Last data update: 2014.03.03

R: Run 'JAGS' from R
jagsR Documentation

Run ‘JAGS’ from R

Description

The jags function takes data and starting values as input. It automatically writes a jags script, calls the model, and saves the simulations for easy access in R.

Usage

jags(data, inits, parameters.to.save, model.file="model.bug",
  n.chains=3, n.iter=2000, n.burnin=floor(n.iter/2),
  n.thin=max(1, floor((n.iter - n.burnin) / 1000)),
  DIC=TRUE, working.directory=NULL, jags.seed = 123,
  refresh = n.iter/50, progress.bar = "text", digits=5,
  RNGname = c("Wichmann-Hill", "Marsaglia-Multicarry",
              "Super-Duper", "Mersenne-Twister"),
  jags.module = c("glm","dic")
  )

jags.parallel(data, inits, parameters.to.save, model.file = "model.bug",
             n.chains = 2, n.iter = 2000, n.burnin = floor(n.iter/2),
             n.thin = max(1, floor((n.iter - n.burnin)/1000)),
             n.cluster= n.chains, DIC = TRUE,
             working.directory = NULL, jags.seed = 123, digits=5,
             RNGname = c("Wichmann-Hill", "Marsaglia-Multicarry",
              "Super-Duper", "Mersenne-Twister"),
             jags.module = c("glm","dic"),
             export_obj_names=NULL,
             envir = .GlobalEnv
             )

jags2(data, inits, parameters.to.save, model.file="model.bug",
  n.chains=3, n.iter=2000, n.burnin=floor(n.iter/2),
  n.thin=max(1, floor((n.iter - n.burnin) / 1000)),
  DIC=TRUE, jags.path="",
  working.directory=NULL, clearWD=TRUE,
  refresh = n.iter/50)

Arguments

data

(1) a vector or list of the names of the data objects used by the model, (2) a (named) list of the data objects themselves, or (3) the name of a "dump" format file containing the data objects, which must end in ".txt", see example below for details.

inits

a list with n.chains elements; each element of the list is itself a list of starting values for the BUGS model, or a function creating (possibly random) initial values. If inits is NULL, JAGS will generate initial values for parameters.

parameters.to.save

character vector of the names of the parameters to save which should be monitored.

model.file

file containing the model written in BUGS code. Alternatively, as in R2WinBUGS, model.file can be an R function that contains a BUGS model that is written to a temporary model file (see tempfile) using write.model

n.chains

number of Markov chains (default: 3)

n.iter

number of total iterations per chain (including burn in; default: 2000)

n.burnin

length of burn in, i.e. number of iterations to discard at the beginning. Default is n.iter/2, that is, discarding the first half of the simulations. If n.burnin is 0, jags() will run 100 iterations for adaption.

n.cluster

number of clusters to use to run parallel chains. Default equals n.chains.

n.thin

thinning rate. Must be a positive integer. Set n.thin > 1 to save memory and computation time if n.iter is large. Default is max(1, floor(n.chains * (n.iter-n.burnin) / 1000)) which will only thin if there are at least 2000 simulations.

DIC

logical; if TRUE (default), compute deviance, pD, and DIC. The rule pD=var(deviance) / 2 is used.

working.directory

sets working directory during execution of this function; This should be the directory where model file is.

jags.seed

random seed for JAGS, default is 123. This function is used for jags.parallell() and does not work for jags(). Use set.seed() instead if you want to produce identical result with jags()

.

jags.path

directory that contains the JAGS executable. The default is “”.

clearWD

indicating whether the files ‘data.txt’, ‘inits[1:n.chains].txt’, ‘codaIndex.txt’, ‘jagsscript.txt’, and ‘CODAchain[1:nchains].txt’ should be removed after jags has finished, default=TRUE.

refresh

refresh frequency for progress bar, default is n.iter/50

progress.bar

type of progress bar. Possible values are “text”, “gui”, and “none”. Type “text” is displayed on the R console. Type “gui” is a graphical progress bar in a new window. The progress bar is suppressed if progress.bar is “none”

digits

as in write.model in the R2WinBUGS package: number of significant digits used for BUGS input, see formatC. Only used if specifying a BUGS model as an R function.

RNGname

the name for random number generator used in JAGS. There are four RNGS supplied by the base moduale in JAGS: Wichmann-Hill, Marsaglia-Multicarry, Super-Duper, Mersenne-Twister

jags.module

the vector of jags modules to be loaded. Default are “glm” and “dic”. Input NULL if you don't want to load any jags module.

export_obj_names

character vector of objects to export to the clusters.

envir

default is .GlobalEnv

Details

To run:

  1. Write a BUGS model in an ASCII file.

  2. Go into R.

  3. Prepare the inputs for the jags function and run it (see Example section).

  4. The model will now run in JAGS. It might take awhile. You will see things happening in the R console.

BUGS version support:

  • jags 1.0.3default

Author(s)

Yu-Sung Su suyusung@tsinghua.edu.cn, Masanao Yajima yajima@stat.columbia.edu

References

Plummer, Martyn (2003) “JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling.” http://citeseer.ist.psu.edu/plummer03jags.html.

Gelman, A., Carlin, J. B., Stern, H.S., Rubin, D.B. (2003) Bayesian Data Analysis, 2nd edition, CRC Press.

Sibylle Sturtz and Uwe Ligges and Andrew Gelman. (2005). “R2WinBUGS: A Package for Running WinBUGS from R.” Journal of Statistical Software 3 (12): 1–6.

Examples

  # An example model file is given in:
  model.file <- system.file(package="R2jags", "model", "schools.txt")
  # Let's take a look:
  file.show(model.file)
  # you can also write BUGS model as a R function, see below:

#=================#
# initialization  #
#=================#

  # data
  J <- 8.0
  y <- c(28.4,7.9,-2.8,6.8,-0.6,0.6,18.0,12.2)
  sd <- c(14.9,10.2,16.3,11.0,9.4,11.4,10.4,17.6)


  jags.data <- list("y","sd","J")
  jags.params <- c("mu","sigma","theta")
  jags.inits <- function(){
    list("mu"=rnorm(1),"sigma"=runif(1),"theta"=rnorm(J))
  }

  ## You can input data in 4 ways
  ## 1) data as list of character
  jagsfit <- jags(data=list("y","sd","J"), inits=jags.inits, jags.params,
                n.iter=10, model.file=model.file)

  ## 2) data as character vector of names
  jagsfit <- jags(data=c("y","sd","J"), inits=jags.inits, jags.params,
                n.iter=10, model.file=model.file)

  ## 3) data as named list
  jagsfit <- jags(data=list(y=y,sd=sd,J=J), inits=jags.inits, jags.params,
                n.iter=10, model.file=model.file)

  ## 4) data as a file
  fn <- "tmpbugsdata.txt"
  dump(c("y","sd","J"), file=fn)
  jagsfit <- jags(data=fn, inits=jags.inits, jags.params,
                  n.iter=10, model.file=model.file)
  unlink("tmpbugsdata.txt")

  ## You can write bugs model in R as a function

  schoolsmodel <- function() {
    for (j in 1:J){                     # J=8, the number of schools
      y[j] ~ dnorm (theta[j], tau.y[j]) # data model:  the likelihood
      tau.y[j] <- pow(sd[j], -2)        # tau = 1/sigma^2
    }
    for (j in 1:J){
      theta[j] ~ dnorm (mu, tau)        # hierarchical model for theta
    }
    tau <- pow(sigma, -2)               # tau = 1/sigma^2
    mu ~ dnorm (0.0, 1.0E-6)            # noninformative prior on mu
    sigma ~ dunif (0, 1000)             # noninformative prior on sigma
  }

  jagsfit <- jags(data=jags.data, inits=jags.inits, jags.params,
                n.iter=10, model.file=schoolsmodel)


#===============================#
# RUN jags and postprocessing   #
#===============================#
  jagsfit <- jags(data=jags.data, inits=jags.inits, jags.params,
    n.iter=5000, model.file=model.file)

  # Run jags parallely, no progress bar. R may be frozen for a while,
  # Be patient. Currenlty update afterward does not run parallelly
  #
   jagsfit.p <- jags.parallel(data=jags.data, inits=jags.inits, jags.params,
     n.iter=5000, model.file=model.file)

  # display the output
  print(jagsfit)
  plot(jagsfit)

  # traceplot
  traceplot(jagsfit.p)
  traceplot(jagsfit)

  # or to use some plots in coda
  # use as.mcmmc to convert rjags object into mcmc.list
  jagsfit.mcmc <- as.mcmc(jagsfit.p)
  jagsfit.mcmc <- as.mcmc(jagsfit)
  ## now we can use the plotting methods from coda
  #require(lattice)
  #xyplot(jagsfit.mcmc)
  #densityplot(jagsfit.mcmc)

  # if the model does not converge, update it!
  jagsfit.upd <- update(jagsfit, n.iter=100)
  print(jagsfit.upd)
  print(jagsfit.upd, intervals=c(0.025, 0.5, 0.975))
  plot(jagsfit.upd)

  # before update parallel jags object, do recompile it
  recompile(jagsfit.p)
  jagsfit.upd <- update(jagsfit.p, n.iter=100)



  # or auto update it until it converges! see ?autojags for details
  # recompile(jagsfit.p)
  jagsfit.upd <- autojags(jagsfit.p)
  jagsfit.upd <- autojags(jagsfit)

  # to get DIC or specify DIC=TRUE in jags() or do the following#
  dic.samples(jagsfit.upd$model, n.iter=1000, type="pD")

  # attach jags object into search path see "attach.bugs" for details
  attach.jags(jagsfit.upd)

  # this will show a 3-way array of the bugs.sim object, for example:
  mu

  # detach jags object into search path see "attach.bugs" for details
  detach.jags()

  # to pick up the last save session
  # for example, load("RWorkspace.Rdata")
  recompile(jagsfit)
  jagsfit.upd <- update(jagsfit, n.iter=100)

  recompile(jagsfit.p)
  jagsfit.upd <- update(jagsfit, n.iter=100)

#=============#
# using jags2 #
#=============#
  ## jags can be run and produces coda files, but cannot be updated once it's done
  ## You may need to edit "jags.path" to make this work,
  ## also you need a write access in the working directory:
  ## e.g. setwd("d:/")

  ## NOT RUN HERE
  ## Not run: 
    jagsfit <- jags2(data=jags.data, inits=jags.inits, jags.params,
      n.iter=5000, model.file=model.file)
    print(jagsfit)
    plot(jagsfit)
    # or to use some plots in coda
    # use as.mcmmc to convert rjags object into mcmc.list
    jagsfit.mcmc <- as.mcmc.list(jagsfit)
    traceplot(jagsfit.mcmc)
    #require(lattice)
    #xyplot(jagsfit.mcmc)
    #densityplot(jagsfit.mcmc)
  
## End(Not run)

Results


R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(R2jags)
Loading required package: rjags
Loading required package: coda
Linked to JAGS 4.1.0
Loaded modules: basemod,bugs

Attaching package: 'R2jags'

The following object is masked from 'package:coda':

    traceplot

> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/R2jags/jags.Rd_%03d_medium.png", width=480, height=480)
> ### Name: jags
> ### Title: Run 'JAGS' from R
> ### Aliases: rjags-class rjags.parallel-class jags jags2 jags.parallel
> ### Keywords: interface models
> 
> ### ** Examples
> 
>   # An example model file is given in:
>   model.file <- system.file(package="R2jags", "model", "schools.txt")
>   # Let's take a look:
>   file.show(model.file)
# Bugs model file for 8 schools analysis from Section 5.5 of "Bayesian Data 
# Analysis". Save this into the file "schools.bug" in your R working directory. 
 
model { 
  for (j in 1:J){                      # J=8, the number of schools 
    y[j] ~ dnorm (theta[j], tau.y[j])  # data model:  the likelihood 
    tau.y[j] <- pow(sd[j], -2)         # tau = 1/sigma^2 
  } 
  for (j in 1:J){ 
    theta[j] ~ dnorm (mu, tau)         # hierarchical model for theta 
  } 
  tau <- pow(sigma, -2)                # tau = 1/sigma^2 
  mu ~ dnorm (0.0, 1.0E-6)             # noninformative prior on mu 
  sigma ~ dunif (0, 1000)              # noninformative prior on sigma 
} 

>   # you can also write BUGS model as a R function, see below:
> 
> #=================#
> # initialization  #
> #=================#
> 
>   # data
>   J <- 8.0
>   y <- c(28.4,7.9,-2.8,6.8,-0.6,0.6,18.0,12.2)
>   sd <- c(14.9,10.2,16.3,11.0,9.4,11.4,10.4,17.6)
> 
> 
>   jags.data <- list("y","sd","J")
>   jags.params <- c("mu","sigma","theta")
>   jags.inits <- function(){
+     list("mu"=rnorm(1),"sigma"=runif(1),"theta"=rnorm(J))
+   }
> 
>   ## You can input data in 4 ways
>   ## 1) data as list of character
>   jagsfit <- jags(data=list("y","sd","J"), inits=jags.inits, jags.params,
+                 n.iter=10, model.file=model.file)
module glm loaded
Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 8
   Unobserved stochastic nodes: 10
   Total graph size: 50

Initializing model

> 
>   ## 2) data as character vector of names
>   jagsfit <- jags(data=c("y","sd","J"), inits=jags.inits, jags.params,
+                 n.iter=10, model.file=model.file)
Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 8
   Unobserved stochastic nodes: 10
   Total graph size: 50

Initializing model

> 
>   ## 3) data as named list
>   jagsfit <- jags(data=list(y=y,sd=sd,J=J), inits=jags.inits, jags.params,
+                 n.iter=10, model.file=model.file)
Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 8
   Unobserved stochastic nodes: 10
   Total graph size: 50

Initializing model

> 
>   ## 4) data as a file
>   fn <- "tmpbugsdata.txt"
>   dump(c("y","sd","J"), file=fn)
>   jagsfit <- jags(data=fn, inits=jags.inits, jags.params,
+                   n.iter=10, model.file=model.file)
Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 8
   Unobserved stochastic nodes: 10
   Total graph size: 50

Initializing model

>   unlink("tmpbugsdata.txt")
> 
>   ## You can write bugs model in R as a function
> 
>   schoolsmodel <- function() {
+     for (j in 1:J){                     # J=8, the number of schools
+       y[j] ~ dnorm (theta[j], tau.y[j]) # data model:  the likelihood
+       tau.y[j] <- pow(sd[j], -2)        # tau = 1/sigma^2
+     }
+     for (j in 1:J){
+       theta[j] ~ dnorm (mu, tau)        # hierarchical model for theta
+     }
+     tau <- pow(sigma, -2)               # tau = 1/sigma^2
+     mu ~ dnorm (0.0, 1.0E-6)            # noninformative prior on mu
+     sigma ~ dunif (0, 1000)             # noninformative prior on sigma
+   }
> 
>   jagsfit <- jags(data=jags.data, inits=jags.inits, jags.params,
+                 n.iter=10, model.file=schoolsmodel)
Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 8
   Unobserved stochastic nodes: 10
   Total graph size: 50

Initializing model

> 
> 
> #===============================#
> # RUN jags and postprocessing   #
> #===============================#
>   jagsfit <- jags(data=jags.data, inits=jags.inits, jags.params,
+     n.iter=5000, model.file=model.file)
Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 8
   Unobserved stochastic nodes: 10
   Total graph size: 50

Initializing model

> 
>   # Run jags parallely, no progress bar. R may be frozen for a while,
>   # Be patient. Currenlty update afterward does not run parallelly
>   #
>    jagsfit.p <- jags.parallel(data=jags.data, inits=jags.inits, jags.params,
+      n.iter=5000, model.file=model.file)
> 
>   # display the output
>   print(jagsfit)
Inference for Bugs model at "/home/ddbj/local/lib64/R/library/R2jags/model/schools.txt", fit using jags,
 3 chains, each with 5000 iterations (first 2500 discarded), n.thin = 2
 n.sims = 3750 iterations saved
         mu.vect sd.vect    2.5%    25%    50%    75%  97.5%  Rhat n.eff
mu         8.283   5.250  -2.546  5.090  8.402 11.748 18.141 1.002  1100
sigma      7.265   5.940   0.262  2.874  6.028 10.141 22.009 1.003   760
theta[1]  12.292   8.516  -2.159  6.889 11.255 16.444 32.125 1.003  1000
theta[2]   8.271   6.535  -4.883  4.294  8.490 12.464 21.533 1.003   960
theta[3]   6.198   8.339 -12.728  1.775  7.022 11.637 20.903 1.001  3700
theta[4]   7.877   6.719  -6.389  3.825  8.143 12.145 20.983 1.001  3800
theta[5]   5.284   6.928 -10.283  1.103  6.012 10.189 16.532 1.001  3800
theta[6]   6.245   7.168  -9.519  2.031  6.987 11.191 18.653 1.001  3100
theta[7]  11.189   6.958  -1.956  6.511 10.845 15.111 26.792 1.005   430
theta[8]   9.017   8.180  -7.128  4.358  9.013 13.489 27.227 1.001  3800
deviance  60.453   2.221  56.907 59.069 60.085 61.471 66.066 1.002  1700

For each parameter, n.eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor (at convergence, Rhat=1).

DIC info (using the rule, pD = var(deviance)/2)
pD = 2.5 and DIC = 62.9
DIC is an estimate of expected predictive error (lower deviance is better).
>   plot(jagsfit)
> 
>   # traceplot
>   traceplot(jagsfit.p)
>   traceplot(jagsfit)
> 
>   # or to use some plots in coda
>   # use as.mcmmc to convert rjags object into mcmc.list
>   jagsfit.mcmc <- as.mcmc(jagsfit.p)
>   jagsfit.mcmc <- as.mcmc(jagsfit)
>   ## now we can use the plotting methods from coda
>   #require(lattice)
>   #xyplot(jagsfit.mcmc)
>   #densityplot(jagsfit.mcmc)
> 
>   # if the model does not converge, update it!
>   jagsfit.upd <- update(jagsfit, n.iter=100)
>   print(jagsfit.upd)
Inference for Bugs model at "/home/ddbj/local/lib64/R/library/R2jags/model/schools.txt", fit using jags,
 3 chains, each with 100 iterations (first 0 discarded)
 n.sims = 300 iterations saved
         mu.vect sd.vect    2.5%    25%    50%    75%  97.5%  Rhat n.eff
mu         7.846   5.229  -1.867  3.812  8.146 10.870 17.368 1.052   100
sigma      4.953   5.859   0.181  0.975  2.714  6.147 20.279 1.242    12
theta[1]  10.395   7.811  -1.199  4.954  8.794 15.013 33.524 1.052    54
theta[2]   7.453   6.173  -3.307  3.412  7.256 10.825 18.544 1.039   140
theta[3]   6.636   7.380 -11.524  3.385  7.334 11.051 18.837 1.049   140
theta[4]   8.025   5.771  -2.394  3.919  7.788 11.092 18.337 1.048   200
theta[5]   6.524   6.097  -5.623  3.271  6.864  9.808 17.291 1.032   120
theta[6]   6.926   6.655  -9.884  3.466  7.571 10.718 17.758 1.068    57
theta[7]  10.093   6.240   0.104  5.866  8.882 14.311 24.388 1.048    44
theta[8]   8.559   7.353  -6.273  3.846  8.056 11.874 26.072 1.018   300
deviance  60.619   1.913  57.453 59.395 60.246 61.544 64.660 1.135    19

For each parameter, n.eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor (at convergence, Rhat=1).

DIC info (using the rule, pD = var(deviance)/2)
pD = 1.6 and DIC = 62.3
DIC is an estimate of expected predictive error (lower deviance is better).
>   print(jagsfit.upd, intervals=c(0.025, 0.5, 0.975))
Inference for Bugs model at "/home/ddbj/local/lib64/R/library/R2jags/model/schools.txt", fit using jags,
 3 chains, each with 100 iterations (first 0 discarded)
 n.sims = 300 iterations saved
         mu.vect sd.vect    2.5%    50%  97.5%  Rhat n.eff
mu         7.846   5.229  -1.867  8.146 17.368 1.052   100
sigma      4.953   5.859   0.181  2.714 20.279 1.242    12
theta[1]  10.395   7.811  -1.199  8.794 33.524 1.052    54
theta[2]   7.453   6.173  -3.307  7.256 18.544 1.039   140
theta[3]   6.636   7.380 -11.524  7.334 18.837 1.049   140
theta[4]   8.025   5.771  -2.394  7.788 18.337 1.048   200
theta[5]   6.524   6.097  -5.623  6.864 17.291 1.032   120
theta[6]   6.926   6.655  -9.884  7.571 17.758 1.068    57
theta[7]  10.093   6.240   0.104  8.882 24.388 1.048    44
theta[8]   8.559   7.353  -6.273  8.056 26.072 1.018   300
deviance  60.619   1.913  57.453 60.246 64.660 1.135    19

For each parameter, n.eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor (at convergence, Rhat=1).

DIC info (using the rule, pD = var(deviance)/2)
pD = 1.6 and DIC = 62.3
DIC is an estimate of expected predictive error (lower deviance is better).
>   plot(jagsfit.upd)
> 
>   # before update parallel jags object, do recompile it
>   recompile(jagsfit.p)
Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 8
   Unobserved stochastic nodes: 10
   Total graph size: 50

Initializing model

Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 8
   Unobserved stochastic nodes: 10
   Total graph size: 50

Initializing model

>   jagsfit.upd <- update(jagsfit.p, n.iter=100)
> 
> 
> 
>   # or auto update it until it converges! see ?autojags for details
>   # recompile(jagsfit.p)
>   jagsfit.upd <- autojags(jagsfit.p)
>   jagsfit.upd <- autojags(jagsfit)
> 
>   # to get DIC or specify DIC=TRUE in jags() or do the following#
>   dic.samples(jagsfit.upd$model, n.iter=1000, type="pD")
Mean deviance:  60.52 
penalty 2.855 
Penalized deviance: 63.38 
> 
>   # attach jags object into search path see "attach.bugs" for details
>   attach.jags(jagsfit.upd)
> 
>   # this will show a 3-way array of the bugs.sim object, for example:
>   mu
                 [,1]
   [1,]  14.881464571
   [2,]   3.604098011
   [3,]  15.978212680
   [4,]   4.959063873
   [5,]   3.151922559
   [6,]   5.782175675
   [7,]   2.047465119
   [8,]   8.678511782
   [9,]   5.654278389
  [10,]   7.344493749
  [11,]   5.782657710
  [12,]  11.436401939
  [13,]  14.090212575
  [14,]  14.496716772
  [15,]   8.899458286
  [16,]   3.960753808
  [17,]   7.681111588
  [18,]   8.776018331
  [19,]   7.230958446
  [20,]  15.503152856
  [21,]   3.337025812
  [22,]   7.441212500
  [23,]  11.572551700
  [24,]   7.141389653
  [25,]   5.970881192
  [26,]  11.516674412
  [27,]   9.907720219
  [28,]   7.638678098
  [29,]   7.440112271
  [30,]   7.449259317
  [31,]  12.551402640
  [32,]  20.496298267
  [33,]   7.997844720
  [34,]   2.528411333
  [35,]   4.708914468
  [36,]   8.731331171
  [37,]   6.501914071
  [38,]   5.925110663
  [39,]   5.519476276
  [40,]   7.626505880
  [41,]   7.346637301
  [42,]  14.400361473
  [43,]  10.868374241
  [44,]   4.209502314
  [45,]   3.653545690
  [46,]  12.820979314
  [47,]   5.294160840
  [48,]  11.035957712
  [49,]   9.532682836
  [50,]  17.130237346
  [51,]  14.248520245
  [52,]  16.732664022
  [53,]   3.332926590
  [54,]   5.900265166
  [55,]   7.939522140
  [56,]  15.899177878
  [57,]   9.239121149
  [58,]   2.319339925
  [59,]  11.789126204
  [60,]   7.870244087
  [61,]   1.479283867
  [62,]   8.636186989
  [63,]   4.515985092
  [64,]   7.775149012
  [65,]   7.477989163
  [66,]  10.905278571
  [67,]   6.386348532
  [68,]  10.184845353
  [69,]   6.972813128
  [70,]  -1.204632133
  [71,]  12.290723046
  [72,]   5.336024790
  [73,]   5.629053137
  [74,]  12.911307524
  [75,]   4.263402388
  [76,]  11.756942175
  [77,]   6.851328701
  [78,]   6.694269463
  [79,]  11.231916631
  [80,]  11.420441026
  [81,]  16.056803464
  [82,]   5.439035101
  [83,]  10.717188466
  [84,]   7.267940890
  [85,]   4.094834409
  [86,]   5.360866882
  [87,]  14.522174998
  [88,]   9.857490869
  [89,]   4.176488846
  [90,]  17.942807242
  [91,]  12.066728764
  [92,]   4.979643715
  [93,]   4.127549412
  [94,]   7.703299903
  [95,]   4.715101284
  [96,]   9.053416477
  [97,]  -4.999941251
  [98,]  10.006114228
  [99,]  12.764989490
 [100,]  19.122006828
 [101,]   2.654536643
 [102,]   7.830745126
 [103,]   1.973625602
 [104,]  16.570034980
 [105,]   8.271698087
 [106,]   6.816608991
 [107,]   5.707964386
 [108,]   8.113976474
 [109,]  11.769146129
 [110,]   0.617720007
 [111,]  15.057564291
 [112,]   6.323683265
 [113,]   0.965988055
 [114,]   9.594935048
 [115,]   2.160081592
 [116,]   2.261093221
 [117,]  11.976691215
 [118,]  17.222454300
 [119,]  10.654743549
 [120,]   6.769094417
 [121,]  10.092302882
 [122,]  10.768852550
 [123,]  -2.358310837
 [124,]  10.573310348
 [125,]   6.247757836
 [126,]  12.495278808
 [127,]   7.088953321
 [128,]   6.285821302
 [129,]   5.507057298
 [130,]   7.614299732
 [131,]  10.978317563
 [132,]   3.762842904
 [133,]  13.409898589
 [134,]  11.821535470
 [135,]  11.429157058
 [136,]   7.744475049
 [137,]   6.246927047
 [138,]  11.031233458
 [139,]   6.304796636
 [140,]  19.426646061
 [141,]   9.128101232
 [142,]   5.234263648
 [143,]  13.589464006
 [144,]   1.541153840
 [145,]   7.839288375
 [146,]   2.767323284
 [147,]  17.792663712
 [148,]   6.021471645
 [149,]  12.409795046
 [150,]  12.203883822
 [151,]   4.751318563
 [152,]   9.018151383
 [153,]  11.571750820
 [154,]   5.815662165
 [155,]   1.386828510
 [156,]  13.535240593
 [157,]   4.320370242
 [158,]  13.924488796
 [159,]   1.577965523
 [160,]  -8.322290685
 [161,]   9.868397892
 [162,]   5.000694533
 [163,]  12.340246687
 [164,]   5.666417076
 [165,]   3.934961667
 [166,]   6.626158800
 [167,]   1.262014677
 [168,]  13.619702068
 [169,]   2.314211832
 [170,]  13.836454936
 [171,]  12.266859085
 [172,]   7.843994479
 [173,]  11.120693663
 [174,]   6.744297665
 [175,]   3.585802069
 [176,]   7.321624777
 [177,]   2.094406327
 [178,]  -0.294174117
 [179,]  14.436633549
 [180,]   9.716926697
 [181,]  -3.912464254
 [182,]   3.752578507
 [183,]  14.871971092
 [184,]  11.274895920
 [185,]  12.067925283
 [186,]   4.403516765
 [187,]   9.931941317
 [188,]   4.803427812
 [189,]  26.090362784
 [190,]   9.405920734
 [191,]   9.149483497
 [192,]   8.858134867
 [193,]   4.654785327
 [194,]   2.265116433
 [195,]   6.124460413
 [196,]   2.903906150
 [197,]   8.549873574
 [198,]   9.153340189
 [199,]   7.928051049
 [200,]  17.462782373
 [201,]  11.489485983
 [202,]   8.713461558
 [203,]   5.018729594
 [204,]   6.201147268
 [205,]  -1.830172463
 [206,]   6.122455960
 [207,]   2.823561684
 [208,]   3.787108337
 [209,]  10.938049725
 [210,]   2.726714951
 [211,]   0.323435541
 [212,]   2.190983029
 [213,]   3.498495116
 [214,]   5.490279338
 [215,]  12.117083726
 [216,]  -0.332120694
 [217,]  12.760160054
 [218,]   3.791381720
 [219,]   5.749741987
 [220,]   9.244606323
 [221,]  12.959163176
 [222,]   8.620183863
 [223,]  11.049689548
 [224,]   7.010150137
 [225,]  11.608888234
 [226,]  12.306529689
 [227,]  16.883345260
 [228,]   4.760341128
 [229,]  12.060640793
 [230,]   7.212559660
 [231,]   4.451273383
 [232,]  14.339276288
 [233,]  11.329628574
 [234,]  12.874715685
 [235,]   5.058239480
 [236,]  14.149536546
 [237,]   8.012159132
 [238,]   0.447122460
 [239,]   7.470179113
 [240,]  16.277616227
 [241,]   7.592785040
 [242,]   9.926939575
 [243,]   7.214833384
 [244,]  10.852421869
 [245,]   7.476682012
 [246,]  19.254965525
 [247,]   8.361414087
 [248,]   9.799555634
 [249,]   4.167733203
 [250,]   2.649888908
 [251,]   5.625724768
 [252,]   5.325137197
 [253,]  -1.249486477
 [254,]  12.378116877
 [255,]   8.342229113
 [256,]   5.222929109
 [257,]   3.343783719
 [258,]   9.719874111
 [259,]  11.026153510
 [260,]   5.387207387
 [261,]  13.049198948
 [262,]   6.716633689
 [263,]  12.570195807
 [264,]  13.448441960
 [265,]   6.130981713
 [266,]   2.304380407
 [267,]   9.889528817
 [268,]   1.869260150
 [269,]   7.680910217
 [270,]   8.366676472
 [271,]  10.326436197
 [272,]   5.935243045
 [273,]   6.617063534
 [274,]  13.216972886
 [275,]   5.348969717
 [276,]   7.000491775
 [277,]   9.768365018
 [278,]  12.442448183
 [279,]   4.686781346
 [280,]   7.400069619
 [281,]   2.314524040
 [282,]   3.401011335
 [283,]   7.925307733
 [284,]  11.855121876
 [285,]  13.037268839
 [286,]  -3.375445353
 [287,]   5.980776586
 [288,]   5.278946362
 [289,]   7.651033981
 [290,]   9.123041851
 [291,]  11.961697488
 [292,]  -1.354494199
 [293,]   5.352099491
 [294,]  13.428175053
 [295,]   6.271358051
 [296,]   6.657126507
 [297,]   5.294231872
 [298,]   7.764427544
 [299,]  12.381464820
 [300,]  13.696520727
 [301,]   6.683545422
 [302,]   8.042073253
 [303,]   7.003689649
 [304,]   5.697189170
 [305,]   1.882493175
 [306,]   2.777972105
 [307,]   5.177833511
 [308,]  10.261720923
 [309,]  -3.409046016
 [310,]   7.656419459
 [311,]  14.036140423
 [312,]   5.712859241
 [313,]  11.465519347
 [314,]   7.235838363
 [315,]   5.699701843
 [316,]   4.249170762
 [317,]   8.530844125
 [318,]   7.602598842
 [319,]   2.173865125
 [320,]   7.983254249
 [321,]   3.671330033
 [322,]   7.836147356
 [323,]   2.542621977
 [324,]  20.448186597
 [325,]  18.133688091
 [326,]  15.507082797
 [327,]  -0.436254326
 [328,]  15.210122617
 [329,]  13.861137304
 [330,]  13.660174223
 [331,]   1.017447056
 [332,]  15.774154481
 [333,]  21.217582923
 [334,]  11.148076007
 [335,]   7.567370812
 [336,]  12.355355558
 [337,]  10.448545689
 [338,]  12.436931024
 [339,]   7.356817734
 [340,]  12.665576421
 [341,]  13.892822194
 [342,]   2.104879071
 [343,]  11.465468748
 [344,]   9.093268402
 [345,]  11.155135283
 [346,]   9.582609168
 [347,]   4.911200998
 [348,]   5.626666366
 [349,]   6.683536391
 [350,]   7.602850042
 [351,]   5.518661012
 [352,]  13.028742149
 [353,]  17.167418917
 [354,]  14.395406261
 [355,]   6.029748922
 [356,]  13.754782587
 [357,]   8.004543145
 [358,]   9.821269455
 [359,]   9.358858788
 [360,]  22.917197757
 [361,]   5.420546759
 [362,]  10.686322473
 [363,]  19.293009419
 [364,]   7.681757876
 [365,]  15.610641870
 [366,]   9.352014664
 [367,]  12.941934557
 [368,]   1.533006335
 [369,]   2.585045760
 [370,]   7.728079684
 [371,]  14.586927419
 [372,]   6.025334745
 [373,]   0.557745398
 [374,]   7.920875592
 [375,]  13.569628178
 [376,]  11.850458910
 [377,]   6.860065020
 [378,]   9.607627812
 [379,]  -7.137127369
 [380,]  12.546188289
 [381,]   6.927676826
 [382,]   2.100369214
 [383,]   5.759973972
 [384,]   4.056890143
 [385,]   6.916803700
 [386,]   9.875742858
 [387,]  13.011498909
 [388,]  11.438487944
 [389,]  10.618013723
 [390,]  15.320267458
 [391,]   5.102716259
 [392,]  11.448699455
 [393,]  14.122961044
 [394,]   7.611040776
 [395,]  14.298650226
 [396,]  15.475913789
 [397,]  11.283383082
 [398,]  14.987665259
 [399,]   2.504082500
 [400,]   4.309008086
 [401,]  13.121005700
 [402,]  13.557766910
 [403,]   9.274929128
 [404,]   6.732364129
 [405,]   2.503153126
 [406,]   7.324559615
 [407,]  11.440138509
 [408,]  16.401488955
 [409,]   5.937846711
 [410,]   7.936680398
 [411,]  14.936692870
 [412,]   8.332742272
 [413,]   8.699932097
 [414,]   0.831131555
 [415,]   4.759541896
 [416,]   4.989411251
 [417,]   4.769176076
 [418,]  26.629981642
 [419,]   8.038562179
 [420,]  11.159314230
 [421,]   3.758080513
 [422,]  10.903438265
 [423,]  23.413037799
 [424,]   6.682276975
 [425,]   6.421103831
 [426,]   5.679228453
 [427,]  12.944304215
 [428,]  -1.250197371
 [429,]  12.020139054
 [430,]  12.891633530
 [431,]  11.067919024
 [432,]   3.619425712
 [433,]   8.533998294
 [434,]  -4.085689054
 [435,]  13.533201384
 [436,]  12.377937668
 [437,]   5.422448918
 [438,]  24.571494480
 [439,]   3.386549941
 [440,]   1.178792921
 [441,]   8.162021469
 [442,]   9.581839304
 [443,]  10.199202714
 [444,]  11.731678568
 [445,]   2.115768662
 [446,]   5.625165271
 [447,]   7.800775750
 [448,]   9.955990687
 [449,]   9.536171554
 [450,]   6.612453343
 [451,]   8.476379614
 [452,]   6.169295547
 [453,]   8.679486319
 [454,]  11.640208942
 [455,]   0.338740498
 [456,]   8.454063021
 [457,]   5.079606078
 [458,]  12.530752130
 [459,]   7.425114293
 [460,]  15.950185826
 [461,]  10.184467145
 [462,]  10.374504082
 [463,]   3.094123424
 [464,]  17.881284630
 [465,]   1.844105682
 [466,]   8.684986529
 [467,]  14.273205189
 [468,]  12.378561389
 [469,]  12.186834076
 [470,]   9.995738687
 [471,]   0.794081765
 [472,]  10.057161547
 [473,]  10.200399064
 [474,]  10.436420794
 [475,]   6.228783243
 [476,]  13.499060466
 [477,]  13.607805253
 [478,]   0.022269523
 [479,]   4.811820217
 [480,]   7.246329643
 [481,]  15.124237612
 [482,]   7.178434873
 [483,]   4.920856523
 [484,]   2.598446732
 [485,]  11.254361874
 [486,]  10.601025474
 [487,]   4.788200665
 [488,]  16.554904888
 [489,]  11.656189255
 [490,]  10.662066257
 [491,]  11.802073872
 [492,]   9.770672333
 [493,]   8.035370884
 [494,]   8.582979788
 [495,]   6.524186865
 [496,]  14.732673723
 [497,]   5.127567394
 [498,]   8.472008603
 [499,]   2.214199088
 [500,]  11.913525868
 [501,]  11.778254646
 [502,]   6.661852485
 [503,]   7.738821093
 [504,]   5.805445548
 [505,]   2.617156449
 [506,]   7.914772628
 [507,]   5.700056585
 [508,]   7.451601498
 [509,]   9.320569788
 [510,]   6.375759296
 [511,]   9.672845650
 [512,]   2.043586534
 [513,]   0.627798354
 [514,]  12.881791362
 [515,]   9.818019451
 [516,]   9.286628447
 [517,]  11.042494966
 [518,]   1.560063887
 [519,]   4.052644088
 [520,]  10.595845441
 [521,]  20.681398388
 [522,]   0.605383085
 [523,]   9.277854964
 [524,]   4.735300383
 [525,]   6.745252365
 [526,]   6.394711339
 [527,]   1.636869928
 [528,]  13.884589885
 [529,]  10.919537139
 [530,]   7.393643089
 [531,]   5.845272266
 [532,]  17.281982740
 [533,]  10.676928907
 [534,]   2.142095689
 [535,]   9.262469302
 [536,]  11.461951764
 [537,]   5.947172023
 [538,]   5.895700559
 [539,]   7.993652879
 [540,]   9.076046941
 [541,]  -0.093103703
 [542,]   5.674132809
 [543,]   7.130745765
 [544,]  16.290016055
 [545,]   3.379247554
 [546,]   3.616586833
 [547,]   9.276043872
 [548,]  10.995677253
 [549,]   9.728204655
 [550,]  13.267519673
 [551,]   1.424369604
 [552,]  14.996763445
 [553,]   1.209443414
 [554,]   1.983833231
 [555,]  16.341578472
 [556,]   8.226945041
 [557,]   8.401420401
 [558,]   7.673930558
 [559,]   5.409550036
 [560,]   8.812233306
 [561,]  10.867706666
 [562,]   8.529668391
 [563,]  10.830518747
 [564,]  15.112474713
 [565,]   3.390827747
 [566,]   8.737295450
 [567,]  11.274031968
 [568,]  18.385668743
 [569,]   3.507016068
 [570,]  26.211277477
 [571,]  -1.774539074
 [572,]   0.776595844
 [573,]   9.942932652
 [574,]   6.201369326
 [575,]   7.888315311
 [576,]  16.508842536
 [577,]   4.200559892
 [578,]   4.270422625
 [579,]  11.984110608
 [580,]  14.213580023
 [581,]  15.199666738
 [582,]   5.967781863
 [583,]   4.615361567
 [584,]   5.999844475
 [585,]   7.707405540
 [586,]  -1.254633377
 [587,]  14.053097044
 [588,]   9.656391238
 [589,]  11.719370600
 [590,]  11.666034019
 [591,]   7.086450661
 [592,]   5.023164282
 [593,]   5.610870617
 [594,]   4.005109092
 [595,]   8.875269892
 [596,]   5.231874586
 [597,]   7.210014731
 [598,]  15.682360786
 [599,]   4.815310151
 [600,]   7.964103700
 [601,]   7.544628714
 [602,]  17.042083123
 [603,]  10.165947316
 [604,]   8.837507314
 [605,]  14.523010617
 [606,]   3.434021875
 [607,]   3.107220312
 [608,]  10.738500107
 [609,]   3.813530079
 [610,]   5.118344639
 [611,]   8.528388945
 [612,]   3.665594359
 [613,]  11.952188115
 [614,]   8.001419255
 [615,]   5.644260319
 [616,]  11.473914077
 [617,]   5.449813680
 [618,]   3.536837796
 [619,]   4.762791635
 [620,]  12.141190546
 [621,]  16.955866831
 [622,]   9.267367039
 [623,]   5.741625326
 [624,]   7.387333814
 [625,]   3.891973247
 [626,]  11.079357204
 [627,]   2.722792592
 [628,]  -2.644887865
 [629,]   8.102374572
 [630,]   7.250094734
 [631,]   8.306452651
 [632,]  12.017946462
 [633,]   3.411310292
 [634,]   3.896680084
 [635,]  -0.238738010
 [636,]   2.582267497
 [637,]  12.080039207
 [638,]  10.646076612
 [639,]   1.970855296
 [640,]  11.399163738
 [641,]   3.756856934
 [642,]   3.924265077
 [643,]  12.891191607
 [644,]   5.018937475
 [645,]   6.104651558
 [646,]   3.597690553
 [647,]   5.290735201
 [648,]   5.003611672
 [649,]  18.458585744
 [650,]   3.211312972
 [651,]  10.239747655
 [652,]   3.865744229
 [653,]   8.887659181
 [654,]  11.369419934
 [655,]   8.631379995
 [656,]  10.189062868
 [657,]  11.244488099
 [658,]  11.611384960
 [659,]  10.452198026
 [660,]  11.355928740
 [661,]  11.659766349
 [662,]   8.227677978
 [663,]   9.208289599
 [664,]  12.350326165
 [665,]  16.549342667
 [666,]   1.031480688
 [667,]   7.621502163
 [668,]   6.962513919
 [669,]   9.059113940
 [670,]  11.274366802
 [671,]  19.698765641
 [672,]  11.564018102
 [673,]  14.095402712
 [674,]   8.736078669
 [675,]   3.771886568
 [676,]  14.225913709
 [677,]   6.000859518
 [678,]  10.799012928
 [679,]   5.720568373
 [680,]   8.578982383
 [681,]  13.219865162
 [682,]  11.994854644
 [683,]   7.443446090
 [684,]  12.411146767
 [685,]   3.579284037
 [686,]  14.546064723
 [687,]   5.412138565
 [688,]   1.685665331
 [689,]  17.705538254
 [690,]  11.615315149
 [691,]   1.565180246
 [692,]   3.206342124
 [693,]   3.592574650
 [694,]   8.956911321
 [695,]   0.856765284
 [696,]  13.559461212
 [697,]  12.483666479
 [698,]  15.882991689
 [699,]   7.884286303
 [700,]   9.026663956
 [701,]  13.024867684
 [702,]   4.334371521
 [703,]  -1.148700911
 [704,]  16.168987235
 [705,]   7.286899885
 [706,]   5.482892900
 [707,]   1.467328605
 [708,]  10.429209707
 [709,]   6.090774298
 [710,]  -1.618932241
 [711,]   7.607696214
 [712,]  13.965484823
 [713,]   5.604435529
 [714,]   3.380809973
 [715,]  15.797034396
 [716,]   6.097476754
 [717,]  22.590235953
 [718,]   0.358035149
 [719,]   5.133468286
 [720,]  12.194265924
 [721,]  11.161464840
 [722,]   3.456676608
 [723,]  12.545786152
 [724,]   4.438254834
 [725,]  11.359782299
 [726,]   3.305581255
 [727,]   6.334427277
 [728,]   0.108671823
 [729,]   1.746430636
 [730,]   3.146932091
 [731,]   7.354197120
 [732,]   7.590574100
 [733,]   6.271658206
 [734,]  10.211537712
 [735,]   5.964114576
 [736,]   0.568334732
 [737,]   9.814777783
 [738,]  14.761609410
 [739,]   6.571341244
 [740,]   4.718197830
 [741,]  11.182109339
 [742,]  12.425014140
 [743,]  16.774735388
 [744,]   0.816120332
 [745,]   9.747954547
 [746,]   8.991334913
 [747,]  16.926710979
 [748,]   5.386785702
 [749,]   3.417410083
 [750,]   6.476752064
 [751,]   5.579845718
 [752,]   7.210506958
 [753,]   0.935288725
 [754,]   7.542397133
 [755,]   7.625789136
 [756,]  10.581327234
 [757,]   1.536763575
 [758,]  15.258422498
 [759,]   5.263272192
 [760,]   8.073158525
 [761,]   3.159309190
 [762,]   2.574223917
 [763,]   7.929814331
 [764,]   3.011157539
 [765,]   6.296495683
 [766,]   6.585702211
 [767,]   6.783069539
 [768,]   3.221778893
 [769,]  11.474452515
 [770,]   4.126616984
 [771,]   4.253921766
 [772,]   8.195884171
 [773,]  12.123992693
 [774,]   1.137776022
 [775,]   7.123041194
 [776,]   7.521782402
 [777,]   2.658547556
 [778,]  15.050852674
 [779,]   3.537197563
 [780,]   6.259237658
 [781,]   5.747775896
 [782,]   4.365861477
 [783,]   7.586411182
 [784,]   6.622277740
 [785,]   9.761850210
 [786,]   7.919871056
 [787,]   7.469328708
 [788,]   7.768287160
 [789,]   2.494205166
 [790,]   3.464163793
 [791,]  10.888763507
 [792,]   9.367421084
 [793,]  11.958863109
 [794,]   8.824784183
 [795,]  15.979097313
 [796,]   2.280724018
 [797,]  12.738249689
 [798,]   1.892844749
 [799,]   6.596093435
 [800,]   3.303340077
 [801,]   7.005810274
 [802,]  23.697903160
 [803,]  10.897914691
 [804,]   3.662798535
 [805,]   4.237600564
 [806,]  11.276775511
 [807,]   3.217512194
 [808,]  20.936084210
 [809,]   7.970274697
 [810,]   4.655135985
 [811,]  13.672232057
 [812,]  11.551954505
 [813,]  11.438172864
 [814,]   0.704317144
 [815,]   3.354270790
 [816,]   2.816482558
 [817,]  11.579452667
 [818,]   6.162640060
 [819,]  22.577311677
 [820,]   3.998182162
 [821,]   2.936981656
 [822,]  16.742296334
 [823,]  -0.032155622
 [824,]   8.002442833
 [825,]  12.229217975
 [826,]   6.115814733
 [827,]   0.983513556
 [828,]  10.927471492
 [829,]   1.347029302
 [830,]   3.737813315
 [831,]   3.595823942
 [832,]  17.622686392
 [833,]  11.756241135
 [834,]   8.486570911
 [835,]  11.395336778
 [836,]  14.776155738
 [837,]   9.609598068
 [838,]  13.330041840
 [839,]  11.183233939
 [840,]   1.562574336
 [841,]   9.488757390
 [842,]  10.301142922
 [843,]  12.697205257
 [844,]   4.262750270
 [845,]   3.281692458
 [846,]  21.5