start:hype_file_reference:info.txt:criteria_equations
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start:hype_file_reference:info.txt:criteria_equations [2019/08/28 14:57] cpers [Criteria equations for a model domain (several stations)] |
start:hype_file_reference:info.txt:criteria_equations [2024/01/25 11:37] |
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| - | ====== Criteria equations ====== | ||
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| - | 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. | ||
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| - | ===== Code to equation coupling ===== | ||
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| - | Equation IDs for subbasin assessment criteria ([[start:HYPE_file_reference:subassX.txt|subassX.txt]]): | ||
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| - | <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> | ||
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| - | Equation IDs for simulation assessment criteria ([[start:HYPE_file_reference:simass.txt|simass.txt]]): | ||
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| - | <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//| | ||
| - | |Spatial NSE|''SR2''|//SPATNSE//| | ||
| - | |Spatial RA|''RRA''|//SPATRA//| | ||
| - | |Spatial RE|''-''|//SPATRB//| | ||
| - | |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> | ||
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| - | Equation IDs for calibration simulation assessment criteria ([[start:HYPE_file_reference:bestsims.txt|bestsims.txt]] and [[start:HYPE_file_reference:allsim.txt|allsim.txt]]): | ||
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| - | <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 efficiencies 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//| | ||
| - | |''mare''|average of absolute relative bias for subbasins (Note: fraction. not %) (MAR in [[start:hype_file_reference:info.txt|info.txt]])|//AVARB//| | ||
| - | |''mnr''|median of normalised RMSE for subbasins|//MEDNE//| | ||
| - | |''mnw''|average of Nash-Sutcliffe efficiencies adjusted for bias for subbasins|//AVNSEW//| | ||
| - | </sortable> | ||
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| - | Equation IDs for performance criteria set in info.txt are tabled [[start:HYPE_file_reference:info.txt:criteria|here]]. | ||
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| - | ===== Equation definitions ===== | ||
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| - | ==== Denotations ==== | ||
| - | <sortable> | ||
| - | |//c//|computed value| | ||
| - | |//r//|recorded value| | ||
| - | |//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| | ||
| - | </sortable> | ||
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| - | ==== Basic equations ==== | ||
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| - | Average value for a time series of a station: | ||
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| - | <m> xm = {1/mi} sum{i=1}{mi}{x_{i}} </m> //x=r// or //c// | ||
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| - | Standard deviation of a time series of a station: | ||
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| - | <m> xd = sqrt{{1/mi} sum{i=1}{mi}{{x_{i}}^2}-xm^2} </m> //x=r// or //c// | ||
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| - | ====Criteria equations for a time series of a station==== | ||
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| - | Nash-Sutcliffe Efficiency (//NSE// or R2): | ||
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| - | <m> NSE = 1-{sum{i=1}{mi}{(c_{i}-r_{i})^2}}/{sum{i=1}{mi}{(r_{i}-rm)^2}} </m> | ||
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| - | Efficiency with coefficient a (//RA//): | ||
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| - | <m> RA = 1-{sum{i=1}{mi}{delim{|} {c_{i}-r_{i}} {|}^a}}/{sum{i=1}{mi}{delim{|} {r_{i}-rm} {|}^a}} </m> | ||
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| - | Bias: | ||
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| - | <m> Bias = {sum{i=1}{mi}{(c_{i}-r_{i})}}/mi </m> | ||
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| - | Relative bias (//RB// or RE): | ||
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| - | <m> RB = {sum{i=1}{mi}{(c_{i}-r_{i})}}/{delim{|}{sum{i=1}{mi}{r_{i}}}{|}} </m> | ||
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| - | Relative bias in percent (//RE%//): | ||
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| - | <m> RE% = RB*100 = {{sum{i=1}{mi}{(c_{i}-r_{i})}}/{delim{|}{sum{i=1}{mi}{r_{i}}}{|}}}*100 </m> | ||
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| - | Error of standard deviation (//ES//): | ||
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| - | <m> ES = {cd-rd} </m> | ||
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| - | Relative error of standard deviation (//RS//): | ||
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| - | <m> RS = {{cd-rd}/rd} </m> | ||
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| - | Relative error of standard deviation in percent (//RS%//): | ||
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| - | <m> RS% = RS*100 = {{cd-rd}/rd}*100 </m> | ||
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| - | Mean absolute error (//MAE//): | ||
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| - | <m> MAE = {sum{i=1}{mi}{delim{|}{c_{i}-r_{i}}{|}}}/mi </m> | ||
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| - | Kling-Gupta efficiency (//KGE//): | ||
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| - | <m> KGE = 1-sqrt{(CC-1)^2+(cd/rd-1)^2+(cm/rm-1)^2} </m> | ||
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| - | Pearson correlation coefficient, Kling-Gupta efficiency part 1 (//CC//): | ||
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| - | <m> CC = {{1/mi} sum{i=1}{mi}{({r_{i}}*{c_{i}})}-cm*rm}/{cd*rd} </m> | ||
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| - | Kling-Gupta efficiency part 2 (//KGESD//): | ||
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| - | <m> KGESD = cd/rd </m> | ||
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| - | Kling-Gupta efficiency part 3 (//KGEM//): | ||
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| - | <m> KGEM = cm/rm </m> | ||
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| - | Root mean square error (//RMSE//): | ||
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| - | <m> RMSE = sqrt{{1/mi sum{i=1}{mi}{({c_{i}}-{r_{i}})^2}}} </m> | ||
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| - | Normalised root mean square error (//NE//): | ||
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| - | <m> NE = sqrt{{1/mi sum{i=1}{mi}{({c_{i}}-{r_{i}})^2}}}/{max{(r_{i}})} </m> | ||
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| - | Kendalls rank correlation coefficient, tau-b, with adjustments for ties (//TAU//): | ||
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| - | <m> TAU = {n_{c}-n_{d}}/{sqrt{(n_{0}-n_{1})(n_{0}-n_{2})}} </m> | ||
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| - | Nash-Sutcliffe Efficiency adjusted for bias (//NSEW//). Introduced in Lindström (2016): | ||
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| - | <m> NSEW = NSE-Bias^2/rd^2 </m> | ||
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| - | where | ||
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| - | <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>) | ||
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| - | <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>) | ||
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| - | <m> n_{0} </m> = number of compared pairs | ||
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| - | <m> n_{1} </m> = number of compared pairs that ties in the computed values | ||
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| - | <m> n_{2} </m> = number of compared pairs that ties in the recorded values | ||
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| - | ====Criteria equations for a model domain (several stations)==== | ||
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| - | Average Nash-Sutcliffe efficiency (//AVNSE//): | ||
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| - | <m> AVNSE = {1/mj sum{j=1}{mj}{NSE_{j}}} </m> | ||
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| - | Median Nash-Sutcliffe efficiency (//MEDNSE//): | ||
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| - | <m> MEDNSE = median delim{lbrace}{{NSE_{j}},{j=1..mj}}{rbrace} </m> | ||
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| - | Spatial Nash-Sutcliffe efficiency (//SPATNSE//): | ||
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| - | <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> | ||
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| - | Regional Nash-Sutcliffe efficiency (//REGNSE//): | ||
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| - | <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> | ||
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| - | Average efficiency with coefficient a (//AVRA//): | ||
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| - | <m> AVRA = {1/mj sum{j=1}{mj}{RA_{j}}} </m> | ||
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| - | Median efficiency with coefficient a (//MEDRA//): | ||
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| - | <m> MEDRA = median delim{lbrace}{{RA_{j}},{j=1..mj}}{rbrace} </m> | ||
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| - | Spatial efficiency with coefficient a (//SPATRA//): | ||
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| - | <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> | ||
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| - | Regional efficiency with coefficient a (//REGRA//): | ||
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| - | <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> | ||
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| - | Average relative bias (//AVRB//): | ||
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| - | <m> AVRB = {1/mj sum{j=1}{mj}{RB_{j}}} </m> | ||
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| - | Regional relative bias (//REGRB//): | ||
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| - | <m> REGRB = {sum{ij=1}{mij}{(c_{ij}-r_{ij})}}/{delim{|}{sum{ij=1}{mij}{r_{ij}}}{|}} </m> | ||
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| - | Spatial relative bias (//SPATRB//): | ||
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| - | <m> SPATRB = {sum{j=1}{mj}{(cm_{j}-rm_{j})}}/{delim{|}{sum{j=1}{mj}{rm_{j}}}{|}} </m> | ||
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| - | Average Kling-Gupta efficiency (//AVKGE//): | ||
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| - | <m> AVKGE = {1/mj sum{j=1}{mj}{KGE_{j}}} </m> | ||
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| - | Median Kling-Gupta efficiency (//MEDKGE//): | ||
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| - | <m> MEDKGE = median delim{lbrace}{{KGE_{j}},{j=1..mj}}{rbrace} </m> | ||
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| - | Median of Normalised root mean square error (//MEDNE//): | ||
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| - | <m> MEDNE = median delim{lbrace}{{NE_{j}},{j=1..mj}}{rbrace} </m> | ||
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| - | Average of absolute relative bias (//AVARB//): | ||
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| - | <m> AVARB = {1/mj sum{j=1}{mj}{delim{|}{RB_{j}}{|}}} </m> | ||
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| - | Average Pearson correlation coefficient (//AVCC//): | ||
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| - | <m> AVCC = {1/mj sum{j=1}{mj}{CC_{j}}} </m> | ||
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| - | Average relative error of standard deviation (//AVRSB//): | ||
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| - | <m> AVRSB = {1/mj sum{j=1}{mj}{RS_{j}}} </m> | ||
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| - | Average Kendalls rank correlation coefficient (//AVTAU//): | ||
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| - | <m> AVTAU = {1/mj sum{j=1}{mj}{TAU_{j}}} </m> | ||
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| - | Regional mean absolute error (//REGMAE//): | ||
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| - | <m> REGMAE = {sum{ij=1}{mij}{delim{|}{c_{ij}-r_{ij}}{|}}}/mij </m> | ||
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| - | Average Nash-Sutcliffe efficiency adjusted for bias (//AVNSEW//): | ||
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| - | <m> AVNSEW = {1/mj sum{j=1}{mj}{NSEW_{j}}} </m> | ||
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| - | ==== 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. | ||
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start/hype_file_reference/info.txt/criteria_equations.txt · Last modified: 2024/01/25 11:37 (external edit)
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