KGE: Kling-Gupta Efficiency

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 (KGE') to ensure that the bias and variability ratios are not cross-correlated.

Tang et al. (2021) proposed a revised version of this index (KGE'') to avoid the anomalously negative KGE' or KGE values when the mean value is close to zero.

For a short description of its three components and the numeric range of varios, pleae see Details.

Usage

KGE(sim, obs, ...)
# S3 method for default
KGE(sim, obs, s=c(1,1,1), na.rm=TRUE, method=c("2009", "2012", "2021"), 
             out.type=c("single", "full"), fun=NULL, ...,
             epsilon.type=c("none", "Pushpalatha2012", "otherFactor", "otherValue"), 
             epsilon.value=NA)
# S3 method for data.frame
KGE(sim, obs, s=c(1,1,1), na.rm=TRUE, method=c("2009", "2012", "2021"), 
             out.type=c("single", "full"), fun=NULL, ...,
             epsilon.type=c("none", "Pushpalatha2012", "otherFactor", "otherValue"), 
             epsilon.value=NA)
# S3 method for matrix
KGE(sim, obs, s=c(1,1,1), na.rm=TRUE, method=c("2009", "2012", "2021"), 
             out.type=c("single", "full"), fun=NULL, ...,
             epsilon.type=c("none", "Pushpalatha2012", "otherFactor", "otherValue"), 
             epsilon.value=NA)
             
# S3 method for zoo
KGE(sim, obs, s=c(1,1,1), na.rm=TRUE, method=c("2009", "2012", "2021"), 
             out.type=c("single", "full"), fun=NULL, ...,
             epsilon.type=c("none", "Pushpalatha2012", "otherFactor", "otherValue"), 
             epsilon.value=NA)

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

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).

-) 2021: the bias is defined as ‘Beta’, the ratio of mean(sim) minus mean(obs) to the standard deviation of obs. The variability is defined as ‘Alpha’, the ratio of the standard deviation of sim values to the standard deviation of obs. See Tang et al. (2021).

out.type

character, indicating the whether the output of the function has to include each one of the three terms used in the computation of the Kling-Gupta efficiency or not. 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).

fun

function to be applied to sim and obs in order to obtain transformed values thereof before computing the Kling-Gupta efficiency.

The first argument MUST BE a numeric vector with any name (e.g., x), and additional arguments are passed using ....

...

arguments passed to fun, in addition to the mandatory first numeric vector.

epsilon.type

argument used to define a numeric value to be added to both sim and obs before applying fun.

It is was designed to allow the use of logarithm and other similar functions that do not work with zero values.

Valid values of epsilon.type are:

1) "none": sim and obs are used by fun without the addition of any nummeric value.

2) "Pushpalatha2012": one hundredth (1/100) of the mean observed values is added to both sim and obs before applying fun, as described in Pushpalatha et al. (2012).

3) "otherFactor": the numeric value defined in the epsilon.value argument is used to multiply the the mean observed values, instead of the one hundredth (1/100) described in Pushpalatha et al. (2012). The resulting value is then added to both sim and obs, before applying fun.

4) "otherValue": the numeric value defined in the epsilon.value argument is directly added to both sim and obs, before applying fun.

epsilon.value

-) when epsilon.type="otherValue" it represents the numeric value to be added to both sim and obs before applying fun.
-) when epsilon.type="otherFactor" it represents the numeric factor used to multiply the mean of the observed values, instead of the one hundredth (1/100) described in Pushpalatha et al. (2012). The resulting value is then added to both sim and obs before applying fun.

Author

Mauricio Zambrano-Bigiarini <mzb.devel@gmail.com>

Details

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. Its 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. Its ideal value is Gamma=1.

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

Kling-Gupta efficiencies range from -Inf to 1. Essentially, the closer to 1, the more similar sim and obs are.

Knoben et al. (2019) showed that KGE values greater than -0.41 indicate that a model improves upon the mean flow benchmark, even if the model's KGE value is negative.

$K G E = 1 - E D$ $E D = \sqrt{(s [1] * (r - 1))^{2} + (s [2] * (v r - 1))^{2} + (s [3] * (β - 1))^{2}}$ $r = P e a r s o n p r o d u c t - m o m e n t c o r r e l a t i o n c o e f f i c i e n t$ $v r = {\begin{array}{cc} α & , m e t h o d = 2009 \\ γ & , m e t h o d = 2012 \end{array}$ $β = μ_{s} / μ_{o}$ $α = σ_{s} / σ_{o}$ $γ = \frac{C V_{s}}{C V_{o}} = \frac{σ_{s} / μ_{s}}{σ_{o} / μ_{o}}$

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.

Tang, G.; Clark, M.P.; Papalexiou, S.M. (2021). SC-earth: a station-based serially complete earth dataset from 1950 to 2019. Journal of Climate, 34(16), 6493-6511. doi:10.1175/JCLI-D-21-0067.1.

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.

Cinkus, G.; Mazzilli, N.; Jourde, H.; Wunsch, A.; Liesch, T.; Ravbar, N.; Chen, Z.; and Goldscheider, N. (2023). When best is the enemy of good - critical evaluation of performance criteria in hydrological models. Hydrology and Earth System Sciences 27, 2397-2411, doi:10.5194/hess-27-2397-2023.

Examples

Run this code

# 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 

# Traditional Kling-Gupta eficiency (Gupta and Kling, 2009)
KGE(sim=sim, obs=obs, method="2009", out.type="full")

# KGE': Kling-Gupta eficiency 2012 (Kling et al.,2012) 
KGE(sim=sim, obs=obs, method="2012", out.type="full")

# KGE'': Kling-Gupta eficiency 2021 (Tang et al.,2021) 
KGE(sim=sim, obs=obs, method="2021", 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)

# Traditional Kling-Gupta eficiency (Gupta and Kling, 2009)
KGE(sim=sim, obs=obs, method="2009", out.type="full")

# KGE': Kling-Gupta eficiency 2012 (Kling et al.,2012) 
KGE(sim=sim, obs=obs, method="2012", out.type="full")

# KGE'': Kling-Gupta eficiency 2021 (Tang et al.,2021) 
KGE(sim=sim, obs=obs, method="2021", out.type="full")

##################
# Example 4: KGE for simulated values equal to observations plus random noise 
#            on the first half of the observed values and applying (natural) 
#            logarithm to 'sim' and 'obs' during computations.

KGE(sim=sim, obs=obs, fun=log)

# Verifying the previous value:
lsim <- log(sim)
lobs <- log(obs)
KGE(sim=lsim, obs=lobs)

##################
# Example 5: KGE for simulated values equal to observations plus random noise 
#            on the first half of the observed values and applying (natural) 
#            logarithm to 'sim' and 'obs' and adding the Pushpalatha2012 constant
#            during computations

KGE(sim=sim, obs=obs, fun=log, epsilon.type="Pushpalatha2012")

# Verifying the previous value, with the epsilon value following Pushpalatha2012
eps  <- mean(obs, na.rm=TRUE)/100
lsim <- log(sim+eps)
lobs <- log(obs+eps)
KGE(sim=lsim, obs=lobs)

##################
# Example 6: KGE for simulated values equal to observations plus random noise 
#            on the first half of the observed values and applying (natural) 
#            logarithm to 'sim' and 'obs' and adding a user-defined constant
#            during computations

eps <- 0.01
KGE(sim=sim, obs=obs, fun=log, epsilon.type="otherValue", epsilon.value=eps)

# Verifying the previous value:
lsim <- log(sim+eps)
lobs <- log(obs+eps)
KGE(sim=lsim, obs=lobs)

##################
# Example 7: KGE for simulated values equal to observations plus random noise 
#            on the first half of the observed values and applying (natural) 
#            logarithm to 'sim' and 'obs' and using a user-defined factor
#            to multiply the mean of the observed values to obtain the constant
#            to be added to 'sim' and 'obs' during computations

fact <- 1/50
KGE(sim=sim, obs=obs, fun=log, epsilon.type="otherFactor", epsilon.value=fact)

# Verifying the previous value:
eps  <- fact*mean(obs, na.rm=TRUE)
lsim <- log(sim+eps)
lobs <- log(obs+eps)
KGE(sim=lsim, obs=lobs)

##################
# Example 8: KGE for simulated values equal to observations plus random noise 
#            on the first half of the observed values and applying a 
#            user-defined function to 'sim' and 'obs' during computations

fun1 <- function(x) {sqrt(x+1)}

KGE(sim=sim, obs=obs, fun=fun1)

# Verifying the previous value, with the epsilon value following Pushpalatha2012
sim1 <- sqrt(sim+1)
obs1 <- sqrt(obs+1)
KGE(sim=sim1, obs=obs1)

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