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| %% File KGEkm.Rd | ||
| %% Part of the hydroGOF R package, https://github.com/hzambran/hydroGOF ; | ||
| %% https://cran.r-project.org/package=hydroGOF | ||
| %% http://www.rforge.net/hydroGOF/ | ||
| %% Copyright 2011-2024 Mauricio Zambrano-Bigiarini | ||
| %% Distributed under GPL 2 or later | ||
|
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| \name{KGEkm} | ||
| \Rdversion{1.1} | ||
| \alias{KGEkm} | ||
| \alias{KGEkm.default} | ||
| \alias{KGEkm.matrix} | ||
| \alias{KGEkm.data.frame} | ||
| \alias{KGEkm.zoo} | ||
| %- Also NEED an '\alias' for EACH other topic documented here. | ||
| \title{ | ||
| Kling-Gupta Efficiency with knowable-moments | ||
| } | ||
| \description{ | ||
| Kling-Gupta efficiency between \code{sim} and \code{obs}, with use of knowable-moments and treatment of missing values. | ||
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| 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, ...) | ||
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| \method{KGEkm}{default}(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) | ||
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| \method{KGEkm}{data.frame}(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) | ||
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| \method{KGEkm}{matrix}(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) | ||
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| \method{KGEkm}{zoo}(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) | ||
| } | ||
| %- maybe also 'usage' for other objects documented here. | ||
| \arguments{ | ||
| \item{sim}{ | ||
| numeric, zoo, matrix or data.frame with simulated values | ||
| } | ||
| \item{obs}{ | ||
| numeric, zoo, matrix or data.frame with observed values | ||
| } | ||
| \item{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., \code{s} elements are used for adjusting the emphasis on different components. | ||
| The first elements is used for rescaling the Pearson product-moment correlation coefficient (\code{r}), the second element is used for rescaling \code{Alpha} and the third element is used for re-scaling \code{Beta} | ||
| } | ||
| \item{na.rm}{ | ||
| a logical value indicating whether 'NA' should be stripped before the computation proceeds. \cr | ||
| When an 'NA' value is found at the i-th position in \code{obs} \bold{OR} \code{sim}, the i-th value of \code{obs} \bold{AND} \code{sim} are removed before the computation. | ||
| } | ||
| \item{method}{ | ||
| character, indicating the formula used to compute the variability ratio in the Kling-Gupta efficiency. Valid values are: | ||
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| -) \kbd{2012}: the variability is defined as \sQuote{Gamma}, the ratio of the coefficient of variation of \code{sim} values to the coefficient of variation of \code{obs}. See Pizarro and Jorquera (2024) and Kling et al. (2012). | ||
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| -) \kbd{2009}: the variability is defined as \sQuote{Alpha}, the ratio of the standard deviation of \code{sim} values to the standard deviation of \code{obs}. This is the default option. See Gupta et al. (2009). | ||
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| -) \kbd{2021}: the bias is defined as \sQuote{Beta}, the ratio of \code{mean(sim)} minus \code{mean(obs)} to the standard deviation of \code{obs}. The variability is defined as \sQuote{Alpha}, the ratio of the standard deviation of \code{sim} values to the standard deviation of \code{obs}. See Tang et al. (2021). | ||
| } | ||
| \item{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: | ||
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| -) \kbd{single}: the output is a numeric with the Kling-Gupta efficiency only. | ||
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| -) \kbd{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 (\sQuote{r}), the ratio between the mean of the simulated values to the mean of observations (\sQuote{Beta}), and the variability measure (\sQuote{Gamma} or \sQuote{Alpha}, depending on the value of \code{method}). | ||
| } | ||
| \item{fun}{ | ||
| function to be applied to \code{sim} and \code{obs} in order to obtain transformed values thereof before computing the Kling-Gupta efficiency. | ||
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| The first argument MUST BE a numeric vector with any name (e.g., \code{x}), and additional arguments are passed using \code{\dots}. | ||
| } | ||
| \item{\dots}{ | ||
| arguments passed to \code{fun}, in addition to the mandatory first numeric vector. | ||
| } | ||
| \item{epsilon.type}{ | ||
| argument used to define a numeric value to be added to both \code{sim} and \code{obs} before applying \code{fun}. | ||
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| It is was designed to allow the use of logarithm and other similar functions that do not work with zero values. | ||
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| Valid values of \code{epsilon.type} are: | ||
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| 1) \kbd{"none"}: \code{sim} and \code{obs} are used by \code{fun} without the addition of any nummeric value. | ||
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| 2) \kbd{"Pushpalatha2012"}: one hundredth (1/100) of the mean observed values is added to both \code{sim} and \code{obs} before applying \code{fun}, as described in Pushpalatha et al. (2012). | ||
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| 3) \kbd{"otherFactor"}: the numeric value defined in the \code{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 \code{sim} and \code{obs}, before applying \code{fun}. | ||
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| 4) \kbd{"otherValue"}: the numeric value defined in the \code{epsilon.value} argument is directly added to both \code{sim} and \code{obs}, before applying \code{fun}. | ||
| } | ||
| \item{epsilon.value}{ | ||
| -) when \code{epsilon.type="otherValue"} it represents the numeric value to be added to both \code{sim} and \code{obs} before applying \code{fun}. \cr | ||
| -) when \code{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 \code{sim} and \code{obs} before applying \code{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. | ||
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| In the same line that the traditional Kling-Gupta efficiency, the KGEkm ranges from -Inf to 1. Essentially, the closer to 1, the more similar \code{sim} and \code{obs} are. | ||
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| In the computation of this index, there are three main components involved: | ||
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| 1) \kbd{r} : the Pearson product-moment correlation coefficient. Ideal value is r=1. | ||
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| 2) \kbd{Beta} : the ratio between the mean of the simulated values and the mean of the observed ones. Ideal value is Beta=1. | ||
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| 3) \kbd{vr} : variability ratio, which could be computed using the standard deviation (\kbd{Alpha}) or the coefficient of variation (\kbd{Gamma}) of \code{sim} and \code{obs}, depending on the value of \code{method}: | ||
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| 3.1) \kbd{Alpha}: the ratio between the standard deviation of the simulated values and the standard deviation of the observed ones. Its ideal value is \kbd{Alpha=1}. | ||
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| 3.2) \kbd{Gamma}: the ratio between the coefficient of variation (\kbd{CV}) of the simulated values to the coefficient of variation of the observed ones. Its ideal value is \kbd{Gamma=1}. | ||
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| \deqn{KGEkm = 1 - ED} | ||
| \deqn{ ED = \sqrt{ (s[1]*(r-1))^2 +(s[2]*(vr-1))^2 + (s[3]*(\beta-1))^2 } }{% | ||
| KGEkm = 1 - sqrt[ (s[1]*(r-1))^2 + (s[2]*(vr-1))^2 + (s[3]*(Beta-1))^2] | ||
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| r = Pearson product-moment correlation coefficient | ||
| beta = mu_s/mu_o ; | ||
| alpha = sigma_s/sigma_o | ||
| gamma = CV_s/CV_o; | ||
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| method="2009": vr=alfa | ||
| method="2012": vr=gamma } | ||
| \deqn{r=Pearson product-moment correlation coefficient} | ||
| \deqn{vr= \left\{ | ||
| \begin{array}{cc} | ||
| \alpha & , \: method=2009 \\ | ||
| \gamma & , \: method=2012 | ||
| \end{array} | ||
| \right.} | ||
| \deqn{\beta=\mu_s/\mu_o} | ||
| \deqn{\alpha=\sigma_s/\sigma_o} | ||
| \deqn{\gamma=\frac{CV_s}{CV_o} = \frac{\sigma_s/\mu_s}{\sigma_o/\mu_o}} | ||
| } | ||
| \value{ | ||
| If \code{out.type=single}: numeric with the Kling-Gupta efficiency between \code{sim} and \code{obs}. If \code{sim} and \code{obs} are matrices, the output value is a vector, with the Kling-Gupta efficiency between each column of \code{sim} and \code{obs} | ||
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| If \code{out.type=full}: a list of two elements: | ||
| \item{KGEkm.value}{ | ||
| numeric with the Kling-Gupta efficiency. If \code{sim} and \code{obs} are matrices, the output value is a vector, with the Kling-Gupta efficiency between each column of \code{sim} and \code{obs} | ||
| } | ||
| \item{KGEkm.elements}{ | ||
| numeric with 3 elements: the Pearson product-moment correlation coefficient (\sQuote{r}), the ratio between the mean of the simulated values to the mean of observations (\sQuote{Beta}), and the variability measure (\sQuote{Gamma} or \sQuote{Alpha}, depending on the value of \code{method}). If \code{sim} and \code{obs} are matrices, the output value is a matrix, with the previous three elements computed for each column of \code{sim} and \code{obs}\cr | ||
| } | ||
| } | ||
| \references{ | ||
| \cite{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.} | ||
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| \cite{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.} | ||
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| \cite{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.} | ||
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| \cite{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.} | ||
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| \cite{Santos, L.; Thirel, G.; Perrin, C. (2018). Pitfalls in using log-transformed flows within the KGEkm criterion. doi:10.5194/hess-22-4583-2018.} | ||
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| \cite{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.} | ||
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| \cite{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} | ||
| } | ||
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| \author{ | ||
| Mauricio Zambrano-Bigiarini <mzb.devel@gmail.com> | ||
| } | ||
| \note{ | ||
| \code{obs} and \code{sim} has to have the same length/dimension \cr | ||
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| The missing values in \code{obs} and \code{sim} are removed before the computation proceeds, and only those positions with non-missing values in \code{obs} and \code{sim} are considered in the computation | ||
| } | ||
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| %% ~Make other sections like Warning with \section{Warning }{....} ~ | ||
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| \seealso{ | ||
| \code{\link{KGE}}, \code{\link{KGElf}}, \code{\link{sKGE}}, \code{\link{KGEnp}}, \code{\link{gof}}, \code{\link{ggof}} | ||
| } | ||
| \examples{ | ||
| # Example1: basic ideal case | ||
| obs <- 1:10 | ||
| sim <- 1:10 | ||
| KGEkm(sim, obs) | ||
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| obs <- 1:10 | ||
| sim <- 2:11 | ||
| KGEkm(sim, obs) | ||
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| ################## | ||
| # Example2: Looking at the difference between 'method=2009' and 'method=2012' | ||
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| # Loading daily streamflows of the Ega River (Spain), from 1961 to 1970 | ||
| data(EgaEnEstellaQts) | ||
| obs <- EgaEnEstellaQts | ||
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| # Simulated daily time series, initially equal to twice the observed values | ||
| sim <- 2*obs | ||
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| # KGEkm 2012 (method="2012" is the default option for KGEkm) | ||
| KGEkm(sim=sim, obs=obs, method="2012", out.type="full") | ||
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| # KGEkm 2009 | ||
| KGEkm(sim=sim, obs=obs, method="2009", out.type="full") | ||
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| ################## | ||
| # Example 2: Looking at the difference between 'KGEkm', KGE', 'NSE', 'wNSE', | ||
| # 'wsNSE' and 'APFB' for detecting differences in high flows | ||
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| # Loading daily streamflows of the Ega River (Spain), from 1961 to 1970 | ||
| data(EgaEnEstellaQts) | ||
| obs <- EgaEnEstellaQts | ||
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| # 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)) | ||
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| # KGEkm (Pizarro and Jorquera, 2024; method='2012') | ||
| KGEkm(sim=sim, obs=obs) | ||
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| # KGE': Kling-Gupta eficiency 2012 (Kling et al.,2012) | ||
| KGE(sim=sim, obs=obs, method="2012") | ||
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| # Traditional Kling-Gupta eficiency (Gupta and Kling, 2009) | ||
| KGE(sim=sim, obs=obs) | ||
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| # KGE'': Kling-Gupta eficiency 2021 (Tang et al.,2021) | ||
| KGE(sim=sim, obs=obs, method="2021") | ||
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| # Traditional Nash-Sutcliffe eficiency (Nash and Sutcliffe, 1970) | ||
| NSE(sim=sim, obs=obs) | ||
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| # Weighted Nash-Sutcliffe efficiency (Hundecha and Bardossy, 2004) | ||
| wNSE(sim=sim, obs=obs) | ||
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| # wsNSE (Zambrano-Bigiarini and Bellin, 2012): | ||
| wsNSE(sim=sim, obs=obs) | ||
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| # APFB (Mizukami et al., 2019): | ||
| APFB(sim=sim, obs=obs) | ||
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| ################## | ||
| # Example 4: Looking at the difference between 'KGE', 'NSE', 'wsNSE', | ||
| # 'dr', 'rd', 'md', and 'KGElf' for detecting | ||
| # differences in low flows | ||
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| # Loading daily streamflows of the Ega River (Spain), from 1961 to 1970 | ||
| data(EgaEnEstellaQts) | ||
| obs <- EgaEnEstellaQts | ||
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| # 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)) | ||
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| # KGEkm (Pizarro and Jorquera, 2024; method='2012') | ||
| KGEkm(sim=sim, obs=obs) | ||
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| # KGE': Kling-Gupta eficiency 2012 (Kling et al.,2012) | ||
| KGE(sim=sim, obs=obs, method="2012") | ||
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| # Traditional Kling-Gupta eficiency (Gupta and Kling, 2009) | ||
| KGE(sim=sim, obs=obs) | ||
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| # KGE'': Kling-Gupta eficiency 2021 (Tang et al.,2021) | ||
| KGE(sim=sim, obs=obs, method="2021") | ||
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| # Traditional Nash-Sutcliffe eficiency (Nash and Sutcliffe, 1970) | ||
| NSE(sim=sim, obs=obs) | ||
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| # Weighted seasonal Nash-Sutcliffe efficiency (Zambrano-Bigiarini and Bellin, 2012): | ||
| wsNSE(sim=sim, obs=obs) | ||
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| # Refined Index of Agreement (Willmott et al., 2012): | ||
| dr(sim=sim, obs=obs) | ||
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| # Relative Index of Agreement (Krause et al., 2005): | ||
| rd(sim=sim, obs=obs) | ||
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| # Modified Index of Agreement (Krause et al., 2005): | ||
| md(sim=sim, obs=obs) | ||
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| # KGElf (Garcia et al., 2017): | ||
| KGElf(sim=sim, obs=obs) | ||
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| ################## | ||
| # 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. | ||
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| KGEkm(sim=sim, obs=obs, fun=log) | ||
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| # Verifying the previous value: | ||
| lsim <- log(sim) | ||
| lobs <- log(obs) | ||
| KGEkm(sim=lsim, obs=lobs) | ||
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| ################## | ||
| # 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 | ||
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| KGEkm(sim=sim, obs=obs, fun=log, epsilon.type="Pushpalatha2012") | ||
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| # 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) | ||
|
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| ################## | ||
| # 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 | ||
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| eps <- 0.01 | ||
| KGEkm(sim=sim, obs=obs, fun=log, epsilon.type="otherValue", epsilon.value=eps) | ||
|
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| # Verifying the previous value: | ||
| lsim <- log(sim+eps) | ||
| lobs <- log(obs+eps) | ||
| KGEkm(sim=lsim, obs=lobs) | ||
|
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| ################## | ||
| # 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 | ||
|
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| fact <- 1/50 | ||
| KGEkm(sim=sim, obs=obs, fun=log, epsilon.type="otherFactor", epsilon.value=fact) | ||
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| # Verifying the previous value: | ||
| eps <- fact*mean(obs, na.rm=TRUE) | ||
| lsim <- log(sim+eps) | ||
| lobs <- log(obs+eps) | ||
| KGEkm(sim=lsim, obs=lobs) | ||
|
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| ################## | ||
| # 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 | ||
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| fun1 <- function(x) {sqrt(x+1)} | ||
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| KGEkm(sim=sim, obs=obs, fun=fun1) | ||
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| # Verifying the previous value, with the epsilon value following Pushpalatha2012 | ||
| sim1 <- sqrt(sim+1) | ||
| obs1 <- sqrt(obs+1) | ||
| KGEkm(sim=sim1, obs=obs1) | ||
| } | ||
| % Add one or more standard keywords, see file 'KEYWORDS' in the | ||
| % R documentation directory. | ||
| \keyword{ math } |