KGE: Kling-Gupta Efficiency

In hydroGOF: Goodness-of-Fit Functions for Comparison of Simulated and Observed Hydrological Time Series

Description

Kling-Gupta efficiency between sim and obs, with treatment of missing values.

This goodness-of-fit measure was developed by Gupta et al. (2009) to provide a diagnostically interesting decomposition of the Nash-Sutcliffe efficiency (and hence MSE), which facilitates the analysis of the relative importance of its different components (correlation, bias and variability) in the context of hydrological modelling
Kling et al. (2012), proposed a revised version of this index, to ensure that the bias and variability ratios are not cross-correlated

In the computation of this index, there are three main components involved:
1) r : the Pearson product-moment correlation coefficient. Ideal value is r=1
2) Beta : the ratio between the mean of the simulated values and the mean of the observed ones. Ideal value is Beta=1
3) vr : variability ratio, which could be computed using the standard deviation (Alpha) or the coefficient of variation (Gamma) of sim and obs, depending on the value of method

3.1) Alpha: the ratio between the standard deviation of the simulated values and the standard deviation of the observed ones. Ideal value is Alpha=1.
3.2) Gamma: the ratio between the coefficient of variation (CV) of the simulated values to the coefficient of variation of the observed ones. Ideal value is Gamma=1.

For a full discussion pf the Kling-Gupta index, and its advantages over the Nash-Sutcliffe efficiency (NSE) see Gupta et al. (2009).

Usage

KGE(sim, obs, ...)

## Default S3 method:
KGE(sim, obs, s=c(1,1,1), na.rm=TRUE, method=c("2009", "2012"), 
             out.type=c("single", "full"), ...)

## S3 method for class 'data.frame'
KGE(sim, obs, s=c(1,1,1), na.rm=TRUE, method=c("2009", "2012"), 
             out.type=c("single", "full"), ...)

## S3 method for class 'matrix'
KGE(sim, obs, s=c(1,1,1), na.rm=TRUE, method=c("2009", "2012"), 
             out.type=c("single", "full"), ...)
             
## S3 method for class 'zoo'
KGE(sim, obs, s=c(1,1,1), na.rm=TRUE, method=c("2009", "2012"), 
             out.type=c("single", "full"), ...)

Arguments

sim	numeric, zoo, matrix or data.frame with simulated values
obs	numeric, zoo, matrix or data.frame with observed values
s	numeric of length 3, representing the scaling factors to be used for re-scaling the criteria space before computing the Euclidean distance from the ideal point c(1,1,1), i.e., s elements are used for adjusting the emphasis on different components. The first elements is used for rescaling the Pearson product-moment correlation coefficient (r), the second element is used for rescaling Alpha and the third element is used for re-scaling Beta
na.rm	a logical value indicating whether 'NA' should be stripped before the computation proceeds. When an 'NA' value is found at the i-th position in obs OR sim, the i-th value of obs AND sim are removed before the computation.
method	character, indicating the formula used to compute the variability ratio in the Kling-Gupta efficiency. Valid values are: -) 2009: the variability is defined as ‘Alpha’, the ratio of the standard deviation of sim values to the standard deviation of obs. This is the default option. See Gupta et al. 2009 -) 2012: the variability is defined as ‘Gamma’, the ratio of the coefficient of variation of sim values to the coefficient of variation of obs. See Kling et al. 2012.
out.type	character, indicating the if the output of the function has to include or not each one of the three terms used in the computation of the Kling-Gupta efficiency. Valid values are: -) single: the output is a numeric with the Kling-Gupta efficiency only -) full: the output is a list of two elements: the first one with the Kling-Gupta efficiency, and the second is a numeric with 3 elements: the Pearson product-moment correlation coefficient (‘r’), the ratio between the mean of the simulated values to the mean of observations (‘Beta’), and the variability measure (‘Gamma’ or ‘Alpha’, depending on the value of method)
...	further arguments passed to or from other methods.

Details

KGE = 1 - ED

ED = √{ (s[1]*(r-1))^2 +(s[2]*(vr-1))^2 + (s[3]*(β-1))^2 }

r=\textrm{Pearson product-moment correlation coefficient}

β=μ_s/μ_o

vr= ≤ft\{ \begin{array}{cc} α & , \: \textrm{method="2009"} \\ γ & , \: \textrm{method="2012"} \end{array} \right.

α=σ_s/σ_o

KGE = 1 - sqrt[ (s[1]*(r-1))^2 + (s[2]*(vr-1))^2 + (s[3]*(Beta-1))^2] ; r=Pearson product-moment correlation coefficient ; alpha=sigma_s/sigma_o ; beta=mu_s/mu_o ; gamma=CV_s/CV_o

Kling-Gupta efficiencies range from -Inf to 1. Essentially, the closer to 1, the more accurate the model is.

Value

If out.type=single: numeric with the Kling-Gupta efficiency between sim and obs. If sim and obs are matrices, the output value is a vector, with the Kling-Gupta efficiency between each column of sim and obs
If out.type=full: a list of two elements:

KGE.value	numeric with the Kling-Gupta efficiency. If sim and obs are matrices, the output value is a vector, with the Kling-Gupta efficiency between each column of sim and obs
KGE.elements	numeric with 3 elements: the Pearson product-moment correlation coefficient (‘r’), the ratio between the mean of the simulated values to the mean of observations (‘Beta’), and the variability measure (‘Gamma’ or ‘Alpha’, depending on the value of method). If sim and obs are matrices, the output value is a matrix, with the previous three elements computed for each column of sim and obs

Note

obs and sim has to have the same length/dimension

The missing values in obs and sim are removed before the computation proceeds, and only those positions with non-missing values in obs and sim are considered in the computation

Author(s)

Mauricio Zambrano-Bigiarini <[e-mail]>

References

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

Kling, H., Fuchs, M., & Paulin, M. (2012). Runoff conditions in the upper Danube basin under an ensemble of climate change scenarios. Journal of Hydrology, 424, 264-277, doi:10.1016/j.jhydrol.2012.01.011

Santos, L., Thirel, G., & Perrin, C. (2018). Pitfalls in using log-transformed flows within the KGE criterion. doi:10.5194/hess-22-4583-2018

Knoben, W. J., Freer, J. E., & Woods, R. A. (2019). Inherent benchmark or not? Comparing Nash-Sutcliffe and Kling-Gupta efficiency scores. Hydrology and Earth System Sciences, 23(10), 4323-4331. doi:10.5194/hess-23-4323-2019

Mizukami, N., Rakovec, O., Newman, A. J., Clark, M. P., Wood, A. W., Gupta, H. V., & Kumar, R. (2019). On the choice of calibration metrics for "high-flow" estimation using hydrologic models. doi:10.5194/hess-23-2601-2019

See Also

NSE, gof, ggof

Examples

# Example1: basic ideal case
obs <- 1:10
sim <- 1:10
KGE(sim, obs)

obs <- 1:10
sim <- 2:11
KGE(sim, obs)

##################
# Example2: Looking at the difference between 'method=2009' and 'method=2012'
# Loading daily streamflows of the Ega River (Spain), from 1961 to 1970
data(EgaEnEstellaQts)
obs <- EgaEnEstellaQts

# Simulated daily time series, initially equal to twice the observed values
sim <- 2*obs 

# KGE 2009
KGE(sim=sim, obs=obs, method="2009", out.type="full")

# KGE 2012
KGE(sim=sim, obs=obs, method="2012", out.type="full")

##################
# Example3: KGE for simulated values equal to observations plus random noise 
#           on the first half of the observed values
# Randomly changing the first 1826 elements of 'sim', by using a normal distribution 
# with mean 10 and standard deviation equal to 1 (default of 'rnorm').
sim <- obs 
sim[1:1826] <- obs[1:1826] + rnorm(1826, mean=10)

# Computing the new 'KGE'
KGE(sim=sim, obs=obs)

# Randomly changing the first 2000 elements of 'sim', by using a normal distribution 
# with mean 10 and standard deviation equal to 1 (default of 'rnorm').
sim[1:2000] <- obs[1:2000] + rnorm(2000, mean=10)

# Computing the new 'KGE'
KGE(sim=sim, obs=obs)

Example output

Loading required package: zoo

Attaching package: 'zoo'

The following objects are masked from 'package:base':

    as.Date, as.Date.numeric

[1] 1
[1] 0.8181818
$KGE.value
[1] -0.4142136

$KGE.elements
    r  Beta Alpha 
    1     2     2 

$KGE.value
[1] 0

$KGE.elements
    r  Beta Gamma 
    1     2     1 

[1] 0.6803623
[1] 0.6504909

hydroGOF documentation built on March 14, 2020, 1:07 a.m.