SUPPLEMENT MATERIALS
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The materials have three parts: 1) the comparison of Poisson model and Negative binomial model; 2)
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Schematic illustration of three probability distribution models; 3) analysis and classification of BCI species that fail
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the chi-square test.
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SECTION 1. POISSON MODEL vs. NEGATIVE BINOMIAL MODEL
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Poisson distribution has the following frequency function:
p(N(A) = x) =
(λA)x e−λA
x!
, x = 1,2, ⋯
(S1)
The frequency distribution function of negative binomial distribution (NBD) is
p(N(A) = x) =
Γ(k+x)
x!Γ(k)
k
−x
(1 + λA)
(1 +
λA −k
k
) , x = 0,1,2, ⋯
(S2)
Sometimes, 1/𝑘 is used as an index of aggregation in such a way: small 1/k (= 0) suggests a random
pattern, while large 1/𝑘 suggests aggregation.
The flexibility and generality of the NBD is obvious from the following results:
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(1) 𝑘 = 1, the NBD turns to be a geometric distribution;
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(2) 𝑘 → 0, it tends to a logarithmic series distribution;
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(3) 𝑘 → +∞, it is a Poisson distribution.
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In particular, the frequency of Poisson distribution and that of negative binomial distribution are identical
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when parameter 𝑘 is large than 20. In Figure S1, we compare their frequency distribution for different values
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of parameter 𝑘.
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In this paper, we have presented three probability distribution models: 𝑝𝑛 (𝑟),
𝑔𝑛 (𝑟), and 𝑓𝑛 (𝑟). 𝑝n (r)
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corresponds to the Poisson case, while 𝑔𝑛 (𝑟), and 𝑓𝑛 (𝑟) correspond to negative binomial case. If we take
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Poisson model and negative binomial model as null models, we can further derive their theoretical Ripley’s
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K-functions. For Poisson case, the theoretical Ripley’s K-function is 𝐾(𝑟) = 𝜋𝑟 2 . For the negative binomial
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case, the theoretical Ripley’s K-function is 𝐾(𝑟) = 𝜋𝑟 2 (1 + 𝑘). The relationships among these probability
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models and the second-order statistics are shown in Figure S2.
=0.0001,A=50*50
0.9
0.8
0.7
Frequency
0.6
0.5
0.4
0.3
0.2
0.1
0
-2
0
2
4
6
8
10
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=0.001,A=50*50
0.8
Poisson
Poisson
NBD(k=0.1)
NBD(k=1)
NBD(k=10)
NBD(k=50)
NBD(k=50)
0.7
Frequency
0.6
0.5
0.4
0.3
0.2
0.1
0
-2
0
2
4
6
8
10
x
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Figure S1: The frequency distribution of Poisson distribution and negative binomial distribution.
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2
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𝑘 → +∞
Quadrat sampling:
Poisson Model
Point-to-Event
Negative Binomial Model
Event-to-Event
Point-to-Event
𝑘 → +∞
Distance sampling:
𝑔𝑛 (𝑟)
𝑝𝑛 (𝑟)
Event-to-Event
𝑓𝑛 (𝑟)
𝑘 → +∞
Ripley’s K-function:
𝑘 → +∞
𝐾(𝑟) = 𝜋𝑟 2
1
𝐾(𝑟) = 𝜋𝑟 2 (1 + )
𝑘
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Figure S2: The relationship among probability models used in this paper and the second-order statistics
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(Ripley’s K-function). Solid lines represent derivations and dashed lines represent limits.
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SECTION 2.
0.1
0.15
0.1
k=0.1;n=1
0.05
0
20
40
r
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Probability density
Probability density
k=1;n=1
0.04
0.03
0.02
0.01
0
20
40
r
60
0.02
0
20
40
r
60
Probability density
k=20;n=1
0.03
0.02
0.01
0
20
40
r
60
80
k=1;n=2
0.03
0.02
0.01
0
20
0.02
0
50
100
r
150
200
40
r
60
150
200
150
200
0.04
k=1;n=6
0.03
0.02
0.01
0
80
0
50
100
r
0.05
0.04
k=20;n=2
0.03
0.02
0.01
0
k=0.1;n=6
0.04
0
80
0.05
0.04
0.06
0.05
0.04
0
80
0.05
Probability density
k=0.1;n=2
0.04
0.05
0.05
0
0.06
0
80
0.06
0
0.08
Probability density
0
0.08
Probability density
Poisson
NBD(Point-to-Event)
NBD(Event-to-Event)
Probability density
Probability density
0.2
Probability density
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0
20
40
r
60
80
0.04
k=20;n=6
0.03
0.02
0.01
0
0
50
100
r
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Figure S3: Probability density functions of n-th NND. For Poisson case, the pdf of Point-to-Event and
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Event-to-Event nearest neighbor distances are the same (𝑝𝑛 (𝑟), green dashed lines) . For NBD case, 𝑓𝑛 (𝑟)
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(red solid lines) and
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parameter 𝑘 become large. Other parameter: λ = 0.001.
𝑔𝑛 (𝑟) (blue dotted lines) are different but both approximate the pdf in Poisson case as
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SECTION 3.
As mentioned in text, we use the two probability density model to fit n-th NND data of 183 species
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(with more than 50 individuals in the inner rectangular range). We classified these species into four
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categories: pass two test, pass 𝑔𝑛 (𝑟) test but not 𝑓𝑛 (𝑟) test, pass
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pass these two tests either.
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𝑓𝑛 (𝑟) test but not 𝑔𝑛 (𝑟) test, cannot
(1) The first category has been demonstrated in the text.
(2) Species belong to second category are Alchornea costaricensis, Croton billbergianus, Adelia
triloba , Luehea seemannii,
Miconia argentea, Senna dariensis,
Spondias
mombin,
Zanthoxylum
ekmanii, etc. Spatial distributions of these classes of species and their fit of the two probability distribution
models are shown in Fig.S4.
(3) Species belong to third category are Anaxagorea panamensis, Licania hypoleuca, Astronium
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graveolens, Gustavia superba, Nectandra cissiflora, Trophis caucana, Pourouma bicolor, Socratea
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exorrhiza, etc. Spatial distributions of these classes of species and their fit of the two probability distribution
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models are shown in Fig.S5.
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(4) Species belong to fourth category are Inga pezizifera, Miconia nervosa, Ocotea oblonga, Oenocarpus
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mapora, Palicourea guianensis, Zanthoxylum acuminatum , etc. Spatial distributions of these classes of
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species and their fit of the two probability distribution models are shown in Fig.S6.
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Luehea seemannii
500
400
400
400
300
300
300
200
100
100
0
200
400
600
X (m)
800
0
1000
0
200
400
600
X (m)
0.005
0
100
200
0.015
0.01
0.005
0
300
0.03
0.02
0.01
200
300
0
50
100
150
200
0
250
0
100
200
r
300
400
0.04
0.03
0.02
0.01
0
1000
0.005
1
1
0.04
800
0.01
Probability density, f (r)
Probability density, f (r)
1
400
600
X (m)
0.015
0.04
100
200
r
0.05
0
0
0.02
r
Probability density, f (r)
0
1000
0.02
1
0.01
0
800
1
Probability density, g (r)
0.015
200
100
0.025
0.02
1
Probability density, g (r)
0.025
0
200
Probability density, g (r)
0
Senna dariensis
Y (m)
500
Y (m)
Y (m)
Alchornea costaricensis
500
0
100
r
200
r
300
0.03
0.02
0.01
0
0
100
200
300
r
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Figure S4: Spatial distributions of three Barro Colorada Island, Panama (BCI) species that have passed the
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chi-square test of 𝑔𝑛 (𝑟) but not 𝑓𝑛 (𝑟), their frequency distribution of Point-to-Event NND and Event-to-Event
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NND (from top to down), and the fitted probability density functions.
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Pourouma bicolor
Trophis caucana
500
400
400
400
300
300
300
200
100
0
Y (m)
500
Y (m)
Y (m)
Anaxagorea panamensis
500
200
100
0
200
400
600
X (m)
800
0
1000
200
100
0
200
400
600
X (m)
800
0
1000
0
200
400
600
X (m)
800
1000
-3
0.01
0.005
0
500
1000
r
1500
1
0.01
4
2
0
200
400
r
600
0.008
0.006
0.004
0.002
0
800
0.4
0.2
0
10
20
r
30
40
0.04
0.03
0.02
0.01
0
200
400
600
0.04
1
1
0.6
0
r
0.05
Probability density, f (r)
1
Probability density, f (r)
0.8
0
6
0
2000
0.012
Probability density, f (r)
0
x 10
Probability density, g (r)
1
Probability density, g (r)
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1
Probability density, g (r)
0.015
0
100
200
300
400
r
500
0.03
0.02
0.01
0
0
100
200
300
400
500
r
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Figure S5: Spatial distributions of three Barro Colorada Island, Panama (BCI) species that have passed the
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chi-square test of 𝑓𝑛 (𝑟) but not 𝑔𝑛 (𝑟), their frequency distribution of Point-to-Event NND and Event-to-Event
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NND (from top to down), and the fitted probability density functions.
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400
400
400
300
300
300
200
200
100
100
0
200
400
600
X (m)
800
0
1000
0.005
800
0
1000
0
200
400
r
600
0.01
0
0
50
100
r
150
0.03
0.02
0.01
0
200
400
600
800
1000
0.3
0.2
0.1
0
800
1000
0.005
0
100
200
r
300
400
0
100
200
r
300
400
0.02
3
1
0.04
400
600
X (m)
0.01
0
200
0.4
0.05
200
3
0.02
800
0
0.015
0.03
Probability density, f (r)
1
Probability density, f (r)
400
600
X (m)
0.04
0.06
0
200
Probability density, g (r)
1
Probability density, g (r)
0.01
0
0
0.05
1
Probability density, g (r)
0.015
200
100
Probability density, f (r)
0
Zanthoxylum acuminatum
500
Y (m)
Palicourea guianensis
500
Y (m)
Inga pezizifera
Y (m)
500
0
50
r
100
150
0.015
0.01
0.005
0
r
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Figure S6: Spatial distributions of three Barro Colorada Island, Panama (BCI) species that have passed the
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chi-square test of neither 𝑔𝑛 (𝑟) nor 𝑓𝑛 (𝑟), their frequency distribution of Point-to-Event NND and
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Event-to-Event NND (from top to down), and the fitted probability density functions.
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