SIBECOL-AEET Conference, 2-6th June, Pontevedra, Galicia
Sonia Illanas (IREC-CSIC,UCLM,JCCM), Javier Fernández-López (IREC-CSIC,UCLM,JCCM)
2025-06-02
Sonia Illanas & Javier Fernández-López
\[\begin{equation} X \sim Name(parameter_1, parameter_2 ) \end{equation}\]
Support: \(X \in (-\infty,\infty)\); \(parameter_1 \in (-\infty,\infty)\); \(parameter_2 > 0\) (real)
In R/NIMBLE: dname(parameter1, parameter2)
\[\begin{equation} X \sim Normal(\mu,\sigma ) \end{equation}\]
Support: \(X \in (-\infty,\infty)\); \(\mu \in (-\infty,\infty)\); \(\sigma > 0\) (real)
In R/NIMBLE: dnorm(mean, sd)
\[\begin{equation} X \sim Normal(\mu,\tau ) \end{equation}\] \[\begin{equation} \tau = 1/\sigma \end{equation}\]
Support: \(X \in (-\infty,\infty)\); \(\mu \in (-\infty,\infty)\); \(\sigma > 0\) (real)
In R/NIMBLE: dnorm(mean, sd)
\[\begin{equation} X \sim Beta(\alpha,\beta ) \end{equation}\]
Support: \(X \in (0,1)\); \(\alpha > 0\) (real); \(\beta > 0\) (real)
In R/NIMBLE: dbeta(alpha, beta)
\[\begin{equation} X \sim Beta(\mu,\sigma ) \end{equation}\] \[\begin{equation} \mu = \frac{\alpha}{\alpha + \beta} \text{ ; }\sigma = \sqrt{\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}} \end{equation}\]
Support: \(X \in (0,1)\); \(\alpha > 0\) (real); \(\beta > 0\) (real)
In R/NIMBLE: dbeta(alpha, beta)
\[\begin{equation} X \sim Gamma(\alpha,\beta) \end{equation}\]
Support: \(X \in (0,\infty)\); \(\alpha > 0\) (real); \(\beta > 0\) (real)
In R/NIMBLE: dgamma(shape, rate)
\[\begin{equation} X \sim Gamma(\mu,\sigma) \end{equation}\] \[\begin{equation} \mu = \frac{\alpha}{\beta} \text{ ; }\sigma = \sqrt(\frac{\alpha}{\beta^2}) \end{equation}\]
Support: \(X \in (0,\infty)\); \(\alpha > 0\) (real); \(\beta > 0\) (real)
In R/NIMBLE: dgamma(shape, rate)
\[\begin{equation} X \sim Uniform(min,max) \end{equation}\]
Support: \(X \in (-\infty,\infty)\); \(min \in (-\infty,\infty)\) (real); \(max \in (-\infty,\infty)\) (real)
In R/NIMBLE: dunif(min, max)
\[\begin{equation} X \sim Poisson(\lambda) \end{equation}\]
Support: \(X \in (0,\infty)\) (natural); \(\lambda \in (0,\infty)\) (real)
In R/NIMBLE: dpois(lambda)
\[\begin{equation} X \sim Bernoulli(p) \end{equation}\]
Support: \(X = \{0,1\}\); \(p \in (0,1)\)
In R/NIMBLE: dbinom(1, prob)
/ dbern(prob)
\[\begin{equation} X \sim Binomial(n, p) \end{equation}\]
Support: \(X = \{0,1\}\);\(n \in (0,\infty)\) (natural) \(p \in (0,1)\)
In R/NIMBLE: dbinom(size, prob)
/ dbinom(prob, size)
\[\begin{equation} weight(kg) \sim Normal(mean=27, sd=3) \end{equation}\]
\[\begin{equation} Y \sim Normal(\mu,\sigma ) \end{equation}\]
\[\begin{equation} Y \sim Normal(27,3) \end{equation}\]
\[\begin{equation} weight_i \sim Normal(\mu_i,\sigma ) \end{equation}\]
\[\begin{equation} \mu_i = \beta_0 + \beta_1X_i \end{equation}\]
\[\begin{equation} \mu_i = \beta_0 + \beta_1Latitude_i \end{equation}\]
\[\begin{equation} weight_i \sim Normal(\mu_i,\sigma ) \end{equation}\] \[\begin{equation} \mu_i = \beta_0 + \beta_1Latitude_i \end{equation}\]
\[\begin{equation} weight_i \sim Normal(\mu_i,\sigma ) \end{equation}\] \[\begin{equation} \mu_i = \beta_0 + \beta_1Latitude_i \end{equation}\]
\[\begin{equation} weight_i \sim Normal(\mu_i,\sigma ) \end{equation}\] \[\begin{equation} \mu_i = \beta_0 + \beta_1Latitude_i \end{equation}\]
\[\begin{equation} weight_i \sim Normal(\mu_i,\sigma ) \end{equation}\] \[\begin{equation} \mu_i = \beta_0 + \beta_1Latitude_i \end{equation}\]
Distributions | Links | Formula | Inverse |
---|---|---|---|
Gaussian(\(\mu,\sigma\)) | Identity | \(\mu = \beta X\) | \(\mu = \beta X\) |
Poisson(\(\lambda\)) | Log | \(log(\lambda) = \beta X\) | \(\lambda = e^{(\beta X)}\) |
Bernoulli(p) | Logit, probit, cloglog | \(log(\frac{p}{1-p}) = \beta X\) | \(p = \frac{e^{(\beta X)}}{1 + e^{(\beta X)}}\) |
Binomial(n,p) | Logit, probit, cloglog | \(log(\frac{p}{1-p}) = \beta X\) | \(p = \frac{e^{(\beta X)}}{1 + e^{(\beta X)}}\) |
We sampled 100 sites across the Iberian Peninsula, looking for roe deer droppings and recording the temperature at each site: \[~\]
conteo temp
1 6 1.2
2 8 2.5
3 28 4.9
4 220 8.9
5 6 0.4
6 225 8.8
7 294 9.3
8 50 5.9
9 46 5.5
10 1 -1.3
11 2 0.5
12 4 0.1
13 59 6.2
14 7 2.6
15 92 7.2
16 13 4.0
17 86 6.6
18 386 9.9
19 17 2.6
20 104 7.3
21 274 9.2
22 2 0.5
23 53 5.8
24 1 -0.5
25 6 1.2
26 6 2.6
27 2 -1.8
28 11 2.6
29 183 8.4
30 7 2.1
31 18 3.8
32 37 5.2
33 16 3.9
34 5 0.2
35 139 7.9
36 52 6.0
37 121 7.5
38 2 -0.7
39 86 6.7
40 10 2.9
41 156 7.9
42 53 5.8
43 123 7.4
44 22 4.6
45 30 4.4
46 123 7.5
47 3 -1.7
48 17 3.7
49 78 6.8
50 58 6.3
51 21 3.7
52 182 8.3
53 11 3.3
54 2 0.9
55 1 -1.2
56 2 -0.8
57 4 1.8
58 19 4.2
59 42 5.9
60 14 2.9
61 250 9.0
62 10 1.5
63 14 3.5
64 5 2.0
65 46 5.8
66 5 1.1
67 14 3.7
68 109 7.2
69 0 -1.0
70 186 8.5
71 9 2.1
72 162 8.1
73 7 2.2
74 7 2.0
75 15 3.7
76 195 8.7
77 178 8.4
78 15 2.7
79 119 7.3
80 318 9.5
81 8 3.2
82 70 6.6
83 9 2.8
84 9 1.9
85 99 7.1
86 1 0.4
87 75 6.5
88 1 -0.5
89 1 0.9
90 1 -0.3
91 3 0.9
92 0 -1.3
93 43 5.7
94 177 8.5
95 85 7.3
96 117 7.6
97 13 3.5
98 8 2.9
99 151 7.7
100 38 5.3
\[~\] We can create a model to relate the number of roe deer droppings to the temperature in each grid cell. What probability distribution could we use?
\[\begin{equation} N_i \sim Poisson(\lambda_i) \end{equation}\]
\[\begin{equation} N_i \sim Poisson(\lambda_i) \end{equation}\]
\[\begin{equation} \lambda_i = \beta_0 + \beta_1Temperature_i \end{equation}\]
conteo temp
1 6 1.2
2 8 2.5
3 28 4.9
4 220 8.9
5 6 0.4
6 225 8.8
7 294 9.3
8 50 5.9
9 46 5.5
10 1 -1.3
11 2 0.5
12 4 0.1
13 59 6.2
14 7 2.6
15 92 7.2
16 13 4.0
17 86 6.6
18 386 9.9
19 17 2.6
20 104 7.3
21 274 9.2
22 2 0.5
23 53 5.8
24 1 -0.5
25 6 1.2
26 6 2.6
27 2 -1.8
28 11 2.6
29 183 8.4
30 7 2.1
31 18 3.8
32 37 5.2
33 16 3.9
34 5 0.2
35 139 7.9
36 52 6.0
37 121 7.5
38 2 -0.7
39 86 6.7
40 10 2.9
41 156 7.9
42 53 5.8
43 123 7.4
44 22 4.6
45 30 4.4
46 123 7.5
47 3 -1.7
48 17 3.7
49 78 6.8
50 58 6.3
51 21 3.7
52 182 8.3
53 11 3.3
54 2 0.9
55 1 -1.2
56 2 -0.8
57 4 1.8
58 19 4.2
59 42 5.9
60 14 2.9
61 250 9.0
62 10 1.5
63 14 3.5
64 5 2.0
65 46 5.8
66 5 1.1
67 14 3.7
68 109 7.2
69 0 -1.0
70 186 8.5
71 9 2.1
72 162 8.1
73 7 2.2
74 7 2.0
75 15 3.7
76 195 8.7
77 178 8.4
78 15 2.7
79 119 7.3
80 318 9.5
81 8 3.2
82 70 6.6
83 9 2.8
84 9 1.9
85 99 7.1
86 1 0.4
87 75 6.5
88 1 -0.5
89 1 0.9
90 1 -0.3
91 3 0.9
92 0 -1.3
93 43 5.7
94 177 8.5
95 85 7.3
96 117 7.6
97 13 3.5
98 8 2.9
99 151 7.7
100 38 5.3
\[\begin{equation} X \sim Poisson(\lambda) \end{equation}\]
Support: \(X \in (0,\infty)\) (natural); \(\lambda \in (0,\infty)\) (real)
En R/NIMBLE: dpois(lambda)
\[\begin{equation} N_i \sim Poisson(\lambda_i) \end{equation}\] \[\begin{equation} log(\lambda_i) = \beta_0 + \beta_1Temperature_i \end{equation}\]
datos <- read.csv("roe_deer.csv")
mod <- glm(conteo ~ temp, family = poisson(link = "log"))
summary(mod)
Call:
glm(formula = conteo ~ temp, family = poisson(link = "log"))
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.930115 0.055375 16.80 <2e-16 ***
temp 0.509376 0.007002 72.75 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 9052.170 on 99 degrees of freedom
Residual deviance: 79.327 on 98 degrees of freedom
AIC: 568.48
Number of Fisher Scoring iterations: 4
\[\begin{equation} N_i \sim Poisson(\lambda_i) \end{equation}\] \[\begin{equation} log(\lambda_i) = 0.93 + 0.5*Temperatura_i \end{equation}\]
\[~~\]
The approach is based upon the idea that the experimenter begins with some prior beliefs about the system.
You never start from scratch.
And then iteratively updates these beliefs on the basis of observed data.
This updating procedure is based upon the Bayes’ Theorem.
Over multiple iterations your a priori beliefs converge to an a posteriori estimation.
The approach is based upon the idea that the experimenter begins with some prior beliefs about the system.
You never start from scratch.
And then iteratively updates these beliefs on the basis of observed data.
This updating procedure is based upon the Bayes’ Theorem.
Over multiple iterations your a priori beliefs converge to an a posteriori estimation.
Both are strategy to fit models. They are not statistical models
Bayes’ formula \(\Pr(B \mid A) = \displaystyle{\frac{ \Pr(A \mid B) \; \Pr(B)}{\Pr(A)}}\)
\[{\color{red}{\Pr(\text{Hypothesis} \mid \text{data})}} = \frac{\color{blue}{\Pr(\text{data} \mid \text{Hypothesis})} \; \color{green}{\Pr(\text{Hypothesis})}}{\color{orange}{\Pr(\text{data})}}\]
For regression models, the “hypothesis” is a parameter (intercept, slopes or error terms).
\(\color{red}{\text{Posterior distribution}}\): Represents what you know after having seen the data. The basis for inference, a distribution, possibly multivariate if more than one parameter.
\(\color{blue}{\text{Likelihood}}\): This quantity is the same as in the frequentist approach.
\(\color{green}{\text{Prior distribution}}\): Represents what you know before seeing the data. The source of much discussion about the Bayesian approach.
\(\color{orange}{\Pr(\text{data}) = \int{L(\text{data} \mid \theta)\Pr(\theta) d\theta}}\) is a \(N\)-dimensional integral if \(\theta = \theta_1, \ldots, \theta_N\).
Difficult if not impossible to calculate. This is one of the reasons why we need simulation (MCMC) methods.
Use stochastic simulations to avoid the calculation of multidimensional integrals.
Several ways of constructing these chains: e.g., Metropolis-Hastings, Gibbs sampler.
For example if you write \(\theta \sim \mbox{Uniform}(0,1)\), the model will only consider values between 0 and 1 for \(\theta\).
If you write \(\theta \sim \mbox{Normal}(0,1000)\), the model will explore a high range of value without limits with a higher probability around 0.
Over this example we define the prior distribution as \(\mbox{Normal}(0,1)\).
\[~~\]
Then, the chains of values explore the space of possibilities and converge to a stationary distribution, i.e: the posterior distribution.
Then, the chains of values explore the space of possibilities and converge to a stationary distribution, i.e: the posterior distribution.
The chains should converge to same target distribution.
We discard the MCMC iterations before convergence is achieved, i.e. burnin, and just use observations from the chain once it has converged.
Once discarded, explore efficiently: The post-convergence sample size required for suitable numerical summaries, named ESS or Effective Sample Size.
Simplest method to determine length of burn-in period is to look at trace plots.
Gimenez et al. 2025
We discard the MCMC iterations before convergence is achieved, i.e. burnin
\[~~\]
and just use observations from the chain once it has converged.
Rhat
to assess the mixing of chains. Values near \(1\) indicates likely convergence, a value of \(≤1.1\) is considered acceptable.\[~\]
Named a density plot.
\[~~\]
\[~~\]
Gimenez et al. 2025
\[~\]
Named a density plot.
You can extract the parameter value: mean, median
the precision associated: standard deviation, credible interval
The Bayesian pipeline
The Bayesian pipeline
\[~~\]
Oh yes, we also need to write a model! 😇
Numerical Inference for statistical Models using Bayesian and Likelihood Estimation.
Uses almost the same model code as BUGS and JAGS. Belongs to a library of other Bayesian methods (Stan), and Laplace approximation (TMB).
\[~~\]
nimble
packageInstall NIMBLE package.
Download
Xcode
for MacOs orRtools
for Windows.
Visit https://r-nimble.org/download
and good luck ! 😬
\[~~\]
We will use the GLM example of part 1.
glm()
conteo temp
1 6 1.2
2 8 2.5
3 28 4.9
4 220 8.9
5 6 0.4
6 225 8.8
7 294 9.3
8 50 5.9
9 46 5.5
10 1 -1.3
11 2 0.5
12 4 0.1
13 59 6.2
14 7 2.6
15 92 7.2
16 13 4.0
17 86 6.6
18 386 9.9
19 17 2.6
20 104 7.3
21 274 9.2
22 2 0.5
23 53 5.8
24 1 -0.5
25 6 1.2
26 6 2.6
27 2 -1.8
28 11 2.6
29 183 8.4
30 7 2.1
31 18 3.8
32 37 5.2
33 16 3.9
34 5 0.2
35 139 7.9
36 52 6.0
37 121 7.5
38 2 -0.7
39 86 6.7
40 10 2.9
41 156 7.9
42 53 5.8
43 123 7.4
44 22 4.6
45 30 4.4
46 123 7.5
47 3 -1.7
48 17 3.7
49 78 6.8
50 58 6.3
51 21 3.7
52 182 8.3
53 11 3.3
54 2 0.9
55 1 -1.2
56 2 -0.8
57 4 1.8
58 19 4.2
59 42 5.9
60 14 2.9
61 250 9.0
62 10 1.5
63 14 3.5
64 5 2.0
65 46 5.8
66 5 1.1
67 14 3.7
68 109 7.2
69 0 -1.0
70 186 8.5
71 9 2.1
72 162 8.1
73 7 2.2
74 7 2.0
75 15 3.7
76 195 8.7
77 178 8.4
78 15 2.7
79 119 7.3
80 318 9.5
81 8 3.2
82 70 6.6
83 9 2.8
84 9 1.9
85 99 7.1
86 1 0.4
87 75 6.5
88 1 -0.5
89 1 0.9
90 1 -0.3
91 3 0.9
92 0 -1.3
93 43 5.7
94 177 8.5
95 85 7.3
96 117 7.6
97 13 3.5
98 8 2.9
99 151 7.7
100 38 5.3
It’s the key step of Bayesian statistics. With nimble
the sky is the limit! A model is made of likelihood and priors.
You can modify, adapt every model.
nimble
.\[N_i \sim \mbox{Poisson}(\lambda_i)\]
\[\mbox{log}(\lambda_i) = \beta_0 + \beta_1 \mbox{Temperature}_i\] for every site \(i\) in nsites
.
To Nimble, not all “data” is data…
Constants:
Data:
Now, you’re ready to run! 🎉 💪
glm.rd <- nimbleCode({
# priors
b0 ~ dnorm(0, 1)
b1 ~ dnorm(0, 1)
# likelihood
for(i in 1:nsites){
log(lambda[i]) <- b0 + b1 * temp[i]
conteo[i] ~ dpois(lambda[i])
}
})
my.data <- list(conteo = datos$conteo)
my.constants <- list(temp = datos$temp,
nsites = nrow(datos))
initial.values <- list(b0 = rnorm(1,0,1),
b1 = rnorm(1, 0, 1))
parameters.to.save <- c("b0", "b1")
n.iter <- 5000
n.burnin <- 1000
n.chains <- 2
n.thin <- 1
List of 2
$ chain1: num [1:4000, 1:2] 0.942 0.947 0.947 0.947 0.947 ...
..- attr(*, "dimnames")=List of 2
.. ..$ : NULL
.. ..$ : chr [1:2] "b0" "b1"
$ chain2: num [1:4000, 1:2] 0.979 0.963 0.927 0.927 0.918 ...
..- attr(*, "dimnames")=List of 2
.. ..$ : NULL
.. ..$ : chr [1:2] "b0" "b1"
mean sd 2.5% 50% 97.5% Rhat n.eff
b0 0.92 0.06 0.81 0.92 1.02 1.05 56
b1 0.51 0.01 0.50 0.51 0.52 1.05 56
glm()
Call:
glm(formula = conteo ~ temp, family = poisson(link = "log"),
data = datos)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.930115 0.055375 16.80 <2e-16 ***
temp 0.509376 0.007002 72.75 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 9052.170 on 99 degrees of freedom
Residual deviance: 79.327 on 98 degrees of freedom
AIC: 568.48
Number of Fisher Scoring iterations: 4
Here, we want to predict the relation between \(\lambda\), the number of roe deer poop, and the temperature.
Here, we want to predict the relation between \(\lambda\), the number of roe deer poop, and the temperature.
b0 b1
[1,] 0.9423483 0.5076950
[2,] 0.9465896 0.5076950
[3,] 0.9465896 0.5076950
[4,] 0.9465881 0.5076950
[5,] 0.9465881 0.5076950
[6,] 0.9465881 0.5076950
[7,] 0.9465881 0.5076950
[8,] 0.9465881 0.5126490
[9,] 0.9274358 0.5103473
[10,] 0.8971905 0.5103473
[11,] 0.8884327 0.5183355
[12,] 0.8836065 0.5179455
[13,] 0.8836065 0.5179455
[14,] 0.8506656 0.5179455
[15,] 0.8506656 0.5179455
[16,] 0.8506656 0.5179455
[17,] 0.8506656 0.5179455
[18,] 0.8506656 0.5179455
[19,] 0.8506656 0.5179086
[20,] 0.8506656 0.5179086
[21,] 0.8562760 0.5192127
[22,] 0.8562760 0.5192127
[23,] 0.8562760 0.5192127
[24,] 0.8562760 0.5192127
[25,] 0.8562760 0.5192127
[26,] 0.8562760 0.5190260
[27,] 0.8745997 0.5190260
[28,] 0.8745997 0.5190260
[29,] 0.8745997 0.5123200
[30,] 0.8771304 0.5123200
[31,] 0.8771304 0.5120637
[32,] 0.9110096 0.5120637
[33,] 0.9110096 0.5120637
[34,] 0.9110096 0.5120637
[35,] 0.9110096 0.5109360
[36,] 0.9110096 0.5109360
[37,] 0.9302906 0.5063999
[38,] 0.9326690 0.5079975
[39,] 0.9326690 0.5079975
[40,] 0.9326690 0.5079975
[41,] 0.9513089 0.5079975
[42,] 0.9310447 0.5079975
[43,] 0.9574473 0.5079975
[44,] 0.9574473 0.5079975
[45,] 0.9574473 0.5079975
[46,] 0.9532816 0.5079975
[47,] 0.9532816 0.5079975
[48,] 0.9585889 0.5079975
[49,] 0.9393235 0.5090864
[50,] 0.9393235 0.5090864
[51,] 0.9518320 0.5093975
[52,] 0.9006476 0.5148062
[53,] 0.8805303 0.5148062
[54,] 0.8805303 0.5148062
[55,] 0.8786149 0.5148062
[56,] 0.8945883 0.5132421
[57,] 0.8945883 0.5132421
[58,] 0.8941929 0.5132421
[59,] 0.9011707 0.5132421
[60,] 0.9011707 0.5132421
[61,] 0.9011707 0.5132421
[62,] 0.9011707 0.5132421
[63,] 0.9058737 0.5133782
[64,] 0.8822067 0.5133782
[65,] 0.8822067 0.5133782
[66,] 0.8822067 0.5142912
[67,] 0.8822067 0.5142912
[68,] 0.8679826 0.5142912
[69,] 0.8679826 0.5142912
[70,] 0.8679826 0.5142912
[71,] 0.8679826 0.5163255
[72,] 0.8679826 0.5163255
[73,] 0.8725215 0.5163255
[74,] 0.8725215 0.5183491
[75,] 0.8741254 0.5183491
[76,] 0.8741254 0.5142623
[77,] 0.8850317 0.5142623
[78,] 0.8712827 0.5142623
[79,] 0.9094121 0.5142623
[80,] 0.9134583 0.5142623
[81,] 0.9134583 0.5105454
[82,] 0.9134583 0.5105454
[83,] 0.9134583 0.5105454
[84,] 0.9134583 0.5105454
[85,] 0.9199818 0.5105454
[86,] 0.9199818 0.5105454
[87,] 0.9199818 0.5089314
[88,] 0.9199818 0.5089314
[89,] 0.9147638 0.5089314
[90,] 0.9147638 0.5089314
[91,] 0.9147638 0.5089314
[92,] 0.9082217 0.5116220
[93,] 0.9042647 0.5124005
[94,] 0.9042647 0.5124005
[95,] 0.9042647 0.5124005
[96,] 0.9042647 0.5139008
[97,] 0.9042647 0.5139008
[98,] 0.8867146 0.5139008
[99,] 0.9100597 0.5138860
[100,] 0.9078876 0.5138860
[101,] 0.8890892 0.5138860
[102,] 0.8890892 0.5138860
[103,] 0.8798377 0.5138860
[104,] 0.8798377 0.5151709
[105,] 0.8798377 0.5151709
[106,] 0.8798377 0.5151709
[107,] 0.8798377 0.5129983
[108,] 0.8798377 0.5129983
[109,] 0.8798377 0.5129983
[110,] 0.8898554 0.5129983
[111,] 0.8840528 0.5173929
[112,] 0.8776466 0.5173929
[113,] 0.8804926 0.5173929
[114,] 0.8804926 0.5164441
[115,] 0.8804926 0.5164441
[116,] 0.8961189 0.5164441
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[1587,] 0.9210246 0.5104850
[1588,] 0.9210246 0.5111041
[1589,] 0.9210246 0.5130900
[1590,] 0.9210246 0.5138998
[1591,] 0.9210246 0.5138998
[1592,] 0.9221143 0.5138998
[1593,] 0.9127101 0.5138998
[1594,] 0.9127101 0.5145804
[1595,] 0.9127101 0.5135604
[1596,] 0.9127101 0.5135604
[1597,] 0.9127101 0.5135604
[1598,] 0.9127101 0.5135604
[1599,] 0.9127101 0.5135604
[1600,] 0.9127101 0.5135604
[1601,] 0.9127101 0.5135604
[1602,] 0.9127101 0.5135604
[1603,] 0.9127101 0.5135604
[1604,] 0.9127101 0.5135604
[1605,] 0.9127101 0.5135604
[1606,] 0.9127101 0.5110914
[1607,] 0.9127101 0.5116290
[1608,] 0.9127101 0.5146020
[1609,] 0.9127101 0.5100094
[1610,] 0.9127101 0.5100094
[1611,] 0.9127101 0.5100094
[1612,] 0.9254379 0.5100094
[1613,] 0.9349873 0.5103708
[1614,] 0.9178522 0.5103708
[1615,] 0.9178522 0.5103559
[1616,] 0.9178522 0.5080479
[1617,] 0.9178522 0.5080479
[1618,] 0.9178522 0.5079091
[1619,] 0.9178522 0.5079707
[1620,] 0.9281525 0.5079707
[1621,] 0.9315335 0.5079707
[1622,] 0.9315335 0.5089788
[1623,] 0.9315335 0.5089788
[1624,] 0.9315228 0.5089788
[1625,] 0.9315228 0.5089788
[1626,] 0.9233732 0.5089788
[1627,] 0.9304565 0.5126691
[1628,] 0.9278712 0.5077131
[1629,] 0.9278712 0.5081224
[1630,] 0.9278712 0.5081224
[1631,] 0.9372075 0.5081224
[1632,] 0.9333395 0.5081224
[1633,] 0.9448047 0.5069194
[1634,] 0.9448047 0.5069194
[1635,] 0.9448047 0.5069194
[1636,] 0.9448047 0.5104967
[1637,] 0.9286618 0.5060796
[1638,] 0.9286618 0.5060796
[1639,] 0.9294543 0.5122494
[1640,] 0.9294543 0.5122494
[1641,] 0.8878598 0.5134018
[1642,] 0.8878598 0.5134018
[1643,] 0.8878598 0.5129463
[1644,] 0.8878598 0.5129463
[1645,] 0.8846040 0.5126681
[1646,] 0.9280905 0.5126681
[1647,] 0.9057774 0.5126681
[1648,] 0.9057774 0.5126681
[1649,] 0.9057774 0.5126681
[1650,] 0.9196900 0.5109284
[1651,] 0.9330387 0.5080502
[1652,] 0.9330387 0.5080502
[1653,] 0.9330387 0.5080502
[1654,] 0.9330387 0.5080502
[1655,] 0.9330387 0.5067630
[1656,] 0.9330387 0.5067630
[1657,] 0.9330387 0.5072309
[1658,] 0.9330387 0.5072309
[1659,] 0.9330387 0.5106026
[1660,] 0.9330387 0.5106026
[1661,] 0.9330387 0.5077538
[1662,] 0.9330387 0.5077538
[1663,] 0.9407449 0.5073731
[1664,] 0.9475335 0.5053883
[1665,] 0.9475335 0.5089858
[1666,] 0.9475335 0.5089858
[1667,] 0.9479768 0.5063725
[1668,] 0.9479768 0.5063725
[1669,] 0.9479768 0.5040903
[1670,] 0.9479768 0.5033010
[1671,] 0.9479768 0.5033010
[1672,] 0.9479768 0.5069657
[1673,] 0.9479768 0.5069657
[1674,] 0.9479768 0.5069657
[1675,] 0.9476226 0.5050556
[1676,] 0.9476226 0.5050556
[1677,] 0.9476226 0.5048296
[1678,] 0.9806273 0.5048296
[1679,] 0.9689460 0.5048296
[1680,] 0.9689460 0.5048296
[1681,] 0.9575739 0.5063636
[1682,] 0.9516260 0.5063636
[1683,] 0.9631273 0.5063636
[1684,] 0.9647570 0.5043359
[1685,] 0.9621106 0.5043359
[1686,] 0.9621106 0.5050007
[1687,] 0.9684468 0.5062779
[1688,] 0.9684468 0.5067091
[1689,] 0.9626930 0.5067091
[1690,] 0.9362043 0.5110330
[1691,] 0.9135281 0.5112458
[1692,] 0.9135281 0.5112458
[1693,] 0.9135281 0.5112458
[1694,] 0.9135281 0.5112458
[1695,] 0.9310407 0.5112458
[1696,] 0.8992826 0.5112458
[1697,] 0.9011538 0.5156735
[1698,] 0.9011538 0.5156735
[1699,] 0.8758226 0.5157793
[1700,] 0.8758226 0.5169155
[1701,] 0.8758226 0.5169155
[1702,] 0.8758226 0.5155573
[1703,] 0.8758226 0.5190186
[1704,] 0.8758226 0.5166354
[1705,] 0.8879586 0.5166354
[1706,] 0.8851931 0.5147600
[1707,] 0.8851931 0.5147600
[1708,] 0.9026936 0.5147600
[1709,] 0.9026936 0.5147032
[1710,] 0.8626262 0.5147032
[1711,] 0.8626120 0.5202049
[1712,] 0.8626120 0.5202049
[1713,] 0.8442949 0.5202049
[1714,] 0.8426059 0.5228987
[1715,] 0.8282754 0.5228987
[1716,] 0.8282754 0.5228987
[1717,] 0.8282754 0.5240749
[1718,] 0.8282754 0.5240749
[1719,] 0.8282754 0.5234096
[1720,] 0.8203351 0.5233581
[1721,] 0.8203351 0.5233581
[1722,] 0.8203351 0.5233581
[1723,] 0.8203351 0.5233581
[1724,] 0.8203351 0.5237852
[1725,] 0.8203351 0.5222690
[1726,] 0.8417492 0.5222690
[1727,] 0.8544070 0.5222690
[1728,] 0.8544070 0.5173019
[1729,] 0.8544070 0.5173471
[1730,] 0.8544070 0.5173471
[1731,] 0.8544070 0.5173471
[1732,] 0.8749798 0.5173056
[1733,] 0.8749798 0.5169690
[1734,] 0.8749798 0.5169690
[1735,] 0.8749798 0.5146800
[1736,] 0.8730257 0.5165313
[1737,] 0.8719349 0.5165313
[1738,] 0.8760946 0.5165313
[1739,] 0.8760792 0.5165313
[1740,] 0.8735045 0.5165313
[1741,] 0.8735045 0.5145498
[1742,] 0.8735045 0.5145498
[1743,] 0.8660668 0.5145498
[1744,] 0.9018700 0.5145498
[1745,] 0.9018700 0.5142480
[1746,] 0.9018700 0.5142480
[1747,] 0.9018700 0.5142480
[1748,] 0.8906568 0.5142480
[1749,] 0.8906568 0.5142939
[1750,] 0.8906568 0.5168311
[1751,] 0.8906568 0.5168311
[1752,] 0.8807225 0.5168311
[1753,] 0.8807225 0.5108069
[1754,] 0.9024451 0.5108069
[1755,] 0.9029418 0.5108069
[1756,] 0.8957859 0.5113392
[1757,] 0.8957859 0.5113392
[1758,] 0.8908955 0.5113392
[1759,] 0.8908955 0.5113392
[1760,] 0.9062727 0.5126728
[1761,] 0.9062727 0.5126728
[1762,] 0.9062727 0.5126728
[1763,] 0.8807402 0.5133564
[1764,] 0.8853271 0.5133564
[1765,] 0.9166086 0.5131540
[1766,] 0.8938264 0.5131540
[1767,] 0.8904157 0.5158120
[1768,] 0.8690924 0.5172136
[1769,] 0.8641732 0.5187498
[1770,] 0.8631456 0.5178749
[1771,] 0.8673123 0.5178749
[1772,] 0.8670869 0.5178749
[1773,] 0.8749094 0.5147381
[1774,] 0.8702865 0.5147381
[1775,] 0.8917898 0.5147381
[1776,] 0.8846930 0.5147381
[1777,] 0.9110178 0.5121058
[1778,] 0.9110178 0.5145062
[1779,] 0.9110178 0.5140233
[1780,] 0.8678369 0.5140233
[1781,] 0.8678369 0.5152770
[1782,] 0.8678369 0.5170945
[1783,] 0.8678369 0.5204140
[1784,] 0.8594114 0.5190274
[1785,] 0.8498293 0.5192372
[1786,] 0.8498293 0.5169108
[1787,] 0.8409787 0.5200420
[1788,] 0.8361784 0.5180166
[1789,] 0.8498153 0.5203113
[1790,] 0.8383139 0.5203113
[1791,] 0.8383139 0.5203113
[1792,] 0.8361969 0.5189558
[1793,] 0.8361969 0.5189558
[1794,] 0.8469385 0.5185055
[1795,] 0.8469385 0.5202926
[1796,] 0.8469385 0.5202926
[1797,] 0.8418442 0.5202312
[1798,] 0.8418442 0.5202312
[1799,] 0.8418442 0.5202312
[1800,] 0.8418442 0.5202312
[1801,] 0.8418442 0.5178787
[1802,] 0.8418442 0.5188593
[1803,] 0.8418442 0.5182277
[1804,] 0.8637004 0.5146493
[1805,] 0.8637004 0.5179660
[1806,] 0.8637004 0.5179660
[1807,] 0.8637004 0.5179660
[1808,] 0.8674246 0.5179660
[1809,] 0.8674246 0.5179660
[1810,] 0.8674246 0.5179660
[1811,] 0.8674246 0.5179660
[1812,] 0.8630784 0.5179660
[1813,] 0.8691189 0.5179660
[1814,] 0.8821025 0.5160851
[1815,] 0.8810212 0.5160851
[1816,] 0.8790722 0.5160851
[1817,] 0.8653362 0.5159905
[1818,] 0.8653362 0.5159905
[1819,] 0.8653362 0.5159905
[1820,] 0.8764274 0.5159905
[1821,] 0.8764274 0.5151579
[1822,] 0.8764274 0.5151579
[1823,] 0.8764274 0.5151579
[1824,] 0.8764274 0.5151579
[1825,] 0.8764274 0.5178981
[1826,] 0.8764274 0.5175755
[1827,] 0.8764274 0.5175755
[1828,] 0.8490300 0.5172866
[1829,] 0.8399494 0.5158742
[1830,] 0.8684534 0.5171547
[1831,] 0.8694000 0.5188435
[1832,] 0.8638547 0.5188435
[1833,] 0.8638547 0.5188435
[1834,] 0.8270290 0.5219018
[1835,] 0.8270290 0.5229482
[1836,] 0.8105596 0.5245519
[1837,] 0.8105596 0.5232236
[1838,] 0.8105596 0.5232236
[1839,] 0.8105596 0.5232236
[1840,] 0.8225105 0.5261398
[1841,] 0.8123393 0.5251642
[1842,] 0.8232391 0.5240995
[1843,] 0.7972557 0.5271413
[1844,] 0.7972557 0.5256165
[1845,] 0.7934016 0.5273597
[1846,] 0.7936323 0.5273597
[1847,] 0.7961837 0.5262874
[1848,] 0.7961837 0.5262874
[1849,] 0.7961837 0.5262874
[1850,] 0.7961837 0.5262874
[1851,] 0.7961837 0.5262874
[1852,] 0.7961837 0.5249021
[1853,] 0.7908430 0.5249021
[1854,] 0.7908430 0.5249021
[1855,] 0.7908430 0.5261320
[1856,] 0.7948430 0.5261320
[1857,] 0.8194390 0.5211077
[1858,] 0.8194390 0.5211077
[1859,] 0.8420951 0.5211077
[1860,] 0.8430953 0.5229593
[1861,] 0.8407040 0.5225185
[1862,] 0.8509282 0.5217005
[1863,] 0.8524586 0.5212984
[1864,] 0.8524586 0.5175336
[1865,] 0.8538654 0.5175336
[1866,] 0.8538654 0.5175336
[1867,] 0.8538654 0.5175336
[1868,] 0.8802674 0.5175336
[1869,] 0.8463375 0.5195155
[1870,] 0.8463375 0.5202033
[1871,] 0.8463375 0.5202033
[1872,] 0.8278939 0.5202033
[1873,] 0.8278939 0.5202033
[1874,] 0.8385447 0.5202033
[1875,] 0.8385447 0.5209199
[1876,] 0.8442372 0.5209199
[1877,] 0.8442372 0.5203448
[1878,] 0.8442372 0.5203448
[1879,] 0.8436158 0.5166625
[1880,] 0.8436158 0.5225318
[1881,] 0.8436158 0.5225318
[1882,] 0.8078077 0.5225318
[1883,] 0.8078077 0.5225318
[1884,] 0.8366197 0.5225318
[1885,] 0.8366197 0.5225318
[1886,] 0.8183462 0.5244699
[1887,] 0.8183462 0.5244699
[1888,] 0.8183462 0.5244699
[1889,] 0.8183462 0.5225105
[1890,] 0.8183462 0.5250963
[1891,] 0.8183462 0.5241300
[1892,] 0.8166255 0.5241300
[1893,] 0.8166255 0.5241300
[1894,] 0.8203657 0.5241300
[1895,] 0.8374623 0.5241300
[1896,] 0.8374623 0.5225975
[1897,] 0.8374623 0.5225975
[1898,] 0.8374623 0.5194224
[1899,] 0.8602805 0.5194224
[1900,] 0.8602805 0.5194224
[1901,] 0.8602805 0.5194224
[1902,] 0.8602805 0.5194622
[1903,] 0.8566204 0.5194622
[1904,] 0.8566204 0.5194622
[1905,] 0.8566204 0.5174349
[1906,] 0.8564608 0.5174349
[1907,] 0.8564608 0.5174349
[1908,] 0.8564608 0.5185896
[1909,] 0.8494714 0.5185896
[1910,] 0.8639003 0.5176634
[1911,] 0.8639003 0.5176634
[1912,] 0.8639003 0.5182171
[1913,] 0.8450400 0.5182171
[1914,] 0.8450400 0.5182171
[1915,] 0.8383502 0.5215141
[1916,] 0.8563080 0.5215141
[1917,] 0.8343537 0.5215141
[1918,] 0.8158146 0.5215141
[1919,] 0.8158146 0.5245210
[1920,] 0.8063867 0.5245210
[1921,] 0.8063867 0.5234311
[1922,] 0.8215003 0.5240549
[1923,] 0.8149074 0.5240549
[1924,] 0.8336799 0.5235871
[1925,] 0.8138315 0.5235871
[1926,] 0.8138315 0.5233410
[1927,] 0.8339627 0.5233410
[1928,] 0.8339627 0.5190304
[1929,] 0.8339627 0.5219992
[1930,] 0.8316162 0.5219992
[1931,] 0.8325861 0.5240116
[1932,] 0.8323612 0.5240116
[1933,] 0.8323612 0.5234336
[1934,] 0.8323612 0.5234336
[1935,] 0.8323612 0.5227324
[1936,] 0.8288808 0.5235251
[1937,] 0.8251321 0.5194726
[1938,] 0.8542701 0.5194726
[1939,] 0.8542701 0.5194726
[1940,] 0.8542701 0.5178052
[1941,] 0.8542701 0.5184920
[1942,] 0.8542701 0.5179119
[1943,] 0.8714684 0.5184439
[1944,] 0.8714684 0.5184439
[1945,] 0.8794922 0.5184439
[1946,] 0.8893004 0.5181436
[1947,] 0.8893004 0.5181436
[1948,] 0.8893004 0.5181436
[1949,] 0.8587197 0.5196044
[1950,] 0.8587197 0.5202416
[1951,] 0.8564801 0.5202416
[1952,] 0.8564801 0.5202416
[1953,] 0.8666142 0.5202416
[1954,] 0.8666142 0.5189938
[1955,] 0.8666142 0.5189938
[1956,] 0.8604992 0.5189938
[1957,] 0.8348566 0.5189938
[1958,] 0.8597555 0.5189938
[1959,] 0.8597555 0.5189938
[1960,] 0.8597555 0.5171471
[1961,] 0.8854593 0.5171471
[1962,] 0.8588029 0.5180364
[1963,] 0.8588029 0.5180364
[1964,] 0.8588029 0.5180364
[1965,] 0.8580829 0.5180364
[1966,] 0.8594724 0.5180364
[1967,] 0.8665079 0.5180364
[1968,] 0.8677548 0.5180364
[1969,] 0.8677548 0.5180364
[1970,] 0.8677548 0.5180364
[1971,] 0.8677548 0.5180364
[1972,] 0.8677548 0.5195541
[1973,] 0.8677548 0.5195541
[1974,] 0.8677548 0.5195541
[1975,] 0.8677548 0.5153880
[1976,] 0.8969281 0.5153880
[1977,] 0.8763617 0.5153246
[1978,] 0.8763617 0.5176680
[1979,] 0.8763617 0.5158091
[1980,] 0.8763617 0.5158091
[1981,] 0.8763617 0.5177265
[1982,] 0.8763617 0.5154710
[1983,] 0.8937614 0.5154710
[1984,] 0.8937614 0.5174797
[1985,] 0.8937614 0.5174797
[1986,] 0.8937614 0.5174797
[1987,] 0.8937614 0.5155292
[1988,] 0.8937614 0.5157282
[1989,] 0.8884666 0.5122876
[1990,] 0.8884666 0.5122876
[1991,] 0.8884666 0.5147162
[1992,] 0.8884666 0.5150402
[1993,] 0.8884666 0.5150402
[1994,] 0.8945286 0.5150402
[1995,] 0.8945286 0.5136387
[1996,] 0.8945286 0.5136387
[1997,] 0.8945286 0.5118845
[1998,] 0.8896564 0.5118845
[1999,] 0.8985495 0.5118845
[2000,] 0.8985495 0.5118845
[2001,] 0.8985495 0.5126454
[2002,] 0.8985495 0.5126454
[2003,] 0.8902427 0.5126454
[2004,] 0.8902427 0.5126454
[2005,] 0.9238658 0.5089929
[2006,] 0.9238658 0.5105444
[2007,] 0.9238658 0.5114831
[2008,] 0.9238658 0.5088430
[2009,] 0.9238658 0.5088430
[2010,] 0.9238658 0.5103266
[2011,] 0.9238658 0.5103266
[2012,] 0.9238658 0.5106332
[2013,] 0.9402143 0.5106332
[2014,] 0.9402143 0.5106332
[2015,] 0.9402143 0.5093328
[2016,] 0.9402143 0.5093328
[2017,] 0.9402143 0.5081768
[2018,] 0.9453047 0.5081768
[2019,] 0.9453047 0.5081768
[2020,] 0.9245082 0.5066951
[2021,] 0.9245082 0.5066951
[2022,] 0.9245082 0.5066951
[2023,] 0.9298385 0.5100088
[2024,] 0.9428156 0.5079295
[2025,] 0.9428156 0.5067715
[2026,] 0.9428156 0.5067715
[2027,] 0.9581810 0.5070661
[2028,] 0.9656497 0.5070661
[2029,] 0.9656497 0.5056580
[2030,] 0.9656497 0.5056580
[2031,] 0.9621921 0.5056580
[2032,] 0.9621921 0.5031944
[2033,] 0.9627280 0.5031944
[2034,] 0.9627280 0.5031944
[2035,] 0.9627280 0.5045915
[2036,] 0.9482520 0.5047840
[2037,] 0.9482520 0.5047840
[2038,] 0.9482520 0.5047840
[2039,] 0.9482520 0.5047840
[2040,] 0.9482520 0.5047840
[2041,] 0.9727675 0.5047840
[2042,] 0.9408585 0.5039089
[2043,] 0.9507298 0.5054548
[2044,] 0.9507298 0.5054548
[2045,] 0.9560486 0.5056606
[2046,] 0.9560486 0.5056606
[2047,] 0.9440505 0.5056606
[2048,] 0.9440505 0.5056606
[2049,] 0.9440505 0.5063353
[2050,] 0.9440505 0.5063353
[2051,] 0.9557857 0.5063353
[2052,] 0.9557857 0.5050522
[2053,] 0.9557857 0.5065351
[2054,] 0.9551245 0.5074361
[2055,] 0.9551245 0.5074361
[2056,] 0.9461508 0.5094094
[2057,] 0.9461508 0.5068608
[2058,] 0.9461508 0.5075003
[2059,] 0.9406259 0.5081918
[2060,] 0.9355187 0.5081918
[2061,] 0.9381771 0.5081918
[2062,] 0.9381771 0.5081918
[2063,] 0.9381771 0.5081918
[2064,] 0.9336543 0.5081918
[2065,] 0.9595835 0.5088055
[2066,] 0.9419552 0.5098238
[2067,] 0.9226274 0.5098238
[2068,] 0.9226274 0.5098238
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[3710,] 0.8503438 0.5208737
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[3713,] 0.8793800 0.5167350
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[3810,] 0.9056455 0.5129509
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[3812,] 0.9049838 0.5129509
[3813,] 0.8983109 0.5129509
[3814,] 0.9083149 0.5111079
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[3818,] 0.9170192 0.5108377
[3819,] 0.9068394 0.5091584
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[3822,] 0.9400801 0.5100741
[3823,] 0.9023094 0.5100741
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[3840,] 0.9196774 0.5120531
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[3851,] 0.9151767 0.5112927
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[3855,] 0.9242624 0.5112927
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[3863,] 0.9403023 0.5105619
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[3884,] 0.9050769 0.5124861
[3885,] 0.9050769 0.5124861
[3886,] 0.8982500 0.5124861
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[3909,] 0.9051393 0.5095187
[3910,] 0.9051393 0.5113073
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[3912,] 0.9051393 0.5101665
[3913,] 0.9184892 0.5101665
[3914,] 0.9184892 0.5101665
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[3919,] 0.9250692 0.5072878
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[3935,] 0.9213866 0.5125545
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[3945,] 0.8915321 0.5148561
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[3949,] 0.9102704 0.5140760
[3950,] 0.9102704 0.5135121
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[3955,] 0.9302708 0.5107324
[3956,] 0.9302708 0.5107324
[3957,] 0.9210979 0.5107324
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[3965,] 0.9304556 0.5102331
[3966,] 0.9146227 0.5100342
[3967,] 0.9146227 0.5100342
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[4760,] 0.9682380 0.5016122
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[4814,] 0.9665228 0.5034795
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[4823,] 0.9311502 0.5061943
[4824,] 0.9311502 0.5061943
[4825,] 0.9214192 0.5061943
[4826,] 0.9214192 0.5082636
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[4829,] 0.9477554 0.5082821
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[4835,] 0.9572433 0.5061553
[4836,] 0.9572433 0.5061553
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[4838,] 0.9346745 0.5061553
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[4841,] 0.9382860 0.5069233
[4842,] 0.9382860 0.5069233
[4843,] 0.9382860 0.5069233
[4844,] 0.9504550 0.5064857
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[4847,] 0.9363231 0.5075972
[4848,] 0.9491658 0.5075972
[4849,] 0.9186232 0.5075972
[4850,] 0.9186232 0.5109809
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[4854,] 0.9212369 0.5096041
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[4858,] 0.9303334 0.5083455
[4859,] 0.9303334 0.5083455
[4860,] 0.9303334 0.5083455
[4861,] 0.9359968 0.5083455
[4862,] 0.9359968 0.5083455
[4863,] 0.9353881 0.5083455
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[4870,] 0.9312780 0.5106763
[4871,] 0.9312780 0.5106763
[4872,] 0.9312780 0.5085866
[4873,] 0.9312780 0.5059604
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[4875,] 0.9171181 0.5105021
[4876,] 0.9171181 0.5105021
[4877,] 0.9171181 0.5105021
[4878,] 0.9047867 0.5105021
[4879,] 0.9047867 0.5105021
[4880,] 0.9206705 0.5105021
[4881,] 0.9184159 0.5105021
[4882,] 0.9184159 0.5105021
[4883,] 0.8902106 0.5151818
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[4885,] 0.8735124 0.5148534
[4886,] 0.9085971 0.5106898
[4887,] 0.9085971 0.5126556
[4888,] 0.9085971 0.5112564
[4889,] 0.9190305 0.5112564
[4890,] 0.9190305 0.5112564
[4891,] 0.9190305 0.5127407
[4892,] 0.9190305 0.5132291
[4893,] 0.9214739 0.5132291
[4894,] 0.8996190 0.5136247
[4895,] 0.8996190 0.5136247
[4896,] 0.8996190 0.5136247
[4897,] 0.9002019 0.5145115
[4898,] 0.8724640 0.5146744
[4899,] 0.8724640 0.5146744
[4900,] 0.9062028 0.5136182
[4901,] 0.9062028 0.5096059
[4902,] 0.9062028 0.5131715
[4903,] 0.9269330 0.5085918
[4904,] 0.9269330 0.5085918
[4905,] 0.9489999 0.5100768
[4906,] 0.9489999 0.5100768
[4907,] 0.9489999 0.5100768
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[4909,] 0.9220373 0.5107954
[4910,] 0.9220373 0.5107954
[4911,] 0.9142470 0.5107954
[4912,] 0.9142470 0.5107954
[4913,] 0.9147476 0.5107954
[4914,] 0.9147476 0.5117844
[4915,] 0.9147476 0.5117844
[4916,] 0.8981450 0.5117844
[4917,] 0.9061719 0.5117844
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[7770,] 0.9180972 0.5108488
[7771,] 0.9180972 0.5108488
[7772,] 0.9180972 0.5108488
[7773,] 0.9180972 0.5108488
[7774,] 0.9180972 0.5108488
[7775,] 0.9180972 0.5108488
[7776,] 0.9180972 0.5108488
[7777,] 0.9180972 0.5106026
[7778,] 0.9184708 0.5106026
[7779,] 0.9184708 0.5106026
[7780,] 0.9184708 0.5079031
[7781,] 0.9548549 0.5079031
[7782,] 0.9548549 0.5045833
[7783,] 0.9709552 0.5040748
[7784,] 0.9709552 0.5040748
[7785,] 0.9705892 0.5040748
[7786,] 0.9718294 0.5049400
[7787,] 0.9718294 0.5049400
[7788,] 0.9573946 0.5049400
[7789,] 0.9573946 0.5068563
[7790,] 0.9434607 0.5068563
[7791,] 0.9434607 0.5068563
[7792,] 0.9597077 0.5068563
[7793,] 0.9597077 0.5065920
[7794,] 0.9597077 0.5072304
[7795,] 0.9630255 0.5062635
[7796,] 0.9414067 0.5061269
[7797,] 0.9453066 0.5061269
[7798,] 0.9453066 0.5061269
[7799,] 0.9453066 0.5061269
[7800,] 0.9453066 0.5061269
[7801,] 0.9453066 0.5079709
[7802,] 0.9482391 0.5079709
[7803,] 0.9487674 0.5051136
[7804,] 0.9447046 0.5066382
[7805,] 0.9447046 0.5066382
[7806,] 0.9620488 0.5066382
[7807,] 0.9427840 0.5066382
[7808,] 0.9667547 0.5066382
[7809,] 0.9330803 0.5087739
[7810,] 0.9330803 0.5090524
[7811,] 0.9244308 0.5122152
[7812,] 0.9244308 0.5122152
[7813,] 0.9267409 0.5076628
[7814,] 0.9267409 0.5078718
[7815,] 0.9267409 0.5078718
[7816,] 0.9267409 0.5078718
[7817,] 0.9267409 0.5078718
[7818,] 0.9342596 0.5098630
[7819,] 0.9342596 0.5098630
[7820,] 0.9342596 0.5098630
[7821,] 0.9342596 0.5098630
[7822,] 0.9342596 0.5068071
[7823,] 0.9342596 0.5094215
[7824,] 0.9280779 0.5094215
[7825,] 0.9124465 0.5087147
[7826,] 0.9252518 0.5099528
[7827,] 0.9309461 0.5115034
[7828,] 0.9309461 0.5115034
[7829,] 0.9252891 0.5115034
[7830,] 0.9088112 0.5115034
[7831,] 0.9088112 0.5117858
[7832,] 0.9088112 0.5117858
[7833,] 0.9164599 0.5117858
[7834,] 0.9164599 0.5117858
[7835,] 0.9164599 0.5117858
[7836,] 0.8950087 0.5117858
[7837,] 0.9177387 0.5117858
[7838,] 0.9177387 0.5117858
[7839,] 0.9177387 0.5121816
[7840,] 0.8975773 0.5120425
[7841,] 0.9129586 0.5081357
[7842,] 0.9260066 0.5081357
[7843,] 0.9260066 0.5072733
[7844,] 0.9345123 0.5095749
[7845,] 0.9368519 0.5095749
[7846,] 0.9406262 0.5095749
[7847,] 0.9406262 0.5091054
[7848,] 0.9217797 0.5091054
[7849,] 0.9317049 0.5091054
[7850,] 0.9329940 0.5091054
[7851,] 0.9329940 0.5091054
[7852,] 0.9374203 0.5094581
[7853,] 0.9373584 0.5094581
[7854,] 0.9328381 0.5094581
[7855,] 0.9322659 0.5094581
[7856,] 0.9322659 0.5094581
[7857,] 0.9322659 0.5088815
[7858,] 0.9322659 0.5088815
[7859,] 0.9294028 0.5115098
[7860,] 0.9294028 0.5079556
[7861,] 0.9294028 0.5079556
[7862,] 0.9519526 0.5079556
[7863,] 0.9519526 0.5079556
[7864,] 0.9592049 0.5069039
[7865,] 0.9592049 0.5071529
[7866,] 0.9592049 0.5071529
[7867,] 0.9592049 0.5071529
[7868,] 0.9592049 0.5077148
[7869,] 0.9592049 0.5055117
[7870,] 0.9685742 0.5063450
[7871,] 0.9528020 0.5051290
[7872,] 0.9327399 0.5051290
[7873,] 0.9403317 0.5086515
[7874,] 0.9403317 0.5076389
[7875,] 0.9403317 0.5076756
[7876,] 0.9403317 0.5076756
[7877,] 0.9403317 0.5071395
[7878,] 0.9590381 0.5071395
[7879,] 0.9590381 0.5071395
[7880,] 0.9361178 0.5071839
[7881,] 0.9267836 0.5071839
[7882,] 0.9267836 0.5071839
[7883,] 0.9267836 0.5071839
[7884,] 0.9267836 0.5078492
[7885,] 0.9267836 0.5078492
[7886,] 0.9386700 0.5078492
[7887,] 0.9386700 0.5078492
[7888,] 0.9386700 0.5078492
[7889,] 0.9332223 0.5087760
[7890,] 0.9236456 0.5106912
[7891,] 0.9236456 0.5110811
[7892,] 0.9236456 0.5110811
[7893,] 0.9236456 0.5101236
[7894,] 0.9236456 0.5101236
[7895,] 0.9236456 0.5101236
[7896,] 0.9236456 0.5101236
[7897,] 0.9236456 0.5101236
[7898,] 0.9236456 0.5112311
[7899,] 0.9215586 0.5112311
[7900,] 0.9215586 0.5112311
[7901,] 0.9206189 0.5093890
[7902,] 0.9343860 0.5100052
[7903,] 0.9343860 0.5100052
[7904,] 0.9446670 0.5100052
[7905,] 0.9488439 0.5056505
[7906,] 0.9583662 0.5052159
[7907,] 0.9583662 0.5052159
[7908,] 0.9583662 0.5052159
[7909,] 0.9564242 0.5052159
[7910,] 0.9487160 0.5052159
[7911,] 0.9487160 0.5056192
[7912,] 0.9487160 0.5056192
[7913,] 0.9689284 0.5056192
[7914,] 0.9689284 0.5055349
[7915,] 0.9578883 0.5055349
[7916,] 0.9578883 0.5055349
[7917,] 0.9582472 0.5055349
[7918,] 0.9582472 0.5055349
[7919,] 0.9582472 0.5084598
[7920,] 0.9582472 0.5027016
[7921,] 0.9582472 0.5027016
[7922,] 0.9582472 0.5048057
[7923,] 0.9632401 0.5069947
[7924,] 0.9565520 0.5042932
[7925,] 0.9477792 0.5042932
[7926,] 0.9477792 0.5077036
[7927,] 0.9325919 0.5077036
[7928,] 0.9325919 0.5093950
[7929,] 0.9168660 0.5093950
[7930,] 0.9168660 0.5093950
[7931,] 0.9168660 0.5093950
[7932,] 0.9168660 0.5131632
[7933,] 0.8903426 0.5134330
[7934,] 0.8903426 0.5134330
[7935,] 0.9150612 0.5142093
[7936,] 0.9150612 0.5142093
[7937,] 0.9150612 0.5109322
[7938,] 0.9150612 0.5140539
[7939,] 0.8822194 0.5140539
[7940,] 0.8793604 0.5172733
[7941,] 0.8612706 0.5172733
[7942,] 0.8612706 0.5172733
[7943,] 0.8612706 0.5172701
[7944,] 0.8612706 0.5172701
[7945,] 0.8612706 0.5172701
[7946,] 0.8612706 0.5182104
[7947,] 0.8569495 0.5182104
[7948,] 0.8625191 0.5181321
[7949,] 0.8453908 0.5181321
[7950,] 0.8453908 0.5181321
[7951,] 0.8453908 0.5199155
[7952,] 0.8486879 0.5210142
[7953,] 0.8486879 0.5195833
[7954,] 0.8486879 0.5195833
[7955,] 0.8486879 0.5195833
[7956,] 0.8572929 0.5195833
[7957,] 0.8418647 0.5185752
[7958,] 0.8418647 0.5188427
[7959,] 0.8585824 0.5182065
[7960,] 0.8585824 0.5183103
[7961,] 0.8591027 0.5214675
[7962,] 0.8591027 0.5214675
[7963,] 0.8522691 0.5187744
[7964,] 0.8541751 0.5187744
[7965,] 0.8541751 0.5187744
[7966,] 0.8541751 0.5187744
[7967,] 0.8578754 0.5187744
[7968,] 0.8485238 0.5187744
[7969,] 0.8546532 0.5218276
[7970,] 0.8481971 0.5226568
[7971,] 0.8481971 0.5191498
[7972,] 0.8481971 0.5193383
[7973,] 0.8481971 0.5193383
[7974,] 0.8481971 0.5193383
[7975,] 0.8447557 0.5203432
[7976,] 0.8449503 0.5175652
[7977,] 0.8671043 0.5175652
[7978,] 0.8868762 0.5175652
[7979,] 0.8833133 0.5123828
[7980,] 0.8833133 0.5123828
[7981,] 0.8831231 0.5123828
[7982,] 0.8831231 0.5123828
[7983,] 0.8831231 0.5160508
[7984,] 0.8773605 0.5156547
[7985,] 0.8773605 0.5156547
[7986,] 0.8773605 0.5156547
[7987,] 0.8730917 0.5156547
[7988,] 0.8730917 0.5156547
[7989,] 0.8774022 0.5156547
[7990,] 0.8774022 0.5165970
[7991,] 0.8774022 0.5148121
[7992,] 0.8774022 0.5148121
[7993,] 0.8934060 0.5148121
[7994,] 0.8934060 0.5148121
[7995,] 0.8934060 0.5128597
[7996,] 0.8934060 0.5152220
[7997,] 0.8860336 0.5152220
[7998,] 0.8860336 0.5152220
[7999,] 0.8938462 0.5152220
[8000,] 0.8938462 0.5141825
lambda
mcmc.bind
), we calculate lambda
with the b0
and b1
estimated, as \(\lambda = \mbox{exp}( \beta_0 + \beta1 \mbox{Temperature})\)\[~~\]
lambda
for each temperature and we can extract mean, median, standard deviation, credible interval, etc.\[~~\]
\[~~\]
# A tibble: 141 × 5
temp mean sd cinf csup
<dbl> <dbl> <dbl> <dbl> <dbl>
1 -5 0.196 0.0175 0.173 0.219
2 -4.9 0.206 0.0182 0.183 0.230
3 -4.8 0.217 0.0190 0.192 0.242
4 -4.7 0.228 0.0199 0.203 0.254
5 -4.6 0.240 0.0207 0.214 0.267
6 -4.5 0.252 0.0217 0.225 0.281
7 -4.4 0.266 0.0226 0.237 0.295
8 -4.3 0.280 0.0236 0.250 0.311
9 -4.2 0.294 0.0246 0.263 0.327
10 -4.1 0.310 0.0257 0.277 0.343
# ℹ 131 more rows
We can plot the prediction.
Bayesian statistics are tools to fit ecological models.
It’s not always a good idea to use Bayesian inference if there are simpler tools available (both to use and to communicate).
Learning Bayesian inference is valuable not only for applying it in our own work, but also for understanding what others are doing!
Bayesian statistics are tools to fit ecological models.
It’s not always a good idea to use Bayesian inference if there are simpler tools available (both to use and to communicate).
Learning Bayesian inference is valuable not only for applying it in our own work, but also for understanding what others are doing!
Because it helps us to understand the statistics and models.
Using nimble
allow to formally write the statistical model you want.
It is useful to adapt existing models to fancy (or shitty) data sets.
Olivier Gimenez’s Youtube course about Bayesian statistics with R
NIMBLE Official website https://r-nimble.org
User Manual https://r-nimble.org/html_manual/cha-welcome-nimble.html and cheatsheet.
About priors: The use of Bayesian priors in Ecology: The good, the bad and the not great
Reference to cite when using nimble in a publication:
de Valpine, P., D. Turek, C. J. Paciorek, C. Anderson-Bergman, D. Temple Lang, and R. Bodik (2017). Programming With Models: Writing Statistical Algorithms for General Model Structures With NIMBLE. Journal of Computational and Graphical Statistics 26 (2): 403–13.
Introduction to Bayesian Inference in Ecology (R & Nimble)