KGEnp {hydroGOF}

Non-parametric version of the Kling-Gupta Efficiency

Description

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

This goodness-of-fit measure was developed by Pool et al. (2018), as a non-parametric alternative to the original Kling-Gupta efficiency (KGE) proposed by Gupta et al. (2009). See Details.

Usage

KGEnp(sim, obs, ...)

## Default S3 method:
KGEnp(sim, obs, na.rm=TRUE, out.type=c("single", "full"), fun=NULL, ..., 
               epsilon.type=c("none", "Pushpalatha2012", "otherFactor", "otherValue"), 
               epsilon.value=NA)

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

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

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.

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 Spearman rank correlation coefficient (‘rSpearman’), the ratio between the mean of the simulated values to the mean of observations (‘Beta’), and the variability measure (‘Alpha’).

fun

function to be applied to sim and obs in order to obtain transformed values thereof before computing this goodness-of-fit index.

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 numeric value. This is the default option.

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

This non-paramettric verison of the Kling-Gupta efficiency keeps the bias term Alpha (mean(sim) / mean(obs)), but for correlation uses the Spearman rank coefficient instead of the Pearson product-moment coefficient; and for variability it uses the normalized flow-duration curve instead of the standard deviation (or coefficient of variation).

The proposed non-parametric based multi-objective function can be seen as a useful alternative to existing performance measures when aiming at acceptable simulations of multiple hydrograph aspects (Pool et al., 2018).

KGE_{np} = 1 - ED

ED = \sqrt{ ((\rho-1)^2 + (\alpha-1)^2 + (\beta-1)^2 }

\rho = \textrm{Spearman rank correlation coefficient}

\alpha = 1 - 0.5*sum( sim(I(k)) / (n*\mu_s) - obs(J(k)) / (n*\mu_o) )

\beta = \mu_s/\mu_o

Traditional Kling-Gupta efficiencies (Gupta et al., 2009; Kling et al., 2012) range from -Inf to 1, and therefore KGEnp should do so. Essentially, the closer to 1, the more similar sim and obs are.

Knoben et al. (2019) showed that traditional Kling-Gupta (Gupta et al., 2009; Kling et al., 2012) values greater than -0.41 indicate that a model improves upon the mean flow benchmark, even if the model's KGE value is negative.

Value

numeric with the non-parametric Kling-Gupta efficiency between sim and obs.
If sim and obs are matrices, the output value is a vector, with the non-parametric Kling-Gupta efficiency between 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

Pool, S.; Vis, M.; Seibert, J. (2018). Evaluating model performance: towards a non-parametric variant of the Kling-Gupta efficiency. Hydrological Sciences Journal, 63(13-14), pp.1941-1953. doi:/10.1080/02626667.2018.1552002.

Garcia, F.; Folton, N.; Oudin, L. (2017). Which objective function to calibrate rainfall-runoff models for low-flow index simulations?. Hydrological sciences journal, 62(7), 1149-1166. doi:10.1080/02626667.2017.1308511.

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.

See Also

KGE, KGElf, sKGE, gof, ggof

Examples

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

obs <- 1:10
sim <- 2:11
KGEnp(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")

# KGEnp (Pool et al., 2018):
KGEnp(sim=sim, obs=obs)

##################
# Example3: KGEnp 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 'KGEnp'
KGEnp(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 'KGEnp'
KGEnp(sim=sim, obs=obs)

##################
# Example 4: KGEnp 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.

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

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

##################
# Example 5: KGEnp 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

KGEnp(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)
KGEnp(sim=lsim, obs=lobs)

##################
# Example 6: KGEnp 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
KGEnp(sim=sim, obs=obs, fun=log, epsilon.type="otherValue", epsilon.value=eps)

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

##################
# Example 7: KGEnp 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
KGEnp(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)
KGEnp(sim=lsim, obs=lobs)

##################
# Example 8: KGEnp 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)}

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

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