Parameter values to empirical cumulative distribution functions
params2ecdf.RdComputes empirical cumulative distribution functions (ECDFs) for model parameters from an ensemble of calibrated or behavioural parameter sets, with optional weighting and plotting.
This function is especially useful in hydrology when analysing the parameter space sampled during calibration or uncertainty analysis with hydroPSO. The resulting ECDFs help identify well-constrained parameters, parameter ranges that are consistently supported by behavioural simulations, and the extent to which prior expectations or reference values agree with the calibrated ensemble.
Usage
params2ecdf(params, ...)
# Default S3 method
params2ecdf(params, param.names=NULL, gofs=NULL, MinMax=NULL,
beh.thr=NA, weights=NULL, byrow=FALSE, plot=TRUE, obs=NULL, main=NULL,
nrows="auto", ylab="Probability", col="blue", leg.cex=1.2,
leg.pos="topleft", cex.axis=1.2, cex.main=1.2, cex.lab=1.2,
verbose=TRUE, ..., do.png=FALSE, png.width=1500, png.height=900,
png.res=90, png.fname="Params_ECDFs.png")
# S3 method for class 'matrix'
params2ecdf(params, param.names=colnames(params), gofs=NULL,
MinMax=NULL, beh.thr=NA, weights=NULL, byrow=FALSE, plot=TRUE,
obs=NULL, main=NULL, nrows="auto", ylab="Probability", col="blue",
leg.cex=1.2, leg.pos="topleft", cex.axis=1.2, cex.main=1.2,
cex.lab=1.2, verbose=TRUE, ..., do.png=FALSE, png.width=1500,
png.height=900, png.res=90, png.fname="Params_ECDFs.png")
# S3 method for class 'data.frame'
params2ecdf(params, param.names=colnames(params), gofs=NULL,
MinMax=NULL, beh.thr=NA, weights=NULL, byrow=FALSE, plot=TRUE,
obs=NULL, main=NULL, nrows="auto", ylab="Probability", col="blue",
leg.cex=1.2, leg.pos="topleft", cex.axis=1.2, cex.main=1.2,
cex.lab=1.2, verbose=TRUE, ..., do.png=FALSE, png.width=1500,
png.height=900, png.res=90, png.fname="Params_ECDFs.png")Arguments
- params
A matrix or data frame with parameter values from multiple model runs. Each row should contain one parameter set, and each column should contain the sampled values of one model parameter.
- param.names
Character vector with the names to be used for the parameters in
params. By default, the column names ofparamsare used.- gofs
Optional numeric vector with one goodness-of-fit value for each parameter set in
params. This is required whenbeh.thris used to retain only behavioural parameter sets.- MinMax
Optional character string indicating whether better model performance corresponds to minimising or maximising the objective function in
gofs. This argument is only used together withbeh.thrto select behavioural parameter sets.Valid values are
"min"and"max".- beh.thr
Optional numeric threshold used to retain only behavioural parameter sets. If
MinMax="max", parameter sets withgofs >= beh.thrare retained. IfMinMax="min", parameter sets withgofs <= beh.thrare retained.The default
beh.thr=NAmeans that all parameter sets are used.- weights
Optional numeric vector with one weight for each row of
params. These weights are passed towtd.Ecdfto compute weighted ECDFs.Omitting
weights, or providingNULLor a zero-length vector, yields the usual unweighted ECDFs.- byrow
Logical value kept for interface compatibility. In the current implementation, the calculations are performed column-wise, so
paramsshould be organised with parameter sets in rows and parameters in columns.- plot
Logical value indicating whether a plot of the ECDF for each parameter should be produced.
- obs
Optional numeric or
zooobject with one reference value for each parameter. Whenplot=TRUE, these values are drawn as vertical reference lines to facilitate visual comparison between observed, target, or preferred parameter values and the parameter ensemble.- main
Optional overall title for the plot.
- nrows
Optional number of rows in the plotting layout when
plot=TRUE. Ifnrows="auto", the layout is chosen automatically from the number of parameters.- ylab
Optional label for the y axis. See
plot.- col
Optional plotting colour used for the ECDF curves. See
par.- leg.cex
Optional character expansion factor for legends when
plot=TRUE.- leg.pos
Optional keyword controlling the legend position. See
legend.- cex.axis
Optional magnification for axis annotation relative to the current
cexsetting.- cex.main
Optional magnification for main titles relative to the current
cexsetting.- cex.lab
Optional magnification for x and y labels relative to the current
cexsetting.- verbose
Logical value. If
TRUE, progress messages are printed.- ...
Further arguments passed to
plotor to other methods.- do.png
Logical value indicating whether the figures should be written to PNG files instead of the active graphics device.
- png.width
Optional width of the PNG device in pixels when
do.png=TRUE. Seepng.- png.height
Optional height of the PNG device in pixels when
do.png=TRUE. Seepng.- png.res
Optional nominal resolution in ppi for the PNG device when
do.png=TRUE. Seepng.- png.fname
Optional file name used to store the PNG output when
do.png=TRUE.
Value
A list with one element per parameter, where the number of parameters is the number of columns in params.
Each list element is the object returned by wtd.Ecdf for the corresponding parameter. These objects contain the information required to inspect, plot, or interpolate the empirical cumulative distribution of each parameter.
If plot=TRUE, one or more figures are also produced. At most 21 parameters are shown per figure, so the number of figures is ceiling(nparam/21).
References
In hydrological calibration and uncertainty analysis, empirical cumulative distributions are useful for diagnosing parameter identifiability, comparing prior and posterior parameter support, and summarising behavioural ensembles.
Author
Mauricio Zambrano-Bigiarini, mzb.devel@gmail.com
Examples
# Example 1: ECDFs of calibrated parameters from an ensemble of behavioural runs
set.seed(123)
params <- matrix(runif(600), nrow=100, ncol=6)
colnames(params) <- c("Kq", "Ks", "Alpha", "Beta", "Kperc", "Smax")
# Unweighted ECDFs for all parameters
pe <- params2ecdf(params, plot=FALSE)
#> [ Computing the ECDF for 'Kq ' , 1/6 => 16.67% ]
#> [ Computing the ECDF for 'Ks ' , 2/6 => 33.33% ]
#> [ Computing the ECDF for 'Alpha' , 3/6 => 50.00% ]
#> [ Computing the ECDF for 'Beta ' , 4/6 => 66.67% ]
#> [ Computing the ECDF for 'Kperc' , 5/6 => 83.33% ]
#> [ Computing the ECDF for 'Smax ' , 6/6 => 100.00% ]
str(pe[[1]])
#> List of 2
#> $ x : num [1:101] 0.000625 0.000625 0.024614 0.04206 0.045556 ...
#> $ ecdf: Named num [1:101] 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 ...
#> ..- attr(*, "names")= chr [1:101] "" "0.000624773325398564" "0.0246136845089495" "0.0420595335308462" ...
# \donttest{
# Example 2: Weighted ECDFs using likelihood-based weights
# Useful when some parameter sets are more plausible than others
w <- rexp(100)
w <- w / sum(w)
params2ecdf(params, weights=w, plot=TRUE,
main="Weighted ECDFs of behavioural parameter sets")
#> [ Computing the weighted ECDF for 'Kq ' , 1/6 => 16.67% ]
#> [ Computing the weighted ECDF for 'Ks ' , 2/6 => 33.33% ]
#> [ Computing the weighted ECDF for 'Alpha' , 3/6 => 50.00% ]
#> [ Computing the weighted ECDF for 'Beta ' , 4/6 => 66.67% ]
#> [ Computing the weighted ECDF for 'Kperc' , 5/6 => 83.33% ]
#> [ Computing the weighted ECDF for 'Smax ' , 6/6 => 100.00% ]
#> $Kq
#> $Kq$x
#> [1] 0.0006247733 0.0006247733 0.0246136845 0.0420595335 0.0455564994
#> [6] 0.0458311667 0.0935949867 0.0948406609 0.1028646443 0.1029246827
#> [11] 0.1111354243 0.1218992600 0.1275316502 0.1306956916 0.1388060634
#> [16] 0.1428000224 0.1471136473 0.1524447477 0.1750526503 0.1876911193
#> [21] 0.2065313896 0.2164079358 0.2201188852 0.2316257854 0.2330340995
#> [26] 0.2436194727 0.2460877344 0.2659726404 0.2743836446 0.2875775201
#> [31] 0.2891597373 0.3181810076 0.3203732425 0.3279207193 0.3435164723
#> [36] 0.3517979092 0.3688454509 0.3744627759 0.3798165377 0.3839696378
#> [41] 0.4089769218 0.4137243263 0.4145463358 0.4176467797 0.4348927415
#> [46] 0.4398316876 0.4422000742 0.4485163414 0.4533341562 0.4566147353
#> [51] 0.4659624503 0.4667790416 0.4753165741 0.4777959711 0.5115054599
#> [56] 0.5281054880 0.5440660247 0.5514350145 0.5609479838 0.5726334020
#> [61] 0.5941420204 0.6127710033 0.6292211316 0.6405068138 0.6531019250
#> [66] 0.6557057991 0.6567581280 0.6651151946 0.6680555874 0.6775706355
#> [71] 0.6907052784 0.6928034062 0.7085304682 0.7101824014 0.7533078643
#> [76] 0.7544751586 0.7584595375 0.7822943013 0.7881958340 0.7883051354
#> [81] 0.7943423211 0.7954674177 0.7989248456 0.8100643530 0.8123895095
#> [86] 0.8146400389 0.8578277153 0.8830174040 0.8864690608 0.8895393161
#> [91] 0.8924190444 0.8930511144 0.8950453592 0.8998249704 0.9022990451
#> [96] 0.9404672843 0.9545036491 0.9568333453 0.9630242325 0.9849569800
#> [101] 0.9942697766
#>
#> $Kq$ecdf
#> [1] 0.00000000 0.01157799 0.01646310 0.02757627 0.03380921 0.04708646
#> [7] 0.04875591 0.05135944 0.05569259 0.06101180 0.06173396 0.08106969
#> [13] 0.08746690 0.11983945 0.13697513 0.14364919 0.14607779 0.14712983
#> [19] 0.16232134 0.18339636 0.18754413 0.18792153 0.19810160 0.20867914
#> [25] 0.24489735 0.24896624 0.27909644 0.30285151 0.31740576 0.33230472
#> [31] 0.33290931 0.33296548 0.33867125 0.34628241 0.35573139 0.36244975
#> [37] 0.38445557 0.40137344 0.40445635 0.40522043 0.42239212 0.44404805
#> [43] 0.44996653 0.48790332 0.50493914 0.51202424 0.51370993 0.51531358
#> [49] 0.51922229 0.54457746 0.54498911 0.56103686 0.57912706 0.58231159
#> [55] 0.60380341 0.61362588 0.61543828 0.61794417 0.62763761 0.62793312
#> [61] 0.63370458 0.63813445 0.64037408 0.64776446 0.64964474 0.65580502
#> [67] 0.65634444 0.66330443 0.66740619 0.67232865 0.68097374 0.68763386
#> [73] 0.70080844 0.70170511 0.70539078 0.74350669 0.74807676 0.75184338
#> [79] 0.76498464 0.77432415 0.78339921 0.78561186 0.79153534 0.80801498
#> [85] 0.83331876 0.84131467 0.84928319 0.85510498 0.85511296 0.86011789
#> [91] 0.87160799 0.87748438 0.87791577 0.88950813 0.91312422 0.92815579
#> [97] 0.93019388 0.96759856 0.97234276 0.98157868 1.00000000
#>
#>
#> $Ks
#> $Ks$x
#> [1] 0.01046711 0.01046711 0.06072057 0.07669117 0.09104400 0.11109646
#> [7] 0.14190691 0.14229430 0.14709469 0.15234662 0.15420230 0.18382849
#> [13] 0.18384952 0.21689276 0.21799067 0.21976763 0.22110294 0.23116178
#> [19] 0.23909996 0.24572368 0.24644899 0.26501781 0.26503273 0.26568669
#> [25] 0.27416662 0.30122890 0.30772001 0.31170220 0.31244816 0.31982062
#> [31] 0.32148276 0.33282354 0.35390457 0.35544538 0.36948887 0.37223806
#> [37] 0.37471396 0.38790903 0.38999444 0.40451028 0.40683259 0.40947495
#> [43] 0.41068978 0.41274612 0.43943154 0.44476800 0.46653270 0.48128980
#> [49] 0.48290240 0.48861303 0.49752727 0.50229956 0.52113573 0.52983569
#> [55] 0.53368795 0.54928466 0.56459043 0.57193531 0.57286706 0.58175010
#> [61] 0.58548335 0.59434319 0.59998896 0.60873498 0.61801787 0.61925648
#> [67] 0.61999331 0.62997305 0.64789348 0.64998516 0.65923032 0.65983845
#> [73] 0.66828465 0.67299909 0.69000710 0.70829032 0.72059627 0.73213521
#> [79] 0.73707774 0.74033436 0.74656800 0.78628155 0.82180546 0.83976776
#> [85] 0.84272932 0.84745317 0.86364411 0.87468234 0.89035022 0.89139412
#> [91] 0.90187439 0.91318822 0.91443819 0.93529980 0.93731409 0.94772694
#> [97] 0.95409124 0.95447383 0.97982192 0.98421920 0.98564088
#>
#> $Ks$ecdf
#> [1] 0.00000000 0.02165593 0.02657839 0.02699004 0.03276151 0.04245495
#> [7] 0.04305954 0.05465189 0.08000706 0.09605481 0.09786720 0.10010683
#> [13] 0.12211266 0.12579832 0.14271618 0.16113751 0.18644128 0.20357696
#> [19] 0.23979518 0.26355025 0.28288598 0.29602724 0.30526315 0.31234825
#> [25] 0.34472081 0.34862952 0.35601990 0.36269395 0.36679571 0.37345583
#> [31] 0.37533611 0.38467562 0.38727915 0.39527506 0.40143535 0.40451827
#> [37] 0.41907251 0.42499599 0.42914376 0.43118184 0.43494846 0.44086693
#> [43] 0.44337283 0.45244789 0.46302543 0.46345682 0.48453184 0.50156766
#> [49] 0.51659923 0.53377092 0.53545661 0.54241659 0.54730171 0.55173157
#> [55] 0.55333522 0.58346543 0.58934181 0.59573902 0.61723083 0.61795299
#> [61] 0.62556416 0.62989731 0.64479628 0.65628638 0.66646645 0.67121065
#> [67] 0.67175007 0.70986599 0.71487091 0.71563499 0.71730444 0.72048898
#> [73] 0.73857918 0.74722427 0.74965287 0.78758967 0.79290888 0.80087739
#> [79] 0.80309004 0.81956968 0.83114767 0.83152507 0.83609515 0.84016404
#> [85] 0.84121607 0.85449332 0.85538999 0.86210835 0.86834130 0.89195740
#> [91] 0.90714891 0.90715689 0.91697937 0.95438405 0.96008981 0.96038532
#> [97] 0.97149848 0.97732028 0.97737645 0.99055103 1.00000000
#>
#>
#> $Alpha
#> $Alpha$x
#> [1] 0.006300784 0.006300784 0.020024301 0.047663627 0.052843943 0.054629109
#> [7] 0.057958561 0.058478489 0.064928300 0.071904097 0.072057124 0.074384513
#> [13] 0.099041829 0.100721582 0.103599645 0.137106081 0.152887223 0.164211225
#> [19] 0.168060811 0.170645235 0.171698494 0.172171746 0.197744737 0.216254790
#> [25] 0.221787935 0.224985329 0.225886433 0.238726027 0.252964929 0.257216746
#> [31] 0.260856857 0.267521432 0.282528330 0.288239281 0.297741783 0.311869747
#> [37] 0.336331206 0.347324631 0.351888638 0.364091865 0.395220135 0.395892688
#> [43] 0.396584595 0.397151953 0.398939825 0.402573342 0.404399181 0.408943998
#> [49] 0.428421509 0.466472377 0.471576278 0.477845380 0.482042606 0.502813046
#> [55] 0.515029727 0.516444894 0.531070399 0.542080367 0.546826157 0.560253264
#> [61] 0.576519021 0.593045652 0.601365726 0.618351227 0.633055360 0.648818124
#> [67] 0.662317642 0.670282040 0.674186843 0.674376388 0.686375082 0.698261595
#> [73] 0.700853087 0.724554346 0.726702539 0.728394428 0.735184306 0.754247402
#> [79] 0.758593170 0.770334074 0.785291671 0.798831611 0.803418542 0.816605888
#> [85] 0.820951324 0.831927563 0.868106807 0.871043414 0.880246541 0.880440569
#> [91] 0.881977559 0.915683538 0.918857348 0.925707956 0.950166979 0.961104794
#> [97] 0.962358936 0.967398371 0.969356411 0.971875636 0.982140374
#>
#> $Alpha$ecdf
#> [1] 0.000000000 0.002239633 0.011314699 0.016633906 0.022794194 0.022802170
#> [7] 0.027135319 0.029347966 0.038583880 0.048032851 0.048929525 0.055647890
#> [13] 0.071695637 0.073575917 0.075245371 0.079347137 0.079403307 0.090981295
#> [19] 0.097378499 0.099884395 0.121540326 0.146895496 0.151465569 0.156388026
#> [25] 0.158073711 0.171214972 0.177091362 0.191990331 0.195899045 0.209176289
#> [31] 0.214061406 0.233397138 0.238402064 0.249892170 0.282264722 0.283316757
#> [37] 0.299796396 0.325100170 0.355230373 0.365052852 0.378227432 0.395263251
#> [43] 0.433200049 0.436384582 0.472602794 0.487634364 0.491320029 0.502433192
#> [49] 0.526049289 0.530479154 0.530910546 0.532722938 0.570127617 0.577212717
#> [55] 0.583034512 0.591030430 0.600723871 0.609368964 0.616043016 0.621814479
#> [61] 0.623418127 0.629341607 0.646513297 0.651257502 0.673263326 0.673985488
#> [67] 0.679903961 0.695095476 0.695859553 0.696155060 0.714576384 0.715180974
#> [73] 0.726773328 0.743909012 0.751877527 0.759267902 0.769447971 0.775153736
#> [79] 0.779222624 0.797312824 0.801460587 0.822952397 0.833529934 0.854604955
#> [85] 0.862216121 0.862593527 0.879511389 0.917627302 0.923860248 0.924399667
#> [91] 0.927003197 0.929431803 0.931469887 0.938429871 0.952984115 0.959644232
#> [97] 0.968983737 0.992738808 0.993150463 0.996233381 1.000000000
#>
#>
#> $Beta
#> $Beta$x
#> [1] 0.008215520 0.008215520 0.009429905 0.017765812 0.033727773 0.034874804
#> [7] 0.055301926 0.094962413 0.101058494 0.103806688 0.105417746 0.117913841
#> [13] 0.119404780 0.136540197 0.143817044 0.147048325 0.156636851 0.163070329
#> [19] 0.185296442 0.192815947 0.193378680 0.208876349 0.210487079 0.225073351
#> [25] 0.229244840 0.237749405 0.239594629 0.270544872 0.282218417 0.286285349
#> [31] 0.295895455 0.296302175 0.308119554 0.308786833 0.314270780 0.323344993
#> [37] 0.338391284 0.342808802 0.353986103 0.354783098 0.363300544 0.370214798
#> [43] 0.372709394 0.388319115 0.389587112 0.406607866 0.414735334 0.418915897
#> [49] 0.421844950 0.452458394 0.455108051 0.464665942 0.480910830 0.483167669
#> [55] 0.486411764 0.492293329 0.492566548 0.515071808 0.526029660 0.533764874
#> [61] 0.537677183 0.584900093 0.617235270 0.621902295 0.623297546 0.630131853
#> [67] 0.641462354 0.643922915 0.668771517 0.697373766 0.703992061 0.713279001
#> [73] 0.725983027 0.729390652 0.737797403 0.760823625 0.769193911 0.774591542
#> [79] 0.779065883 0.783946479 0.784575267 0.785687834 0.802812273 0.834054309
#> [85] 0.835255320 0.835708837 0.853763374 0.856885420 0.866483315 0.896738683
#> [91] 0.913995450 0.915063762 0.963843332 0.967469494 0.971342450 0.977098996
#> [97] 0.977853405 0.983350181 0.985954165 0.991236556 0.999404528
#>
#> $Beta$ecdf
#> [1] 0.00000000 0.01149011 0.02082961 0.04613338 0.05930796 0.06507943
#> [7] 0.06537493 0.06538291 0.06705236 0.07321265 0.07891842 0.09496616
#> [13] 0.10088464 0.11416188 0.12112186 0.12851224 0.13833472 0.15767045
#> [19] 0.16690637 0.16950989 0.18598953 0.19041940 0.20561091 0.22761674
#> [25] 0.25137181 0.26248497 0.28355999 0.28732661 0.29320300 0.31681910
#> [31] 0.32347922 0.32535950 0.33991374 0.34663211 0.35151722 0.35194861
#> [37] 0.35255320 0.35479284 0.36240400 0.36772321 0.37571913 0.39285481
#> [43] 0.43079161 0.43120326 0.46860794 0.47768301 0.48435706 0.48804273
#> [49] 0.52615864 0.52866454 0.54024252 0.54415124 0.55038418 0.55746928
#> [55] 0.55852132 0.58387649 0.58706102 0.59298450 0.61464043 0.63273063
#> [61] 0.63706378 0.64113267 0.64587687 0.65557031 0.66353883 0.67857040
#> [67] 0.67894780 0.67900398 0.68540118 0.68997125 0.70839258 0.71331503
#> [73] 0.72276400 0.72858580 0.73723089 0.74133266 0.75447392 0.75755684
#> [79] 0.77472853 0.77633218 0.79123114 0.79177056 0.80336292 0.80557557
#> [85] 0.82249343 0.85262363 0.85762856 0.85966664 0.86056332 0.86132739
#> [91] 0.87836321 0.88079182 0.89097189 0.89265757 0.89337973 0.92575229
#> [97] 0.93632982 0.97254803 0.97669580 0.99818761 1.00000000
#>
#>
#> $Kperc
#> $Kperc$x
#> [1] 0.0004653491 0.0004653491 0.0279925719 0.0495384356 0.0581917963
#> [6] 0.0704889777 0.0825027458 0.0869401679 0.0939166071 0.0994458233
#> [11] 0.1081785294 0.1157796648 0.1247420886 0.1277056879 0.1370674714
#> [16] 0.1470841086 0.1549117258 0.1604750925 0.1611657627 0.1652980812
#> [21] 0.1655209111 0.1920233187 0.2151664882 0.2151720827 0.2172435140
#> [26] 0.2300969204 0.2348196758 0.2366197421 0.2405307016 0.2530989940
#> [31] 0.2707780194 0.2760496726 0.2828523996 0.2829928708 0.3029049153
#> [36] 0.3077638743 0.3169492481 0.3217255378 0.3219373757 0.3295474134
#> [41] 0.3393125802 0.3498017928 0.3580569634 0.3882046638 0.3954488591
#> [46] 0.4274282807 0.4289485959 0.4361299961 0.4445273969 0.4498024841
#> [51] 0.4597851932 0.4640166268 0.4784563838 0.4870325699 0.4979489362
#> [56] 0.5238226017 0.5474594680 0.5636446835 0.5763018376 0.5838723360
#> [61] 0.5849365573 0.5962635446 0.6126719653 0.6262569474 0.6271461844
#> [66] 0.6442402205 0.6623855066 0.6733324814 0.6782238574 0.6807875512
#> [71] 0.6863249468 0.6877426857 0.6912078315 0.7015851184 0.7048722457
#> [76] 0.7065019011 0.7085741372 0.7317076589 0.7475163222 0.7558872597
#> [81] 0.7603995362 0.7869273527 0.8103980662 0.8118274161 0.8139337213
#> [86] 0.8220300712 0.8315685980 0.8344009779 0.8625518975 0.8664677632
#> [91] 0.8900779132 0.8911143125 0.9053095817 0.9326810790 0.9506212641
#> [96] 0.9531012138 0.9714380668 0.9747258085 0.9860542973 0.9929440161
#> [101] 0.9966172187
#>
#> $Kperc$ecdf
#> [1] 0.000000000 0.004429865 0.006467949 0.008137403 0.027473135 0.029353415
#> [7] 0.040843521 0.046549286 0.084665199 0.084721370 0.089291442 0.106327262
#> [13] 0.123245124 0.132481037 0.141820542 0.142232197 0.142663589 0.143268178
#> [19] 0.153845714 0.168399958 0.190405782 0.201998136 0.206067024 0.210989482
#> [25] 0.217707847 0.234187486 0.239072602 0.248766043 0.254689523 0.258456143
#> [31] 0.260059791 0.265378998 0.274454065 0.281128117 0.282940509 0.320877306
#> [37] 0.358281986 0.369395149 0.387816473 0.396461565 0.398967461 0.400653146
#> [43] 0.424408217 0.430179679 0.445211249 0.446107923 0.446115900 0.448719430
#> [49] 0.458899498 0.465132444 0.474581415 0.475345493 0.482956658 0.483334064
#> [55] 0.483629571 0.496906816 0.501911742 0.525527840 0.531349634 0.537268107
#> [61] 0.545264025 0.552654401 0.555838934 0.569013513 0.572096431 0.578756549
#> [67] 0.596846749 0.604815264 0.610691653 0.636046822 0.652094570 0.665235831
#> [73] 0.690539605 0.691079024 0.691801187 0.701623666 0.718759350 0.740415281
#> [79] 0.744100947 0.755678935 0.791897147 0.813388958 0.820474057 0.826871261
#> [85] 0.830973026 0.833212659 0.837121373 0.852312888 0.854741493 0.855793528
#> [91] 0.859941291 0.866101578 0.883273269 0.887606418 0.917736621 0.922480826
#> [97] 0.954853378 0.961813362 0.976712331 0.997787353 1.000000000
#>
#>
#> $Smax
#> $Smax$x
#> [1] 0.01616048 0.01616048 0.01836408 0.02552670 0.03197163 0.03736996
#> [7] 0.03949920 0.04770998 0.06078222 0.06265000 0.07997291 0.08146656
#> [13] 0.08253118 0.08836330 0.09164469 0.12023713 0.15133817 0.17174341
#> [19] 0.17801381 0.20108062 0.20493870 0.27454536 0.27685507 0.28181167
#> [25] 0.28710013 0.30114252 0.30288762 0.31211279 0.32759740 0.33749500
#> [31] 0.34132128 0.34871949 0.35360608 0.36467103 0.36545427 0.36644144
#> [37] 0.36789595 0.38517360 0.39525376 0.39566064 0.41330541 0.42346365
#> [43] 0.44703699 0.45476163 0.46471377 0.49006345 0.50394871 0.50837063
#> [49] 0.50984290 0.51880492 0.52659008 0.52971409 0.52989246 0.53554139
#> [55] 0.53605372 0.54702342 0.55331399 0.56673408 0.56938266 0.58273974
#> [61] 0.58695866 0.58746137 0.65026150 0.65376668 0.66573514 0.66670516
#> [67] 0.67901342 0.68760758 0.70903790 0.73589935 0.73903169 0.75311010
#> [73] 0.76928554 0.77020476 0.80094821 0.80223822 0.81230483 0.81285269
#> [79] 0.81508150 0.81647814 0.82925082 0.83210777 0.85409998 0.85436475
#> [85] 0.86010685 0.87394991 0.87867402 0.88076411 0.88513265 0.88805519
#> [91] 0.89433463 0.90323356 0.90604811 0.92935068 0.93913840 0.94503510
#> [97] 0.94958532 0.96415257 0.96903886 0.97738358 0.98907827
#>
#> $Smax$ecdf
#> [1] 0.00000000 0.01691786 0.02134773 0.02711919 0.04861100 0.06703232
#> [7] 0.07191744 0.07887742 0.08075770 0.08836887 0.09419066 0.13040888
#> [13] 0.13431759 0.14002335 0.14594183 0.14648125 0.16351707 0.17510942
#> [19] 0.18134237 0.19589661 0.19758230 0.20815983 0.23016566 0.23940157
#> [25] 0.25657326 0.26157818 0.26400679 0.27139717 0.28467441 0.28634386
#> [31] 0.31169903 0.31626911 0.33116808 0.33782819 0.35285976 0.36219927
#> [37] 0.37534053 0.38330904 0.39978868 0.40008419 0.40316711 0.41474510
#> [43] 0.42339019 0.45352039 0.49092507 0.49164724 0.50313734 0.50690396
#> [49] 0.50733535 0.51349564 0.54586819 0.55556163 0.55924730 0.55930347
#> [55] 0.56912595 0.57820101 0.61631693 0.62303529 0.62895877 0.64415029
#> [61] 0.64829805 0.64919472 0.65719064 0.65879429 0.66519149 0.67106788
#> [67] 0.68424246 0.70785856 0.71803863 0.72335784 0.72654237 0.72759440
#> [73] 0.72835848 0.73947164 0.74614570 0.76990077 0.79520454 0.79930631
#> [79] 0.80134439 0.80355704 0.80393444 0.82559037 0.86352717 0.86393883
#> [85] 0.86886128 0.87319443 0.87726332 0.89833834 0.90542344 0.92475918
#> [91] 0.94080692 0.94261932 0.94485895 0.94746248 0.95220668 0.95471258
#> [97] 0.97280278 0.98993846 0.99938743 0.99939541 1.00000000
#>
#>
# Example 3: Retaining only behavioural parameter sets above a threshold
gofs <- runif(100, min=0.4, max=0.95)
params2ecdf(params, gofs=gofs, MinMax="max", beh.thr=0.75,
plot=TRUE, main="Parameter ECDFs for behavioural simulations")
#> [ Number of behavioural parameter sets: 35 ]
#> [ Computing the ECDF for 'Kq ' , 1/6 => 16.67% ]
#> [ Computing the ECDF for 'Ks ' , 2/6 => 33.33% ]
#> [ Computing the ECDF for 'Alpha' , 3/6 => 50.00% ]
#> [ Computing the ECDF for 'Beta ' , 4/6 => 66.67% ]
#> [ Computing the ECDF for 'Kperc' , 5/6 => 83.33% ]
#> [ Computing the ECDF for 'Smax ' , 6/6 => 100.00% ]
#> $Kq
#> $Kq$x
#> [1] 0.04205953 0.04205953 0.09359499 0.10292468 0.11113542 0.14280002
#> [7] 0.14711365 0.20653139 0.22011889 0.24361947 0.24608773 0.32792072
#> [13] 0.34351647 0.37446278 0.41764678 0.43983169 0.44220007 0.46677904
#> [19] 0.47779597 0.51150546 0.56094798 0.59414202 0.61277100 0.66511519
#> [25] 0.70853047 0.75330786 0.75447516 0.75845954 0.78819583 0.81006435
#> [31] 0.88301740 0.89305111 0.89982497 0.95450365 0.96302423 0.98495698
#>
#> $Kq$ecdf
#> 0.0420595335308462 0.0935949867125601 0.102924682665616
#> 0.00000000 0.02857143 0.05714286 0.08571429
#> 0.111135424347594 0.142800022382289 0.147113647311926 0.20653138961643
#> 0.11428571 0.14285714 0.17142857 0.20000000
#> 0.220118885161355 0.243619472719729 0.24608773435466 0.327920719282702
#> 0.22857143 0.25714286 0.28571429 0.31428571
#> 0.343516472261399 0.374462775886059 0.417646779678762 0.439831687603146
#> 0.34285714 0.37142857 0.40000000 0.42857143
#> 0.442200074205175 0.46677904156968 0.477795971091837 0.511505459900945
#> 0.45714286 0.48571429 0.51428571 0.54285714
#> 0.560947983758524 0.59414202044718 0.612771003274247 0.665115194628015
#> 0.57142857 0.60000000 0.62857143 0.65714286
#> 0.708530468167737 0.753307864302769 0.754475158639252 0.758459537522867
#> 0.68571429 0.71428571 0.74285714 0.77142857
#> 0.788195834029466 0.810064353048801 0.883017404004931 0.893051114398986
#> 0.80000000 0.82857143 0.85714286 0.88571429
#> 0.899824970401824 0.954503649147227 0.963024232536554 0.984956979984418
#> 0.91428571 0.94285714 0.97142857 1.00000000
#>
#>
#> $Ks
#> $Ks$x
#> [1] 0.0910440 0.0910440 0.1110965 0.1422943 0.1523466 0.2168928 0.2179907
#> [8] 0.2650178 0.2650327 0.2656867 0.3117022 0.3899944 0.4045103 0.4975273
#> [15] 0.5022996 0.5298357 0.5492847 0.5645904 0.5728671 0.5817501 0.5854834
#> [22] 0.6180179 0.6192565 0.6299731 0.6592303 0.6598384 0.6900071 0.7082903
#> [29] 0.7205963 0.7403344 0.8218055 0.8397678 0.9540912 0.9544738 0.9842192
#> [36] 0.9856409
#>
#> $Ks$ecdf
#> 0.0910439998842776 0.111096461303532 0.142294295597821
#> 0.00000000 0.02857143 0.05714286 0.08571429
#> 0.152346616843715 0.216892762808129 0.21799066872336 0.265017806086689
#> 0.11428571 0.14285714 0.17142857 0.20000000
#> 0.26503273146227 0.265686686849222 0.31170220207423 0.389994435245171
#> 0.22857143 0.25714286 0.28571429 0.31428571
#> 0.404510281747207 0.497527267085388 0.502299563260749 0.529835685854778
#> 0.34285714 0.37142857 0.40000000 0.42857143
#> 0.549284656066447 0.564590434776619 0.572867058217525 0.581750099780038
#> 0.45714286 0.48571429 0.51428571 0.54285714
#> 0.585483353119344 0.618017873261124 0.619256483390927 0.629973053466529
#> 0.57142857 0.60000000 0.62857143 0.65714286
#> 0.659230324206874 0.65983844967559 0.690007101511583 0.70829032221809
#> 0.68571429 0.71428571 0.74285714 0.77142857
#> 0.720596273429692 0.740334360394627 0.821805460145697 0.839767764788121
#> 0.80000000 0.82857143 0.85714286 0.88571429
#> 0.95409123855643 0.954473827499896 0.984219203470275 0.985640884377062
#> 0.91428571 0.94285714 0.97142857 1.00000000
#>
#>
#> $Alpha
#> $Alpha$x
#> [1] 0.04766363 0.04766363 0.06492830 0.07190410 0.09904183 0.10359964
#> [7] 0.19774474 0.22178793 0.22498533 0.22588643 0.33633121 0.35188864
#> [13] 0.39522013 0.39658459 0.39715195 0.40439918 0.40894400 0.46647238
#> [19] 0.50281305 0.51502973 0.53107040 0.54682616 0.56025326 0.61835123
#> [25] 0.64881812 0.70085309 0.73518431 0.75859317 0.78529167 0.79883161
#> [31] 0.82095132 0.86810681 0.87104341 0.91568354 0.91885735 0.92570796
#>
#> $Alpha$ecdf
#> 0.0476636274252087 0.0649282999802381 0.0719040972180665
#> 0.00000000 0.02857143 0.05714286 0.08571429
#> 0.0990418293513358 0.103599644964561 0.197744736680761 0.221787934657186
#> 0.11428571 0.14285714 0.17142857 0.20000000
#> 0.224985329201445 0.225886432919651 0.336331205908209 0.351888638455421
#> 0.22857143 0.25714286 0.28571429 0.31428571
#> 0.395220134640113 0.396584594622254 0.397151953307912 0.404399181017652
#> 0.34285714 0.37142857 0.40000000 0.42857143
#> 0.408943997928873 0.466472376836464 0.502813045633957 0.515029727248475
#> 0.45714286 0.48571429 0.51428571 0.54285714
#> 0.531070398865268 0.546826156554744 0.56025326368399 0.618351227371022
#> 0.57142857 0.60000000 0.62857143 0.65714286
#> 0.648818124318495 0.700853087473661 0.735184305813164 0.758593169506639
#> 0.68571429 0.71428571 0.74285714 0.77142857
#> 0.785291671287268 0.798831611173227 0.820951323956251 0.868106806883588
#> 0.80000000 0.82857143 0.85714286 0.88571429
#> 0.871043413644657 0.915683538420126 0.918857348151505 0.9257079558447
#> 0.91428571 0.94285714 0.97142857 1.00000000
#>
#>
#> $Beta
#> $Beta$x
#> [1] 0.03372777 0.03372777 0.03487480 0.10105849 0.11791384 0.14381704
#> [7] 0.18529644 0.19337868 0.20887635 0.23774940 0.28221842 0.35398610
#> [13] 0.35478310 0.37270939 0.41473533 0.41891590 0.42184495 0.48316767
#> [19] 0.49256655 0.58490009 0.61723527 0.62190230 0.69737377 0.72598303
#> [25] 0.72939065 0.76919391 0.80281227 0.83525532 0.83570884 0.85688542
#> [31] 0.91506376 0.96384333 0.96746949 0.97134245 0.98595417 0.99123656
#>
#> $Beta$ecdf
#> 0.0337277729995549 0.0348748040851206 0.10105849406682
#> 0.00000000 0.02857143 0.05714286 0.08571429
#> 0.11791384126991 0.143817043630406 0.185296441894025 0.193378679919988
#> 0.11428571 0.14285714 0.17142857 0.20000000
#> 0.208876348566264 0.237749404972419 0.282218417385593 0.353986102622002
#> 0.22857143 0.25714286 0.28571429 0.31428571
#> 0.354783097514883 0.372709394432604 0.414735334459692 0.418915896909311
#> 0.34285714 0.37142857 0.40000000 0.42857143
#> 0.421844950178638 0.483167669270188 0.492566547822207 0.584900093497708
#> 0.45714286 0.48571429 0.51428571 0.54285714
#> 0.617235269630328 0.621902295155451 0.697373766219243 0.725983026903123
#> 0.57142857 0.60000000 0.62857143 0.65714286
#> 0.729390651918948 0.769193911226466 0.802812272915617 0.835255319951102
#> 0.68571429 0.71428571 0.74285714 0.77142857
#> 0.835708837024868 0.856885420158505 0.91506376164034 0.963843331905082
#> 0.80000000 0.82857143 0.85714286 0.88571429
#> 0.967469494324178 0.971342450240627 0.985954165225849 0.991236555855721
#> 0.91428571 0.94285714 0.97142857 1.00000000
#>
#>
#> $Kperc
#> $Kperc$x
#> [1] 0.0004653491 0.0004653491 0.0279925719 0.0495384356 0.0939166071
#> [6] 0.1081785294 0.1247420886 0.1277056879 0.1920233187 0.2151664882
#> [11] 0.2300969204 0.2366197421 0.2760496726 0.2829928708 0.3077638743
#> [16] 0.3217255378 0.3498017928 0.3882046638 0.4445273969 0.4597851932
#> [21] 0.4784563838 0.5763018376 0.6126719653 0.6262569474 0.6782238574
#> [26] 0.6863249468 0.6877426857 0.7048722457 0.7475163222 0.7869273527
#> [31] 0.8103980662 0.8625518975 0.8900779132 0.9506212641 0.9531012138
#> [36] 0.9747258085
#>
#> $Kperc$ecdf
#> 0.000465349061414599 0.0279925719369203
#> 0.00000000 0.02857143 0.05714286
#> 0.0495384356472641 0.0939166070893407 0.10817852942273
#> 0.08571429 0.11428571 0.14285714
#> 0.124742088606581 0.127705687889829 0.192023318726569
#> 0.17142857 0.20000000 0.22857143
#> 0.215166488196701 0.230096920393407 0.236619742121547
#> 0.25714286 0.28571429 0.31428571
#> 0.276049672625959 0.282992870779708 0.307763874297962
#> 0.34285714 0.37142857 0.40000000
#> 0.321725537767634 0.349801792763174 0.388204663759097
#> 0.42857143 0.45714286 0.48571429
#> 0.444527396932244 0.45978519320488 0.478456383803859
#> 0.51428571 0.54285714 0.57142857
#> 0.576301837572828 0.612671965267509 0.626256947405636
#> 0.60000000 0.62857143 0.65714286
#> 0.678223857423291 0.686324946815148 0.687742685666308
#> 0.68571429 0.71428571 0.74285714
#> 0.704872245667502 0.747516322182491 0.786927352659404
#> 0.77142857 0.80000000 0.82857143
#> 0.810398066183552 0.862551897531375 0.890077913179994
#> 0.85714286 0.88571429 0.91428571
#> 0.950621264055371 0.953101213788614 0.974725808482617
#> 0.94285714 0.97142857 1.00000000
#>
#>
#> $Smax
#> $Smax$x
#> [1] 0.01616048 0.01616048 0.01836408 0.02552670 0.03197163 0.04770998
#> [7] 0.06265000 0.07997291 0.17174341 0.20493870 0.28181167 0.30288762
#> [13] 0.33749500 0.34871949 0.36789595 0.39525376 0.45476163 0.49006345
#> [19] 0.52971409 0.52989246 0.55331399 0.58695866 0.66670516 0.67901342
#> [25] 0.70903790 0.73589935 0.73903169 0.77020476 0.80094821 0.81508150
#> [31] 0.85409998 0.87867402 0.88513265 0.89433463 0.93913840 0.96903886
#>
#> $Smax$ecdf
#> 0.0161604809109122 0.0183640837203711 0.0255266984459013
#> 0.00000000 0.02857143 0.05714286 0.08571429
#> 0.0319716280791909 0.0477099849376827 0.0626499983482063 0.0799729123245925
#> 0.11428571 0.14285714 0.17142857 0.20000000
#> 0.171743405982852 0.204938702983782 0.28181166970171 0.302887624595314
#> 0.22857143 0.25714286 0.28571429 0.31428571
#> 0.337494997307658 0.348719493718818 0.367895947769284 0.395253761438653
#> 0.34285714 0.37142857 0.40000000 0.42857143
#> 0.454761631321162 0.490063453093171 0.529714091448113 0.529892458580434
#> 0.45714286 0.48571429 0.51428571 0.54285714
#> 0.553313987562433 0.58695865678601 0.666705157840624 0.679013415705413
#> 0.57142857 0.60000000 0.62857143 0.65714286
#> 0.709037903463468 0.735899350373074 0.739031685516238 0.770204759435728
#> 0.68571429 0.71428571 0.74285714 0.77142857
#> 0.800948211923242 0.815081495093182 0.854099977528676 0.87867402122356
#> 0.80000000 0.82857143 0.85714286 0.88571429
#> 0.885132651543245 0.894334625452757 0.939138400601223 0.969038863666356
#> 0.91428571 0.94285714 0.97142857 1.00000000
#>
#>
# }