Kling-Gupta Efficiency with knowable-moments

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

This goodness-of-fit measure was developed by Pizarro and Jorquera (2024), as a modification to the original Kling-Gupta efficiency (KGE) proposed by Gupta et al. (2009). See Details.

Usage

KGEkm(sim, obs, ...)

# Default S3 method
KGEkm(sim, obs, s=c(1,1,1), na.rm=TRUE, method=c("2012", "2009", "2021"), 
             out.type=c("single", "full"), fun=NULL, ...,
             epsilon.type=c("none", "Pushpalatha2012", "otherFactor", "otherValue"), 
             epsilon.value=NA)

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

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

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:

-) 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 Pizarro and Jorquera (2024) and Kling et al. (2012).

-) 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).

-) 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.

Details

Traditional objective functions, such as Nash-Sutcliffe Efficiency (NSE) and Kling-Gupta Efficiency (KGE), often make assumptions about data distribution and are sensitive to outliers. The Kling-Gupta Efficiency with knowable-moments (KGEkm) goodness-of-fit measure was developed by Pizarro and Jorquera (2024) to provide a reliable estimation and effective description of high-order statistics from typical hydrological samples and, therefore, reducing uncertainty in their estimation and computation of the KGE.

In the $KGE_{km}$, the dispersion is quantified using knowable moments (computed over ordered values of the samples in ascending order) instead of the standard deviation, while retaining the decomposition into correlation, variability, and bias components.

The general formulation of Kling–Gupta Efficiency with knowable moments ($KGE_{km}$) is:

$$ KGE_{km} = 1 - \sqrt{ \left[ s_1 (r - 1) \right]^2 + \left[ s_2 (vr - 1) \right]^2 + \left[ s_3 (br - 1) \right]^2 } $$

where $r$ is the Pearson product–moment correlation coefficient between simulated ($Q^{sim}_t$) and observed ($Q^{obs}_t$) values, $vr$ is the variability ratio, $br$ is the bias ratio, and $s = (s_1, s_2, s_3)$ is a vector of non-negative scaling factors that control the relative importance of each component.

Dispersion is computed from the second knowable moment. For a sample $x_1, x_2, \ldots, x_n$, the second knowable moment is defined as:

$$ K_2 = \frac{1}{n(n-1)} \sum_{i=1}^{n} 2 (i-1) x_{(i)} $$

where $x_{(i)}$ denotes the ordered values of the sample in ascending order. The corresponding dispersion measure is:

$$ \sigma_{km} = \sqrt{ 2 K_2 } $$

The variability ratio depends on the selected method:

For method = "2009", variability is defined as the ratio of knowable-moment dispersions:

$$ vr = \alpha = \frac{\sigma^{sim}_{km}}{\sigma^{obs}_{km}} $$
For method = "2012", variability is defined as the ratio of coefficients of variation:

$$ vr = \gamma = \frac{ \sigma^{sim}_{km} / \mu^{sim} }{ \sigma^{obs}_{km} / \mu^{obs} } $$
For method = "2021", variability is defined as in the 2009 formulation:

$$ vr = \alpha = \frac{\sigma^{sim}_{km}}{\sigma^{obs}_{km}} $$

The bias component also depends on the selected method:

For method = "2009" and method = "2012":

$$ br = \beta = \frac{\mu^{sim}}{\mu^{obs}} $$
For method = "2021":

$$ br = \beta_{2021} = \frac{ \mu^{sim} - \mu^{obs} }{ \sigma^{obs}_{km} } $$

In the same line that the traditional Kling-Gupta efficiency, the ($KGE_{km}$) ranges from -Inf to 1. Essentially, the closer to 1, the more similar sim and obs are.

As with other KGE-type metrics, the statistic integrates information about correlation, variability, and bias into a single performance measure while allowing explicit control over the relative contribution of each component through the scaling factors $s$.

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:

KGEkm.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
KGEkm.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

References

Pizarro, A.; Jorquera, J. (2024). Advancing objective functions in hydrological modelling: Integrating knowable moments for improved simulation accuracy. Journal of Hydrology, 634, 131071. doi:10.1016/j.jhydrol.2024.131071.

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.

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.

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 KGEkm 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.

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

Author

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

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

Examples

# Example1: basic ideal case
obs <- 1:10
sim <- 1:10
KGEkm(sim, obs)
#> [1] 1

obs <- 1:10
sim <- 2:11
KGEkm(sim, obs)
#> [1] 0.793454

##################
# 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 

# KGEkm 2012 (method="2012" is the default option for KGEkm)
KGEkm(sim=sim, obs=obs, method="2012", out.type="full")
#> $KGEkm.value
#> [1] -0.04201077
#> 
#> $KGEkm.elements
#>         r      Beta     Gamma 
#> 1.0000000 2.0000000 0.7071068 
#> 

# KGEkm 2009
KGEkm(sim=sim, obs=obs, method="2009", out.type="full")
#> $KGEkm.value
#> [1] -0.0823922
#> 
#> $KGEkm.elements
#>        r     Beta    Alpha 
#> 1.000000 2.000000 1.414214 
#> 


##################
# Example 2: Looking at the difference between 'KGEkm', KGE', 'NSE', 'wNSE', 
#            'wsNSE' and 'APFB' for detecting differences in high flows

# Loading daily streamflows of the Ega River (Spain), from 1961 to 1970
data(EgaEnEstellaQts)
obs <- EgaEnEstellaQts

# Simulated daily time series, created equal to the observed values and then 
# random noise is added only to high flows, i.e., those equal or higher than 
# the quantile 0.9 of the observed values.
sim      <- obs
hQ.thr   <- quantile(obs, probs=0.9, na.rm=TRUE)
hQ.index <- which(obs >= hQ.thr)
hQ.n     <- length(hQ.index)
sim[hQ.index] <- sim[hQ.index] + rnorm(hQ.n, mean=mean(sim[hQ.index], na.rm=TRUE))

# KGEkm (Pizarro and Jorquera, 2024; method='2012')
KGEkm(sim=sim, obs=obs)
#> [1] 0.578109

# KGE': Kling-Gupta eficiency 2012 (Kling et al.,2012) 
KGE(sim=sim, obs=obs, method="2012")
#> [1] 0.4887559

# Traditional Kling-Gupta eficiency (Gupta and Kling, 2009)
KGE(sim=sim, obs=obs)
#> [1] 0.06833165

# KGE'': Kling-Gupta eficiency 2021 (Tang et al.,2021) 
KGE(sim=sim, obs=obs, method="2021")
#> [1] 0.1010568

# Traditional Nash-Sutcliffe eficiency (Nash and Sutcliffe, 1970)
NSE(sim=sim, obs=obs)
#> [1] 0.004794342

# Weighted Nash-Sutcliffe efficiency (Hundecha and Bardossy, 2004)
wNSE(sim=sim, obs=obs)
#> [1] 0.281951

# wsNSE (Zambrano-Bigiarini and Bellin, 2012):
wsNSE(sim=sim, obs=obs)
#> [1] -0.2213234

# APFB (Mizukami et al., 2019):
APFB(sim=sim, obs=obs)
#> [1] 0.2918397


##################
# Example 4: Looking at the difference between 'KGE', 'NSE', 'wsNSE',
#            'dr', 'rd', 'md', and 'KGElf' for detecting 
#            differences in low flows

# Loading daily streamflows of the Ega River (Spain), from 1961 to 1970
data(EgaEnEstellaQts)
obs <- EgaEnEstellaQts

# Simulated daily time series, created equal to the observed values and then 
# random noise is added only to low flows, i.e., those equal or lower than 
# the quantile 0.4 of the observed values.
sim      <- obs
lQ.thr   <- quantile(obs, probs=0.4, na.rm=TRUE)
lQ.index <- which(obs <= lQ.thr)
lQ.n     <- length(lQ.index)
sim[lQ.index] <- sim[lQ.index] + rnorm(lQ.n, mean=mean(sim[lQ.index], na.rm=TRUE))

# KGEkm (Pizarro and Jorquera, 2024; method='2012')
KGEkm(sim=sim, obs=obs)
#> [1] 0.8774177

# KGE': Kling-Gupta eficiency 2012 (Kling et al.,2012) 
KGE(sim=sim, obs=obs, method="2012")
#> [1] 0.8405711

# Traditional Kling-Gupta eficiency (Gupta and Kling, 2009)
KGE(sim=sim, obs=obs)
#> [1] 0.8943441

# KGE'': Kling-Gupta eficiency 2021 (Tang et al.,2021) 
KGE(sim=sim, obs=obs, method="2021")
#> [1] 0.912638

# Traditional Nash-Sutcliffe eficiency (Nash and Sutcliffe, 1970)
NSE(sim=sim, obs=obs)
#> [1] 0.9844205

# Weighted seasonal Nash-Sutcliffe efficiency (Zambrano-Bigiarini and Bellin, 2012):
wsNSE(sim=sim, obs=obs, lambda=0.05, j=1/2)
#> [1] 0.6815896

# Refined Index of Agreement (Willmott et al., 2012):
dr(sim=sim, obs=obs)
#> [1] 0.9394451

# Relative Index of Agreement (Krause et al., 2005):
rd(sim=sim, obs=obs)
#> [1] 0.9077374

# Modified Index of Agreement (Krause et al., 2005):
md(sim=sim, obs=obs)
#> [1] 0.9355419

# KGElf (Garcia et al., 2017):
KGElf(sim=sim, obs=obs)
#> [1] 0.5862099


##################
# Example 5: KGEkm 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.

KGEkm(sim=sim, obs=obs, fun=log)
#> [1] 0.8341867

# Verifying the previous value:
lsim <- log(sim)
lobs <- log(obs)
KGEkm(sim=lsim, obs=lobs)
#> [1] 0.8341867

##################
# Example 6: KGEkm 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

KGEkm(sim=sim, obs=obs, fun=log, epsilon.type="Pushpalatha2012")
#> [1] 0.8408504

# 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)
KGEkm(sim=lsim, obs=lobs)
#> [1] 0.8408504

##################
# Example 7: KGEkm 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
KGEkm(sim=sim, obs=obs, fun=log, epsilon.type="otherValue", epsilon.value=eps)
#> [1] 0.8346255

# Verifying the previous value:
lsim <- log(sim+eps)
lobs <- log(obs+eps)
KGEkm(sim=lsim, obs=lobs)
#> [1] 0.8346255

##################
# Example 8: KGEkm 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
KGEkm(sim=sim, obs=obs, fun=log, epsilon.type="otherFactor", epsilon.value=fact)
#> [1] 0.8469799

# Verifying the previous value:
eps  <- fact*mean(obs, na.rm=TRUE)
lsim <- log(sim+eps)
lobs <- log(obs+eps)
KGEkm(sim=lsim, obs=lobs)
#> [1] 0.8469799

##################
# Example 9: KGEkm 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)}

KGEkm(sim=sim, obs=obs, fun=fun1)
#> [1] 0.8960538

# Verifying the previous value, with the epsilon value following Pushpalatha2012
sim1 <- sqrt(sim+1)
obs1 <- sqrt(obs+1)
KGEkm(sim=sim1, obs=obs1)
#> [1] 0.8960538