}{\sum_{t=1}^T\left(ln(Q_o^t)-\overline{ln(Q_o+\epsilon)}\right)^2} }$

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)[3].

### Kling-Gupta efficiency

$\displaystyle{ \text{KGE} = 1 - \sqrt{ (r - 1)^2 + (\beta - 1)^2 + (\gamma - 1)^2 } }$
with
r: correlation coefficient
β: bias ratio
γ: variability ratio

-∞ ≤ KGE ≤ 1. Larger is better.

Reference: Kling et al. 2012[4]

### Hydrologic deviation

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

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[5]

#### Rating

Deviation Rating
0 - 3 very good
3 - 10 good
10 - 18 usable
— Schultz (1968)[5]

## 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.

## Literature

1. Nash, J. E. and Sutcliffe, J. V. (1970): River flow forecasting through conceptual models part I — A discussion of principles, Journal of Hydrology, 10 (3), 282–290, DOI:10.1016/0022-1694(70)90255-6.
2. Wikipedia contributors: "Nash-Sutcliffe efficiency coefficient," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Nash-Sutcliffe_efficiency_coefficient&oldid=231196847 (accessed September 18, 2008).
3. Pushpalatha, Raji, Perrin, Charles, Le Moine, Nicolas, Andréassian, Vazken (2012): A review of efficiency criteria suitable for evaluating low-flow simulations, Journal of Hydrology, Volumes 420–421, 2012, Pages 171-182, ISSN 0022-1694, https://doi.org/10.1016/j.jhydrol.2011.11.055.
4. Kling, H., Fuchs, M., Paulin, M. (2012): Runoff conditions in the upper Danube basin under an ensemble of climate change scenarios, Journal of Hydrology, Volumes 424–425, Pages 264-277, ISSN 0022-1694, https://doi.org/10.1016/j.jhydrol.2012.01.011.
5. Schultz, G. A. (1968): Bestimmung theoretischer Abflußganglinien durch elektronische Berechnung von Niederschlagskonzentration und Retention (HYREUN-Verfahren), Versuchsanstalt für Wasserbau der Technischen Hochschule München, Bericht Nr. 11, [IHWB-Signatur: 10 WBW 11]