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Criteria equations

Performance criteria are used in several files. Different criterion is given in subass.txt and simass.txt files. In addition criteria can be selected in info.txt. Below is listed the code/heading used in each file together with the equation identificator. Further down all the equations are defined.

Code to equation coupling

Equation IDs for subbasin assessment criteria (subassX.txt):

Heading Description Equation ID
NSENash-Sutcliffe efficiencyNSE
CCPearson correlation coefficient (Kling-Gupta efficiency, part 1)CC
RE(%)relative bias in percentRE%
RSDE(%)relative error in standard deviation in percentRS%
Simaverage of simulated variablecm
Recaverage of observed variablerm
SDSimstandard deviation of simulated variablecd
SDRecstandard deviation of observed variablerd
MAEmean absolute error MAE
RMSEroot mean square error RMSE
SDEError of standard deviationES
KGEKling-Gupta efficiency KGE
KGESDKling-Gupta efficiency, part 2KGESD
KGEMKling-Gupta efficiency, part 3KGEM
NRMSEnormalised root mean square error NE
NSEWNash-Sutcliffe efficiency adjusted for biasNSEW

Equation IDs for simulation assessment criteria (simass.txt):

Name Code Equation ID
Kendalls TauTAUAVTAU

Equation IDs for calibration simulation assessment criteria (bestsims.txt and allsim.txt):

Heading Description Equation ID
rr2regional Nash-Sutcliffe efficiency (data from all subbasins combined in one data series)REGNSE
sr2spatial Nash-Sutcliffe efficiency, calculated using annual means for all subbasins (requires at least 5 years and 5 subbasins with data) to form one data series to calculate the Nash-Sutcliffe efficiency onSPATNSE
mr2average of Nash-Sutcliffe efficiencies for subbasinsAVNSE
rmaeregional mean absolute error (data from all subbasins combined in one data series)REGMAE
srespatial relative bias (calculated on annual means for all subbasins)SPATRB
rreregional relative bias (data from all subbasins combined in one data series)REGRB
mreaverage of the relative bias for all subbasins (Note: fraction, not %)AVRB
rraregional RA, similar to regional NSE, RA is a Nash-Sutcliffe like criterion where the square in the Nash-Sutcliffe formula is exchanged with a coefficient valueREGRA
sraspatial RA, similar to spatial NSE, RA is a Nash-Sutcliffe like criterion where the square in the Nash-Sutcliffe formula is exchanged for a coefficient valueSPATRA
mraaverage value of RA for subbasins, RA is a Nash-Sutcliffe like criterion where the square in the Nash-Sutcliffe formula is exchanged with a coefficient valueAVRA
tauaverage of Kendall's Tau value for subbasinsAVTAU
md2median of Nash-Sutcliffe efficiency for subbasinsMEDNSE
mdamedian of all subbasins’ RA (Nash-Sutcliffe like criteria where the square is exchanged with a coefficient value)MEDRA
mrsaverage of error in standard deviation for subbasinsAVRSB
mccPearson correlation coefficient, average of all subbasins with observationsAVCC
mdkgmedian of Kling-Gupta efficiency (MKG in info.txt) for subbasinsMEDKGE
mareaverage of absolute relative bias for subbasins (Note: fraction. not %) (MAR in info.txt)AVARB
mnrmedian of normalised RMSE for subbasinsMEDNE
mnwaverage of Nash-Sutcliffe efficiencies adjusted for bias for subbasinsAVNSEW

Equation IDs for performance criteria set in info.txt are tabled here.

Equation definitions


ccomputed value
rrecorded value
iindex for time steps with observations in a time series of a station
minumber of values in a time series of a station
jindex of stations
mjnumber of stations
ijindex over time steps with observations for all stations
mijnumber of time steps with obsevations for all stations
cmaverage value of c_{i}, i=1,mi for a station
rmaverage value of r_{i}, i=1,mi for a station
cdstandard deviation of c_{i}, i=1,mi for a station
rdstandard deviation of r_{i}, i=1,mi for a station

Basic equations

Average value for a time series of a station:

xm = {1/mi} sum{i=1}{mi}{x_{i}} x=r or c

Standard deviation of a time series of a station:

xd = sqrt{{1/mi} sum{i=1}{mi}{{x_{i}}^2}-xm^2} x=r or c

Criteria equations for a time series of a station

Nash-Sutcliffe Efficiency (NSE or R2):

NSE = 1-{sum{i=1}{mi}{(c_{i}-r_{i})^2}}/{sum{i=1}{mi}{(r_{i}-rm)^2}}

Efficiency with coefficient a (RA):

RA = 1-{sum{i=1}{mi}{delim{|} {c_{i}-r_{i}} {|}^a}}/{sum{i=1}{mi}{delim{|} {r_{i}-rm} {|}^a}}


Bias = {sum{i=1}{mi}{(c_{i}-r_{i})}}/mi

Relative bias (RB or RE):

RB = {sum{i=1}{mi}{(c_{i}-r_{i})}}/{delim{|}{sum{i=1}{mi}{r_{i}}}{|}}

Relative bias in percent (RE%):

RE% = RB*100 = {{sum{i=1}{mi}{(c_{i}-r_{i})}}/{delim{|}{sum{i=1}{mi}{r_{i}}}{|}}}*100

Error of standard deviation (ES):

ES = {cd-rd}

Relative error of standard deviation (RS):

RS = {{cd-rd}/rd}

Relative error of standard deviation in percent (RS%):

RS% = RS*100 = {{cd-rd}/rd}*100

Mean absolute error (MAE):

MAE = {sum{i=1}{mi}{delim{|}{c_{i}-r_{i}}{|}}}/mi

Kling-Gupta efficiency (KGE):

KGE = 1-sqrt{(CC-1)^2+(cd/rd-1)^2+(cm/rm-1)^2}

Pearson correlation coefficient, Kling-Gupta efficiency part 1 (CC):

CC = {{1/mi} sum{i=1}{mi}{({r_{i}}*{c_{i}})}-cm*rm}/{cd*rd}

Kling-Gupta efficiency part 2 (KGESD):

KGESD = cd/rd

Kling-Gupta efficiency part 3 (KGEM):

KGEM = cm/rm

Root mean square error (RMSE):

RMSE = sqrt{{1/mi sum{i=1}{mi}{({c_{i}}-{r_{i}})^2}}}

Normalised root mean square error (NE):

NE = sqrt{{1/mi sum{i=1}{mi}{({c_{i}}-{r_{i}})^2}}}/{max{(r_{i}})}

Kendalls rank correlation coefficient, tau-b, with adjustments for ties (TAU):

TAU = {n_{c}-n_{d}}/{sqrt{(n_{0}-n_{1})(n_{0}-n_{2})}}

Nash-Sutcliffe Efficiency adjusted for bias (NSEW). Introduced in Lindström (2016):

NSEW = NSE-Bias^2/rd^2


n_{c} = number of concordant pairs (c_{i}<c_{k} and r_{i}<r_{k} or c_{i}>c_{k} and r_{i}>r_{k}, i=1,mi k=1,mi)

n_{d} = number of discordant pairs (c_{i}<c_{k} and r_{i}>r_{k} or c_{i}>c_{k} and r_{i}<r_{k}, i=1,mi k=1,mi)

n_{0} = number of compared pairs

n_{1} = number of compared pairs that ties in the computed values

n_{2} = number of compared pairs that ties in the recorded values

Criteria equations for a model domain (several stations)

Average Nash-Sutcliffe efficiency (AVNSE):

AVNSE = {1/mj sum{j=1}{mj}{NSE_{j}}}

Median Nash-Sutcliffe efficiency (MEDNSE):

MEDNSE = median delim{lbrace}{{NSE_{j}},{j=1..mj}}{rbrace}

Spatial Nash-Sutcliffe efficiency (SPATNSE):

SPATNSE = 1-{sum{j=1}{mj}{(cm_{j}-rm_{j})^2}}/{sum{j=1}{mj}{(rm_{j}-{1/mj} sum{j=1}{mj}{rm_{j}})^2}}

Regional Nash-Sutcliffe efficiency (REGNSE):

REGNSE = 1-{sum{ij=1}{mij}{(c_{ij}-r_{ij})^2}}/{sum{ij=1}{mij}{(r_{ij}-{1/mij} sum{ij=1}{mij}{r_{ij}})^2}}

Average efficiency with coefficient a (AVRA):

AVRA = {1/mj sum{j=1}{mj}{RA_{j}}}

Median efficiency with coefficient a (MEDRA):

MEDRA = median delim{lbrace}{{RA_{j}},{j=1..mj}}{rbrace}

Spatial efficiency with coefficient a (SPATRA):

SPATRA = 1-{sum{j=1}{mj}{delim{|}{cm_{j}-rm_{j}}{|}^a}}/{sum{j=1}{mj}{delim{|}{rm_{j}-{1/mj} sum{j=1}{mj}{rm_{j}}}{|}^a}}

Regional efficiency with coefficient a (REGRA):

REGRA = 1-{sum{ij=1}{mij}{delim{|}{c_{ij}-r_{ij}}{|}^a}}/{sum{ij=1}{mij}{delim{|}{r_{ij}-{1/mij} sum{ij=1}{mij}{r_{ij}}}{|}^a}}

Average relative bias (AVRB):

AVRB = {1/mj sum{j=1}{mj}{RB_{j}}}

Regional relative bias (REGRB):

REGRB = {sum{ij=1}{mij}{(c_{ij}-r_{ij})}}/{delim{|}{sum{ij=1}{mij}{r_{ij}}}{|}}

Spatial relative bias (SPATRB):

SPATRB = {sum{j=1}{mj}{(cm_{j}-rm_{j})}}/{delim{|}{sum{j=1}{mj}{rm_{j}}}{|}}

Average Kling-Gupta efficiency (AVKGE):

AVKGE = {1/mj sum{j=1}{mj}{KGE_{j}}}

Median Kling-Gupta efficiency (MEDKGE):

MEDKGE = median delim{lbrace}{{KGE_{j}},{j=1..mj}}{rbrace}

Median of Normalised root mean square error (MEDNE):

MEDNE = median delim{lbrace}{{NE_{j}},{j=1..mj}}{rbrace}

Average of absolute relative bias (AVARB):

AVARB = {1/mj sum{j=1}{mj}{delim{|}{RB_{j}}{|}}}

Average Pearson correlation coefficient (AVCC):

AVCC = {1/mj sum{j=1}{mj}{CC_{j}}}

Average relative error of standard deviation (AVRSB):

AVRSB = {1/mj sum{j=1}{mj}{RS_{j}}}

Average Kendalls rank correlation coefficient (AVTAU):

AVTAU = {1/mj sum{j=1}{mj}{TAU_{j}}}

Regional mean absolute error (REGMAE):

REGMAE = {sum{ij=1}{mij}{delim{|}{c_{ij}-r_{ij}}{|}}}/mij

Average Nash-Sutcliffe efficiency adjusted for bias (AVNSEW):

AVNSEW = {1/mj sum{j=1}{mj}{NSEW_{j}}}


Lindström, G., 2016. Lake water levels for calibration of the S-HYPE model. Hydrology Research 47.4:672-682. doi: 10.2166/nh.2016.019.

start/hype_file_reference/info.txt/criteria_equations.txt · Last modified: 2019/08/28 14:57 by cpers