Wave:GoodnessOfFit

From BlueM
Jump to navigation Jump to search

Wave.png Wave | Development

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

: 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)
: 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] [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

[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[3]

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[4]

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[5]

Rating

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

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.

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. BWK: "Detaillierte Nachweisführung immissionsorientierter Anforderungen an Misch- und Niederschlagswassereinleitungen gemäß BWK - Merkblatt 3", Okt. 2008
  4. Gupta, H. V., Kling, H., Yilmaz, K. K., & Martinez, G. F. (2009): Decomposition of the mean squared error and NSE performance criteria: Implications for improving hydrological modelling. Journal of hydrology, 377(1-2), 80-91. doi:10.1016/j.jhydrol.2009.08.003. ISSN 0022-1694
  5. 5.0 5.1 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]