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- | ====== Criteria equations ====== | ||
- | |||
- | Performance criteria are used in several files. Different criterion is given in [[start:HYPE_file_reference:subassX.txt|subass.txt]] and [[start:HYPE_file_reference:simass.txt|simass.txt]] files. In addition criteria can be selected in [[start:hype_file_reference:info.txt#performance_criteria_options|info.txt]]. [[start:hype_file_reference:info.txt:criteria_equations#code_to_equation_coupling|Below]] is listed the code/heading used in each file together with the equation identificator. [[start:hype_file_reference:info.txt:criteria_equations#equation_definitions|Further down]] all the equations are defined. | ||
- | |||
- | |||
- | |||
- | ===== Code to equation coupling ===== | ||
- | |||
- | Equation IDs for subbasin assessment criteria ([[start:HYPE_file_reference:subassX.txt|subassX.txt]]): | ||
- | |||
- | <sortable> | ||
- | ^ Heading ^ Description ^ Equation ID ^ | ||
- | |''NSE''|Nash-Sutcliffe efficiency|//NSE//| | ||
- | |''CC''|Pearson correlation coefficient (Kling-Gupta efficiency, part 1)|//CC//| | ||
- | |''RE(%)''|relative bias in percent|//RE%//| | ||
- | |''RSDE(%)''|relative error in standard deviation in percent|//RS%//| | ||
- | |''Sim''|average of simulated variable|//cm//| | ||
- | |''Rec''|average of observed variable|//rm//| | ||
- | |''SDSim''|standard deviation of simulated variable|//cd//| | ||
- | |''SDRec''|standard deviation of observed variable|//rd//| | ||
- | |''MAE''|mean absolute error |//MAE//| | ||
- | |''RMSE''|root mean square error |//RMSE//| | ||
- | |''Bias''|bias|//Bias//| | ||
- | |''SDE''|Error of standard deviation|//ES//| | ||
- | |''KGE''|Kling-Gupta efficiency |//KGE//| | ||
- | |''KGESD''|Kling-Gupta efficiency, part 2|//KGESD//| | ||
- | |''KGEM''|Kling-Gupta efficiency, part 3|//KGEM//| | ||
- | |''NRMSE''|normalised root mean square error |//NE//| | ||
- | |''NSEW''|Nash-Sutcliffe efficiency adjusted for bias|//NSEW//| | ||
- | </sortable> | ||
- | |||
- | Equation IDs for simulation assessment criteria ([[start:HYPE_file_reference:simass.txt|simass.txt]]): | ||
- | |||
- | <sortable> | ||
- | ^ Name ^ Code ^ Equation ID ^ | ||
- | | Regional NSE | ''RR2'' | //REGNSE// | | ||
- | | Regional RA | ''RRA'' | //REGRA// | | ||
- | | Regional RE | ''RRE'' | //REGRB// | | ||
- | | Regional MAE | ''-'' | //REGMAE// | | ||
- | | Average NSE | ''MR2'' | //AVNSE// | | ||
- | | Average RA | ''MRA'' | //AVRA// | | ||
- | | Average RE | ''MRE'' | //AVRB// | | ||
- | | Average RSDE | ''MRS'' | //AVRSB// | | ||
- | | Average CC | ''MCC'' | //AVCC// | | ||
- | | Average ARE | ''MAR'' | //AVARB// | | ||
- | | Average KGE | ''AKG'' | //AVKGE// | | ||
- | | Aver scalKGE | ''ASK'' | //ASCKGE// | | ||
- | | Spatial NSE | ''SR2'' | //SPATNSE// | | ||
- | | Spatial RA | ''RRA'' | //SPATRA// | | ||
- | | Spatial RE | ''-'' | //SPATRB// | | ||
- | | Spatial Bias | ''SMB'' | //SPATASB// | | ||
- | | Spatial RMSE | ''SNR'' | //SPATRMSE// | | ||
- | | Kendalls Tau | ''TAU'' | //AVTAU// | | ||
- | | Median NSE | ''MD2'' | //MEDNSE// | | ||
- | | Median RA | ''MDA'' | //MEDRA// | | ||
- | | Median KGE | ''MKG'' | //MEDKGE// | | ||
- | | Median NRMSE | ''MNR'' | //MEDNE// | | ||
- | | Mean NSEW | ''MNW'' | //AVNSEW// | | ||
- | </sortable> | ||
- | |||
- | Equation IDs for calibration simulation assessment criteria ([[start:HYPE_file_reference:bestsims.txt|bestsims.txt]] and [[start:HYPE_file_reference:allsim.txt|allsim.txt]]): | ||
- | |||
- | <sortable> | ||
- | ^ Heading ^ Description ^ Equation ID ^ | ||
- | |''rr2''|regional Nash-Sutcliffe efficiency (data from all subbasins combined in one data series)|//REGNSE//| | ||
- | |''sr2''|spatial 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 on|//SPATNSE//| | ||
- | |''mr2''|average of Nash-Sutcliffe efficiency for subbasins|//AVNSE//| | ||
- | |''rmae''|regional mean absolute error (data from all subbasins combined in one data series)|//REGMAE//| | ||
- | |''sre''|spatial relative bias (calculated on annual means for all subbasins)|//SPATRB//| | ||
- | |''rre''|regional relative bias (data from all subbasins combined in one data series)|//REGRB//| | ||
- | |''mre''|average of the relative bias for all subbasins (Note: fraction, not %)|//AVRB//| | ||
- | |''rra''|regional 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 value|//REGRA//| | ||
- | |''sra''|spatial 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 value|//SPATRA//| | ||
- | |''mra''|average 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 value|//AVRA//| | ||
- | |''tau''|average of Kendall's Tau value for subbasins|//AVTAU//| | ||
- | |''md2''|median of Nash-Sutcliffe efficiency for subbasins|//MEDNSE//| | ||
- | |''mda''|median of all subbasins’ RA (Nash-Sutcliffe like criteria where the square is exchanged with a coefficient value)|//MEDRA//| | ||
- | |''mrs''|average of error in standard deviation for subbasins|//AVRSB//| | ||
- | |''mcc''|Pearson correlation coefficient, average of all subbasins with observations|//AVCC//| | ||
- | |''mdkg''|median of Kling-Gupta efficiency (MKG in [[start:hype_file_reference:info.txt|info.txt]]) for subbasins|//MEDKGE//| | ||
- | |''akg''|average of Kling-Gupta efficiency for subbasins|//AVKGE//| | ||
- | |''asckg''|average of Kling-Gupta efficiency rescaled to interval [-1,1] (C2M criteria applied to KGE, Mathevet et al. 2006)|//ASCKGE//| | ||
- | |''mare''|average of absolute relative bias for subbasins (Note: fraction. not %) (MAR in [[start:hype_file_reference:info.txt|info.txt]])|//AVARB//| | ||
- | |''mdnr''|median of normalised RMSE for subbasins|//MEDNE//| | ||
- | |''mnw''|average of Nash-Sutcliffe efficiencies adjusted for bias for subbasins|//AVNSEW//| | ||
- | |''snr''|spatial root mean square error|//SPATRMSE//| | ||
- | |''smb''|spatial mean absolute scaled bias on natural log transformed values|//SPATASB//| | ||
- | </sortable> | ||
- | |||
- | Equation IDs for performance criteria set in info.txt are tabled [[start:HYPE_file_reference:info.txt:criteria|here]]. | ||
- | |||
- | |||
- | ===== Equation definitions ===== | ||
- | |||
- | ==== Denotations ==== | ||
- | <sortable> | ||
- | |//c//|computed value| | ||
- | |//r//|recorded value| | ||
- | |//cl//|log transform of computed value, natural logarithm| | ||
- | |//rl//|log transform of recorded value, natural logarithm| | ||
- | |//i//|index for time steps with observations in a time series of a station| | ||
- | |//mi//|number of values in a time series of a station| | ||
- | |//j//|index of stations| | ||
- | |//mj//|number of stations| | ||
- | |//ij//|index over time steps with observations for all stations| | ||
- | |//mij//|number of time steps with obsevations for all stations| | ||
- | |//cm//|average value of <m> c_{i}, i=1,mi </m> for a station| | ||
- | |//rm//|average value of <m> r_{i}, i=1,mi </m> for a station| | ||
- | |//cd//|standard deviation of <m> c_{i}, i=1,mi </m> for a station| | ||
- | |//rd//|standard deviation of <m> r_{i}, i=1,mi </m> for a station| | ||
- | |//cmax//|maximum value of <m> c_{i}, i=1,mi </m> for a station| | ||
- | |//rmax//|maximum value of <m> r_{i}, i=1,mi </m> for a station| | ||
- | |//cmin//|minimum value of <m> c_{i}, i=1,mi </m> for a station| | ||
- | |//rmin//|minimum value of <m> r_{i}, i=1,mi </m> for a station| | ||
- | |//w//|weight of station| | ||
- | </sortable> | ||
- | |||
- | ==== Basic equations ==== | ||
- | |||
- | Average value for a time series of a station: | ||
- | |||
- | <m> xm = {1/mi} sum{i=1}{mi}{x_{i}} </m> //x=r// or //c// | ||
- | |||
- | Standard deviation of a time series of a station: | ||
- | |||
- | <m> xd = sqrt{{1/mi} sum{i=1}{mi}{{x_{i}}^2}-xm^2} </m> //x=r// or //c// | ||
- | |||
- | Natural logaritm of value: | ||
- | |||
- | <m> xl = LN(x) </m> //x=r// or //c// or //rm// or //cm//, //x>0// | ||
- | |||
- | |||
- | ====Criteria equations for a time series of a station==== | ||
- | |||
- | Nash-Sutcliffe Efficiency (//NSE// or R2): | ||
- | |||
- | <m> NSE = 1-{sum{i=1}{mi}{(c_{i}-r_{i})^2}}/{sum{i=1}{mi}{(r_{i}-rm)^2}} </m> | ||
- | |||
- | Efficiency with coefficient a (//RA//): | ||
- | |||
- | <m> RA = 1-{sum{i=1}{mi}{delim{|} {c_{i}-r_{i}} {|}^a}}/{sum{i=1}{mi}{delim{|} {r_{i}-rm} {|}^a}} </m> | ||
- | |||
- | Bias: | ||
- | |||
- | <m> Bias = {sum{i=1}{mi}{(c_{i}-r_{i})}}/mi </m> | ||
- | |||
- | Relative bias (//RB// or RE): | ||
- | |||
- | <m> RB = {sum{i=1}{mi}{(c_{i}-r_{i})}}/{delim{|}{sum{i=1}{mi}{r_{i}}}{|}} </m> | ||
- | |||
- | Relative bias in percent (//RE%//): | ||
- | |||
- | <m> RE% = RB*100 = {{sum{i=1}{mi}{(c_{i}-r_{i})}}/{delim{|}{sum{i=1}{mi}{r_{i}}}{|}}}*100 </m> | ||
- | |||
- | Error of standard deviation (//ES//): | ||
- | |||
- | <m> ES = {cd-rd} </m> | ||
- | |||
- | Relative error of standard deviation (//RS//): | ||
- | |||
- | <m> RS = {{cd-rd}/rd} </m> | ||
- | |||
- | Relative error of standard deviation in percent (//RS%//): | ||
- | |||
- | <m> RS% = RS*100 = {{cd-rd}/rd}*100 </m> | ||
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- | Mean absolute error (//MAE//): | ||
- | |||
- | <m> MAE = {sum{i=1}{mi}{delim{|}{c_{i}-r_{i}}{|}}}/mi </m> | ||
- | |||
- | Kling-Gupta efficiency (//KGE//): | ||
- | |||
- | <m> KGE = 1-sqrt{(CC-1)^2+(cd/rd-1)^2+(cm/rm-1)^2} </m> //cm>0// and //rm>0// and //cd>0// and //rd>0// | ||
- | |||
- | Pearson correlation coefficient, Kling-Gupta efficiency part 1 (//CC//): | ||
- | |||
- | <m> CC = {{1/mi} sum{i=1}{mi}{({r_{i}}*{c_{i}})}-cm*rm}/{cd*rd} </m> | ||
- | |||
- | Kling-Gupta efficiency part 2 (//KGESD//): | ||
- | |||
- | <m> KGESD = cd/rd </m> | ||
- | |||
- | Kling-Gupta efficiency part 3 (//KGEM//): | ||
- | |||
- | <m> KGEM = cm/rm </m> | ||
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- | Root mean square error (//RMSE//): | ||
- | |||
- | <m> RMSE = sqrt{{1/mi sum{i=1}{mi}{({c_{i}}-{r_{i}})^2}}} </m> | ||
- | |||
- | Normalised root mean square error (//NE//): | ||
- | |||
- | <m> NE = sqrt{{1/mi sum{i=1}{mi}{({c_{i}}-{r_{i}})^2}}}/{max{(r_{i}})} </m> | ||
- | |||
- | Kendalls rank correlation coefficient, tau-b, with adjustments for ties (//TAU//): | ||
- | |||
- | <m> TAU = {n_{c}-n_{d}}/{sqrt{(n_{0}-n_{1})(n_{0}-n_{2})}} </m> | ||
- | |||
- | Nash-Sutcliffe Efficiency adjusted for bias (//NSEW//). Introduced in Lindström (2016): | ||
- | |||
- | <m> NSEW = NSE+Bias^2/rd^2 </m> | ||
- | |||
- | where | ||
- | |||
- | <m> n_{c} </m> = number of concordant pairs (<m> 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 </m>) | ||
- | |||
- | <m> n_{d} </m> = number of discordant pairs (<m> 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 </m>) | ||
- | |||
- | <m> n_{0} </m> = number of compared pairs | ||
- | |||
- | <m> n_{1} </m> = number of compared pairs that ties in the computed values | ||
- | |||
- | <m> n_{2} </m> = number of compared pairs that ties in the recorded values | ||
- | |||
- | Scaled bias (//ScBias//): | ||
- | |||
- | <m> ScBias = {sum{i=1}{mi}{delim{|}{{(c_{i}-r_{i})}/{(c_{i}+r_{i})}}{|}}}/mi </m> | ||
- | |||
- | Scaled KGE (//SCKGE//): | ||
- | |||
- | <m> SCKGE = KGE/{2-KGE} </m> | ||
- | |||
- | ====Criteria equations for a model domain (several stations)==== | ||
- | |||
- | Average Nash-Sutcliffe efficiency (//AVNSE//): | ||
- | |||
- | //AVNSE// arithmetric mean | ||
- | |||
- | <m> AVNSE = {1/mj sum{j=1}{mj}{NSE_{j}}} </m> | ||
- | |||
- | or //AVNSE// weighted average | ||
- | |||
- | <m> AVNSE = {sum{j=1}{mj}{w_{j}*NSE_{j}}}/{sum{j=1}{mj}{w_{j}}} </m> | ||
- | |||
- | Median Nash-Sutcliffe efficiency (//MEDNSE//): | ||
- | |||
- | <m> MEDNSE = median delim{lbrace}{{NSE_{j}},{j=1..mj}}{rbrace} </m> | ||
- | |||
- | Spatial Nash-Sutcliffe efficiency (//SPATNSE//): | ||
- | |||
- | <m> 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}} </m> | ||
- | |||
- | Regional Nash-Sutcliffe efficiency (//REGNSE//): | ||
- | |||
- | <m> 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}} </m> | ||
- | |||
- | Average Nash-Sutcliffe efficiency adjusted for bias (//AVNSEW//): | ||
- | |||
- | //AVNSEW// arithmetric mean | ||
- | |||
- | <m> AVNSEW = {1/mj sum{j=1}{mj}{NSEW_{j}}} </m> | ||
- | |||
- | or //AVNSEW// weighted average | ||
- | |||
- | <m> AVNSEW = {sum{j=1}{mj}{w_{j}*NSEW_{j}}}/{sum{j=1}{mj}{w_{j}}} </m> | ||
- | |||
- | Average efficiency with coefficient a (//AVRA//): | ||
- | |||
- | //AVRA// arithmetric mean | ||
- | |||
- | <m> AVRA = {1/mj sum{j=1}{mj}{RA_{j}}} </m> | ||
- | |||
- | or //AVRA// weighted average | ||
- | |||
- | <m> AVRA = {sum{j=1}{mj}{w_{j}*RA_{j}}}/{sum{j=1}{mj}{w_{j}}} </m> | ||
- | |||
- | Median efficiency with coefficient a (//MEDRA//): | ||
- | |||
- | <m> MEDRA = median delim{lbrace}{{RA_{j}},{j=1..mj}}{rbrace} </m> | ||
- | |||
- | Spatial efficiency with coefficient a (//SPATRA//): | ||
- | |||
- | <m> 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}} </m> | ||
- | |||
- | Regional efficiency with coefficient a (//REGRA//): | ||
- | |||
- | <m> 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}} </m> | ||
- | |||
- | Average relative bias (//AVRB//): | ||
- | |||
- | //AVRB// arithmetric mean | ||
- | |||
- | <m> AVRB = {1/mj sum{j=1}{mj}{RB_{j}}} </m> | ||
- | |||
- | or //AVRB// weighted average | ||
- | |||
- | <m> AVRB = {sum{j=1}{mj}{w_{j}*RB_{j}}}/{sum{j=1}{mj}{w_{j}}} </m> | ||
- | |||
- | Regional relative bias (//REGRB//): | ||
- | |||
- | <m> REGRB = {sum{ij=1}{mij}{(c_{ij}-r_{ij})}}/{delim{|}{sum{ij=1}{mij}{r_{ij}}}{|}} </m> | ||
- | |||
- | Spatial relative bias (//SPATRB//): | ||
- | |||
- | <m> SPATRB = {sum{j=1}{mj}{(cm_{j}-rm_{j})}}/{delim{|}{sum{j=1}{mj}{rm_{j}}}{|}} </m> | ||
- | |||
- | Average Kling-Gupta efficiency (//AVKGE//): | ||
- | |||
- | //AVKGE// arithmetric mean | ||
- | |||
- | <m> AVKGE = {1/mj sum{j=1}{mj}{KGE_{j}}} </m> | ||
- | |||
- | or //AVKGE// weighted average | ||
- | |||
- | <m> AVKGE = {sum{j=1}{mj}{w_{j}*KGE_{j}}}/{sum{j=1}{mj}{w_{j}}} </m> | ||
- | |||
- | Median Kling-Gupta efficiency (//MEDKGE//): | ||
- | |||
- | <m> MEDKGE = median delim{lbrace}{{KGE_{j}},{j=1..mj}}{rbrace} </m> | ||
- | |||
- | Average scaled Kling-Gupta efficiency (//ASCKGE//): | ||
- | |||
- | //ASCKGE// arithmetric mean | ||
- | |||
- | <m> ASCKGE = {1/mj sum{j=1}{mj}{SCKGE_{j}}} </m> | ||
- | |||
- | or //ASCKGE// weighted average | ||
- | |||
- | <m> ASCKGE = {sum{j=1}{mj}{w_{j}*SCKGE_{j}}}/{sum{j=1}{mj}{w_{j}}} </m> | ||
- | |||
- | Spatial root mean square error (//SPATRMSE//): | ||
- | |||
- | <m> SPATRMSE = sqrt{{1/mj sum{j=1}{mj}{({cm_{j}}-{rm_{j}})^2}}} </m> | ||
- | |||
- | Median of Normalised root mean square error (//MEDNE//): | ||
- | |||
- | <m> MEDNE = median delim{lbrace}{{NE_{j}},{j=1..mj}}{rbrace} </m> | ||
- | |||
- | Average of absolute relative bias (//AVARB//): | ||
- | |||
- | //AVARB// arithmetric mean | ||
- | |||
- | <m> AVARB = {1/mj sum{j=1}{mj}{delim{|}{RB_{j}}{|}}} </m> | ||
- | |||
- | or //AVARB// weighted average | ||
- | |||
- | <m> AVARB = {sum{j=1}{mj}{w_{j}*{delim{|}{RB_{j}}{|}}}}/{sum{j=1}{mj}{w_{j}}} </m> | ||
- | |||
- | Average Pearson correlation coefficient (//AVCC//): | ||
- | |||
- | //AVCC// arithmetric mean | ||
- | |||
- | <m> AVCC = {1/mj sum{j=1}{mj}{CC_{j}}} </m> | ||
- | |||
- | or //AVCC// weighted average | ||
- | |||
- | <m> AVCC = {sum{j=1}{mj}{w_{j}*CC_{j}}}/{sum{j=1}{mj}{w_{j}}} </m> | ||
- | |||
- | Average relative error of standard deviation (//AVRSB//): | ||
- | |||
- | //AVRSB// arithmetric mean | ||
- | |||
- | <m> AVRSB = {1/mj sum{j=1}{mj}{RS_{j}}} </m> | ||
- | |||
- | or //AVRSB// weighted average | ||
- | |||
- | <m> AVRSB = {sum{j=1}{mj}{w_{j}*RS_{j}}}/{sum{j=1}{mj}{w_{j}}} </m> | ||
- | |||
- | Average Kendalls rank correlation coefficient (//AVTAU//): | ||
- | |||
- | //AVTAU// arithmetric mean | ||
- | |||
- | <m> AVTAU = {1/mj sum{j=1}{mj}{TAU_{j}}} </m> | ||
- | |||
- | or //AVTAU// weighted average | ||
- | |||
- | <m> AVTAU = {sum{j=1}{mj}{w_{j}*TAU_{j}}}/{sum{j=1}{mj}{w_{j}}} </m> | ||
- | |||
- | Regional mean absolute error (//REGMAE//): | ||
- | |||
- | <m> REGMAE = {sum{ij=1}{mij}{delim{|}{c_{ij}-r_{ij}}{|}}}/mij </m> | ||
- | |||
- | Spatial mean absolute scaled bias on log transformed values (//SPATASB//): | ||
- | |||
- | <m> SPATASB = {sum{j=1}{mj}{delim{|}{{cml_{j}-rml_{j}}/{cml_{j}+rml_{j}}}{|}}}/{mj} </m> | ||
- | |||
- | |||
- | ==== References ==== | ||
- | 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. | ||
- | |||
- | Mathevet et al. 2006. A bounded version of the Nash-Sutcliffe criterion for better model assessment on large sets of basins. In: Large Sample Basin Experiments for Hydrological Model Parameterization: Results of the Model Parameter Experiment–MOPEX. IAHS Publ. 307, 2006, p. 211-219. | ||