beals: Beals Smoothing and Degree of Absence
In vegan: Community Ecology Package
Description
Usage
Arguments
Details
Value
Author(s)
References
See Also
Examples
Description
Beals smoothing replaces each entry in the community data with a
probability of a target species occurring in that particular site, based
on the joint occurrences of the target species with the species that
actually occur in the site. Swan's (1970) degree of absence applies
Beals smoothing to zero items so long that all zeros are replaced
with smoothed values.
Usage
1
2
Arguments
x
Community data frame or matrix.
species
Column index used to compute Beals function for a single species.
The default (NA) indicates that the function will be computed for all species.
reference
Community data frame or matrix to be used to compute
joint occurrences. By default, x is used as reference to
compute the joint occurrences.
type
Numeric. Specifies if and how abundance values have to be
used in function beals. See details for more explanation.
include
This logical flag indicates whether the target species has to be
included when computing the mean of the conditioned probabilities. The
original Beals (1984) definition is equivalent to include=TRUE,
while the formulation of Münzbergová and Herben is
equal to include=FALSE.
maxit
Maximum number of iterations. The default Inf
means that iterations are continued until there are no zeros or
the number of zeros does not change. Probably only
maxit = 1 makes sense in addition to the default.
Details
Beals smoothing is the estimated probability p[ij] that
species j occurs at site i. It is defined as p[ij] = 1/S[i]
Sum(k) N[jk] I[ik] / N[k], where S[i] is the number of
species at site i, N[jk] is the number of joint
occurrences of species j and k, N[k] is the
number of occurrences of species k, and I is the incidence
(0 or 1) of species (this last term is usually omitted from the
equation, but it is necessary). As N[jk] can be
interpreted as a mean of conditional probability, the beals
function can be interpreted as a mean of conditioned probabilities (De
Cáceres & Legendre 2008). The present function is
generalized to abundance values (De Cáceres & Legendre
2008).
The type argument specifies if and how abundance values have to be
used. type = 0 presence/absence mode. type = 1
abundances in reference (or x) are used to compute
conditioned probabilities. type = 2 abundances in x are
used to compute weighted averages of conditioned
probabilities. type = 3 abundances are used to compute both
conditioned probabilities and weighted averages.
Beals smoothing was originally suggested as a method of data
transformation to remove excessive zeros (Beals 1984, McCune 1994).
However, it is not a suitable method for this purpose since it does
not maintain the information on species presences: a species may have
a higher probability of occurrence at a site where it does not occur
than at sites where it occurs. Moreover, it regularizes data too
strongly. The method may be useful in identifying species that belong
to the species pool (Ewald 2002) or to identify suitable unoccupied
patches in metapopulation analysis (Münzbergová &
Herben 2004). In this case, the function should be called with
include=FALSE for cross-validation smoothing for species;
argument species can be used if only one species is studied.
Swan (1970) suggested replacing zero values with degrees of absence of
a species in a community data matrix. Swan expressed the method in
terms of a similarity matrix, but it is equivalent to applying Beals
smoothing to zero values, at each step shifting the smallest initially
non-zero item to value one, and repeating this so many times that
there are no zeros left in the data. This is actually very similar to
extended dissimilarities (implemented in function
stepacross), but very rarely used.
Value
The function returns a transformed data matrix or a vector if Beals smoothing
is requested for a single species.
Author(s)
Miquel De Cáceres and Jari Oksanen
References
Beals, E.W. 1984. Bray-Curtis ordination: an effective strategy for
analysis of multivariate ecological data. Pp. 1–55 in: MacFadyen, A. &
E.D. Ford [eds.] Advances in Ecological Research, 14. Academic
Press, London.
De Cáceres, M. & Legendre, P. 2008. Beals smoothing
revisited. Oecologia 156: 657–669.
Ewald, J. 2002. A probabilistic approach to estimating species pools
from large compositional matrices. J. Veg. Sci. 13: 191–198.
McCune, B. 1994. Improving community ordination with the Beals smoothing
function. Ecoscience 1: 82–86.
Münzbergová, Z. & Herben, T. 2004. Identification of
suitable unoccupied
habitats in metapopulation studies using co-occurrence of species. Oikos
105: 408–414.
Swan, J.M.A. 1970. An examination of some ordination problems by use of
simulated vegetational data. Ecology 51: 89–102.
See Also
decostand for proper standardization methods,
specpool for an attempt to assess the size of species
pool. Function indpower assesses the power of each species
to estimate the probabilities predicted by beals.
Examples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
data(dune)
## Default
x <- beals(dune)
## Remove target species
x <- beals(dune, include = FALSE)
## Smoothed values against presence or absence of species
pa <- decostand(dune, "pa")
boxplot(as.vector(x) ~ unlist(pa), xlab="Presence", ylab="Beals")
## Remove the bias of tarbet species: Yields lower values.
beals(dune, type =3, include = FALSE)
## Uses abundance information.
## Vector with beals smoothing values corresponding to the first species
## in dune.
beals(dune, species=1, include=TRUE)
Example output
Loading required package: permute
Loading required package: lattice
This is vegan 2.5-4
Achimill Agrostol Airaprae Alopgeni Anthodor Bellpere Bromhord
1 0.49590853 0.38333415 0.01157407 0.4923280 0.30827883 0.4935662 0.43263047
2 0.47083676 0.39501120 0.03361524 0.4718807 0.34723984 0.4917791 0.42000984
3 0.34063019 0.52738394 0.01520046 0.5309152 0.21609954 0.4033301 0.33010938
4 0.30816435 0.51198853 0.02876960 0.5971801 0.21542662 0.4398775 0.35732610
5 0.59949785 0.27622698 0.06632771 0.3349203 0.48876285 0.4322142 0.44309579
6 0.58819821 0.26299306 0.05967771 0.2700508 0.53154426 0.3696613 0.39760652
7 0.56496165 0.29412293 0.05329633 0.3403047 0.48010987 0.4051777 0.40471531
8 0.21230502 0.66906674 0.02588333 0.5187956 0.16247716 0.2720122 0.21219877
9 0.30323659 0.59744543 0.02213662 0.5792855 0.21896113 0.3292320 0.28613526
10 0.54083871 0.26902092 0.07349127 0.3372958 0.42671693 0.4705094 0.42934344
11 0.40509331 0.31656550 0.10259239 0.3185489 0.38766111 0.3713794 0.31413659
12 0.21008725 0.66278454 0.03625297 0.5753377 0.20078932 0.2802946 0.22974415
13 0.21850759 0.68239707 0.02191119 0.6404427 0.16737280 0.2939740 0.24942466
14 0.13570397 0.76284476 0.02298398 0.4107645 0.12128973 0.1682755 0.13757552
15 0.09168815 0.79412733 0.02538032 0.4505613 0.10117099 0.1420251 0.09794548
16 0.06335463 0.87877202 0.00742115 0.5232448 0.05538377 0.1516354 0.09458531
17 0.55254140 0.07330247 0.29233391 0.1013889 0.69331132 0.3129358 0.34982363
18 0.37751017 0.34451209 0.08535723 0.2838834 0.36918166 0.3676424 0.30478244
19 0.29826049 0.25952255 0.35137675 0.1934048 0.51929869 0.2237843 0.18074796
20 0.05429986 0.76675441 0.06144615 0.4063662 0.10738280 0.1450721 0.06706410
Chenalbu Cirsarve Comapalu Eleopalu Elymrepe Empenigr
1 0.025132275 0.09504980 0.000000000 0.05592045 0.4667439 0.00000000
2 0.043866562 0.08570299 0.026548839 0.08656209 0.4407282 0.01829337
3 0.065338638 0.08967477 0.031898812 0.16099072 0.4137888 0.01074444
4 0.057970906 0.12920228 0.039859621 0.16112450 0.4399661 0.02527165
5 0.026434737 0.05520104 0.015892090 0.05419613 0.3575948 0.03029752
6 0.021256367 0.03223112 0.030347896 0.08784329 0.3138879 0.03093489
7 0.038467708 0.04706743 0.017083997 0.06694311 0.3586644 0.02304603
8 0.063278453 0.06688407 0.100703044 0.29777644 0.3046956 0.02102222
9 0.069879277 0.07647268 0.045830682 0.19018562 0.3523460 0.01838883
10 0.025686639 0.06037513 0.029746617 0.07787078 0.3736128 0.03425596
11 0.021234732 0.05778318 0.035740922 0.11146095 0.2884798 0.07310076
12 0.103543341 0.07799259 0.045375827 0.19518888 0.3354080 0.03413656
13 0.122547745 0.07905124 0.056084315 0.22437598 0.3511708 0.01840390
14 0.042990591 0.03618335 0.241811837 0.55982776 0.1428372 0.01989756
15 0.035609053 0.04022968 0.198176675 0.53973883 0.1462975 0.02215971
16 0.056246994 0.05184498 0.201352298 0.51523810 0.1832397 0.00742115
17 0.007716049 0.01049383 0.009876543 0.02777778 0.1929470 0.21968254
18 0.014640428 0.04454602 0.042890320 0.17341352 0.2651538 0.06763669
19 0.019591245 0.03668466 0.031845637 0.12592768 0.1422725 0.26011417
20 0.037623741 0.03453783 0.185726965 0.58476297 0.1168700 0.05905666
Hyporadi Juncarti Juncbufo Lolipere Planlanc Poaprat Poatriv
1 0.07702746 0.14794933 0.1987270 0.9226190 0.40103107 0.9863946 0.8826329
2 0.07454127 0.13017869 0.2070478 0.8272395 0.40700777 0.8972046 0.8288385
3 0.05562332 0.22291082 0.2544828 0.7205525 0.27933493 0.8083020 0.8383185
4 0.06985986 0.21320122 0.2318440 0.7197924 0.25797285 0.7940926 0.8197302
5 0.10245961 0.10406655 0.2164230 0.8380779 0.52628928 0.9035899 0.8094632
6 0.11463153 0.11631772 0.2166255 0.8000021 0.58765018 0.8666677 0.7782619
7 0.10837376 0.11293676 0.2110045 0.8053380 0.51905808 0.8925059 0.8018775
8 0.06550319 0.33219882 0.2323566 0.5403355 0.20596764 0.6160461 0.7101299
9 0.05343787 0.23134366 0.2675624 0.6874068 0.25274756 0.7523318 0.8247374
10 0.13692492 0.09080902 0.1678040 0.8102783 0.52588347 0.8915882 0.7543592
11 0.18108995 0.13478872 0.1656396 0.7180948 0.47012501 0.8062720 0.6404351
12 0.06777311 0.27306206 0.3231724 0.5875943 0.22110550 0.6932541 0.8199960
13 0.04250245 0.28204736 0.3339728 0.5714581 0.18153869 0.7063028 0.7993754
14 0.04665747 0.46685537 0.1206518 0.3356311 0.14342002 0.3817081 0.5090703
15 0.05040404 0.51561767 0.1370235 0.3689922 0.13523214 0.4078219 0.5263520
16 0.01731602 0.54304667 0.1776781 0.3561752 0.07269979 0.4124222 0.6071083
17 0.36492870 0.03333333 0.1038156 0.5858415 0.59641331 0.7434618 0.5036834
18 0.17491099 0.18956922 0.1376386 0.7124388 0.45087176 0.7368632 0.5859071
19 0.39145281 0.13543701 0.1127832 0.4289185 0.40784415 0.5548077 0.3605827
20 0.07795311 0.53056145 0.1192488 0.3262685 0.13059496 0.3662817 0.4523029
Ranuflam Rumeacet Sagiproc Salirepe Scorautu Trifprat Trifrepe
1 0.08105273 0.3160963 0.3371121 0.02729885 0.8898317 0.21701279 0.8782576
2 0.13042865 0.3031318 0.3302063 0.05983781 0.9349640 0.20673650 0.9125666
3 0.22632936 0.2909068 0.4204104 0.06065155 0.9036443 0.14654749 0.8817430
4 0.21909541 0.2610006 0.4191908 0.07579199 0.9204237 0.12896524 0.8943213
5 0.08063087 0.3979230 0.2612828 0.07589611 0.9576838 0.34808957 0.9142360
6 0.10909966 0.4330705 0.2539380 0.08921540 0.9590466 0.35423465 0.9110822
7 0.10541081 0.4113622 0.2954682 0.07094548 0.9550487 0.32489503 0.9171688
8 0.40134447 0.2331043 0.4009544 0.11569906 0.8755515 0.09897600 0.8002526
9 0.26006489 0.3464870 0.4531178 0.07351827 0.9145996 0.16269563 0.8714833
10 0.10355742 0.3226025 0.2732735 0.09037489 0.9568824 0.26807372 0.9003730
11 0.13269569 0.2753878 0.3673397 0.16465286 0.9442707 0.19982976 0.8979262
12 0.29873222 0.3507140 0.5122033 0.08041977 0.9377963 0.13849854 0.9079979
13 0.33309468 0.3107471 0.5131337 0.06572594 0.9255312 0.11199114 0.8841739
14 0.64674225 0.1241545 0.2528665 0.15917563 0.8477706 0.06176123 0.6485949
15 0.64449081 0.1459458 0.3151199 0.17750323 0.8430677 0.05831084 0.7170446
16 0.66893881 0.1508409 0.3480368 0.15783292 0.8131968 0.03916718 0.6776273
17 0.03549383 0.2913631 0.3292030 0.25651777 0.9839744 0.27593101 0.8141660
18 0.18805395 0.2668100 0.3154533 0.17191937 0.9554011 0.20461193 0.8600701
19 0.14551892 0.1831168 0.4798245 0.36429493 0.9902041 0.11966159 0.8147968
20 0.62796060 0.1098600 0.3105433 0.21674174 0.8457313 0.03708580 0.6350394
Vicilath Bracruta Callcusp
1 0.17244420 0.7476589 0.003527337
2 0.18494940 0.7415172 0.034597921
3 0.12833142 0.7666969 0.075789630
4 0.12550967 0.7919786 0.081110164
5 0.16693075 0.8079786 0.023129027
6 0.18035860 0.8387650 0.040981168
7 0.19027523 0.8089116 0.024070054
8 0.10213052 0.8109194 0.201958942
9 0.08630413 0.7972178 0.092982775
10 0.23383453 0.7660374 0.033527777
11 0.24317802 0.8182692 0.043950322
12 0.08049055 0.8061715 0.100954127
13 0.06604026 0.7465509 0.122856392
14 0.07857237 0.7238162 0.347514804
15 0.07370069 0.7997141 0.381379395
16 0.03353260 0.8029953 0.364128496
17 0.20728700 0.7635487 0.009876543
18 0.26222869 0.8397471 0.109959916
19 0.18188455 0.8161275 0.082361157
20 0.09111967 0.8397124 0.397924041
1 2 3 4 5 6 7 8
0.5923077 0.5032372 0.3499038 0.3306953 0.5944041 0.5928780 0.5824352 0.2082532
9 10 11 12 13 14 15 16
0.2960799 0.5462492 0.3659392 0.2610043 0.1982372 0.0922619 0.1140625 0.1066506
17 18 19 20
0.6020408 0.3844577 0.2865741 0.0750000
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Description Usage Arguments Details Value Author(s) References See Also Examples
Description
Beals smoothing replaces each entry in the community data with a probability of a target species occurring in that particular site, based on the joint occurrences of the target species with the species that actually occur in the site. Swan's (1970) degree of absence applies Beals smoothing to zero items so long that all zeros are replaced with smoothed values.
Usage
1 2 |
Arguments
x |
Community data frame or matrix. |
species |
Column index used to compute Beals function for a single species.
The default ( |
reference |
Community data frame or matrix to be used to compute
joint occurrences. By default, |
type |
Numeric. Specifies if and how abundance values have to be
used in function |
include |
This logical flag indicates whether the target species has to be
included when computing the mean of the conditioned probabilities. The
original Beals (1984) definition is equivalent to |
maxit |
Maximum number of iterations. The default |
Details
Beals smoothing is the estimated probability p[ij] that
species j occurs at site i. It is defined as p[ij] = 1/S[i]
Sum(k) N[jk] I[ik] / N[k], where S[i] is the number of
species at site i, N[jk] is the number of joint
occurrences of species j and k, N[k] is the
number of occurrences of species k, and I is the incidence
(0 or 1) of species (this last term is usually omitted from the
equation, but it is necessary). As N[jk] can be
interpreted as a mean of conditional probability, the beals
function can be interpreted as a mean of conditioned probabilities (De
Cáceres & Legendre 2008). The present function is
generalized to abundance values (De Cáceres & Legendre
2008).
The type argument specifies if and how abundance values have to be
used. type = 0 presence/absence mode. type = 1
abundances in reference (or x) are used to compute
conditioned probabilities. type = 2 abundances in x are
used to compute weighted averages of conditioned
probabilities. type = 3 abundances are used to compute both
conditioned probabilities and weighted averages.
Beals smoothing was originally suggested as a method of data
transformation to remove excessive zeros (Beals 1984, McCune 1994).
However, it is not a suitable method for this purpose since it does
not maintain the information on species presences: a species may have
a higher probability of occurrence at a site where it does not occur
than at sites where it occurs. Moreover, it regularizes data too
strongly. The method may be useful in identifying species that belong
to the species pool (Ewald 2002) or to identify suitable unoccupied
patches in metapopulation analysis (Münzbergová &
Herben 2004). In this case, the function should be called with
include=FALSE for cross-validation smoothing for species;
argument species can be used if only one species is studied.
Swan (1970) suggested replacing zero values with degrees of absence of
a species in a community data matrix. Swan expressed the method in
terms of a similarity matrix, but it is equivalent to applying Beals
smoothing to zero values, at each step shifting the smallest initially
non-zero item to value one, and repeating this so many times that
there are no zeros left in the data. This is actually very similar to
extended dissimilarities (implemented in function
stepacross), but very rarely used.
Value
The function returns a transformed data matrix or a vector if Beals smoothing is requested for a single species.
Author(s)
Miquel De Cáceres and Jari Oksanen
References
Beals, E.W. 1984. Bray-Curtis ordination: an effective strategy for analysis of multivariate ecological data. Pp. 1–55 in: MacFadyen, A. & E.D. Ford [eds.] Advances in Ecological Research, 14. Academic Press, London.
De Cáceres, M. & Legendre, P. 2008. Beals smoothing revisited. Oecologia 156: 657–669.
Ewald, J. 2002. A probabilistic approach to estimating species pools from large compositional matrices. J. Veg. Sci. 13: 191–198.
McCune, B. 1994. Improving community ordination with the Beals smoothing function. Ecoscience 1: 82–86.
Münzbergová, Z. & Herben, T. 2004. Identification of suitable unoccupied habitats in metapopulation studies using co-occurrence of species. Oikos 105: 408–414.
Swan, J.M.A. 1970. An examination of some ordination problems by use of simulated vegetational data. Ecology 51: 89–102.
See Also
decostand for proper standardization methods,
specpool for an attempt to assess the size of species
pool. Function indpower assesses the power of each species
to estimate the probabilities predicted by beals.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 | data(dune)
## Default
x <- beals(dune)
## Remove target species
x <- beals(dune, include = FALSE)
## Smoothed values against presence or absence of species
pa <- decostand(dune, "pa")
boxplot(as.vector(x) ~ unlist(pa), xlab="Presence", ylab="Beals")
## Remove the bias of tarbet species: Yields lower values.
beals(dune, type =3, include = FALSE)
## Uses abundance information.
## Vector with beals smoothing values corresponding to the first species
## in dune.
beals(dune, species=1, include=TRUE)
|
Example output
Loading required package: permute
Loading required package: lattice
This is vegan 2.5-4
Achimill Agrostol Airaprae Alopgeni Anthodor Bellpere Bromhord
1 0.49590853 0.38333415 0.01157407 0.4923280 0.30827883 0.4935662 0.43263047
2 0.47083676 0.39501120 0.03361524 0.4718807 0.34723984 0.4917791 0.42000984
3 0.34063019 0.52738394 0.01520046 0.5309152 0.21609954 0.4033301 0.33010938
4 0.30816435 0.51198853 0.02876960 0.5971801 0.21542662 0.4398775 0.35732610
5 0.59949785 0.27622698 0.06632771 0.3349203 0.48876285 0.4322142 0.44309579
6 0.58819821 0.26299306 0.05967771 0.2700508 0.53154426 0.3696613 0.39760652
7 0.56496165 0.29412293 0.05329633 0.3403047 0.48010987 0.4051777 0.40471531
8 0.21230502 0.66906674 0.02588333 0.5187956 0.16247716 0.2720122 0.21219877
9 0.30323659 0.59744543 0.02213662 0.5792855 0.21896113 0.3292320 0.28613526
10 0.54083871 0.26902092 0.07349127 0.3372958 0.42671693 0.4705094 0.42934344
11 0.40509331 0.31656550 0.10259239 0.3185489 0.38766111 0.3713794 0.31413659
12 0.21008725 0.66278454 0.03625297 0.5753377 0.20078932 0.2802946 0.22974415
13 0.21850759 0.68239707 0.02191119 0.6404427 0.16737280 0.2939740 0.24942466
14 0.13570397 0.76284476 0.02298398 0.4107645 0.12128973 0.1682755 0.13757552
15 0.09168815 0.79412733 0.02538032 0.4505613 0.10117099 0.1420251 0.09794548
16 0.06335463 0.87877202 0.00742115 0.5232448 0.05538377 0.1516354 0.09458531
17 0.55254140 0.07330247 0.29233391 0.1013889 0.69331132 0.3129358 0.34982363
18 0.37751017 0.34451209 0.08535723 0.2838834 0.36918166 0.3676424 0.30478244
19 0.29826049 0.25952255 0.35137675 0.1934048 0.51929869 0.2237843 0.18074796
20 0.05429986 0.76675441 0.06144615 0.4063662 0.10738280 0.1450721 0.06706410
Chenalbu Cirsarve Comapalu Eleopalu Elymrepe Empenigr
1 0.025132275 0.09504980 0.000000000 0.05592045 0.4667439 0.00000000
2 0.043866562 0.08570299 0.026548839 0.08656209 0.4407282 0.01829337
3 0.065338638 0.08967477 0.031898812 0.16099072 0.4137888 0.01074444
4 0.057970906 0.12920228 0.039859621 0.16112450 0.4399661 0.02527165
5 0.026434737 0.05520104 0.015892090 0.05419613 0.3575948 0.03029752
6 0.021256367 0.03223112 0.030347896 0.08784329 0.3138879 0.03093489
7 0.038467708 0.04706743 0.017083997 0.06694311 0.3586644 0.02304603
8 0.063278453 0.06688407 0.100703044 0.29777644 0.3046956 0.02102222
9 0.069879277 0.07647268 0.045830682 0.19018562 0.3523460 0.01838883
10 0.025686639 0.06037513 0.029746617 0.07787078 0.3736128 0.03425596
11 0.021234732 0.05778318 0.035740922 0.11146095 0.2884798 0.07310076
12 0.103543341 0.07799259 0.045375827 0.19518888 0.3354080 0.03413656
13 0.122547745 0.07905124 0.056084315 0.22437598 0.3511708 0.01840390
14 0.042990591 0.03618335 0.241811837 0.55982776 0.1428372 0.01989756
15 0.035609053 0.04022968 0.198176675 0.53973883 0.1462975 0.02215971
16 0.056246994 0.05184498 0.201352298 0.51523810 0.1832397 0.00742115
17 0.007716049 0.01049383 0.009876543 0.02777778 0.1929470 0.21968254
18 0.014640428 0.04454602 0.042890320 0.17341352 0.2651538 0.06763669
19 0.019591245 0.03668466 0.031845637 0.12592768 0.1422725 0.26011417
20 0.037623741 0.03453783 0.185726965 0.58476297 0.1168700 0.05905666
Hyporadi Juncarti Juncbufo Lolipere Planlanc Poaprat Poatriv
1 0.07702746 0.14794933 0.1987270 0.9226190 0.40103107 0.9863946 0.8826329
2 0.07454127 0.13017869 0.2070478 0.8272395 0.40700777 0.8972046 0.8288385
3 0.05562332 0.22291082 0.2544828 0.7205525 0.27933493 0.8083020 0.8383185
4 0.06985986 0.21320122 0.2318440 0.7197924 0.25797285 0.7940926 0.8197302
5 0.10245961 0.10406655 0.2164230 0.8380779 0.52628928 0.9035899 0.8094632
6 0.11463153 0.11631772 0.2166255 0.8000021 0.58765018 0.8666677 0.7782619
7 0.10837376 0.11293676 0.2110045 0.8053380 0.51905808 0.8925059 0.8018775
8 0.06550319 0.33219882 0.2323566 0.5403355 0.20596764 0.6160461 0.7101299
9 0.05343787 0.23134366 0.2675624 0.6874068 0.25274756 0.7523318 0.8247374
10 0.13692492 0.09080902 0.1678040 0.8102783 0.52588347 0.8915882 0.7543592
11 0.18108995 0.13478872 0.1656396 0.7180948 0.47012501 0.8062720 0.6404351
12 0.06777311 0.27306206 0.3231724 0.5875943 0.22110550 0.6932541 0.8199960
13 0.04250245 0.28204736 0.3339728 0.5714581 0.18153869 0.7063028 0.7993754
14 0.04665747 0.46685537 0.1206518 0.3356311 0.14342002 0.3817081 0.5090703
15 0.05040404 0.51561767 0.1370235 0.3689922 0.13523214 0.4078219 0.5263520
16 0.01731602 0.54304667 0.1776781 0.3561752 0.07269979 0.4124222 0.6071083
17 0.36492870 0.03333333 0.1038156 0.5858415 0.59641331 0.7434618 0.5036834
18 0.17491099 0.18956922 0.1376386 0.7124388 0.45087176 0.7368632 0.5859071
19 0.39145281 0.13543701 0.1127832 0.4289185 0.40784415 0.5548077 0.3605827
20 0.07795311 0.53056145 0.1192488 0.3262685 0.13059496 0.3662817 0.4523029
Ranuflam Rumeacet Sagiproc Salirepe Scorautu Trifprat Trifrepe
1 0.08105273 0.3160963 0.3371121 0.02729885 0.8898317 0.21701279 0.8782576
2 0.13042865 0.3031318 0.3302063 0.05983781 0.9349640 0.20673650 0.9125666
3 0.22632936 0.2909068 0.4204104 0.06065155 0.9036443 0.14654749 0.8817430
4 0.21909541 0.2610006 0.4191908 0.07579199 0.9204237 0.12896524 0.8943213
5 0.08063087 0.3979230 0.2612828 0.07589611 0.9576838 0.34808957 0.9142360
6 0.10909966 0.4330705 0.2539380 0.08921540 0.9590466 0.35423465 0.9110822
7 0.10541081 0.4113622 0.2954682 0.07094548 0.9550487 0.32489503 0.9171688
8 0.40134447 0.2331043 0.4009544 0.11569906 0.8755515 0.09897600 0.8002526
9 0.26006489 0.3464870 0.4531178 0.07351827 0.9145996 0.16269563 0.8714833
10 0.10355742 0.3226025 0.2732735 0.09037489 0.9568824 0.26807372 0.9003730
11 0.13269569 0.2753878 0.3673397 0.16465286 0.9442707 0.19982976 0.8979262
12 0.29873222 0.3507140 0.5122033 0.08041977 0.9377963 0.13849854 0.9079979
13 0.33309468 0.3107471 0.5131337 0.06572594 0.9255312 0.11199114 0.8841739
14 0.64674225 0.1241545 0.2528665 0.15917563 0.8477706 0.06176123 0.6485949
15 0.64449081 0.1459458 0.3151199 0.17750323 0.8430677 0.05831084 0.7170446
16 0.66893881 0.1508409 0.3480368 0.15783292 0.8131968 0.03916718 0.6776273
17 0.03549383 0.2913631 0.3292030 0.25651777 0.9839744 0.27593101 0.8141660
18 0.18805395 0.2668100 0.3154533 0.17191937 0.9554011 0.20461193 0.8600701
19 0.14551892 0.1831168 0.4798245 0.36429493 0.9902041 0.11966159 0.8147968
20 0.62796060 0.1098600 0.3105433 0.21674174 0.8457313 0.03708580 0.6350394
Vicilath Bracruta Callcusp
1 0.17244420 0.7476589 0.003527337
2 0.18494940 0.7415172 0.034597921
3 0.12833142 0.7666969 0.075789630
4 0.12550967 0.7919786 0.081110164
5 0.16693075 0.8079786 0.023129027
6 0.18035860 0.8387650 0.040981168
7 0.19027523 0.8089116 0.024070054
8 0.10213052 0.8109194 0.201958942
9 0.08630413 0.7972178 0.092982775
10 0.23383453 0.7660374 0.033527777
11 0.24317802 0.8182692 0.043950322
12 0.08049055 0.8061715 0.100954127
13 0.06604026 0.7465509 0.122856392
14 0.07857237 0.7238162 0.347514804
15 0.07370069 0.7997141 0.381379395
16 0.03353260 0.8029953 0.364128496
17 0.20728700 0.7635487 0.009876543
18 0.26222869 0.8397471 0.109959916
19 0.18188455 0.8161275 0.082361157
20 0.09111967 0.8397124 0.397924041
1 2 3 4 5 6 7 8
0.5923077 0.5032372 0.3499038 0.3306953 0.5944041 0.5928780 0.5824352 0.2082532
9 10 11 12 13 14 15 16
0.2960799 0.5462492 0.3659392 0.2610043 0.1982372 0.0922619 0.1140625 0.1066506
17 18 19 20
0.6020408 0.3844577 0.2865741 0.0750000
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