Goodness-of-fit measures to compare observed and simulated time series with hydroGOF

Citation

If you use hydroGOF, please cite it as Zambrano-Bigiarini (2026):

Zambrano-Bigiarini, M. (2026) hydroGOF: Goodness-of-fit functions for comparison of simulated and observed hydrological time series R package version 0.7-0. URL: https://cran.r-project.org/package=hydroGOF. doi:10.32614/CRAN.package.hydroGOF.

Installation

Installing the latest stable version (from CRAN):

install.packages("hydroGOF")

Alternatively, you can also try the under-development version (from Github):

if (!require(devtools)) install.packages("devtools")
library(devtools)
install_github("hzambran/hydroGOF")

Setting up the environment

Loading the hydroGOF package, which contains data and functions used in this analysis:

library(hydroGOF)
## Loading required package: zoo
## 
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

Example using NSE

The following examples use the well-known Nash-Sutcliffe efficiency (NSE), but you can repeat the computations using any of the goodness-of-fit measures included in the hydroGOF package (e.g., KGE, ubRMSE, dr).

Ex 1: ideal integer values

Basic ideal case with a numeric sequence of integers:

obs <- 1:10
sim <- 1:10
NSE(sim, obs)
## [1] 1
obs <- 1:10
sim <- 2:11
NSE(sim, obs)
## [1] 0.8787879

Ex 2: ideal real values

Basic ideal case with a numeric sequence of integers:

obs <- 1:10 + 0.1
sim <- 1:10 +0.1
NSE(sim, obs)
## [1] 1
obs <- 1:10 + 0.1
sim <- 2:11 + 0.1
NSE(sim, obs)
## [1] 0.8787879

Ex 3: ideal daily ts

From this example onwards, a streamflow time series will be used.

First, we load the daily streamflows of the Ega River (Spain), from 1961 to 1970:

data(EgaEnEstellaQts)
obs <- EgaEnEstellaQts

Generating a simulated daily time series, initially equal to the observed series:

sim <- obs

Visualising the observed and simulated time series:

ggof(sim, obs)

Computing the ‘NSE’ for the “best” (unattainable) case

NSE(sim=sim, obs=obs)
## [1] 1

Ex 4: random noise in the first half of the simulated values

NSE for simulated values equal to observations plus random noise on the first half of the observed values.

This random noise has more relative importance for low flows than for medium and high flows.

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[1:1826] <- obs[1:1826] + rnorm(1826, mean=10)
ggof(sim, obs)

NSE(sim=sim, obs=obs)
## [1] 0.8739233

Let’s have a look at other goodness-of-fit measures:

mNSE(sim=sim, obs=obs)                 # Modified NSE
## [1] 0.6049353
rNSE(sim=sim, obs=obs)                 # Relative NSE
## [1] -0.5462133
wNSE(sim=sim, obs=obs)                 # Weighted NSE
## [1] 0.9769437
wsNSE(sim=sim, obs=obs)                # Weighted Seasonal NSE
## [1] 0.9685997
KGE(sim=sim, obs=obs)                  # Kling-Gupta efficiency (KGE), 2009
## [1] 0.6805676
KGE(sim=sim, obs=obs, method="2012")   # Kling-Gupta efficiency (KGE), 2012
## [1] 0.6168535
KGE(sim=sim, obs=obs, method="2021")   # Kling-Gupta efficiency (KGE), 2021
## [1] 0.7461033
KGElf(sim=sim, obs=obs)                # KGE for low flows
## [1] 0.517451
KGEnp(sim=sim, obs=obs)                # Non-parametric KGE
## [1] 0.6337526
sKGE(sim=sim, obs=obs)                 # Split KGE
## [1] 0.6544168
KGEkm(sim=sim, obs=obs)                # Knowable Moments KGE
## [1] 0.6467695
JDKGE(sim=sim, obs=obs)                # Joint Divergence KGE
## [1] 0.6084504
d(sim=sim, obs=obs)                    # Index of Agreement
## [1] 0.9697094
dr(sim=sim, obs=obs)                   # Refined Index of Agreement
## [1] 0.8024676
md(sim=sim, obs=obs)                   # Modified Index of Agreement
## [1] 0.7979285
rd(sim=sim, obs=obs)                   # Relative Index of Agreement
## [1] 0.6285137
VE(sim=sim, obs=obs)                   # Volumetric Efficiency
## [1] 0.6838346
cp(sim=sim, obs=obs)                   # Coefficient of Persistence
## [1] 0.468186
APFB(sim=sim, obs=obs)                 # Annual Peak Flow Bias
## [1] 0.03075188
HFB(sim=sim, obs=obs)                  # High Flow Bias
## [1] 0.08249524
LME(sim=sim, obs=obs)                  # Liu-Mean Efficiency
## [1] 0.6838218
LCE(sim=sim, obs=obs)                  # Lee and Choi Efficiency
## [1] 0.6777607
PMR(sim=sim, obs=obs)                  # Proxy for Model Robustness
## [1] 0.3164055
pbias(sim=sim, obs=obs)                # Percent bias (PBIAS)
## [1] 31.6
pbiasfdc(sim=sim, obs=obs)             # PBIAS in the slope of the midsegment of the FDC
## [Note: 'thr.shw' was set to FALSE to avoid confusing legends...]

## [1] -35.37753
me(sim=sim, obs=obs)                   # Mean Error
## [1] 4.999291
mae(sim=sim, obs=obs)                  # Mean Absolute Error
## [1] 4.999291
mse(sim=sim, obs=obs)                  # Mean Squared Error 
## [1] 50.49381
rmse(sim=sim, obs=obs)                 # Root Mean Square Error (RMSE)
## [1] 7.1059
ubRMSE(sim=sim, obs=obs)               # Unbiased RMSE
## [1] 5.049841
nrmse(sim=sim, obs=obs, norm="sd")     # Normalised Root Mean Square Error
## [1] 35.5
nrmse(sim=sim, obs=obs, norm="maxmin") # Normalised Root Mean Square Error
## [1] 3
nrmse(sim=sim, obs=obs, norm="mean")   # Normalised Root Mean Square Error
## [1] 44.9
nrmse(sim=sim, obs=obs, norm="IQR")    # Normalised Root Mean Square Error
## [1] 47.1
rPearson(sim=sim, obs=obs)             # Pearson correlation coefficient
## [1] 0.969776
rSpearman(sim=sim, obs=obs)            # Spearman rank correlation coefficient
## [1] 0.8357504
R2(sim=sim, obs=obs)                   # Coefficient of determination (R2)
## [1] 0.8739233
br2(sim=sim, obs=obs)                  # R2 multiplied by the slope of the regression line
## [1] 0.778017

Ex 5: random noise and logarithmic transformation

NSE for simulated values equal to observations plus random noise on the first half of the observed values and applying (natural) logarithm to obs against sim during computations.

NSE(sim=sim, obs=obs, fun=log)
## [1] 0.4800871

Verifying the previous value:

lsim <- log(sim)
lobs <- log(obs)
NSE(sim=lsim, obs=lobs)
## [1] 0.4800871

Let’s have a look at other goodness-of-fit measures:

mNSE(sim=sim, obs=obs)                 # Modified NSE
## [1] 0.6049353
rNSE(sim=sim, obs=obs)                 # Relative NSE
## [1] -0.5462133
wNSE(sim=sim, obs=obs)                 # Weighted NSE
## [1] 0.9769437
wsNSE(sim=sim, obs=obs)                # Weighted Seasonal NSE
## [1] 0.9685997
KGE(sim=sim, obs=obs)                  # Kling-Gupta efficiency (KGE), 2009
## [1] 0.6805676
KGE(sim=sim, obs=obs, method="2012")   # Kling-Gupta efficiency (KGE), 2012
## [1] 0.6168535
KGE(sim=sim, obs=obs, method="2021")   # Kling-Gupta efficiency (KGE), 2021
## [1] 0.7461033
KGElf(sim=sim, obs=obs)                # KGE for low flows
## [1] 0.517451
KGEnp(sim=sim, obs=obs)                # Non-parametric KGE
## [1] 0.6337526
sKGE(sim=sim, obs=obs)                 # Split KGE
## [1] 0.6544168
KGEkm(sim=sim, obs=obs)                # Knowable Moments KGE
## [1] 0.6467695
JDKGE(sim=sim, obs=obs)                # Joint Divergence KGE
## [1] 0.6084504
d(sim=sim, obs=obs)                    # Index of Agreement
## [1] 0.9697094
dr(sim=sim, obs=obs)                   # Refined Index of Agreement
## [1] 0.8024676
md(sim=sim, obs=obs)                   # Modified Index of Agreement
## [1] 0.7979285
rd(sim=sim, obs=obs)                   # Relative Index of Agreement
## [1] 0.6285137
VE(sim=sim, obs=obs)                   # Volumetric Efficiency
## [1] 0.6838346
cp(sim=sim, obs=obs)                   # Coefficient of Persistence
## [1] 0.468186
APFB(sim=sim, obs=obs)                 # Annual Peak Flow Bias
## [1] 0.03075188
HFB(sim=sim, obs=obs)                  # High Flow Bias
## [1] 0.08249524
LME(sim=sim, obs=obs)                  # Liu-Mean Efficiency
## [1] 0.6838218
LCE(sim=sim, obs=obs)                  # Lee and Choi Efficiency
## [1] 0.6777607
PMR(sim=sim, obs=obs)                  # Proxy for Model Robustness
## [1] 0.3164055
pbias(sim=sim, obs=obs)                # Percent bias (PBIAS)
## [1] 31.6
pbiasfdc(sim=sim, obs=obs)             # PBIAS in the slope of the midsegment of the FDC
## [Note: 'thr.shw' was set to FALSE to avoid confusing legends...]

## [1] -35.37753
me(sim=sim, obs=obs)                   # Mean Error
## [1] 4.999291
mae(sim=sim, obs=obs)                  # Mean Absolute Error
## [1] 4.999291
mse(sim=sim, obs=obs)                  # Mean Squared Error 
## [1] 50.49381
rmse(sim=sim, obs=obs)                 # Root Mean Square Error (RMSE)
## [1] 7.1059
ubRMSE(sim=sim, obs=obs)               # Unbiased RMSE
## [1] 5.049841
nrmse(sim=sim, obs=obs, norm="sd")     # Normalised Root Mean Square Error
## [1] 35.5
nrmse(sim=sim, obs=obs, norm="maxmin") # Normalised Root Mean Square Error
## [1] 3
nrmse(sim=sim, obs=obs, norm="mean")   # Normalised Root Mean Square Error
## [1] 44.9
nrmse(sim=sim, obs=obs, norm="IQR")    # Normalised Root Mean Square Error
## [1] 47.1
rPearson(sim=sim, obs=obs)             # Pearson correlation coefficient
## [1] 0.969776
rSpearman(sim=sim, obs=obs)            # Spearman rank correlation coefficient
## [1] 0.8357504
R2(sim=sim, obs=obs)                   # Coefficient of determination (R2)
## [1] 0.8739233
br2(sim=sim, obs=obs)                  # R2 multiplied by the slope of the regression line
## [1] 0.778017

Ex 6: random noise, logarithmic transformation and Pushpalatha2012 constant

NSE for simulated values equal to observations plus random noise on the first half of the observed values and applying (natural) logarithm to obs against sim and adding the Pushpalatha2012 constant during computations.

NSE(sim=sim, obs=obs, fun=log, epsilon.type="Pushpalatha2012")
## [1] 0.4872317

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)
NSE(sim=lsim, obs=lobs)
## [1] 0.4872317

Let’s have a look at other goodness-of-fit measures:

gof(sim=sim, obs=obs, fun=log, epsilon.type="Pushpalatha2012", do.spearman=TRUE, do.pbfdc=TRUE, do.pmr=TRUE)
##              [,1]
## ME           0.41
## MAE          0.41
## MSE          0.46
## RMSE         0.68
## ubRMSE       0.54
## NRMSE %     71.60
## PBIAS %     18.20
## RSR          0.72
## rSD          0.89
## NSE          0.49
## mNSE         0.48
## rNSE        -2.00
## wNSE         0.74
## wsNSE        0.78
## d            0.86
## dr           0.74
## md           0.74
## rd           0.20
## cp          -7.68
## r            0.83
## R2           0.49
## bR2          0.44
## VE           0.82
## KGE          0.72
## KGElf        0.52
## KGEnp        0.74
## KGEkm        0.74
## JDKGE        0.72
## LME          0.68
## LCE          0.67
## sKGE         0.53
## APFB         0.01
## HFB          0.02
## rSpearman    0.84
## pbiasFDC % -46.31
## PMR          0.20

Ex 7: random noise, logarithmic transformation and user-defined constant

NSE for simulated values equal to observations plus random noise on the first half of the observed values and applying (natural) logarithm to obs against sim and adding a user-defined constant during computations

eps <- 0.01
NSE(sim=sim, obs=obs, fun=log, epsilon.type="otherValue", epsilon.value=eps)
## [1] 0.4805544

Verifying the previous value:

lsim <- log(sim+eps)
lobs <- log(obs+eps)
NSE(sim=lsim, obs=lobs)
## [1] 0.4805544

Let’s have a look at other goodness-of-fit measures:

gof(sim=sim, obs=obs, fun=log, epsilon.type="otherValue", epsilon.value=eps, do.spearman=TRUE, do.pbfdc=TRUE, do.pmr=TRUE)
##              [,1]
## ME           0.42
## MAE          0.42
## MSE          0.48
## RMSE         0.69
## ubRMSE       0.55
## NRMSE %     72.10
## PBIAS %     18.70
## RSR          0.72
## rSD          0.88
## NSE          0.48
## mNSE         0.48
## rNSE        -3.98
## wNSE         0.74
## wsNSE        0.78
## d            0.86
## dr           0.74
## md           0.74
## rd          -0.33
## cp          -7.93
## r            0.82
## R2           0.48
## bR2          0.43
## VE           0.81
## KGE          0.72
## KGElf        0.51
## KGEnp        0.74
## KGEkm        0.73
## JDKGE        0.71
## LME          0.67
## LCE          0.66
## sKGE         0.48
## APFB         0.01
## HFB          0.02
## rSpearman    0.84
## pbiasFDC % -46.79
## PMR          0.21

Ex 8: random noise, logarithmic transformation and user-defined factor

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

fact <- 1/50
NSE(sim=sim, obs=obs, fun=log, epsilon.type="otherFactor", epsilon.value=fact)
## [1] 0.4939063

Verifying the previous value:

fact <- 1/50
eps  <- fact*mean(obs, na.rm=TRUE)
lsim <- log(sim+eps)
lobs <- log(obs+eps)
NSE(sim=lsim, obs=lobs)
## [1] 0.4939063

Let’s have a look at other goodness-of-fit measures:

gof(sim=sim, obs=obs, fun=log, epsilon.type="otherFactor", epsilon.value=fact, do.spearman=TRUE, do.pbfdc=TRUE, do.pmr=TRUE)
##              [,1]
## ME           0.41
## MAE          0.41
## MSE          0.44
## RMSE         0.66
## ubRMSE       0.52
## NRMSE %     71.10
## PBIAS %     17.60
## RSR          0.71
## rSD          0.89
## NSE          0.49
## mNSE         0.48
## rNSE        -1.29
## wNSE         0.74
## wsNSE        0.78
## d            0.87
## dr           0.74
## md           0.74
## rd           0.39
## cp          -7.43
## r            0.83
## R2           0.49
## bR2          0.44
## VE           0.82
## KGE          0.73
## KGElf        0.53
## KGEnp        0.74
## KGEkm        0.75
## JDKGE        0.72
## LME          0.68
## LCE          0.68
## sKGE         0.56
## APFB         0.01
## HFB          0.02
## rSpearman    0.84
## pbiasFDC % -45.81
## PMR          0.19

Ex 9: random noise, user-defined transformation

NSE for simulated values equal to observations plus random noise on the first half of the observed values and applying a user-defined function to obs against sim during computations:

fun1 <- function(x) {sqrt(x+1)}
NSE(sim=sim, obs=obs, fun=fun1)
## [1] 0.7263728

Verifying the previous value, with the epsilon value following Pushpalatha2012:

sim1 <- sqrt(sim+1)
obs1 <- sqrt(obs+1)
NSE(sim=sim1, obs=obs1)
## [1] 0.7263728
gof(sim=sim, obs=obs, fun=fun1, do.spearman=TRUE, do.pbfdc=TRUE, do.pmr=TRUE)
##              [,1]
## ME           0.65
## MAE          0.65
## MSE          0.92
## RMSE         0.96
## ubRMSE       0.71
## NRMSE %     52.30
## PBIAS %     17.70
## RSR          0.52
## rSD          0.97
## NSE          0.73
## mNSE         0.54
## rNSE         0.34
## wNSE         0.89
## wsNSE        0.76
## d            0.93
## dr           0.77
## md           0.76
## rd           0.83
## cp          -1.17
## r            0.92
## R2           0.73
## bR2          0.65
## VE           0.82
## KGE          0.81
## KGElf        0.51
## KGEnp        0.75
## KGEkm        0.80
## JDKGE        0.79
## LME          0.80
## LCE          0.79
## sKGE         0.84
## APFB         0.02
## HFB          0.04
## rSpearman    0.84
## pbiasFDC % -33.38
## PMR          0.19

A short example from hydrological modelling

Loading observed streamflows of the Ega River (Spain), with daily data from 1961-Jan-01 up to 1970-Dec-31

require(zoo)
data(EgaEnEstellaQts)
obs <- EgaEnEstellaQts

Generating a simulated daily time series, initially equal to the observed values (simulated values are usually read from the output files of the hydrological model)

sim <- obs

Plotting the graphical comparison of obs against sim, along with the default goodness-of-fit measures:

ggof(sim, obs)

Plotting the graphical comparison of obs against sim, along with user-defined goodness-of-fit measures:

ggof(sim, obs, 
     gofs=c( "PBIAS", "dr", "R2",  "NSE", "KGE",  "LCE", "JDKGE", "APFB", "HFB"))

Computing the numeric goodness-of-fit measures for the best (unattainable) case, including the following three measures that -due to slightly high computation time- are not computed by default: i) the Spearman’s Rank Correlation Coefficient (rSpearman), ii) the percent bias in the Slope of the Midsegment of the Flow Duration Curve (pbiasFDC), and iii) the Proxy for Model Robustness (PMR).

gof(sim=sim, obs=obs, do.spearman=TRUE, do.pbfdc=TRUE, do.pmr=TRUE)
##            [,1]
## ME            0
## MAE           0
## MSE           0
## RMSE          0
## ubRMSE        0
## NRMSE %       0
## PBIAS %       0
## RSR           0
## rSD           1
## NSE           1
## mNSE          1
## rNSE          1
## wNSE          1
## wsNSE         1
## d             1
## dr            1
## md            1
## rd            1
## cp            1
## r             1
## R2            1
## bR2           1
## VE            1
## KGE           1
## KGElf         1
## KGEnp         1
## KGEkm         1
## JDKGE         1
## LME           1
## LCE           1
## sKGE          1
## APFB          0
## HFB           0
## rSpearman     1
## pbiasFDC %    0
## PMR           0
  • Randomly changing the first 1826 elements of ‘sim’ (half of the ts), by using a normal distribution with mean 10 and standard deviation equal to 1 (default of ‘rnorm’).
sim[1:1826] <- obs[1:1826] + rnorm(1826, mean=10)

Plotting the graphical comparison of obs against sim, along with the default goodness-of-fit measures for the daily and monthly time series:

ggof(sim=sim, obs=obs, ftype="dm", FUN=mean)

Plotting the graphical comparison of obs against sim, along with user-defined goodness-of-fit measures for the daily and monthly time series:

ggof(sim=sim, obs=obs, ftype="dm", FUN=mean,
     gofs=c( "PBIAS", "dr", "R2",  "NSE", "KGE",  "LCE", "JDKGE", "APFB", "HFB"))

Removing warm-up period

Using the first two years (1961-1962) as warm-up period, and removing the corresponding observed and simulated values from the computation of the goodness-of-fit measures:

ggof(sim=sim, obs=obs, ftype="dm", FUN=mean, cal.ini="1963-01-01")

Verification of the goodness-of-fit measures for the daily values after removing the warm-up period:

sim <- window(sim, start="1963-01-01")
obs <- window(obs, start="1963-01-01")

gof(sim, obs)
##          [,1]
## ME       3.74
## MAE      3.74
## MSE     37.77
## RMSE     6.15
## ubRMSE   4.87
## NRMSE % 33.70
## PBIAS % 25.20
## RSR      0.34
## rSD      1.02
## NSE      0.89
## mNSE     0.68
## rNSE    -0.48
## wNSE     0.98
## wsNSE    0.82
## d        0.97
## dr       0.84
## md       0.84
## rd       0.64
## cp       0.52
## r        0.97
## R2       0.89
## bR2      0.81
## VE       0.75
## KGE      0.74
## KGElf    0.57
## KGEnp    0.69
## KGEkm    0.73
## JDKGE    0.74
## LME      0.75
## LCE      0.74
## sKGE     0.70
## APFB     0.02
## HFB      0.00

Plotting uncertainty bands

Generating fictitious lower and upper uncertainty bounds:

lband <- obs - 5
uband <- obs + 5

Plotting the previously generated uncertainty bands:

plotbands(obs, lband, uband)

Randomly generating a simulated time series:

sim <- obs + rnorm(length(obs), mean=3)

Plotting the previously generated simulated time series along the observations and the uncertainty bounds:

plotbands(obs, lband, uband, sim)

Computing P-factor

The P-factor quantifies the percentage of observed values that fall within the prediction uncertainty band defined by the lower and upper bounds. It is a measure of the coverage of the uncertainty interval.

The P-factor ranges from 0 to 1. A value of 1 indicates that all observations are bracketed by the uncertainty bounds, whereas a value of 0 indicates that none of the observations fall within the bounds.

Computing the P-factor:

( pfactor(x=obs, lband=lband, uband=uband) )
## [1] 0.9993155

Computing R-factor

The R-factor quantifies the average width of the prediction uncertainty band relative to the variability of the observed data. It is a measure of the magnitude of predictive uncertainty associated with a model simulation.

The R-factor ranges from 0 to infinity, with an optimal value of 0 indicating perfect agreement between simulated and observed values (i.e., zero prediction uncertainty). In practical applications, the R-factor represents the width of the uncertainty interval and should be as small as possible. Values close to or smaller than 1 are commonly considered indicative of an acceptable level of predictive uncertainty, although acceptable thresholds depend on the quality of observations and the modelling context.

Computing the R-factor:

( rfactor(x=obs, lband=lband, uband=uband) )
## [1] 0.5488274

Important note:

Because a larger fraction of observations can often be bracketed by widening the uncertainty bounds, the R-factor is typically interpreted jointly with the P-factor. A balance between a high P-factor (good coverage) and a low R-factor (narrow uncertainty bounds) is therefore sought during model calibration and uncertainty analysis.

Analysis of the residuals

Computing the daily residuals (even if this is a dummy example, it is enough for illustrating the capability)

r <- sim-obs

Summarizing and plotting the residuals (it requires the hydroTSM package):

library(hydroTSM)
smry(r) 
##               Index         r
## Min.     1963-01-01   -1.6160
## 1st Qu.  1964-12-31    2.3300
## Median   1966-12-31    3.0360
## Mean     1966-12-31    3.0330
## 3rd Qu.  1968-12-30    3.7470
## Max.     1970-12-31    7.3350
## IQR            <NA>    1.4163
## sd             <NA>    1.0413
## cv             <NA>    0.3433
## Skewness       <NA>   -0.0089
## Kurtosis       <NA>    0.1889
## NA's           <NA>    2.0000
## n              <NA> 2922.0000
# daily, monthly and annual plots, boxplots and histograms
hydroplot(r, FUN=mean)

Seasonal plots and boxplots

# daily, monthly and annual plots, boxplots and histograms
hydroplot(r, FUN=mean, pfreq="seasonal")

Software details

This tutorial was built under:

## [1] "x86_64-pc-linux-gnu"
## [1] "R version 4.6.0 (2026-04-24)"
## [1] "hydroGOF 0.7-0"

Version history of this vignette

  • v0.4: 30-Apr-2026
  • v0.3: 21-Jan-2024
  • v0.2: Mar-2020
  • v0.1: Aug 2011

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