Bloque 1 Modelos con R y Nimble
Part 2 Introduccion al Bayesiano y Nimble
© Bea Rubio

IREC, 13/05/2024
Valentin Lauret, Javi Fernandez-Lopez
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5/13/24

Que hacemos hoy?

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The theory: Bayes formula about conditional probability, a rapid overview.
In practice: Bayesian statistics and MCMC toolkit for ecologists.
Nimble: a flexible package to fit all the models you want.
Roe deer (chevreuil) poop GLM example with NIMBLE

The Theory

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Bayes’ formula about conditionnal probability \[\Pr(B \mid A) = \displaystyle{\frac{ \Pr(A \mid B) \; \Pr(B)}{\Pr(A)}}\]

What is the Bayesian approach?

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.

What is the Bayesian approach?

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.

Frequentist and Bayesian

Both are strategy to fit models. They are not statistical models

Typical stats problems involve estimating parameter \(\theta\) with available data.
The frequentist approach (maximum likelihood estimation – MLE) assumes that the parameters are fixed, but have unknown values to be estimated.
Classical estimates are generally point estimates of the parameters of interest.
The Bayesian approach assumes that the parameters are not fixed but have some fixed unknown distribution.

The Theory

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.

Bayesian Computation

Use stochastic simulations to avoid the calculation of multidimensional integrals.

Markov Chain Monte Carlo (MCMC) methods.

A Markov chain is a discrete sequence of states, in which the probability of an event depends only on the state in the previous event.
A Markov chain has an equilibrium (aka stationary) distribution that is the desired posterior distribution!

Several ways of constructing these chains: e.g., Metropolis-Hastings, Gibbs sampler.

Where everything begins: the prior distribution

For \(\theta\), the parameter you want to estimate, you must define an a priori distribution (see cheatsheet).
The prior distribution will define the possible values of \(\theta\) the model will explore.

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.

Where everything begins: the prior distribution

Over this example we define the prior distribution as \(\mbox{Normal}(0,1)\).

Then we start: MCMC functioning

Within the prior distribution, we define any possible initial value of the parameter to be estimated.

To decide where to visit next, the algorithm (called a sampler) propose to move away from the current value of the parameter to a candidate value.

We compute the ratio of the probabilities at the candidate and current locations \(R\). This is where \(\Pr(\text{data})\), the denominator of the Bayes theorem, cancels out.

MCMC functioning

If the candidate value improves the likelihood we accept it, otherwise we remain at the current location.

We repeat steps 2-4 a number of times or steps, iterations.

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Then, the chains of values explore the space of possibilities and converge to a stationary distribution, i.e: the posterior distribution.

MCMC functioning

Then, the chains of values explore the space of possibilities and converge to a stationary distribution, i.e: the posterior distribution.

Parametrize MCMC

3 chains starting from different initial values

Now with 3000 iterations.

MCMC functionning

Assessing convergence

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.

Burnin

We discard the MCMC iterations before convergence is achieved, i.e. burnin

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and just use observations from the chain once it has converged.

Efficient sample size and convergence

Efficient sample size (ESS). MCMC chains are autocorrelated, successive steps are near each other, and are not independant. ESS reflects the number of independant iterations, i.e. chain length while taking into account the autocorrelation. Check the ESS of any parameter of interest.
We need \(ess > 100\) independant steps.

coda::effectiveSize()

      b0       b1 
58.69595 59.15920

Gelman-Rubin test \(\hat{R}\) or Rhat to assess the mixing of chains. Values near \(1\) indicates likely convergence, a value of \(≤1.1\) is considered acceptable.

coda::gelman.diag()

What if we have issues of convergence ?

Increase burnin.
Sample more.
Consider informative priors.
Change initial values (e.g. estimates from simpler models).
Change the sampler.
Go for a simpler model.

Plot posterior distribution

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Named a density plot.

Plot posterior distribution

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Prior distribution

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Posterior distribution

Plot prior and posterior distribution

Plot posterior distribution

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Named a density plot.

You can extract the parameter value: mean, median

the precision associated: standard deviation, credible interval

To sum up with Bayesian statistics and MCMC

The Bayesian pipeline

Priors: define a prior distribution.
Initial values: pick starting values for the chain.
MCMC parameters: define nb of chains, nb of iterations, burnin, (thinning).
Check convergence: trace and density plots, Rhat, effective size.
Interpret the results: work with posterior distribution.

To sum up with Bayesian statistics and MCMC

The Bayesian pipeline

Priors: define a prior distribution.
Initial values: pick starting values for the chain.
MCMC parameters: define nb of chains, nb of iterations, burnin, (thinning).
Check convergence: trace and density plots, Rhat, effective size.
Interpret the results: work with posterior distribution.

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Oh yes, we also need to write a model! 😇

NIMBLE

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).

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A flexible programming system to write new analysis methods.

Load `nimble` package

Install NIMBLE package.

Download Xcode for MacOs or Rtools for Windows.
Visit https://r-nimble.org/download
and good luck ! 😬

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Then load NIMBLE

library(nimble)

The fictive roe deer poop example

We will use the GLM example of part 1.

Roe deer counts

In frequentist with glm()

mod <- glm(conteo ~ temp, data = datos, family = poisson(link = "log"))

    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

Step 1: write a model

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 (we will see that tomorrow).

In Bayesian with 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.

Code

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])
  }
  
})

Step 2: Read in data

To Nimble, not all “data” is data…

Constants:

Do never change
E.g. vector of known index values, variables used to define for-loops, etc.

Data:

Can be (re-)simulated within a model
E.g. stuff that only appears to the left of a “~”

Code

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))

Step 3: Initials values and parameters to save.

Specify initial values

Initial values can be fixed, then it will the same start for every MCMC chain
or Initial values can be obtain from a draw in a probability distribution.
It must be in the prior distribution

Which parameters to save?

Indicate all the parameter you want to save. Here, we are only interested in \(\beta_0\) and \(\beta_1\).

Code

\[~~\]

initial.values <- list(b0 = rnorm(1,0,1),
                       b1 = rnorm(1, 0, 1))

parameters.to.save <- c("b0", "b1")

Step 4: MCMC details

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Number of iterations
Length of burnin
Number of chains
Ratio of thinning

Code

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n.iter <- 3000
n.burnin <- 200
n.chains <- 3
n.thin <- 1

Sum up

Write a model
Read in data
Pick initial values
Specify parameters to save
Parametrize MCMC

Now, you’re ready to run! 🎉 💪

Code

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

Run model!

mcmc.output <- nimbleMCMC(code = glm.rd,     
                          data = my.data,  
                          constants = my.constants,
                          inits = initial.values,
                          monitors = parameters.to.save,
                          thin = n.thin,
                          niter = n.iter, 
                          nburnin = n.burnin, 
                          nchains = n.chains)

Explore MCMC outputs

str(mcmc.output)

List of 2
 $ chain1: num [1:4000, 1:2] 1.06 1.06 1.06 1.06 1.05 ...
  ..- attr(*, "dimnames")=List of 2
  .. ..$ : NULL
  .. ..$ : chr [1:2] "b0" "b1"
 $ chain2: num [1:4000, 1:2] 0.991 0.97 0.97 0.97 0.97 ...
  ..- attr(*, "dimnames")=List of 2
  .. ..$ : NULL
  .. ..$ : chr [1:2] "b0" "b1"

Explore MCMC outputs

mcmc.output$chain1

               b0        b1
   [1,] 1.0552801 0.4924236
   [2,] 1.0552801 0.4924236
   [3,] 1.0568864 0.4937307
   [4,] 1.0568864 0.4937307
   [5,] 1.0517870 0.4937307
   [6,] 1.0517870 0.4937307
   [7,] 1.0517870 0.4943446
   [8,] 1.0517870 0.4943446
   [9,] 1.0517870 0.4941333
  [10,] 1.0434708 0.4941333
  [11,] 1.0434708 0.4941333
  [12,] 1.0434708 0.4941333
  [13,] 1.0434708 0.4941333
  [14,] 1.0363103 0.4941333
  [15,] 1.0363103 0.4941333
  [16,] 1.0290184 0.4941333
  [17,] 1.0290184 0.4953782
  [18,] 1.0290184 0.4953782
  [19,] 1.0513797 0.4953782
  [20,] 1.0513797 0.4953782
  [21,] 1.0513797 0.4953782
  [22,] 1.0262853 0.4997459
  [23,] 1.0262853 0.4951756
  [24,] 1.0270733 0.4951756
  [25,] 1.0270733 0.4989521
  [26,] 1.0109282 0.4989521
  [27,] 0.9931602 0.4989521
  [28,] 0.9931602 0.5027903
  [29,] 0.9630684 0.5070866
  [30,] 0.9207093 0.5070866
  [31,] 0.9207093 0.5146613
  [32,] 0.9207093 0.5146613
  [33,] 0.8870311 0.5139844
  [34,] 0.8870311 0.5139844
  [35,] 0.8975645 0.5139844
  [36,] 0.8889593 0.5137728
  [37,] 0.8889593 0.5137728
  [38,] 0.8889593 0.5163884
  [39,] 0.8889593 0.5163884
  [40,] 0.8889593 0.5126824
  [41,] 0.8841225 0.5126824
  [42,] 0.9181226 0.5126824
  [43,] 0.8986903 0.5140703
  [44,] 0.8986903 0.5143884
  [45,] 0.8986903 0.5143884
  [46,] 0.8744352 0.5143884
  [47,] 0.8744352 0.5143884
  [48,] 0.8744352 0.5143884
  [49,] 0.8744352 0.5167918
  [50,] 0.8744352 0.5167918
  [51,] 0.8914429 0.5167918
  [52,] 0.8833844 0.5167918
  [53,] 0.8876445 0.5165377
  [54,] 0.8656432 0.5165377
  [55,] 0.8656432 0.5165377
  [56,] 0.8764725 0.5165377
  [57,] 0.8791676 0.5167646
  [58,] 0.8856460 0.5167646
  [59,] 0.8856460 0.5167646
  [60,] 0.8856460 0.5167646
  [61,] 0.8856460 0.5163110
  [62,] 0.8856460 0.5188620
  [63,] 0.8856460 0.5125562
  [64,] 0.8856460 0.5125562
  [65,] 0.8856460 0.5125562
  [66,] 0.8999257 0.5125562
  [67,] 0.8999257 0.5125562
  [68,] 0.8999257 0.5125562
  [69,] 0.8999257 0.5125562
  [70,] 0.8999257 0.5125562
  [71,] 0.8999257 0.5125562
  [72,] 0.8999257 0.5125562
  [73,] 0.8999257 0.5125562
  [74,] 0.8999257 0.5125562
  [75,] 0.8999257 0.5125562
  [76,] 0.9095753 0.5125562
  [77,] 0.9095753 0.5125562
  [78,] 0.9218319 0.5125562
  [79,] 0.9218319 0.5125562
  [80,] 0.9203864 0.5120229
  [81,] 0.9203864 0.5120229
  [82,] 0.9203864 0.5085626
  [83,] 0.9203864 0.5085626
  [84,] 0.9203864 0.5107559
  [85,] 0.9203864 0.5089782
  [86,] 0.9203864 0.5089782
  [87,] 0.9203864 0.5089782
  [88,] 0.9203864 0.5089782
  [89,] 0.9203864 0.5089782
  [90,] 0.9203864 0.5089782
  [91,] 0.9271155 0.5092824
  [92,] 0.9271155 0.5073463
  [93,] 0.9271155 0.5073463
  [94,] 0.9271155 0.5073463
  [95,] 0.9271155 0.5073463
  [96,] 0.9271155 0.5075708
  [97,] 0.9271155 0.5075708
  [98,] 0.9271155 0.5075708
  [99,] 0.9527457 0.5075708
 [100,] 0.9320396 0.5075708
 [101,] 0.9407688 0.5075708
 [102,] 0.9407688 0.5075708
 [103,] 0.9407688 0.5075708
 [104,] 0.9505194 0.5067987
 [105,] 0.9505194 0.5067987
 [106,] 0.9435924 0.5067987
 [107,] 0.9435924 0.5040832
 [108,] 0.9608891 0.5040832
 [109,] 0.9608891 0.5040832
 [110,] 0.9678189 0.5040832
 [111,] 0.9654640 0.5040832
 [112,] 0.9654640 0.5040832
 [113,] 0.9654640 0.5040832
 [114,] 0.9654640 0.5040832
 [115,] 0.9654640 0.5040832
 [116,] 0.9623944 0.5059515
 [117,] 0.9623944 0.5059515
 [118,] 0.9623944 0.5059515
 [119,] 0.9623944 0.5059515
 [120,] 0.9623944 0.5059515
 [121,] 0.9574138 0.5059515
 [122,] 0.9574138 0.5059515
 [123,] 0.9538291 0.5056437
 [124,] 0.9587684 0.5056437
 [125,] 0.9342405 0.5056437
 [126,] 0.9669729 0.5016769
 [127,] 0.9669729 0.5082864
 [128,] 0.9259729 0.5082864
 [129,] 0.9259729 0.5082864
 [130,] 0.9255417 0.5094198
 [131,] 0.9255417 0.5098379
 [132,] 0.9255417 0.5098379
 [133,] 0.9255417 0.5098379
 [134,] 0.9255417 0.5098379
 [135,] 0.9115461 0.5098379
 [136,] 0.9115461 0.5110312
 [137,] 0.9115461 0.5139385
 [138,] 0.9055878 0.5139385
 [139,] 0.8993496 0.5139385
 [140,] 0.9100028 0.5139385
 [141,] 0.9144343 0.5139385
 [142,] 0.9144343 0.5092001
 [143,] 0.9477063 0.5067919
 [144,] 0.9477063 0.5067919
 [145,] 0.9477063 0.5067919
 [146,] 0.9629912 0.5067919
 [147,] 0.9629912 0.5067919
 [148,] 0.9525723 0.5067919
 [149,] 0.9581460 0.5067919
 [150,] 0.9581460 0.5067919
 [151,] 0.9581460 0.5050991
 [152,] 0.9581460 0.5062042
 [153,] 0.9581460 0.5062042
 [154,] 0.9619220 0.5062042
 [155,] 0.9619220 0.5062042
 [156,] 0.9619220 0.5062042
 [157,] 0.9619220 0.5062042
 [158,] 0.9461817 0.5095787
 [159,] 0.9461817 0.5095787
 [160,] 0.9267057 0.5095787
 [161,] 0.9267057 0.5095787
 [162,] 0.9267057 0.5095787
 [163,] 0.9267057 0.5087704
 [164,] 0.9267057 0.5087704
 [165,] 0.9267057 0.5087704
 [166,] 0.9267057 0.5087704
 [167,] 0.9267057 0.5087704
 [168,] 0.9255132 0.5087704
 [169,] 0.9443933 0.5087704
 [170,] 0.9443933 0.5087704
 [171,] 0.9455089 0.5087704
 [172,] 0.9455089 0.5081079
 [173,] 0.9336007 0.5081079
 [174,] 0.9336007 0.5081079
 [175,] 0.9336007 0.5086657
 [176,] 0.9336007 0.5086657
 [177,] 0.9188879 0.5086657
 [178,] 0.9396748 0.5086657
 [179,] 0.9396748 0.5093254
 [180,] 0.9563718 0.5093254
 [181,] 0.9592107 0.5093254
 [182,] 0.9592107 0.5093254
 [183,] 0.9592107 0.5093254
 [184,] 0.9592107 0.5089409
 [185,] 0.9216101 0.5079858
 [186,] 0.9216101 0.5102577
 [187,] 0.9106663 0.5102577
 [188,] 0.9106663 0.5102577
 [189,] 0.9105645 0.5102577
 [190,] 0.9105645 0.5102577
 [191,] 0.9105645 0.5102577
 [192,] 0.9069448 0.5102577
 [193,] 0.9069448 0.5102577
 [194,] 0.9248541 0.5088106
 [195,] 0.9248541 0.5088106
 [196,] 0.9248541 0.5088106
 [197,] 0.9248541 0.5088106
 [198,] 0.9498188 0.5084346
 [199,] 0.9236720 0.5084346
 [200,] 0.9229870 0.5084346
 [201,] 0.9477319 0.5084346
 [202,] 0.9477319 0.5053621
 [203,] 0.9477319 0.5053621
 [204,] 0.9552997 0.5047333
 [205,] 0.9552997 0.5047333
 [206,] 0.9552997 0.5069826
 [207,] 0.9514649 0.5069826
 [208,] 0.9529861 0.5069826
 [209,] 0.9529861 0.5069826
 [210,] 0.9529861 0.5069826
 [211,] 0.9529861 0.5069826
 [212,] 0.9529861 0.5069826
 [213,] 0.9304337 0.5069826
 [214,] 0.9369050 0.5067840
 [215,] 0.9331589 0.5067840
 [216,] 0.9331589 0.5076624
 [217,] 0.9275649 0.5094877
 [218,] 0.9275649 0.5090898
 [219,] 0.9275649 0.5090898
 [220,] 0.9275649 0.5090898
 [221,] 0.9471128 0.5090898
 [222,] 0.9146363 0.5090898
 [223,] 0.9383960 0.5090898
 [224,] 0.9383960 0.5090898
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[1005,] 0.8633034 0.5170336
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[1007,] 0.8567567 0.5170336
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[1102,] 0.8971646 0.5152768
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[1111,] 0.9048000 0.5130811
[1112,] 0.9003705 0.5130811
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[1114,] 0.8920184 0.5122063
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[1687,] 1.0567342 0.4943447
[1688,] 1.0567342 0.4943447
[1689,] 1.0567342 0.4943447
[1690,] 1.0378874 0.4957350
[1691,] 1.0378874 0.4957350
[1692,] 1.0378874 0.4957350
[1693,] 1.0424716 0.4953462
[1694,] 1.0535876 0.4915501
[1695,] 1.0535876 0.4964071
[1696,] 1.0109010 0.5000088
[1697,] 1.0109010 0.5000088
[1698,] 1.0109010 0.5000088
[1699,] 0.9898475 0.5010480
[1700,] 0.9898475 0.5010480
[1701,] 0.9898475 0.5031480
[1702,] 0.9711366 0.5031480
[1703,] 0.9756277 0.5031480
[1704,] 0.9896856 0.5031480
[1705,] 0.9898026 0.5031480
[1706,] 0.9898026 0.5031480
[1707,] 0.9898026 0.5031480
[1708,] 0.9948165 0.5031480
[1709,] 0.9948165 0.5032703
[1710,] 0.9979959 0.4996009
[1711,] 1.0187953 0.4996009
[1712,] 1.0187953 0.4996009
[1713,] 1.0187953 0.4994982
[1714,] 1.0187953 0.4994982
[1715,] 1.0178639 0.4977231
[1716,] 1.0178639 0.5017203
[1717,] 1.0153395 0.4999762
[1718,] 1.0153395 0.4999762
[1719,] 1.0153395 0.4978734
[1720,] 1.0258100 0.4978734
[1721,] 1.0258100 0.4983847
[1722,] 1.0258100 0.4953449
[1723,] 1.0258100 0.4953449
[1724,] 1.0554081 0.4953449
[1725,] 1.0644700 0.4953449
[1726,] 1.0444154 0.4953449
[1727,] 1.0444154 0.4953449
[1728,] 1.0442989 0.4922532
[1729,] 1.0442989 0.4921179
[1730,] 1.0641845 0.4921179
[1731,] 1.0722252 0.4921179
[1732,] 1.0722252 0.4921179
[1733,] 1.0733127 0.4921179
[1734,] 1.0757179 0.4921179
[1735,] 1.0749355 0.4934231
[1736,] 1.0579004 0.4917446
[1737,] 1.0579004 0.4931234
[1738,] 1.0579004 0.4931234
[1739,] 1.0601744 0.4931234
[1740,] 1.0528775 0.4956039
[1741,] 1.0528775 0.4966458
[1742,] 1.0528775 0.4925265
[1743,] 1.0509544 0.4958086
[1744,] 1.0366216 0.4958086
[1745,] 1.0366216 0.4924681
[1746,] 1.0634939 0.4924681
[1747,] 1.0634939 0.4924681
[1748,] 1.0634939 0.4924681
[1749,] 1.0634939 0.4924681
[1750,] 1.0540840 0.4923659
[1751,] 1.0540840 0.4923659
[1752,] 1.0457533 0.4923659
[1753,] 1.0536047 0.4923659
[1754,] 1.0536047 0.4952315
[1755,] 1.0314511 0.4944229
[1756,] 1.0436962 0.4944229
[1757,] 1.0436962 0.4950945
[1758,] 1.0346052 0.4950945
[1759,] 1.0346052 0.4963018
[1760,] 1.0346052 0.4958639
[1761,] 1.0346052 0.4958639
[1762,] 1.0346052 0.4981894
[1763,] 1.0346052 0.4979031
[1764,] 1.0346052 0.4992008
[1765,] 1.0346052 0.4992008
[1766,] 1.0346052 0.4992008
[1767,] 1.0346052 0.4992008
[1768,] 1.0346052 0.4992008
[1769,] 1.0346052 0.4983934
[1770,] 1.0343340 0.4958391
[1771,] 1.0343340 0.4958391
[1772,] 1.0370738 0.4954641
[1773,] 1.0334590 0.4953603
[1774,] 1.0382331 0.4953603
[1775,] 1.0382331 0.4953603
[1776,] 1.0498006 0.4953603
[1777,] 1.0542553 0.4953603
[1778,] 1.0472404 0.4958699
[1779,] 1.0472404 0.4938930
[1780,] 1.0472404 0.4938930
[1781,] 1.0472404 0.4938930
[1782,] 1.0472404 0.4936417
[1783,] 1.0472404 0.4936417
[1784,] 1.0184717 0.4969040
[1785,] 1.0184717 0.4976391
[1786,] 1.0234283 0.4976391
[1787,] 1.0234283 0.4976391
[1788,] 1.0234283 0.4976391
[1789,] 1.0364394 0.4976391
[1790,] 1.0364394 0.4976391
[1791,] 1.0364394 0.4988382
[1792,] 1.0304173 0.4988382
[1793,] 1.0304173 0.4988382
[1794,] 1.0304173 0.4951508
[1795,] 1.0108239 0.4951508
[1796,] 1.0296765 0.4984729
[1797,] 1.0081919 0.4984729
[1798,] 1.0173235 0.4982322
[1799,] 1.0173235 0.4982322
[1800,] 1.0173235 0.4982322
[1801,] 1.0119060 0.4993188
[1802,] 1.0119060 0.4987602
[1803,] 1.0119060 0.4960424
[1804,] 1.0119060 0.4981167
[1805,] 1.0119060 0.4981167
[1806,] 1.0119060 0.4981167
[1807,] 1.0119060 0.4981167
[1808,] 1.0119060 0.4981167
[1809,] 1.0119060 0.4981167
[1810,] 1.0119060 0.4981167
[1811,] 1.0257655 0.4959499
[1812,] 1.0257655 0.4959499
[1813,] 1.0257655 0.4959499
[1814,] 1.0360315 0.4959499
[1815,] 1.0343544 0.4959499
[1816,] 1.0343544 0.4959499
[1817,] 1.0252628 0.4961600
[1818,] 1.0057629 0.5016439
[1819,] 0.9789736 0.5037509
[1820,] 0.9789736 0.5037509
[1821,] 0.9741449 0.5037509
[1822,] 0.9971262 0.5006111
[1823,] 0.9859804 0.5006111
[1824,] 0.9859804 0.5021947
[1825,] 0.9859804 0.5021947
[1826,] 0.9859804 0.5028002
[1827,] 0.9859804 0.5028002
[1828,] 0.9758822 0.5028002
[1829,] 0.9758822 0.5028002
[1830,] 0.9758822 0.5028002
[1831,] 0.9758822 0.5028961
[1832,] 0.9758822 0.5081354
[1833,] 0.9548414 0.5081354
[1834,] 0.9548414 0.5049801
[1835,] 0.9548414 0.5049801
[1836,] 0.9548414 0.5061056
[1837,] 0.9548414 0.5061056
[1838,] 0.9521376 0.5049628
[1839,] 0.9672149 0.5049873
[1840,] 0.9689546 0.5049873
[1841,] 0.9689546 0.5049873
[1842,] 0.9689546 0.5032528
[1843,] 0.9742252 0.5033486
[1844,] 0.9742252 0.5033486
[1845,] 0.9742252 0.5047582
[1846,] 0.9444883 0.5060397
[1847,] 0.9444883 0.5060397
[1848,] 0.9444883 0.5060397
[1849,] 0.9444883 0.5065755
[1850,] 0.9444883 0.5064825
[1851,] 0.9444883 0.5064825
[1852,] 0.9444883 0.5064825
[1853,] 0.9588756 0.5064825
[1854,] 0.9588756 0.5057789
[1855,] 0.9463020 0.5081722
[1856,] 0.9463020 0.5081722
[1857,] 0.9431348 0.5081722
[1858,] 0.9431348 0.5081722
[1859,] 0.9215505 0.5081722
[1860,] 0.9215505 0.5095987
[1861,] 0.9215505 0.5095987
[1862,] 0.9215505 0.5111669
[1863,] 0.9095079 0.5111669
[1864,] 0.9095079 0.5143792
[1865,] 0.9122365 0.5138484
[1866,] 0.9122365 0.5138484
[1867,] 0.8755742 0.5138484
[1868,] 0.8755742 0.5142270
[1869,] 0.8864549 0.5142270
[1870,] 0.8878095 0.5115305
[1871,] 0.9018848 0.5116974
[1872,] 0.9150244 0.5116974
[1873,] 0.9087070 0.5116974
[1874,] 0.9087070 0.5116974
[1875,] 0.9087070 0.5116974
[1876,] 0.9087070 0.5116974
[1877,] 0.9283300 0.5116974
[1878,] 0.8972240 0.5136486
[1879,] 0.9164536 0.5136486
[1880,] 0.9164536 0.5128168
[1881,] 0.9164536 0.5128168
[1882,] 0.8967362 0.5128168
[1883,] 0.8828406 0.5166594
[1884,] 0.8828406 0.5166594
[1885,] 0.8828406 0.5166594
[1886,] 0.8543930 0.5166594
[1887,] 0.8773033 0.5153936
[1888,] 0.8773033 0.5153936
[1889,] 0.8852278 0.5185889
[1890,] 0.8681723 0.5185889
[1891,] 0.8681723 0.5185889
[1892,] 0.8681723 0.5165506
[1893,] 0.8681723 0.5141469
[1894,] 0.8681723 0.5141469
[1895,] 0.8830398 0.5141469
[1896,] 0.9098347 0.5141469
[1897,] 0.9098347 0.5141469
[1898,] 0.9098347 0.5141469
[1899,] 0.8885464 0.5141469
[1900,] 0.8885464 0.5141469
[1901,] 0.8789187 0.5158466
[1902,] 0.8789187 0.5158466
[1903,] 0.8781689 0.5147783
[1904,] 0.8781689 0.5169870
[1905,] 0.8781689 0.5169870
[1906,] 0.8743291 0.5169870
[1907,] 0.8743291 0.5169870
[1908,] 0.8743291 0.5173506
[1909,] 0.8743291 0.5173506
[1910,] 0.8817369 0.5173506
[1911,] 0.8599816 0.5167454
[1912,] 0.8729389 0.5148041
[1913,] 0.8995184 0.5148041
[1914,] 0.8995184 0.5156378
[1915,] 0.8995184 0.5156378
[1916,] 0.8618950 0.5156378
[1917,] 0.8618950 0.5202238
[1918,] 0.8383806 0.5202238
[1919,] 0.8383806 0.5202238
[1920,] 0.8383806 0.5202238
[1921,] 0.8302967 0.5202238
[1922,] 0.8302967 0.5194075
[1923,] 0.8706423 0.5194075
[1924,] 0.8668484 0.5163751
[1925,] 0.8668484 0.5166944
[1926,] 0.8710359 0.5166944
[1927,] 0.8710359 0.5166944
[1928,] 0.8710359 0.5166944
[1929,] 0.8763926 0.5173723
[1930,] 0.8763926 0.5173723
[1931,] 0.8763926 0.5150392
[1932,] 0.8990951 0.5134869
[1933,] 0.8986516 0.5134869
[1934,] 0.8986516 0.5134869
[1935,] 0.8993237 0.5134869
[1936,] 0.8993237 0.5134869
[1937,] 0.8820332 0.5134869
[1938,] 0.8820332 0.5134869
[1939,] 0.8847260 0.5134869
[1940,] 0.8847260 0.5134869
[1941,] 0.8847260 0.5168704
[1942,] 0.8827581 0.5149884
[1943,] 0.8867680 0.5149884
[1944,] 0.8960102 0.5149884
[1945,] 0.8960102 0.5149884
[1946,] 0.8768192 0.5149884
[1947,] 0.8682365 0.5169211
[1948,] 0.8682365 0.5169211
[1949,] 0.8585157 0.5169211
[1950,] 0.8855020 0.5169211
[1951,] 0.8855020 0.5169211
[1952,] 0.8794865 0.5169211
[1953,] 0.8794865 0.5169211
[1954,] 0.8850838 0.5169211
[1955,] 0.8830718 0.5124706
[1956,] 0.8830718 0.5124706
[1957,] 0.9406888 0.5124706
[1958,] 0.9406888 0.5119912
[1959,] 0.9406888 0.5119912
[1960,] 0.9246589 0.5119912
[1961,] 0.9246589 0.5119912
[1962,] 0.9131416 0.5119912
[1963,] 0.9131416 0.5132272
[1964,] 0.8850049 0.5132272
[1965,] 0.8850049 0.5132272
[1966,] 0.8850049 0.5132272
[1967,] 0.8850049 0.5143667
[1968,] 0.8850049 0.5143667
[1969,] 0.8850049 0.5143667
[1970,] 0.8978854 0.5143667
[1971,] 0.8978854 0.5143667
[1972,] 0.8845361 0.5143667
[1973,] 0.8960588 0.5178214
[1974,] 0.8368006 0.5178214
[1975,] 0.8368006 0.5211169
[1976,] 0.8368006 0.5211169
[1977,] 0.8442571 0.5211169
[1978,] 0.8226570 0.5211169
[1979,] 0.8226570 0.5224712
[1980,] 0.8226570 0.5243744
[1981,] 0.8361059 0.5243744
[1982,] 0.8403768 0.5217309
[1983,] 0.8166686 0.5203706
[1984,] 0.8448800 0.5203706
[1985,] 0.8448800 0.5186539
[1986,] 0.8525694 0.5186539
[1987,] 0.8525694 0.5186539
[1988,] 0.8525694 0.5195728
[1989,] 0.8525694 0.5190762
[1990,] 0.8722149 0.5154846
[1991,] 0.8722149 0.5154846
[1992,] 0.8792930 0.5154846
[1993,] 0.8913496 0.5160611
[1994,] 0.8705802 0.5160611
[1995,] 0.8705802 0.5160611
[1996,] 0.8705802 0.5160611
[1997,] 0.8767948 0.5160611
[1998,] 0.8767948 0.5160611
[1999,] 0.8805270 0.5160611
[2000,] 0.8799504 0.5160611
[2001,] 0.8799504 0.5163211
[2002,] 0.8799504 0.5169527
[2003,] 0.8799504 0.5136695
[2004,] 0.8799504 0.5136695
[2005,] 0.8799504 0.5171804
[2006,] 0.8566175 0.5168277
[2007,] 0.8566175 0.5203194
[2008,] 0.8441332 0.5203194
[2009,] 0.8441332 0.5203194
[2010,] 0.8441332 0.5203194
[2011,] 0.8441332 0.5203194
[2012,] 0.8441332 0.5203194
[2013,] 0.8441332 0.5213140
[2014,] 0.8276458 0.5230474
[2015,] 0.8276458 0.5230474
[2016,] 0.8276458 0.5194911
[2017,] 0.8288799 0.5232746
[2018,] 0.8288799 0.5206796
[2019,] 0.8288799 0.5206796
[2020,] 0.8375038 0.5206796
[2021,] 0.8375038 0.5206796
[2022,] 0.8375038 0.5206568
[2023,] 0.8262982 0.5206568
[2024,] 0.8275642 0.5206568
[2025,] 0.8357882 0.5206568
[2026,] 0.8377768 0.5206568
[2027,] 0.8443525 0.5206568
[2028,] 0.8443525 0.5175631
[2029,] 0.8443525 0.5175631
[2030,] 0.8675817 0.5175631
[2031,] 0.8675817 0.5175631
[2032,] 0.8675817 0.5182795
[2033,] 0.8675817 0.5182795
[2034,] 0.8675817 0.5182795
[2035,] 0.8675817 0.5166252
[2036,] 0.8675817 0.5166252
[2037,] 0.8751237 0.5166252
[2038,] 0.8912117 0.5166252
[2039,] 0.8912117 0.5166252
[2040,] 0.8791103 0.5143899
[2041,] 0.8791103 0.5151664
[2042,] 0.8894147 0.5151664
[2043,] 0.8894147 0.5151664
[2044,] 0.8762455 0.5172681
[2045,] 0.8762455 0.5172681
[2046,] 0.8762455 0.5172681
[2047,] 0.8709233 0.5172681
[2048,] 0.8689115 0.5172681
[2049,] 0.8689115 0.5172681
[2050,] 0.8689115 0.5173427
[2051,] 0.8689115 0.5165971
[2052,] 0.8689115 0.5156667
[2053,] 0.8679419 0.5145054
[2054,] 0.8679419 0.5145054
[2055,] 0.8772701 0.5145054
[2056,] 0.8954754 0.5145054
[2057,] 0.8954754 0.5101378
[2058,] 0.8955570 0.5101378
[2059,] 0.8955570 0.5101378
[2060,] 0.8955570 0.5129856
[2061,] 0.8679207 0.5129856
[2062,] 0.9102206 0.5129856
[2063,] 0.9112283 0.5122291
[2064,] 0.9112283 0.5132340
[2065,] 0.9110485 0.5103780
[2066,] 0.9110485 0.5103780
[2067,] 0.9150334 0.5103780
[2068,] 0.9254530 0.5103780
[2069,] 0.9089087 0.5103780
[2070,] 0.9089087 0.5103780
[2071,] 0.9169508 0.5112470
[2072,] 0.9227466 0.5112470
[2073,] 0.9227466 0.5112470
[2074,] 0.8953032 0.5112470
[2075,] 0.8953032 0.5122568
[2076,] 0.8953032 0.5142139
[2077,] 0.8910408 0.5142139
[2078,] 0.8910408 0.5142139
[2079,] 0.8640978 0.5142139
[2080,] 0.8759998 0.5142139
[2081,] 0.8914446 0.5114117
[2082,] 0.8914446 0.5118786
[2083,] 0.9185158 0.5118786
[2084,] 0.9325251 0.5114247
[2085,] 0.9325251 0.5114247
[2086,] 0.8924560 0.5134325
[2087,] 0.8924560 0.5157719
[2088,] 0.8800137 0.5156645
[2089,] 0.9043956 0.5160704
[2090,] 0.8743642 0.5160704
[2091,] 0.8743642 0.5160704
[2092,] 0.8743642 0.5160704
[2093,] 0.8743642 0.5161872
[2094,] 0.8743642 0.5166707
[2095,] 0.8820217 0.5166707
[2096,] 0.8820217 0.5166707
[2097,] 0.8820217 0.5189422
[2098,] 0.8820217 0.5189422
[2099,] 0.8820217 0.5174328
[2100,] 0.8820217 0.5174328
[2101,] 0.8471115 0.5174328
[2102,] 0.8582537 0.5174328
[2103,] 0.8582537 0.5174328
[2104,] 0.8566917 0.5174328
[2105,] 0.8542303 0.5174328
[2106,] 0.8392747 0.5174328
[2107,] 0.8392747 0.5174328
[2108,] 0.8513414 0.5174328
[2109,] 0.8513414 0.5174328
[2110,] 0.8513414 0.5182527
[2111,] 0.8929435 0.5182527
[2112,] 0.8929435 0.5107478
[2113,] 0.8937792 0.5107478
[2114,] 0.8980918 0.5135137
[2115,] 0.8980918 0.5135137
[2116,] 0.8980918 0.5139634
[2117,] 0.8985361 0.5139634
[2118,] 0.8985361 0.5139634
[2119,] 0.8873173 0.5139634
[2120,] 0.8873173 0.5165059
[2121,] 0.8873173 0.5166724
[2122,] 0.8873173 0.5141159
[2123,] 0.8873173 0.5166531
[2124,] 0.8723290 0.5166531
[2125,] 0.8723290 0.5166531
[2126,] 0.8753953 0.5181204
[2127,] 0.8753953 0.5139047
[2128,] 0.8741117 0.5177836
[2129,] 0.8741117 0.5177836
[2130,] 0.8741117 0.5177836
[2131,] 0.8829139 0.5146345
[2132,] 0.8829139 0.5146345
[2133,] 0.8829139 0.5165434
[2134,] 0.8555862 0.5164426
[2135,] 0.8555862 0.5164426
[2136,] 0.8555862 0.5180883
[2137,] 0.8555862 0.5190131
[2138,] 0.8516452 0.5187449
[2139,] 0.8516452 0.5187449
[2140,] 0.8516452 0.5187449
[2141,] 0.8516452 0.5187449
[2142,] 0.8396437 0.5174380
[2143,] 0.8684368 0.5182314
[2144,] 0.8684368 0.5182314
[2145,] 0.8820827 0.5133104
[2146,] 0.8897455 0.5157628
[2147,] 0.8897455 0.5157628
[2148,] 0.8897455 0.5146593
[2149,] 0.8897455 0.5146593
[2150,] 0.9074114 0.5094911
[2151,] 0.9446480 0.5094911
[2152,] 0.9424818 0.5108680
[2153,] 0.9218995 0.5108680
[2154,] 0.8982932 0.5108680
[2155,] 0.8982932 0.5108680
[2156,] 0.8982932 0.5119443
[2157,] 0.8982932 0.5138481
[2158,] 0.8982932 0.5138481
[2159,] 0.8982932 0.5159906
[2160,] 0.8982932 0.5146634
[2161,] 0.8996115 0.5146634
[2162,] 0.8813786 0.5146634
[2163,] 0.8813786 0.5150852
[2164,] 0.8982119 0.5151565
[2165,] 0.8982119 0.5128016
[2166,] 0.8982119 0.5128016
[2167,] 0.9014735 0.5128016
[2168,] 0.9014735 0.5130317
[2169,] 0.9014735 0.5140833
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[2239,] 0.9670480 0.5066168
[2240,] 0.9670480 0.5066168
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[2250,] 0.9612360 0.5066972
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[2299,] 0.9919382 0.5012587
[2300,] 0.9919382 0.5012587
[2301,] 0.9919382 0.5012587
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[2303,] 1.0029592 0.4982584
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[2314,] 0.9832207 0.5020545
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[2321,] 0.9752428 0.5043640
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[2346,] 0.9254569 0.5070834
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[2365,] 0.9145912 0.5100157
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[2414,] 0.8432309 0.5201767
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[2422,] 0.8548614 0.5202816
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[2429,] 0.8626770 0.5199592
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[2453,] 0.8881497 0.5161530
[2454,] 0.8881497 0.5161530
[2455,] 0.8881497 0.5160448
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[2457,] 0.8881497 0.5160448
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[2522,] 0.9873462 0.5040607
[2523,] 0.9873462 0.5040607
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[2592,] 0.9013095 0.5129056
[2593,] 0.9013095 0.5129056
[2594,] 0.9013095 0.5129056
[2595,] 0.9013095 0.5129056
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[2601,] 0.9026933 0.5147753
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[2623,] 0.9303783 0.5127569
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[3596,] 0.9432160 0.5086781
[3597,] 0.9490384 0.5060327
[3598,] 0.9511582 0.5080060
[3599,] 0.9511582 0.5080060
[3600,] 0.9511582 0.5080060
[3601,] 0.9351084 0.5080060
[3602,] 0.9491950 0.5080060
[3603,] 0.9406509 0.5080060
[3604,] 0.9406509 0.5071548
[3605,] 0.9466310 0.5083394
[3606,] 0.9466310 0.5083394
[3607,] 0.9374968 0.5083394
[3608,] 0.9282393 0.5083394
[3609,] 0.9282393 0.5090705
[3610,] 0.9282393 0.5090705
[3611,] 0.9282393 0.5109390
[3612,] 0.9282393 0.5100533
[3613,] 0.9282393 0.5100533
[3614,] 0.9065212 0.5116432
[3615,] 0.9065212 0.5109007
[3616,] 0.9176505 0.5128301
[3617,] 0.9176505 0.5128301
[3618,] 0.9079713 0.5141636
[3619,] 0.9079713 0.5141636
[3620,] 0.9079713 0.5141636
[3621,] 0.8912645 0.5141636
[3622,] 0.8899582 0.5141636
[3623,] 0.8899582 0.5141636
[3624,] 0.8852487 0.5141636
[3625,] 0.8770847 0.5147322
[3626,] 0.8953855 0.5122654
[3627,] 0.9033297 0.5122654
[3628,] 0.9128161 0.5122654
[3629,] 0.9202917 0.5122654
[3630,] 0.9142209 0.5122654
[3631,] 0.9142209 0.5110277
[3632,] 0.9232904 0.5094023
[3633,] 0.9232083 0.5094023
[3634,] 0.9310693 0.5094023
[3635,] 0.9310693 0.5094023
[3636,] 0.9367084 0.5094023
[3637,] 0.9367084 0.5094023
[3638,] 0.9259564 0.5094023
[3639,] 0.9259564 0.5087179
[3640,] 0.9370215 0.5087179
[3641,] 0.9370215 0.5087179
[3642,] 0.9319180 0.5077511
[3643,] 0.9227319 0.5094457
[3644,] 0.9396717 0.5094457
[3645,] 0.9377908 0.5056829
[3646,] 0.9377908 0.5056829
[3647,] 0.9377908 0.5056829
[3648,] 0.9750468 0.5033450
[3649,] 0.9939265 0.5033450
[3650,] 0.9841300 0.5033450
[3651,] 0.9841300 0.5042581
[3652,] 0.9841300 0.5042581
[3653,] 0.9747683 0.5042581
[3654,] 0.9747683 0.5042581
[3655,] 0.9747683 0.5049824
[3656,] 0.9747683 0.5068992
[3657,] 0.9313563 0.5082534
[3658,] 0.9109509 0.5087865
[3659,] 0.9473050 0.5072242
[3660,] 0.9592872 0.5023862
[3661,] 0.9592872 0.5020979
[3662,] 0.9592872 0.5014380
[3663,] 0.9654791 0.5014380
[3664,] 0.9654791 0.5014380
[3665,] 0.9773219 0.5004495
[3666,] 1.0098230 0.5004495
[3667,] 1.0098230 0.4952237
[3668,] 1.0019469 0.4995489
[3669,] 1.0096532 0.4995794
[3670,] 0.9983495 0.4995794
[3671,] 0.9983495 0.5033187
[3672,] 0.9983495 0.5033187
[3673,] 0.9983495 0.5033187
[3674,] 0.9737729 0.5033187
[3675,] 0.9737729 0.5044289
[3676,] 0.9737729 0.5062837
[3677,] 0.9327977 0.5062837
[3678,] 0.9327977 0.5086786
[3679,] 0.9465375 0.5042838
[3680,] 0.9465375 0.5042383
[3681,] 0.9451414 0.5095241
[3682,] 0.9486063 0.5095241
[3683,] 0.9356013 0.5095241
[3684,] 0.9356013 0.5095241
[3685,] 0.9356013 0.5095054
[3686,] 0.9356013 0.5095054
[3687,] 0.9356013 0.5080316
[3688,] 0.9380167 0.5080316
[3689,] 0.9380167 0.5100606
[3690,] 0.9380167 0.5111962
[3691,] 0.9380167 0.5103946
[3692,] 0.9238733 0.5103946
[3693,] 0.9233935 0.5103946
[3694,] 0.9233935 0.5103946
[3695,] 0.9092715 0.5143175
[3696,] 0.8853111 0.5133070
[3697,] 0.8928239 0.5133070
[3698,] 0.9048455 0.5133070
[3699,] 0.9048455 0.5133070
[3700,] 0.9186641 0.5125475
[3701,] 0.8946788 0.5124199
[3702,] 0.9069136 0.5124199
[3703,] 0.9069136 0.5124199
[3704,] 0.8959458 0.5127837
[3705,] 0.8959458 0.5142537
[3706,] 0.8959458 0.5124867
[3707,] 0.8959458 0.5124867
[3708,] 0.8959458 0.5138238
[3709,] 0.8959458 0.5138238
[3710,] 0.9040954 0.5138238
[3711,] 0.9040954 0.5138238
[3712,] 0.8927658 0.5138238
[3713,] 0.8659265 0.5138238
[3714,] 0.8892335 0.5138238
[3715,] 0.8892335 0.5138238
[3716,] 0.8892335 0.5138238
[3717,] 0.9081429 0.5140215
[3718,] 0.9081429 0.5130272
[3719,] 0.8982988 0.5130272
[3720,] 0.8982988 0.5130862
[3721,] 0.8943710 0.5149435
[3722,] 0.8996458 0.5099609
[3723,] 0.8996458 0.5168644
[3724,] 0.8996458 0.5152636
[3725,] 0.9026282 0.5140030
[3726,] 0.9026282 0.5135666
[3727,] 0.9026282 0.5135666
[3728,] 0.9026282 0.5135666
[3729,] 0.9026282 0.5135666
[3730,] 0.9026282 0.5135666
[3731,] 0.8843140 0.5135666
[3732,] 0.9057374 0.5136002
[3733,] 0.8990393 0.5136002
[3734,] 0.9070177 0.5136002
[3735,] 0.9048308 0.5145065
[3736,] 0.9048308 0.5137686
[3737,] 0.8844595 0.5136853
[3738,] 0.9027296 0.5136853
[3739,] 0.9027296 0.5136853
[3740,] 0.9027296 0.5144439
[3741,] 0.8823888 0.5151506
[3742,] 0.8823888 0.5151506
[3743,] 0.8823888 0.5151506
[3744,] 0.8657229 0.5151506
[3745,] 0.8657229 0.5151506
[3746,] 0.8845914 0.5151506
[3747,] 0.8845914 0.5151506
[3748,] 0.9000818 0.5147930
[3749,] 0.9000818 0.5147930
[3750,] 0.9058198 0.5130906
[3751,] 0.9058198 0.5126576
[3752,] 0.9006410 0.5126576
[3753,] 0.9006410 0.5126576
[3754,] 0.9006410 0.5126576
[3755,] 0.9096812 0.5114256
[3756,] 0.9046996 0.5114256
[3757,] 0.9046996 0.5114256
[3758,] 0.9046996 0.5131630
[3759,] 0.9058082 0.5131630
[3760,] 0.9056435 0.5131630
[3761,] 0.9056435 0.5131630
[3762,] 0.9006357 0.5131630
[3763,] 0.8945754 0.5124273
[3764,] 0.8972088 0.5124273
[3765,] 0.8972088 0.5124273
[3766,] 0.9304374 0.5116088
[3767,] 0.9304374 0.5101090
[3768,] 0.9200316 0.5096661
[3769,] 0.9152425 0.5096661
[3770,] 0.9207807 0.5096661
[3771,] 0.9207807 0.5096661
[3772,] 0.9134455 0.5112581
[3773,] 0.9286333 0.5112581
[3774,] 0.8964185 0.5112581
[3775,] 0.8964185 0.5112581
[3776,] 0.9039985 0.5103718
[3777,] 0.9447222 0.5097610
[3778,] 0.9447222 0.5060605
[3779,] 0.9652877 0.5075392
[3780,] 0.9398098 0.5075392
[3781,] 0.9398098 0.5075392
[3782,] 0.9398098 0.5075392
[3783,] 0.9398098 0.5075392
[3784,] 0.9398098 0.5053279
[3785,] 0.9398098 0.5063068
[3786,] 0.9398098 0.5072913
[3787,] 0.9413007 0.5072913
[3788,] 0.9413007 0.5072913
[3789,] 0.9413007 0.5055490
[3790,] 0.9432521 0.5057915
[3791,] 0.9432521 0.5057915
[3792,] 0.9667552 0.5057915
[3793,] 0.9667552 0.5057915
[3794,] 0.9459301 0.5057915
[3795,] 0.9361748 0.5087104
[3796,] 0.9476035 0.5087104
[3797,] 0.9565351 0.5087104
[3798,] 0.9290613 0.5087104
[3799,] 0.9290613 0.5104720
[3800,] 0.9211114 0.5104720
[3801,] 0.9313409 0.5104720
[3802,] 0.9313409 0.5104720
[3803,] 0.9313409 0.5104720
[3804,] 0.9313409 0.5104720
[3805,] 0.9313409 0.5104720
[3806,] 0.9391063 0.5055187
[3807,] 0.9391063 0.5074202
[3808,] 0.9391063 0.5074202
[3809,] 0.9391063 0.5074202
[3810,] 0.9427695 0.5074202
[3811,] 0.9247887 0.5074202
[3812,] 0.9247887 0.5074202
[3813,] 0.9247887 0.5074202
[3814,] 0.9308803 0.5064129
[3815,] 0.9800224 0.5051000
[3816,] 0.9800224 0.5007279
[3817,] 0.9800224 0.5007279
[3818,] 0.9726284 0.5025229
[3819,] 0.9588888 0.5053635
[3820,] 0.9775807 0.5053635
[3821,] 0.9670398 0.5051830
[3822,] 0.9670398 0.5041896
[3823,] 0.9670398 0.5041896
[3824,] 0.9670398 0.5033827
[3825,] 0.9670398 0.5033827
[3826,] 0.9670398 0.5033827
[3827,] 0.9670398 0.5033827
[3828,] 0.9670576 0.5033827
[3829,] 0.9670576 0.5069946
[3830,] 0.9670576 0.5069946
[3831,] 0.9457267 0.5076036
[3832,] 0.9591698 0.5051330
[3833,] 0.9615385 0.5035502
[3834,] 0.9615385 0.5018942
[3835,] 0.9615385 0.5018942
[3836,] 0.9615385 0.5032689
[3837,] 0.9588338 0.5032689
[3838,] 0.9834204 0.5022304
[3839,] 0.9834204 0.5022304
[3840,] 0.9725309 0.5022304
[3841,] 0.9725309 0.5031195
[3842,] 0.9725309 0.5032792
[3843,] 0.9725309 0.5023034
[3844,] 0.9725309 0.5023034
[3845,] 0.9725309 0.5023034
[3846,] 0.9588829 0.5039383
[3847,] 0.9532009 0.5081777
[3848,] 0.9532009 0.5065438
[3849,] 0.9532009 0.5059022
[3850,] 0.9532009 0.5071662
[3851,] 0.9532009 0.5071662
[3852,] 0.9452011 0.5071662
[3853,] 0.9452011 0.5098192
[3854,] 0.9452011 0.5093787
[3855,] 0.9109627 0.5093787
[3856,] 0.9109627 0.5084191
[3857,] 0.9158774 0.5101904
[3858,] 0.9158774 0.5101904
[3859,] 0.9314109 0.5119203
[3860,] 0.9314109 0.5105741
[3861,] 0.9314109 0.5105741
[3862,] 0.9314109 0.5089443
[3863,] 0.9314109 0.5089443
[3864,] 0.9314109 0.5089443
[3865,] 0.9199248 0.5087519
[3866,] 0.9428967 0.5094577
[3867,] 0.9428967 0.5094577
[3868,] 0.9428967 0.5094577
[3869,] 0.9458918 0.5094577
[3870,] 0.9458918 0.5094577
[3871,] 0.9458918 0.5094577
[3872,] 0.9252802 0.5094577
[3873,] 0.9280738 0.5094577
[3874,] 0.9280738 0.5094577
[3875,] 0.9280738 0.5095709
[3876,] 0.9260246 0.5096955
[3877,] 0.9260246 0.5107031
[3878,] 0.9260246 0.5102183
[3879,] 0.9260246 0.5102183
[3880,] 0.9151391 0.5120975
[3881,] 0.9037870 0.5115242
[3882,] 0.9037870 0.5128469
[3883,] 0.9037870 0.5138823
[3884,] 0.9037870 0.5108289
[3885,] 0.9316117 0.5105196
[3886,] 0.9316117 0.5076463
[3887,] 0.9377909 0.5080111
[3888,] 0.9377909 0.5080111
[3889,] 0.9377909 0.5080111
[3890,] 0.9463065 0.5086188
[3891,] 0.9463065 0.5095562
[3892,] 0.9294089 0.5108655
[3893,] 0.9294089 0.5108655
[3894,] 0.9057292 0.5108655
[3895,] 0.9057292 0.5108655
[3896,] 0.9057292 0.5122251
[3897,] 0.9057292 0.5122251
[3898,] 0.9116764 0.5122590
[3899,] 0.9116764 0.5122590
[3900,] 0.9116764 0.5122590
[3901,] 0.9099974 0.5122590
[3902,] 0.9099974 0.5122590
[3903,] 0.9099974 0.5125428
[3904,] 0.9099974 0.5121258
[3905,] 0.9099974 0.5121258
[3906,] 0.9099974 0.5121258
[3907,] 0.8910595 0.5136938
[3908,] 0.8910595 0.5157261
[3909,] 0.8910595 0.5148379
[3910,] 0.8910595 0.5148379
[3911,] 0.8878642 0.5148379
[3912,] 0.8878642 0.5148379
[3913,] 0.8878642 0.5148379
[3914,] 0.8878642 0.5148379
[3915,] 0.8878642 0.5158357
[3916,] 0.8799543 0.5158357
[3917,] 0.8799543 0.5166797
[3918,] 0.8794454 0.5166797
[3919,] 0.8794454 0.5166797
[3920,] 0.8794454 0.5166797
[3921,] 0.8794454 0.5166797
[3922,] 0.8794454 0.5175928
[3923,] 0.8650071 0.5194363
[3924,] 0.8650071 0.5161449
[3925,] 0.8650071 0.5161449
[3926,] 0.8650071 0.5185921
[3927,] 0.8464123 0.5185921
[3928,] 0.8464123 0.5185921
[3929,] 0.8464123 0.5191832
[3930,] 0.8630494 0.5192876
[3931,] 0.8630494 0.5181466
[3932,] 0.8630494 0.5181466
[3933,] 0.8719825 0.5169622
[3934,] 0.8719825 0.5174370
[3935,] 0.8717386 0.5152911
[3936,] 0.9007105 0.5117690
[3937,] 0.9007105 0.5117690
[3938,] 0.9217067 0.5117690
[3939,] 0.8936217 0.5155226
[3940,] 0.8865791 0.5148577
[3941,] 0.8865791 0.5129407
[3942,] 0.8865791 0.5132100
[3943,] 0.8942889 0.5148645
[3944,] 0.8942889 0.5147149
[3945,] 0.9000369 0.5147149
[3946,] 0.9051136 0.5138679
[3947,] 0.9051136 0.5142057
[3948,] 0.9051136 0.5153624
[3949,] 0.9051136 0.5153624
[3950,] 0.9051136 0.5115375
[3951,] 0.9309024 0.5069450
[3952,] 0.9463091 0.5069450
[3953,] 0.9463091 0.5042630
[3954,] 0.9463091 0.5042630
[3955,] 0.9674522 0.5042505
[3956,] 0.9674522 0.5070203
[3957,] 0.9674522 0.5052467
[3958,] 0.9629289 0.5052467
[3959,] 0.9724519 0.5052467
[3960,] 0.9724519 0.5052467
[3961,] 0.9467447 0.5067948
[3962,] 0.9467447 0.5067948
[3963,] 0.9467447 0.5102343
[3964,] 0.9467447 0.5068050
[3965,] 0.9467447 0.5068050
[3966,] 0.9467447 0.5079753
[3967,] 0.9528844 0.5084164
[3968,] 0.9528844 0.5065172
[3969,] 0.9528844 0.5065172
[3970,] 0.9528844 0.5044954
[3971,] 0.9528844 0.5035895
[3972,] 0.9794295 0.5066883
[3973,] 0.9794295 0.5066883
[3974,] 0.9794295 0.5066579
[3975,] 0.9794295 0.5000499
[3976,] 1.0035772 0.5000499
[3977,] 1.0231203 0.5000499
[3978,] 1.0231203 0.5000499
[3979,] 1.0231203 0.5000499
[3980,] 1.0229641 0.5000499
[3981,] 0.9983591 0.5000499
[3982,] 0.9983591 0.4985262
[3983,] 0.9903743 0.5018820
[3984,] 1.0077554 0.5018820
[3985,] 1.0022037 0.5018820
[3986,] 1.0022037 0.5018820
[3987,] 0.9972220 0.5018820
[3988,] 0.9749074 0.5032865
[3989,] 0.9749074 0.5049714
[3990,] 0.9557249 0.5057876
[3991,] 0.9557249 0.5057876
[3992,] 0.9557249 0.5057876
[3993,] 0.9579220 0.5056419
[3994,] 0.9579220 0.5072729
[3995,] 0.9579220 0.5072729
[3996,] 0.9579220 0.5072729
[3997,] 0.9579220 0.5072729
[3998,] 0.9498418 0.5072729
[3999,] 0.9498418 0.5072729
[4000,] 0.9498418 0.5072582

Numerical summaries

Bayesian with Nimble

library(MCMCvis)
MCMCsummary(mcmc.output, round = 2)

   mean   sd 2.5%  50% 97.5% Rhat n.eff
b0 0.94 0.05 0.84 0.94  1.04 1.02    60
b1 0.51 0.01 0.50 0.51  0.52 1.01    63

Frequentist with glm()

summary(mod)


Call:
glm(formula = conteo ~ temp, family = poisson(link = "log"), 
    data = datos)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.9659  -0.6913  -0.1997   0.5955   2.1766  

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

Trace and posterior density

MCMCtrace(mcmc.output,  pdf = F)

Predictions

Here, we want to predict the relation between \(\lambda\), the number of roe deer poop, and the temperature.

Predictions

Here, we want to predict the relation between \(\lambda\), the number of roe deer poop, and the temperature.

One key point of Bayesian statistics is that you obtain posterior distribution of the parameter, not a point estimate.

mcmc.bind <- rbind(mcmc.output$chain1, mcmc.output$chain2)

mcmc.bind

               b0        b1
   [1,] 1.0552801 0.4924236
   [2,] 1.0552801 0.4924236
   [3,] 1.0568864 0.4937307
   [4,] 1.0568864 0.4937307
   [5,] 1.0517870 0.4937307
   [6,] 1.0517870 0.4937307
   [7,] 1.0517870 0.4943446
   [8,] 1.0517870 0.4943446
   [9,] 1.0517870 0.4941333
  [10,] 1.0434708 0.4941333
  [11,] 1.0434708 0.4941333
  [12,] 1.0434708 0.4941333
  [13,] 1.0434708 0.4941333
  [14,] 1.0363103 0.4941333
  [15,] 1.0363103 0.4941333
  [16,] 1.0290184 0.4941333
  [17,] 1.0290184 0.4953782
  [18,] 1.0290184 0.4953782
  [19,] 1.0513797 0.4953782
  [20,] 1.0513797 0.4953782
  [21,] 1.0513797 0.4953782
  [22,] 1.0262853 0.4997459
  [23,] 1.0262853 0.4951756
  [24,] 1.0270733 0.4951756
  [25,] 1.0270733 0.4989521
  [26,] 1.0109282 0.4989521
  [27,] 0.9931602 0.4989521
  [28,] 0.9931602 0.5027903
  [29,] 0.9630684 0.5070866
  [30,] 0.9207093 0.5070866
  [31,] 0.9207093 0.5146613
  [32,] 0.9207093 0.5146613
  [33,] 0.8870311 0.5139844
  [34,] 0.8870311 0.5139844
  [35,] 0.8975645 0.5139844
  [36,] 0.8889593 0.5137728
  [37,] 0.8889593 0.5137728
  [38,] 0.8889593 0.5163884
  [39,] 0.8889593 0.5163884
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[1507,] 0.8951078 0.5119058
[1508,] 0.8951078 0.5119058
[1509,] 0.9276817 0.5117028
[1510,] 0.9002330 0.5140704
[1511,] 0.9002330 0.5140704
[1512,] 0.8987307 0.5120668
[1513,] 0.9074338 0.5129816
[1514,] 0.9074338 0.5122441
[1515,] 0.9074338 0.5089570
[1516,] 0.9074338 0.5095523
[1517,] 0.9074338 0.5107757
[1518,] 0.9074338 0.5131793
[1519,] 0.9074338 0.5130727
[1520,] 0.9033886 0.5130727
[1521,] 0.9033886 0.5124159
[1522,] 0.9059146 0.5123646
[1523,] 0.9174034 0.5102756
[1524,] 0.9174034 0.5111775
[1525,] 0.9020020 0.5111775
[1526,] 0.9137899 0.5104787
[1527,] 0.9137899 0.5102389
[1528,] 0.9137899 0.5094110
[1529,] 0.9137899 0.5094110
[1530,] 0.9238863 0.5101730
[1531,] 0.9238863 0.5116616
[1532,] 0.9238863 0.5116616
[1533,] 0.9238863 0.5140529
[1534,] 0.9238863 0.5134531
[1535,] 0.8981669 0.5134531
[1536,] 0.8919282 0.5134531
[1537,] 0.8919282 0.5136025
[1538,] 0.8919282 0.5136025
[1539,] 0.9087060 0.5136025
[1540,] 0.9087060 0.5136025
[1541,] 0.8930974 0.5136025
[1542,] 0.8770064 0.5164614
[1543,] 0.8770064 0.5164614
[1544,] 0.8648467 0.5176441
[1545,] 0.8648467 0.5176441
[1546,] 0.8710333 0.5149310
[1547,] 0.8710333 0.5149310
[1548,] 0.9068456 0.5127057
[1549,] 0.9149250 0.5127057
[1550,] 0.9128963 0.5123282
[1551,] 0.9128963 0.5123282
[1552,] 0.9165140 0.5123282
[1553,] 0.9048811 0.5123282
[1554,] 0.9048811 0.5123945
[1555,] 0.9048811 0.5159956
[1556,] 0.9120447 0.5134604
[1557,] 0.9120447 0.5120464
[1558,] 0.9120447 0.5112682
[1559,] 0.9120447 0.5112682
[1560,] 0.9120447 0.5112682
[1561,] 0.9252964 0.5112682
[1562,] 0.9252964 0.5112682
[1563,] 0.9283341 0.5112682
[1564,] 0.9283341 0.5091528
[1565,] 0.9283341 0.5091528
[1566,] 0.9283341 0.5091528
[1567,] 0.9427839 0.5091528
[1568,] 0.9170551 0.5091528
[1569,] 0.9287686 0.5082546
[1570,] 0.9287686 0.5082546
[1571,] 0.9287686 0.5070667
[1572,] 0.9637460 0.5070667
[1573,] 0.9637460 0.5051030
[1574,] 0.9637460 0.5051030
[1575,] 0.9637460 0.5051030
[1576,] 0.9637460 0.5051030
[1577,] 0.9637460 0.5051030
[1578,] 0.9637460 0.5059068
[1579,] 0.9455266 0.5059068
[1580,] 0.9455266 0.5059068
[1581,] 0.9455266 0.5059068
[1582,] 0.9407231 0.5082814
[1583,] 0.9407231 0.5082814
[1584,] 0.9594052 0.5047410
[1585,] 0.9594052 0.5047410
[1586,] 0.9856195 0.4996706
[1587,] 0.9856195 0.4996706
[1588,] 0.9901681 0.5034517
[1589,] 0.9815622 0.5034517
[1590,] 0.9815622 0.5006607
[1591,] 0.9787762 0.5006607
[1592,] 0.9986573 0.5006607
[1593,] 0.9986573 0.5006607
[1594,] 0.9811788 0.5006607
[1595,] 0.9811788 0.5006607
[1596,] 1.0090543 0.4975097
[1597,] 1.0314721 0.4970686
[1598,] 1.0366351 0.4970686
[1599,] 1.0366351 0.4970686
[1600,] 1.0366351 0.4985399
[1601,] 1.0366351 0.4977328
[1602,] 1.0366351 0.4959763
[1603,] 1.0366351 0.4954731
[1604,] 1.0366351 0.4954731
[1605,] 1.0366351 0.4954731
[1606,] 1.0384729 0.4966920
[1607,] 1.0384729 0.4965422
[1608,] 1.0384729 0.4957647
[1609,] 1.0384729 0.4957647
[1610,] 1.0391460 0.4936018
[1611,] 1.0391460 0.4932598
[1612,] 1.0391460 0.4928872
[1613,] 1.0690931 0.4928872
[1614,] 1.0690931 0.4930009
[1615,] 1.0690931 0.4930009
[1616,] 1.0690931 0.4930009
[1617,] 1.0749328 0.4912147
[1618,] 1.0749328 0.4918367
[1619,] 1.0749328 0.4925181
[1620,] 1.0502640 0.4925181
[1621,] 1.0502640 0.4958366
[1622,] 1.0502640 0.4916867
[1623,] 1.0502640 0.4916867
[1624,] 1.0547656 0.4936995
[1625,] 1.0547656 0.4939774
[1626,] 1.0593916 0.4939774
[1627,] 1.0548480 0.4939774
[1628,] 1.0548480 0.4939774
[1629,] 1.0548480 0.4939774
[1630,] 1.0548480 0.4942483
[1631,] 1.0474827 0.4942483
[1632,] 1.0474827 0.4942483
[1633,] 1.0474827 0.4942483
[1634,] 1.0474827 0.4942483
[1635,] 1.0503427 0.4942483
[1636,] 1.0503427 0.4942483
[1637,] 1.0503427 0.4942483
[1638,] 1.0586830 0.4920425
[1639,] 1.0586830 0.4927077
[1640,] 1.0489346 0.4911411
[1641,] 1.0489346 0.4937553
[1642,] 1.0505839 0.4945568
[1643,] 1.0160832 0.5002490
[1644,] 1.0160832 0.4959535
[1645,] 1.0160832 0.4959535
[1646,] 1.0192340 0.4958298
[1647,] 1.0326927 0.4973184
[1648,] 1.0267485 0.5000942
[1649,] 1.0078648 0.5000942
[1650,] 1.0037331 0.4990366
[1651,] 1.0037331 0.4990366
[1652,] 1.0037331 0.4990366
[1653,] 1.0037331 0.4990366
[1654,] 1.0273569 0.4981317
[1655,] 1.0185559 0.4981317
[1656,] 1.0185559 0.4981317
[1657,] 1.0185559 0.4981317
[1658,] 1.0139157 0.4981317
[1659,] 1.0094751 0.4969554
[1660,] 1.0094751 0.4969554
[1661,] 1.0447493 0.4969554
[1662,] 1.0282750 0.4986666
[1663,] 1.0282750 0.4963715
[1664,] 1.0313335 0.4969601
[1665,] 1.0355420 0.4969601
[1666,] 1.0253711 0.4950399
[1667,] 1.0536661 0.4950399
[1668,] 1.0530204 0.4950399
[1669,] 1.0339574 0.4950399
[1670,] 1.0339574 0.4968018
[1671,] 1.0339574 0.4973043
[1672,] 1.0519132 0.4973043
[1673,] 1.0519132 0.4977056
[1674,] 1.0519132 0.4974859
[1675,] 1.0128562 0.4974859
[1676,] 1.0334921 0.4947175
[1677,] 1.0334921 0.4947175
[1678,] 1.0334921 0.4947175
[1679,] 1.0334921 0.4976354
[1680,] 1.0334921 0.4976354
[1681,] 1.0334921 0.4953619
[1682,] 1.0421016 0.4953619
[1683,] 1.0421016 0.4953619
[1684,] 1.0421016 0.4946936
[1685,] 1.0421016 0.4953187
[1686,] 1.0421016 0.4943447
[1687,] 1.0567342 0.4943447
[1688,] 1.0567342 0.4943447
[1689,] 1.0567342 0.4943447
[1690,] 1.0378874 0.4957350
[1691,] 1.0378874 0.4957350
[1692,] 1.0378874 0.4957350
[1693,] 1.0424716 0.4953462
[1694,] 1.0535876 0.4915501
[1695,] 1.0535876 0.4964071
[1696,] 1.0109010 0.5000088
[1697,] 1.0109010 0.5000088
[1698,] 1.0109010 0.5000088
[1699,] 0.9898475 0.5010480
[1700,] 0.9898475 0.5010480
[1701,] 0.9898475 0.5031480
[1702,] 0.9711366 0.5031480
[1703,] 0.9756277 0.5031480
[1704,] 0.9896856 0.5031480
[1705,] 0.9898026 0.5031480
[1706,] 0.9898026 0.5031480
[1707,] 0.9898026 0.5031480
[1708,] 0.9948165 0.5031480
[1709,] 0.9948165 0.5032703
[1710,] 0.9979959 0.4996009
[1711,] 1.0187953 0.4996009
[1712,] 1.0187953 0.4996009
[1713,] 1.0187953 0.4994982
[1714,] 1.0187953 0.4994982
[1715,] 1.0178639 0.4977231
[1716,] 1.0178639 0.5017203
[1717,] 1.0153395 0.4999762
[1718,] 1.0153395 0.4999762
[1719,] 1.0153395 0.4978734
[1720,] 1.0258100 0.4978734
[1721,] 1.0258100 0.4983847
[1722,] 1.0258100 0.4953449
[1723,] 1.0258100 0.4953449
[1724,] 1.0554081 0.4953449
[1725,] 1.0644700 0.4953449
[1726,] 1.0444154 0.4953449
[1727,] 1.0444154 0.4953449
[1728,] 1.0442989 0.4922532
[1729,] 1.0442989 0.4921179
[1730,] 1.0641845 0.4921179
[1731,] 1.0722252 0.4921179
[1732,] 1.0722252 0.4921179
[1733,] 1.0733127 0.4921179
[1734,] 1.0757179 0.4921179
[1735,] 1.0749355 0.4934231
[1736,] 1.0579004 0.4917446
[1737,] 1.0579004 0.4931234
[1738,] 1.0579004 0.4931234
[1739,] 1.0601744 0.4931234
[1740,] 1.0528775 0.4956039
[1741,] 1.0528775 0.4966458
[1742,] 1.0528775 0.4925265
[1743,] 1.0509544 0.4958086
[1744,] 1.0366216 0.4958086
[1745,] 1.0366216 0.4924681
[1746,] 1.0634939 0.4924681
[1747,] 1.0634939 0.4924681
[1748,] 1.0634939 0.4924681
[1749,] 1.0634939 0.4924681
[1750,] 1.0540840 0.4923659
[1751,] 1.0540840 0.4923659
[1752,] 1.0457533 0.4923659
[1753,] 1.0536047 0.4923659
[1754,] 1.0536047 0.4952315
[1755,] 1.0314511 0.4944229
[1756,] 1.0436962 0.4944229
[1757,] 1.0436962 0.4950945
[1758,] 1.0346052 0.4950945
[1759,] 1.0346052 0.4963018
[1760,] 1.0346052 0.4958639
[1761,] 1.0346052 0.4958639
[1762,] 1.0346052 0.4981894
[1763,] 1.0346052 0.4979031
[1764,] 1.0346052 0.4992008
[1765,] 1.0346052 0.4992008
[1766,] 1.0346052 0.4992008
[1767,] 1.0346052 0.4992008
[1768,] 1.0346052 0.4992008
[1769,] 1.0346052 0.4983934
[1770,] 1.0343340 0.4958391
[1771,] 1.0343340 0.4958391
[1772,] 1.0370738 0.4954641
[1773,] 1.0334590 0.4953603
[1774,] 1.0382331 0.4953603
[1775,] 1.0382331 0.4953603
[1776,] 1.0498006 0.4953603
[1777,] 1.0542553 0.4953603
[1778,] 1.0472404 0.4958699
[1779,] 1.0472404 0.4938930
[1780,] 1.0472404 0.4938930
[1781,] 1.0472404 0.4938930
[1782,] 1.0472404 0.4936417
[1783,] 1.0472404 0.4936417
[1784,] 1.0184717 0.4969040
[1785,] 1.0184717 0.4976391
[1786,] 1.0234283 0.4976391
[1787,] 1.0234283 0.4976391
[1788,] 1.0234283 0.4976391
[1789,] 1.0364394 0.4976391
[1790,] 1.0364394 0.4976391
[1791,] 1.0364394 0.4988382
[1792,] 1.0304173 0.4988382
[1793,] 1.0304173 0.4988382
[1794,] 1.0304173 0.4951508
[1795,] 1.0108239 0.4951508
[1796,] 1.0296765 0.4984729
[1797,] 1.0081919 0.4984729
[1798,] 1.0173235 0.4982322
[1799,] 1.0173235 0.4982322
[1800,] 1.0173235 0.4982322
[1801,] 1.0119060 0.4993188
[1802,] 1.0119060 0.4987602
[1803,] 1.0119060 0.4960424
[1804,] 1.0119060 0.4981167
[1805,] 1.0119060 0.4981167
[1806,] 1.0119060 0.4981167
[1807,] 1.0119060 0.4981167
[1808,] 1.0119060 0.4981167
[1809,] 1.0119060 0.4981167
[1810,] 1.0119060 0.4981167
[1811,] 1.0257655 0.4959499
[1812,] 1.0257655 0.4959499
[1813,] 1.0257655 0.4959499
[1814,] 1.0360315 0.4959499
[1815,] 1.0343544 0.4959499
[1816,] 1.0343544 0.4959499
[1817,] 1.0252628 0.4961600
[1818,] 1.0057629 0.5016439
[1819,] 0.9789736 0.5037509
[1820,] 0.9789736 0.5037509
[1821,] 0.9741449 0.5037509
[1822,] 0.9971262 0.5006111
[1823,] 0.9859804 0.5006111
[1824,] 0.9859804 0.5021947
[1825,] 0.9859804 0.5021947
[1826,] 0.9859804 0.5028002
[1827,] 0.9859804 0.5028002
[1828,] 0.9758822 0.5028002
[1829,] 0.9758822 0.5028002
[1830,] 0.9758822 0.5028002
[1831,] 0.9758822 0.5028961
[1832,] 0.9758822 0.5081354
[1833,] 0.9548414 0.5081354
[1834,] 0.9548414 0.5049801
[1835,] 0.9548414 0.5049801
[1836,] 0.9548414 0.5061056
[1837,] 0.9548414 0.5061056
[1838,] 0.9521376 0.5049628
[1839,] 0.9672149 0.5049873
[1840,] 0.9689546 0.5049873
[1841,] 0.9689546 0.5049873
[1842,] 0.9689546 0.5032528
[1843,] 0.9742252 0.5033486
[1844,] 0.9742252 0.5033486
[1845,] 0.9742252 0.5047582
[1846,] 0.9444883 0.5060397
[1847,] 0.9444883 0.5060397
[1848,] 0.9444883 0.5060397
[1849,] 0.9444883 0.5065755
[1850,] 0.9444883 0.5064825
[1851,] 0.9444883 0.5064825
[1852,] 0.9444883 0.5064825
[1853,] 0.9588756 0.5064825
[1854,] 0.9588756 0.5057789
[1855,] 0.9463020 0.5081722
[1856,] 0.9463020 0.5081722
[1857,] 0.9431348 0.5081722
[1858,] 0.9431348 0.5081722
[1859,] 0.9215505 0.5081722
[1860,] 0.9215505 0.5095987
[1861,] 0.9215505 0.5095987
[1862,] 0.9215505 0.5111669
[1863,] 0.9095079 0.5111669
[1864,] 0.9095079 0.5143792
[1865,] 0.9122365 0.5138484
[1866,] 0.9122365 0.5138484
[1867,] 0.8755742 0.5138484
[1868,] 0.8755742 0.5142270
[1869,] 0.8864549 0.5142270
[1870,] 0.8878095 0.5115305
[1871,] 0.9018848 0.5116974
[1872,] 0.9150244 0.5116974
[1873,] 0.9087070 0.5116974
[1874,] 0.9087070 0.5116974
[1875,] 0.9087070 0.5116974
[1876,] 0.9087070 0.5116974
[1877,] 0.9283300 0.5116974
[1878,] 0.8972240 0.5136486
[1879,] 0.9164536 0.5136486
[1880,] 0.9164536 0.5128168
[1881,] 0.9164536 0.5128168
[1882,] 0.8967362 0.5128168
[1883,] 0.8828406 0.5166594
[1884,] 0.8828406 0.5166594
[1885,] 0.8828406 0.5166594
[1886,] 0.8543930 0.5166594
[1887,] 0.8773033 0.5153936
[1888,] 0.8773033 0.5153936
[1889,] 0.8852278 0.5185889
[1890,] 0.8681723 0.5185889
[1891,] 0.8681723 0.5185889
[1892,] 0.8681723 0.5165506
[1893,] 0.8681723 0.5141469
[1894,] 0.8681723 0.5141469
[1895,] 0.8830398 0.5141469
[1896,] 0.9098347 0.5141469
[1897,] 0.9098347 0.5141469
[1898,] 0.9098347 0.5141469
[1899,] 0.8885464 0.5141469
[1900,] 0.8885464 0.5141469
[1901,] 0.8789187 0.5158466
[1902,] 0.8789187 0.5158466
[1903,] 0.8781689 0.5147783
[1904,] 0.8781689 0.5169870
[1905,] 0.8781689 0.5169870
[1906,] 0.8743291 0.5169870
[1907,] 0.8743291 0.5169870
[1908,] 0.8743291 0.5173506
[1909,] 0.8743291 0.5173506
[1910,] 0.8817369 0.5173506
[1911,] 0.8599816 0.5167454
[1912,] 0.8729389 0.5148041
[1913,] 0.8995184 0.5148041
[1914,] 0.8995184 0.5156378
[1915,] 0.8995184 0.5156378
[1916,] 0.8618950 0.5156378
[1917,] 0.8618950 0.5202238
[1918,] 0.8383806 0.5202238
[1919,] 0.8383806 0.5202238
[1920,] 0.8383806 0.5202238
[1921,] 0.8302967 0.5202238
[1922,] 0.8302967 0.5194075
[1923,] 0.8706423 0.5194075
[1924,] 0.8668484 0.5163751
[1925,] 0.8668484 0.5166944
[1926,] 0.8710359 0.5166944
[1927,] 0.8710359 0.5166944
[1928,] 0.8710359 0.5166944
[1929,] 0.8763926 0.5173723
[1930,] 0.8763926 0.5173723
[1931,] 0.8763926 0.5150392
[1932,] 0.8990951 0.5134869
[1933,] 0.8986516 0.5134869
[1934,] 0.8986516 0.5134869
[1935,] 0.8993237 0.5134869
[1936,] 0.8993237 0.5134869
[1937,] 0.8820332 0.5134869
[1938,] 0.8820332 0.5134869
[1939,] 0.8847260 0.5134869
[1940,] 0.8847260 0.5134869
[1941,] 0.8847260 0.5168704
[1942,] 0.8827581 0.5149884
[1943,] 0.8867680 0.5149884
[1944,] 0.8960102 0.5149884
[1945,] 0.8960102 0.5149884
[1946,] 0.8768192 0.5149884
[1947,] 0.8682365 0.5169211
[1948,] 0.8682365 0.5169211
[1949,] 0.8585157 0.5169211
[1950,] 0.8855020 0.5169211
[1951,] 0.8855020 0.5169211
[1952,] 0.8794865 0.5169211
[1953,] 0.8794865 0.5169211
[1954,] 0.8850838 0.5169211
[1955,] 0.8830718 0.5124706
[1956,] 0.8830718 0.5124706
[1957,] 0.9406888 0.5124706
[1958,] 0.9406888 0.5119912
[1959,] 0.9406888 0.5119912
[1960,] 0.9246589 0.5119912
[1961,] 0.9246589 0.5119912
[1962,] 0.9131416 0.5119912
[1963,] 0.9131416 0.5132272
[1964,] 0.8850049 0.5132272
[1965,] 0.8850049 0.5132272
[1966,] 0.8850049 0.5132272
[1967,] 0.8850049 0.5143667
[1968,] 0.8850049 0.5143667
[1969,] 0.8850049 0.5143667
[1970,] 0.8978854 0.5143667
[1971,] 0.8978854 0.5143667
[1972,] 0.8845361 0.5143667
[1973,] 0.8960588 0.5178214
[1974,] 0.8368006 0.5178214
[1975,] 0.8368006 0.5211169
[1976,] 0.8368006 0.5211169
[1977,] 0.8442571 0.5211169
[1978,] 0.8226570 0.5211169
[1979,] 0.8226570 0.5224712
[1980,] 0.8226570 0.5243744
[1981,] 0.8361059 0.5243744
[1982,] 0.8403768 0.5217309
[1983,] 0.8166686 0.5203706
[1984,] 0.8448800 0.5203706
[1985,] 0.8448800 0.5186539
[1986,] 0.8525694 0.5186539
[1987,] 0.8525694 0.5186539
[1988,] 0.8525694 0.5195728
[1989,] 0.8525694 0.5190762
[1990,] 0.8722149 0.5154846
[1991,] 0.8722149 0.5154846
[1992,] 0.8792930 0.5154846
[1993,] 0.8913496 0.5160611
[1994,] 0.8705802 0.5160611
[1995,] 0.8705802 0.5160611
[1996,] 0.8705802 0.5160611
[1997,] 0.8767948 0.5160611
[1998,] 0.8767948 0.5160611
[1999,] 0.8805270 0.5160611
[2000,] 0.8799504 0.5160611
[2001,] 0.8799504 0.5163211
[2002,] 0.8799504 0.5169527
[2003,] 0.8799504 0.5136695
[2004,] 0.8799504 0.5136695
[2005,] 0.8799504 0.5171804
[2006,] 0.8566175 0.5168277
[2007,] 0.8566175 0.5203194
[2008,] 0.8441332 0.5203194
[2009,] 0.8441332 0.5203194
[2010,] 0.8441332 0.5203194
[2011,] 0.8441332 0.5203194
[2012,] 0.8441332 0.5203194
[2013,] 0.8441332 0.5213140
[2014,] 0.8276458 0.5230474
[2015,] 0.8276458 0.5230474
[2016,] 0.8276458 0.5194911
[2017,] 0.8288799 0.5232746
[2018,] 0.8288799 0.5206796
[2019,] 0.8288799 0.5206796
[2020,] 0.8375038 0.5206796
[2021,] 0.8375038 0.5206796
[2022,] 0.8375038 0.5206568
[2023,] 0.8262982 0.5206568
[2024,] 0.8275642 0.5206568
[2025,] 0.8357882 0.5206568
[2026,] 0.8377768 0.5206568
[2027,] 0.8443525 0.5206568
[2028,] 0.8443525 0.5175631
[2029,] 0.8443525 0.5175631
[2030,] 0.8675817 0.5175631
[2031,] 0.8675817 0.5175631
[2032,] 0.8675817 0.5182795
[2033,] 0.8675817 0.5182795
[2034,] 0.8675817 0.5182795
[2035,] 0.8675817 0.5166252
[2036,] 0.8675817 0.5166252
[2037,] 0.8751237 0.5166252
[2038,] 0.8912117 0.5166252
[2039,] 0.8912117 0.5166252
[2040,] 0.8791103 0.5143899
[2041,] 0.8791103 0.5151664
[2042,] 0.8894147 0.5151664
[2043,] 0.8894147 0.5151664
[2044,] 0.8762455 0.5172681
[2045,] 0.8762455 0.5172681
[2046,] 0.8762455 0.5172681
[2047,] 0.8709233 0.5172681
[2048,] 0.8689115 0.5172681
[2049,] 0.8689115 0.5172681
[2050,] 0.8689115 0.5173427
[2051,] 0.8689115 0.5165971
[2052,] 0.8689115 0.5156667
[2053,] 0.8679419 0.5145054
[2054,] 0.8679419 0.5145054
[2055,] 0.8772701 0.5145054
[2056,] 0.8954754 0.5145054
[2057,] 0.8954754 0.5101378
[2058,] 0.8955570 0.5101378
[2059,] 0.8955570 0.5101378
[2060,] 0.8955570 0.5129856
[2061,] 0.8679207 0.5129856
[2062,] 0.9102206 0.5129856
[2063,] 0.9112283 0.5122291
[2064,] 0.9112283 0.5132340
[2065,] 0.9110485 0.5103780
[2066,] 0.9110485 0.5103780
[2067,] 0.9150334 0.5103780
[2068,] 0.9254530 0.5103780
[2069,] 0.9089087 0.5103780
[2070,] 0.9089087 0.5103780
[2071,] 0.9169508 0.5112470
[2072,] 0.9227466 0.5112470
[2073,] 0.9227466 0.5112470
[2074,] 0.8953032 0.5112470
[2075,] 0.8953032 0.5122568
[2076,] 0.8953032 0.5142139
[2077,] 0.8910408 0.5142139
[2078,] 0.8910408 0.5142139
[2079,] 0.8640978 0.5142139
[2080,] 0.8759998 0.5142139
[2081,] 0.8914446 0.5114117
[2082,] 0.8914446 0.5118786
[2083,] 0.9185158 0.5118786
[2084,] 0.9325251 0.5114247
[2085,] 0.9325251 0.5114247
[2086,] 0.8924560 0.5134325
[2087,] 0.8924560 0.5157719
[2088,] 0.8800137 0.5156645
[2089,] 0.9043956 0.5160704
[2090,] 0.8743642 0.5160704
[2091,] 0.8743642 0.5160704
[2092,] 0.8743642 0.5160704
[2093,] 0.8743642 0.5161872
[2094,] 0.8743642 0.5166707
[2095,] 0.8820217 0.5166707
[2096,] 0.8820217 0.5166707
[2097,] 0.8820217 0.5189422
[2098,] 0.8820217 0.5189422
[2099,] 0.8820217 0.5174328
[2100,] 0.8820217 0.5174328
[2101,] 0.8471115 0.5174328
[2102,] 0.8582537 0.5174328
[2103,] 0.8582537 0.5174328
[2104,] 0.8566917 0.5174328
[2105,] 0.8542303 0.5174328
[2106,] 0.8392747 0.5174328
[2107,] 0.8392747 0.5174328
[2108,] 0.8513414 0.5174328
[2109,] 0.8513414 0.5174328
[2110,] 0.8513414 0.5182527
[2111,] 0.8929435 0.5182527
[2112,] 0.8929435 0.5107478
[2113,] 0.8937792 0.5107478
[2114,] 0.8980918 0.5135137
[2115,] 0.8980918 0.5135137
[2116,] 0.8980918 0.5139634
[2117,] 0.8985361 0.5139634
[2118,] 0.8985361 0.5139634
[2119,] 0.8873173 0.5139634
[2120,] 0.8873173 0.5165059
[2121,] 0.8873173 0.5166724
[2122,] 0.8873173 0.5141159
[2123,] 0.8873173 0.5166531
[2124,] 0.8723290 0.5166531
[2125,] 0.8723290 0.5166531
[2126,] 0.8753953 0.5181204
[2127,] 0.8753953 0.5139047
[2128,] 0.8741117 0.5177836
[2129,] 0.8741117 0.5177836
[2130,] 0.8741117 0.5177836
[2131,] 0.8829139 0.5146345
[2132,] 0.8829139 0.5146345
[2133,] 0.8829139 0.5165434
[2134,] 0.8555862 0.5164426
[2135,] 0.8555862 0.5164426
[2136,] 0.8555862 0.5180883
[2137,] 0.8555862 0.5190131
[2138,] 0.8516452 0.5187449
[2139,] 0.8516452 0.5187449
[2140,] 0.8516452 0.5187449
[2141,] 0.8516452 0.5187449
[2142,] 0.8396437 0.5174380
[2143,] 0.8684368 0.5182314
[2144,] 0.8684368 0.5182314
[2145,] 0.8820827 0.5133104
[2146,] 0.8897455 0.5157628
[2147,] 0.8897455 0.5157628
[2148,] 0.8897455 0.5146593
[2149,] 0.8897455 0.5146593
[2150,] 0.9074114 0.5094911
[2151,] 0.9446480 0.5094911
[2152,] 0.9424818 0.5108680
[2153,] 0.9218995 0.5108680
[2154,] 0.8982932 0.5108680
[2155,] 0.8982932 0.5108680
[2156,] 0.8982932 0.5119443
[2157,] 0.8982932 0.5138481
[2158,] 0.8982932 0.5138481
[2159,] 0.8982932 0.5159906
[2160,] 0.8982932 0.5146634
[2161,] 0.8996115 0.5146634
[2162,] 0.8813786 0.5146634
[2163,] 0.8813786 0.5150852
[2164,] 0.8982119 0.5151565
[2165,] 0.8982119 0.5128016
[2166,] 0.8982119 0.5128016
[2167,] 0.9014735 0.5128016
[2168,] 0.9014735 0.5130317
[2169,] 0.9014735 0.5140833
[2170,] 0.9014735 0.5123289
[2171,] 0.9014735 0.5123289
[2172,] 0.9159648 0.5135421
[2173,] 0.9159648 0.5096255
[2174,] 0.9159648 0.5096255
[2175,] 0.9170743 0.5096255
[2176,] 0.9128196 0.5141498
[2177,] 0.9128196 0.5131069
[2178,] 0.9011554 0.5131069
[2179,] 0.9011554 0.5131069
[2180,] 0.9011554 0.5131069
[2181,] 0.9128561 0.5095343
[2182,] 0.9313168 0.5095343
[2183,] 0.9345891 0.5118165
[2184,] 0.9345891 0.5100563
[2185,] 0.9147671 0.5103662
[2186,] 0.9147671 0.5103662
[2187,] 0.9147671 0.5099967
[2188,] 0.9147671 0.5099967
[2189,] 0.9147671 0.5109198
[2190,] 0.9147671 0.5111350
[2191,] 0.9147671 0.5102701
[2192,] 0.9243563 0.5100863
[2193,] 0.9243563 0.5081517
[2194,] 0.9243563 0.5104005
[2195,] 0.9243563 0.5104005
[2196,] 0.9243563 0.5104005
[2197,] 0.9243563 0.5104005
[2198,] 0.9252126 0.5107581
[2199,] 0.9295421 0.5089085
[2200,] 0.9210219 0.5073396
[2201,] 0.9461602 0.5073396
[2202,] 0.9461602 0.5073396
[2203,] 0.9443059 0.5073396
[2204,] 0.9443059 0.5073396
[2205,] 0.9443059 0.5073396
[2206,] 0.9443059 0.5073396
[2207,] 0.9443059 0.5073396
[2208,] 0.9443059 0.5069051
[2209,] 0.9443059 0.5069051
[2210,] 0.9443059 0.5085196
[2211,] 0.9443059 0.5068164
[2212,] 0.9443059 0.5068164
[2213,] 0.9343302 0.5081684
[2214,] 0.9547159 0.5081684
[2215,] 0.9547159 0.5087108
[2216,] 0.9547159 0.5087108
[2217,] 0.9395307 0.5100963
[2218,] 0.9380082 0.5085335
[2219,] 0.9133525 0.5085335
[2220,] 0.9133525 0.5141850
[2221,] 0.9133525 0.5102262
[2222,] 0.9135147 0.5111072
[2223,] 0.9135147 0.5111072
[2224,] 0.9135147 0.5111072
[2225,] 0.9135147 0.5092808
[2226,] 0.9407054 0.5061718
[2227,] 0.9407054 0.5061718
[2228,] 0.9407054 0.5061984
[2229,] 0.9483590 0.5061984
[2230,] 0.9483590 0.5070022
[2231,] 0.9483590 0.5070022
[2232,] 0.9620519 0.5070022
[2233,] 0.9634736 0.5070022
[2234,] 0.9438421 0.5056344
[2235,] 0.9565758 0.5056344
[2236,] 0.9506904 0.5056344
[2237,] 0.9506904 0.5066168
[2238,] 0.9670480 0.5066168
[2239,] 0.9670480 0.5066168
[2240,] 0.9670480 0.5066168
[2241,] 0.9461819 0.5066168
[2242,] 0.9529786 0.5066168
[2243,] 0.9529786 0.5084857
[2244,] 0.9565927 0.5068300
[2245,] 0.9484283 0.5068300
[2246,] 0.9484283 0.5068300
[2247,] 0.9484283 0.5066972
[2248,] 0.9719963 0.5066972
[2249,] 0.9719963 0.5066972
[2250,] 0.9612360 0.5066972
[2251,] 0.9529198 0.5067379
[2252,] 0.9529198 0.5067379
[2253,] 0.9529198 0.5067379
[2254,] 0.9653086 0.5067379
[2255,] 0.9750048 0.5040162
[2256,] 0.9631248 0.5040162
[2257,] 0.9631248 0.5038649
[2258,] 0.9631248 0.5080799
[2259,] 0.9631248 0.5080799
[2260,] 0.9561133 0.5059333
[2261,] 0.9561133 0.5059333
[2262,] 0.9561133 0.5059333
[2263,] 0.9561133 0.5059333
[2264,] 0.9727084 0.5059333
[2265,] 0.9727084 0.5064233
[2266,] 0.9823485 0.5025369
[2267,] 0.9834486 0.5025369
[2268,] 0.9939399 0.5013663
[2269,] 0.9939399 0.5025853
[2270,] 0.9687057 0.5025853
[2271,] 0.9687057 0.5025853
[2272,] 0.9687057 0.5025853
[2273,] 0.9687057 0.5026529
[2274,] 0.9687057 0.5026529
[2275,] 0.9934742 0.5026529
[2276,] 0.9934742 0.5033440
[2277,] 0.9818838 0.5033440
[2278,] 0.9809327 0.5033440
[2279,] 0.9974909 0.5033440
[2280,] 0.9963305 0.5033440
[2281,] 1.0027302 0.5011055
[2282,] 1.0027302 0.5009784
[2283,] 1.0027302 0.5009784
[2284,] 1.0027302 0.5018052
[2285,] 1.0027302 0.5018052
[2286,] 1.0027302 0.5015836
[2287,] 0.9906765 0.5015836
[2288,] 0.9906765 0.5015836
[2289,] 0.9770743 0.5073021
[2290,] 0.9770743 0.5032964
[2291,] 0.9790263 0.5032964
[2292,] 0.9790263 0.5040159
[2293,] 0.9521084 0.5040159
[2294,] 0.9521084 0.5040159
[2295,] 0.9794889 0.5032317
[2296,] 0.9919382 0.5007375
[2297,] 0.9919382 0.5012587
[2298,] 0.9919382 0.5012587
[2299,] 0.9919382 0.5012587
[2300,] 0.9919382 0.5012587
[2301,] 0.9919382 0.5012587
[2302,] 0.9919382 0.5033880
[2303,] 1.0029592 0.4982584
[2304,] 1.0029592 0.4982584
[2305,] 1.0029592 0.5016169
[2306,] 0.9929651 0.5016169
[2307,] 0.9929651 0.5020604
[2308,] 0.9929651 0.5020604
[2309,] 0.9929651 0.5015639
[2310,] 0.9832207 0.5029545
[2311,] 0.9832207 0.5020545
[2312,] 0.9832207 0.5020545
[2313,] 0.9832207 0.5020545
[2314,] 0.9832207 0.5020545
[2315,] 0.9832207 0.5026725
[2316,] 0.9832189 0.5049892
[2317,] 0.9355376 0.5049892
[2318,] 0.9752428 0.5049892
[2319,] 0.9752428 0.5043640
[2320,] 0.9752428 0.5043640
[2321,] 0.9752428 0.5043640
[2322,] 0.9752428 0.5043640
[2323,] 0.9747806 0.5043640
[2324,] 0.9588609 0.5066890
[2325,] 0.9588609 0.5066890
[2326,] 0.9588609 0.5066890
[2327,] 0.9601722 0.5066890
[2328,] 0.9601722 0.5072435
[2329,] 0.9283642 0.5072435
[2330,] 0.9365727 0.5072435
[2331,] 0.9577023 0.5078570
[2332,] 0.9577023 0.5078570
[2333,] 0.9577023 0.5074507
[2334,] 0.9577023 0.5070559
[2335,] 0.9455983 0.5060314
[2336,] 0.9489201 0.5062416
[2337,] 0.9657181 0.5062416
[2338,] 0.9529321 0.5041052
[2339,] 0.9414643 0.5055579
[2340,] 0.9414643 0.5062826
[2341,] 0.9414643 0.5099289
[2342,] 0.9414643 0.5099289
[2343,] 0.9204002 0.5070834
[2344,] 0.9204002 0.5070834
[2345,] 0.9254569 0.5070834
[2346,] 0.9254569 0.5070834
[2347,] 0.9254569 0.5070834
[2348,] 0.9344878 0.5070834
[2349,] 0.9344878 0.5070834
[2350,] 0.9344878 0.5070834
[2351,] 0.9344878 0.5106476
[2352,] 0.9344878 0.5106476
[2353,] 0.9197834 0.5106476
[2354,] 0.9197834 0.5106476
[2355,] 0.9413323 0.5106476
[2356,] 0.9413323 0.5109554
[2357,] 0.9413323 0.5080527
[2358,] 0.9256374 0.5080527
[2359,] 0.9387842 0.5080527
[2360,] 0.9387842 0.5095022
[2361,] 0.9387842 0.5094077
[2362,] 0.9387842 0.5060723
[2363,] 0.9475777 0.5100157
[2364,] 0.9475777 0.5100157
[2365,] 0.9145912 0.5100157
[2366,] 0.9145912 0.5100157
[2367,] 0.9145912 0.5100157
[2368,] 0.9145912 0.5118595
[2369,] 0.9119846 0.5118595
[2370,] 0.9145458 0.5118595
[2371,] 0.8958067 0.5118595
[2372,] 0.8958067 0.5118595
[2373,] 0.8909983 0.5113357
[2374,] 0.8977407 0.5149705
[2375,] 0.9044774 0.5149705
[2376,] 0.9044774 0.5149705
[2377,] 0.8981296 0.5146706
[2378,] 0.8981296 0.5146706
[2379,] 0.8728566 0.5146706
[2380,] 0.8941067 0.5146706
[2381,] 0.8941067 0.5126753
[2382,] 0.9123432 0.5134063
[2383,] 0.9024971 0.5134063
[2384,] 0.9024971 0.5134063
[2385,] 0.9070648 0.5113966
[2386,] 0.9070648 0.5113966
[2387,] 0.9070648 0.5103296
[2388,] 0.9070648 0.5152853
[2389,] 0.8821216 0.5152853
[2390,] 0.8821216 0.5153139
[2391,] 0.8848814 0.5150944
[2392,] 0.8910616 0.5161458
[2393,] 0.8862099 0.5161458
[2394,] 0.8862099 0.5161458
[2395,] 0.8862099 0.5134300
[2396,] 0.8862099 0.5134300
[2397,] 0.9296004 0.5098444
[2398,] 0.9117112 0.5098444
[2399,] 0.9117112 0.5135867
[2400,] 0.8848733 0.5135867
[2401,] 0.8881645 0.5135867
[2402,] 0.8881645 0.5135867
[2403,] 0.8881645 0.5135867
[2404,] 0.8881645 0.5139515
[2405,] 0.8777387 0.5167191
[2406,] 0.8435564 0.5209491
[2407,] 0.8343414 0.5209491
[2408,] 0.8343414 0.5195380
[2409,] 0.8343414 0.5194451
[2410,] 0.8343414 0.5225920
[2411,] 0.8343414 0.5225920
[2412,] 0.8343414 0.5201767
[2413,] 0.8432309 0.5201767
[2414,] 0.8432309 0.5201767
[2415,] 0.8128787 0.5201767
[2416,] 0.8354287 0.5201767
[2417,] 0.8575947 0.5194933
[2418,] 0.8548614 0.5183331
[2419,] 0.8548614 0.5183331
[2420,] 0.8548614 0.5202816
[2421,] 0.8548614 0.5202816
[2422,] 0.8548614 0.5202816
[2423,] 0.8548614 0.5169374
[2424,] 0.8548614 0.5177301
[2425,] 0.8548614 0.5199220
[2426,] 0.8390911 0.5199220
[2427,] 0.8390911 0.5199592
[2428,] 0.8390911 0.5199592
[2429,] 0.8626770 0.5199592
[2430,] 0.8626770 0.5171216
[2431,] 0.8626770 0.5171216
[2432,] 0.8824475 0.5181442
[2433,] 0.8824475 0.5127881
[2434,] 0.8824475 0.5174188
[2435,] 0.8824475 0.5173549
[2436,] 0.8824475 0.5173549
[2437,] 0.8824475 0.5159093
[2438,] 0.8824475 0.5159093
[2439,] 0.8824475 0.5159154
[2440,] 0.8824475 0.5133460
[2441,] 0.8932694 0.5133460
[2442,] 0.8788689 0.5165197
[2443,] 0.8779330 0.5165197
[2444,] 0.8779330 0.5160311
[2445,] 0.8790369 0.5160311
[2446,] 0.8790369 0.5160311
[2447,] 0.8790369 0.5161530
[2448,] 0.8770277 0.5161530
[2449,] 0.8705644 0.5161530
[2450,] 0.8908996 0.5161530
[2451,] 0.8881497 0.5161530
[2452,] 0.8881497 0.5161530
[2453,] 0.8881497 0.5161530
[2454,] 0.8881497 0.5161530
[2455,] 0.8881497 0.5160448
[2456,] 0.8881497 0.5160448
[2457,] 0.8881497 0.5160448
[2458,] 0.8833133 0.5142030
[2459,] 0.8833133 0.5142030
[2460,] 0.8674033 0.5154656
[2461,] 0.8674033 0.5168693
[2462,] 0.8674033 0.5168693
[2463,] 0.8674033 0.5168693
[2464,] 0.8674033 0.5168693
[2465,] 0.8674033 0.5168693
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[7701,] 0.9919210 0.5030465
[7702,] 0.9919210 0.5030465
[7703,] 0.9705200 0.5030465
[7704,] 0.9705200 0.5030465
[7705,] 0.9705200 0.5030465
[7706,] 0.9705200 0.5020129
[7707,] 0.9705200 0.5020129
[7708,] 0.9889975 0.5020129
[7709,] 0.9807838 0.5027984
[7710,] 0.9689472 0.5038075
[7711,] 0.9860546 0.5011816
[7712,] 0.9860546 0.5011816
[7713,] 0.9949119 0.5011816
[7714,] 0.9980008 0.5011816
[7715,] 1.0071327 0.5011816
[7716,] 1.0158345 0.4987738
[7717,] 1.0158345 0.4987738
[7718,] 1.0158345 0.4987738
[7719,] 1.0042647 0.4987738
[7720,] 1.0042647 0.4992030
[7721,] 0.9858562 0.4992030
[7722,] 1.0035983 0.4992030
[7723,] 1.0035983 0.5016758
[7724,] 0.9813011 0.5040602
[7725,] 0.9467688 0.5086511
[7726,] 0.9467688 0.5086511
[7727,] 0.9416085 0.5086511
[7728,] 0.9416085 0.5069227
[7729,] 0.9650960 0.5069227
[7730,] 0.9753208 0.5058554
[7731,] 0.9753208 0.5022851
[7732,] 0.9949388 0.5022851
[7733,] 0.9835453 0.5022851
[7734,] 0.9835453 0.5022851
[7735,] 0.9835453 0.5026808
[7736,] 0.9854474 0.5026808
[7737,] 0.9854474 0.5026808
[7738,] 0.9875872 0.5026808
[7739,] 0.9930261 0.5026808
[7740,] 0.9930261 0.5028388
[7741,] 0.9930261 0.5028388
[7742,] 0.9930261 0.5016176
[7743,] 0.9930261 0.5016176
[7744,] 0.9930261 0.5023172
[7745,] 0.9930261 0.5011587
[7746,] 0.9930261 0.5010559
[7747,] 0.9930261 0.5008393
[7748,] 0.9749782 0.5008393
[7749,] 0.9959037 0.5008393
[7750,] 1.0131680 0.5001798
[7751,] 0.9935740 0.5006700
[7752,] 0.9935740 0.5006700
[7753,] 0.9935740 0.5007261
[7754,] 0.9935740 0.5007261
[7755,] 0.9935740 0.5007261
[7756,] 0.9935740 0.5007261
[7757,] 0.9935740 0.5024615
[7758,] 0.9781766 0.5027891
[7759,] 0.9781766 0.5027891
[7760,] 0.9781766 0.5061245
[7761,] 0.9429289 0.5089062
[7762,] 0.9337976 0.5089062
[7763,] 0.9337976 0.5089062
[7764,] 0.9337976 0.5089062
[7765,] 0.9466889 0.5089062
[7766,] 0.9466889 0.5089062
[7767,] 0.9466889 0.5089062
[7768,] 0.9466889 0.5042625
[7769,] 0.9628868 0.5042625
[7770,] 0.9628868 0.5042625
[7771,] 0.9677484 0.5042625
[7772,] 0.9588036 0.5042625
[7773,] 0.9588036 0.5042625
[7774,] 0.9588036 0.5042625
[7775,] 0.9588036 0.5060887
[7776,] 0.9588036 0.5060887
[7777,] 0.9588036 0.5060887
[7778,] 0.9505670 0.5060887
[7779,] 0.9539422 0.5060887
[7780,] 0.9697431 0.5062859
[7781,] 0.9431646 0.5055772
[7782,] 0.9642634 0.5055772
[7783,] 0.9509638 0.5055772
[7784,] 0.9635438 0.5072962
[7785,] 0.9635438 0.5072962
[7786,] 0.9635438 0.5073137
[7787,] 0.9635438 0.5058468
[7788,] 0.9635438 0.5057040
[7789,] 0.9706682 0.5065078
[7790,] 0.9628809 0.5086629
[7791,] 0.9628809 0.5066418
[7792,] 0.9633932 0.5066418
[7793,] 0.9642082 0.5066418
[7794,] 0.9528863 0.5066418
[7795,] 0.9481575 0.5066418
[7796,] 0.9481575 0.5051053
[7797,] 0.9481575 0.5050557
[7798,] 0.9481575 0.5050557
[7799,] 0.9810707 0.5050557
[7800,] 0.9810707 0.5050557
[7801,] 0.9409078 0.5101463
[7802,] 0.9409078 0.5101463
[7803,] 0.9433274 0.5072898
[7804,] 0.9433274 0.5057914
[7805,] 0.9400904 0.5074634
[7806,] 0.9400904 0.5115544
[7807,] 0.9400904 0.5115544
[7808,] 0.9400904 0.5115544
[7809,] 0.9400904 0.5115544
[7810,] 0.9085651 0.5115544
[7811,] 0.9051672 0.5115544
[7812,] 0.9051672 0.5115544
[7813,] 0.8861741 0.5110998
[7814,] 0.9050894 0.5110998
[7815,] 0.9050894 0.5099747
[7816,] 0.9134498 0.5099747
[7817,] 0.9134498 0.5115296
[7818,] 0.9134498 0.5141057
[7819,] 0.9075715 0.5141057
[7820,] 0.9048983 0.5141057
[7821,] 0.9048983 0.5111498
[7822,] 0.9189090 0.5106520
[7823,] 0.9189090 0.5097237
[7824,] 0.9189090 0.5097237
[7825,] 0.9189090 0.5097237
[7826,] 0.9135787 0.5122681
[7827,] 0.9135787 0.5122681
[7828,] 0.9135787 0.5132131
[7829,] 0.9137655 0.5132131
[7830,] 0.9123578 0.5136398
[7831,] 0.8871425 0.5136398
[7832,] 0.8871425 0.5136398
[7833,] 0.8871425 0.5147581
[7834,] 0.8871425 0.5156368
[7835,] 0.8871425 0.5148901
[7836,] 0.8929336 0.5148901
[7837,] 0.8794487 0.5148901
[7838,] 0.8922268 0.5146799
[7839,] 0.8922268 0.5145588
[7840,] 0.8708396 0.5145588
[7841,] 0.8708396 0.5145588
[7842,] 0.8708396 0.5145588
[7843,] 0.8656772 0.5154360
[7844,] 0.8913213 0.5163958
[7845,] 0.8817782 0.5163958
[7846,] 0.8817782 0.5163958
[7847,] 0.8830488 0.5150749
[7848,] 0.8906306 0.5147897
[7849,] 0.8906306 0.5147897
[7850,] 0.8906306 0.5147897
[7851,] 0.8906306 0.5147897
[7852,] 0.8892471 0.5147897
[7853,] 0.8799422 0.5147897
[7854,] 0.8799422 0.5144864
[7855,] 0.8799422 0.5144864
[7856,] 0.8798454 0.5146068
[7857,] 0.8977037 0.5146068
[7858,] 0.8573838 0.5153132
[7859,] 0.8570895 0.5147744
[7860,] 0.8748807 0.5147744
[7861,] 0.9096237 0.5147744
[7862,] 0.9096237 0.5131633
[7863,] 0.9096237 0.5142519
[7864,] 0.8789420 0.5142519
[7865,] 0.8749795 0.5139242
[7866,] 0.9070250 0.5105310
[7867,] 0.9070250 0.5105310
[7868,] 0.8971835 0.5155614
[7869,] 0.8971835 0.5132944
[7870,] 0.8971835 0.5132944
[7871,] 0.9123965 0.5132944
[7872,] 0.9150312 0.5112004
[7873,] 0.9094476 0.5095948
[7874,] 0.9258849 0.5095948
[7875,] 0.9258849 0.5110643
[7876,] 0.9258849 0.5110643
[7877,] 0.9258849 0.5110643
[7878,] 0.9258849 0.5110643
[7879,] 0.9258849 0.5099272
[7880,] 0.9258849 0.5099272
[7881,] 0.9218648 0.5099272
[7882,] 0.9218648 0.5113949
[7883,] 0.9218648 0.5090837
[7884,] 0.9141834 0.5115110
[7885,] 0.8996575 0.5130613
[7886,] 0.8815342 0.5130613
[7887,] 0.8815342 0.5154510
[7888,] 0.8815342 0.5154510
[7889,] 0.8816867 0.5154510
[7890,] 0.8916508 0.5156660
[7891,] 0.8916508 0.5156660
[7892,] 0.8916508 0.5150039
[7893,] 0.8916508 0.5150039
[7894,] 0.8981695 0.5150039
[7895,] 0.8981695 0.5150039
[7896,] 0.8981695 0.5107248
[7897,] 0.8981695 0.5107248
[7898,] 0.8946108 0.5105673
[7899,] 0.8946108 0.5105673
[7900,] 0.8935885 0.5105673
[7901,] 0.8935885 0.5128765
[7902,] 0.8935885 0.5128765
[7903,] 0.8935885 0.5128765
[7904,] 0.9126886 0.5128765
[7905,] 0.8752314 0.5175046
[7906,] 0.8752314 0.5180807
[7907,] 0.8752314 0.5173755
[7908,] 0.8521205 0.5173755
[7909,] 0.8666265 0.5173755
[7910,] 0.8666265 0.5173755
[7911,] 0.8666265 0.5165491
[7912,] 0.8613484 0.5188927
[7913,] 0.8652711 0.5191790
[7914,] 0.8652711 0.5176471
[7915,] 0.8652711 0.5196719
[7916,] 0.8652711 0.5157951
[7917,] 0.8817191 0.5157951
[7918,] 0.8817191 0.5157951
[7919,] 0.8865885 0.5128149
[7920,] 0.8865885 0.5125930
[7921,] 0.8865885 0.5125930
[7922,] 0.8865885 0.5126403
[7923,] 0.8865885 0.5126403
[7924,] 0.8865885 0.5171655
[7925,] 0.8865885 0.5155834
[7926,] 0.8980016 0.5151014
[7927,] 0.8841651 0.5151014
[7928,] 0.8841651 0.5151014
[7929,] 0.8841651 0.5151014
[7930,] 0.8841651 0.5143275
[7931,] 0.8841651 0.5129313
[7932,] 0.8841651 0.5129313
[7933,] 0.8841651 0.5148424
[7934,] 0.8841651 0.5156285
[7935,] 0.8841651 0.5169432
[7936,] 0.8580542 0.5169432
[7937,] 0.8843067 0.5151944
[7938,] 0.8843067 0.5151944
[7939,] 0.8843067 0.5151944
[7940,] 0.8843067 0.5151944
[7941,] 0.8843067 0.5151944
[7942,] 0.8829882 0.5151944
[7943,] 0.8935157 0.5166769
[7944,] 0.8929970 0.5166769
[7945,] 0.8822075 0.5166769
[7946,] 0.8822075 0.5166769
[7947,] 0.8822075 0.5156994
[7948,] 0.8755189 0.5175473
[7949,] 0.8755189 0.5175473
[7950,] 0.8707058 0.5175473
[7951,] 0.8739756 0.5171120
[7952,] 0.8679836 0.5168280
[7953,] 0.8741823 0.5168280
[7954,] 0.8741823 0.5154456
[7955,] 0.8790476 0.5154456
[7956,] 0.8790476 0.5163867
[7957,] 0.8797235 0.5146043
[7958,] 0.9120508 0.5150452
[7959,] 0.9124865 0.5119094
[7960,] 0.9124865 0.5119094
[7961,] 0.9040432 0.5119094
[7962,] 0.9234152 0.5119094
[7963,] 0.9069163 0.5119094
[7964,] 0.9069163 0.5119094
[7965,] 0.8994162 0.5119094
[7966,] 0.9237601 0.5100741
[7967,] 0.9237601 0.5100741
[7968,] 0.9254557 0.5100741
[7969,] 0.9396699 0.5100741
[7970,] 0.9396699 0.5100419
[7971,] 0.9345588 0.5069560
[7972,] 0.9370163 0.5071488
[7973,] 0.9370163 0.5086193
[7974,] 0.9370163 0.5093520
[7975,] 0.9379616 0.5093520
[7976,] 0.9379616 0.5082043
[7977,] 0.9457649 0.5096542
[7978,] 0.9457649 0.5096542
[7979,] 0.9149011 0.5096542
[7980,] 0.9149011 0.5096542
[7981,] 0.9149011 0.5098810
[7982,] 0.9180792 0.5101901
[7983,] 0.9225616 0.5101901
[7984,] 0.9103968 0.5101901
[7985,] 0.9103968 0.5101901
[7986,] 0.9296786 0.5101901
[7987,] 0.9296786 0.5101901
[7988,] 0.9296786 0.5085411
[7989,] 0.9296786 0.5085411
[7990,] 0.9364770 0.5085411
[7991,] 0.9178152 0.5115032
[7992,] 0.9178152 0.5115032
[7993,] 0.9178152 0.5115032
[7994,] 0.9178152 0.5087657
[7995,] 0.9196918 0.5087657
[7996,] 0.9193074 0.5087657
[7997,] 0.9193074 0.5087198
[7998,] 0.9144740 0.5105620
[7999,] 0.9144481 0.5108223
[8000,] 0.9144481 0.5129375

We make a prediction for each iteration!

Predictions

We define a range of temperature in which we want to predict lambda

For each iteration (each line of mcmc.bind), we calculate lambda with the b0 and b1 estimated, as \(\lambda = \mbox{exp}( \beta_0 + \beta1 \mbox{Temperature})\)

\[~~\]

We have the predicted distribution of lambda for each temperature and we can extract mean, median, standard deviation, credible interval, etc.

pred.temp <- seq(from = -5, to = 9, by = 0.1)

\[~~\]

# create a matrix niteration x ntemp
pred.lambda <- matrix(NA, nrow = nrow(mcmc.bind), ncol = length(pred.temp))

for(i in 1:nrow(pred.lambda)){
  pred.lambda[i,] <- exp(mcmc.bind[i,"b0"] + mcmc.bind[i,"b1"] * pred.temp)
}

\[~~\]

mean.lambda <- apply(pred.lambda, 2, mean) # extract mean
sd.lambda <- apply(pred.lambda, 2, sd) # extract sd
ci <- apply(pred.lambda, 2, quantile, c(0.1, 0.9)) # 80% CI

pred.results <- tibble(temp = pred.temp, 
                       mean = mean.lambda,
                       sd = sd.lambda,
                       cinf = ci[1,],
                       csup = ci[2,])

# A tibble: 141 × 5
    temp  mean     sd  cinf  csup
   <dbl> <dbl>  <dbl> <dbl> <dbl>
 1  -5   0.202 0.0176 0.180 0.227
 2  -4.9 0.213 0.0183 0.189 0.239
 3  -4.8 0.224 0.0191 0.199 0.251
 4  -4.7 0.235 0.0200 0.210 0.264
 5  -4.6 0.248 0.0209 0.221 0.277
 6  -4.5 0.260 0.0218 0.233 0.291
 7  -4.4 0.274 0.0227 0.245 0.306
 8  -4.3 0.288 0.0237 0.258 0.322
 9  -4.2 0.303 0.0247 0.272 0.338
10  -4.1 0.319 0.0258 0.286 0.356
# ℹ 131 more rows

Predictions

We can plot the prediction.

Conclusions

Bayesian statistics are tools to fit ecological models.

Conclusions

Bayesian statistics are tools to fit ecological models.

Why I like Bayesian?

Because it helps me to understand the models.
Using nimble allow to formally write the statistical model you want.
It is useful to adapt existing models to fancy (or shitty) datasets.
Or to integrate multiple datasets, although alternative exist {spAbundance}, {spOccupancy}, {pointedSDM}.

Useful resources

Olivier Gimenez’s online classes
Official website https://r-nimble.org
User Manual https://r-nimble.org/html_manual/cha-welcome-nimble.html and cheatsheet.
Users mailing list https://groups.google.com/forum/#!forum/nimble-users
Training material https://github.com/nimble-training
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.

Mañana

We will see how to fit widely used ecological models with nimble.

Occupancy models
N-mixture
Spatial Capture-Recapture

Bloque 1 Modelos con R y Nimble Part 2 Introduccion al Bayesiano y Nimble © Bea Rubio

Que hacemos hoy?

The Theory

What is the Bayesian approach?

What is the Bayesian approach?

Frequentist and Bayesian

The Theory

Bayesian Computation

Markov Chain Monte Carlo (MCMC) methods.

Where everything begins: the prior distribution

Where everything begins: the prior distribution

Then we start: MCMC functioning

MCMC functioning

MCMC functioning

Parametrize MCMC

MCMC functionning

Assessing convergence

Assessing convergence

Burnin

Efficient sample size and convergence

What if we have issues of convergence ?

Plot posterior distribution

Plot posterior distribution

Prior distribution

Posterior distribution

Plot prior and posterior distribution

Plot posterior distribution

To sum up with Bayesian statistics and MCMC

To sum up with Bayesian statistics and MCMC

NIMBLE

Load nimble package

The fictive roe deer poop example

Roe deer counts

Step 1: write a model

Code

Step 2: Read in data

Code

Step 3: Initials values and parameters to save.

Specify initial values

Which parameters to save?

Code

Step 4: MCMC details

Code

Sum up

Code

Run model!

Explore MCMC outputs

Explore MCMC outputs

Numerical summaries

Trace and posterior density

Predictions

Predictions

Predictions

Predictions

Conclusions

Conclusions

Why I like Bayesian?

Useful resources

Mañana

Bloque 1 Modelos con R y Nimble
Part 2 Introduccion al Bayesiano y Nimble
© Bea Rubio

Load `nimble` package