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start:hype_file_reference:info.txt:criteria_equations [2023/10/05 11:09]
cpers [Basic equations]
start:hype_file_reference:info.txt:criteria_equations [2024/01/25 11:37]
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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> 
- 
-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> 
- 
-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. 
  
start/hype_file_reference/info.txt/criteria_equations.txt · Last modified: 2024/01/25 11:37 (external edit)