Data

setwd("c:/users/mark/documents/work/stemibayes")
library(rethinking)
## Loading required package: rstan
## Loading required package: ggplot2
## Loading required package: StanHeaders
## rstan (Version 2.12.1, packaged: 2016-09-11 13:07:50 UTC, GitRev: 85f7a56811da)
## For execution on a local, multicore CPU with excess RAM we recommend calling
## rstan_options(auto_write = TRUE)
## options(mc.cores = parallel::detectCores())
## Loading required package: parallel
## rethinking (Version 1.59)
par(mfrow=c(1,1))
wd<-read.csv("d.csv",header=TRUE)

Model 1

model1<-map2stan(
  alist(
    Count~dpois(lambda),
    log(lambda)~a+day[Day]+log(261),
    a~dnorm(0,5),
    day[Day]~dnorm(0,5)
  ),
  data=wd,
  iter=1e4
)
## 
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##  Elapsed Time: 0.401 seconds (Warm-up)
##                0.501 seconds (Sampling)
##                0.902 seconds (Total)
## 
## 
## SAMPLING FOR MODEL 'Count ~ dpois(lambda)' NOW (CHAIN 1).
## WARNING: No variance estimation is
##          performed for num_warmup < 20
## 
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Model 2

model2<-map2stan(
  alist(
    Count~dpois(m),
    log(m)~a+log(261),
    a~dnorm(0,5)
  ),
  data=wd,
  iter=1e4
  
)
## 
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##  Elapsed Time: 0.062 seconds (Warm-up)
##                0.054 seconds (Sampling)
##                0.116 seconds (Total)
## 
## 
## SAMPLING FOR MODEL 'Count ~ dpois(m)' NOW (CHAIN 1).
## WARNING: No variance estimation is
##          performed for num_warmup < 20
## 
## 
## Chain 1, Iteration: 1 / 1 [100%]  (Sampling)
##  Elapsed Time: 0 seconds (Warm-up)
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## Constructing posterior predictions
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Compare Models

cm<-compare(model1,model2)
plot(cm)

Counter-factuals

## $breaks
## [1] -1  0
## 
## $counts
## [1] 5000
## 
## $density
## [1] 1
## 
## $mids
## [1] -0.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30  35
## 
## $counts
##  [1]    2    7   14  113  370  917 1423 1225  644  212   56   13    4
## 
## $density
##  [1] 0.00008 0.00028 0.00056 0.00452 0.01480 0.03668 0.05692 0.04900
##  [9] 0.02576 0.00848 0.00224 0.00052 0.00016
## 
## $mids
##  [1] -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5
## [12]  27.5  32.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -35 -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30
## 
## $counts
##  [1]    2    8   45  173  517 1050 1313 1036  564  224   54   10    4
## 
## $density
##  [1] 0.00008 0.00032 0.00180 0.00692 0.02068 0.04200 0.05252 0.04144
##  [9] 0.02256 0.00896 0.00216 0.00040 0.00016
## 
## $mids
##  [1] -32.5 -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5
## [12]  22.5  27.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30
## 
## $counts
##  [1]    1   10   66  216  710 1251 1346  893  378  102   21    6
## 
## $density
##  [1] 0.00004 0.00040 0.00264 0.00864 0.02840 0.05004 0.05384 0.03572
##  [9] 0.01512 0.00408 0.00084 0.00024
## 
## $mids
##  [1] -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5
## [12]  27.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30
## 
## $counts
##  [1]   10   37  147  516  968 1325 1145  575  220   45   10    2
## 
## $density
##  [1] 0.00040 0.00148 0.00588 0.02064 0.03872 0.05300 0.04580 0.02300
##  [9] 0.00880 0.00180 0.00040 0.00008
## 
## $mids
##  [1] -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5
## [12]  27.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -45 -40 -35 -30 -25 -20 -15 -10  -5   0   5  10  15  20
## 
## $counts
##  [1]    1    3   17   85  232  597 1044 1272 1003  522  172   41   11
## 
## $density
##  [1] 0.00004 0.00012 0.00068 0.00340 0.00928 0.02388 0.04176 0.05088
##  [9] 0.04012 0.02088 0.00688 0.00164 0.00044
## 
## $mids
##  [1] -42.5 -37.5 -32.5 -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5
## [12]  12.5  17.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -25 -20 -15 -10  -5   0   5  10  15  20  25  30
## 
## $counts
##  [1]    1    8   50  243  686 1273 1396  885  319  111   28
## 
## $density
##  [1] 0.00004 0.00032 0.00200 0.00972 0.02744 0.05092 0.05584 0.03540
##  [9] 0.01276 0.00444 0.00112
## 
## $mids
##  [1] -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5  27.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -35 -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30
## 
## $counts
##  [1]    4   13   56  212  644 1225 1423  917  370  113   14    7    2
## 
## $density
##  [1] 0.00016 0.00052 0.00224 0.00848 0.02576 0.04900 0.05692 0.03668
##  [9] 0.01480 0.00452 0.00056 0.00028 0.00008
## 
## $mids
##  [1] -32.5 -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5
## [12]  22.5  27.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
## [1] -1  0
## 
## $counts
## [1] 5000
## 
## $density
## [1] 1
## 
## $mids
## [1] -0.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -35 -30 -25 -20 -15 -10  -5   0   5  10  15  20
## 
## $counts
##  [1]    5   31  115  387  901 1343 1276  609  253   68   12
## 
## $density
##  [1] 0.00020 0.00124 0.00460 0.01548 0.03604 0.05372 0.05104 0.02436
##  [9] 0.01012 0.00272 0.00048
## 
## $mids
##  [1] -32.5 -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -35 -30 -25 -20 -15 -10  -5   0   5  10  15  20  25
## 
## $counts
##  [1]    2    3   42  179  536 1156 1361 1101  471  116   31    2
## 
## $density
##  [1] 0.00008 0.00012 0.00168 0.00716 0.02144 0.04624 0.05444 0.04404
##  [9] 0.01884 0.00464 0.00124 0.00008
## 
## $mids
##  [1] -32.5 -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5
## [12]  22.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -35 -30 -25 -20 -15 -10  -5   0   5  10  15  20  25
## 
## $counts
##  [1]    3   21  113  366  892 1312 1287  691  235   72    7    1
## 
## $density
##  [1] 0.00012 0.00084 0.00452 0.01464 0.03568 0.05248 0.05148 0.02764
##  [9] 0.00940 0.00288 0.00028 0.00004
## 
## $mids
##  [1] -32.5 -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5
## [12]  22.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -45 -40 -35 -30 -25 -20 -15 -10  -5   0   5  10  15  20
## 
## $counts
##  [1]    6    9   38  186  474  958 1316 1164  578  222   37   11    1
## 
## $density
##  [1] 0.00024 0.00036 0.00152 0.00744 0.01896 0.03832 0.05264 0.04656
##  [9] 0.02312 0.00888 0.00148 0.00044 0.00004
## 
## $mids
##  [1] -42.5 -37.5 -32.5 -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5
## [12]  12.5  17.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30
## 
## $counts
##  [1]    1    4   23  154  540 1213 1464 1011  457  114   16    3
## 
## $density
##  [1] 0.00004 0.00016 0.00092 0.00616 0.02160 0.04852 0.05856 0.04044
##  [9] 0.01828 0.00456 0.00064 0.00012
## 
## $mids
##  [1] -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5
## [12]  27.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30  35
## 
## $counts
##  [1]    4   10   54  224  564 1036 1313 1050  517  173   45    8    2
## 
## $density
##  [1] 0.00016 0.00040 0.00216 0.00896 0.02256 0.04144 0.05252 0.04200
##  [9] 0.02068 0.00692 0.00180 0.00032 0.00008
## 
## $mids
##  [1] -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5
## [12]  27.5  32.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -20 -15 -10  -5   0   5  10  15  20  25  30  35
## 
## $counts
##  [1]   12   68  253  609 1276 1343  901  387  115   31    5
## 
## $density
##  [1] 0.00048 0.00272 0.01012 0.02436 0.05104 0.05372 0.03604 0.01548
##  [9] 0.00460 0.00124 0.00020
## 
## $mids
##  [1] -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5  27.5  32.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
## [1] -1  0
## 
## $counts
## [1] 5000
## 
## $density
## [1] 1
## 
## $mids
## [1] -0.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30  35
## 
## $counts
##  [1]    2    1   33  153  497  988 1314 1146  589  208   57    7    5
## 
## $density
##  [1] 0.00008 0.00004 0.00132 0.00612 0.01988 0.03952 0.05256 0.04584
##  [9] 0.02356 0.00832 0.00228 0.00028 0.00020
## 
## $mids
##  [1] -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5
## [12]  27.5  32.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30  35
## 
## $counts
##  [1]    1   24   85  318  801 1213 1263  788  365  112   26    3    1
## 
## $density
##  [1] 0.00004 0.00096 0.00340 0.01272 0.03204 0.04852 0.05052 0.03152
##  [9] 0.01460 0.00448 0.00104 0.00012 0.00004
## 
## $mids
##  [1] -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5
## [12]  27.5  32.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -40 -35 -30 -25 -20 -15 -10  -5   0   5  10  15  20  25
## 
## $counts
##  [1]    2    7   39  167  446  857 1153 1166  750  313   77   20    3
## 
## $density
##  [1] 0.00008 0.00028 0.00156 0.00668 0.01784 0.03428 0.04612 0.04664
##  [9] 0.03000 0.01252 0.00308 0.00080 0.00012
## 
## $mids
##  [1] -37.5 -32.5 -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5
## [12]  17.5  22.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30  35  40
## 
## $counts
##  [1]    1    1    2   27  142  488 1042 1326 1159  549  208   44   10    1
## 
## $density
##  [1] 0.00004 0.00004 0.00008 0.00108 0.00568 0.01952 0.04168 0.05304
##  [9] 0.04636 0.02196 0.00832 0.00176 0.00040 0.00004
## 
## $mids
##  [1] -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5
## [12]  27.5  32.5  37.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30
## 
## $counts
##  [1]    6   21  102  378  893 1346 1251  710  216   66   10    1
## 
## $density
##  [1] 0.00024 0.00084 0.00408 0.01512 0.03572 0.05384 0.05004 0.02840
##  [9] 0.00864 0.00264 0.00040 0.00004
## 
## $mids
##  [1] -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5
## [12]  27.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -25 -20 -15 -10  -5   0   5  10  15  20  25  30  35
## 
## $counts
##  [1]    2   31  116  471 1101 1361 1156  536  179   42    3    2
## 
## $density
##  [1] 0.00008 0.00124 0.00464 0.01884 0.04404 0.05444 0.04624 0.02144
##  [9] 0.00716 0.00168 0.00012 0.00008
## 
## $mids
##  [1] -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5  27.5
## [12]  32.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -35 -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30
## 
## $counts
##  [1]    5    7   57  208  589 1146 1314  988  497  153   33    1    2
## 
## $density
##  [1] 0.00020 0.00028 0.00228 0.00832 0.02356 0.04584 0.05256 0.03952
##  [9] 0.01988 0.00612 0.00132 0.00004 0.00008
## 
## $mids
##  [1] -32.5 -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5
## [12]  22.5  27.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
## [1] -1  0
## 
## $counts
## [1] 5000
## 
## $density
## [1] 1
## 
## $mids
## [1] -0.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -35 -30 -25 -20 -15 -10  -5   0   5  10  15  20  25
## 
## $counts
##  [1]    3   10   57  195  579 1078 1332 1030  505  174   34    3
## 
## $density
##  [1] 0.00012 0.00040 0.00228 0.00780 0.02316 0.04312 0.05328 0.04120
##  [9] 0.02020 0.00696 0.00136 0.00012
## 
## $mids
##  [1] -32.5 -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5
## [12]  22.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -40 -35 -30 -25 -20 -15 -10  -5   0   5  10  15  20  25
## 
## $counts
##  [1]    5   16   89  279  749 1125 1210  934  420  136   31    5    1
## 
## $density
##  [1] 0.00020 0.00064 0.00356 0.01116 0.02996 0.04500 0.04840 0.03736
##  [9] 0.01680 0.00544 0.00124 0.00020 0.00004
## 
## $mids
##  [1] -37.5 -32.5 -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5
## [12]  17.5  22.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -20 -15 -10  -5   0   5  10  15  20  25  30  35
## 
## $counts
##  [1]    6   60  299  818 1364 1342  774  268   62    5    2
## 
## $density
##  [1] 0.00024 0.00240 0.01196 0.03272 0.05456 0.05368 0.03096 0.01072
##  [9] 0.00248 0.00020 0.00008
## 
## $mids
##  [1] -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5  27.5  32.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30
## 
## $counts
##  [1]    2   10   45  220  575 1145 1325  968  516  147   37   10
## 
## $density
##  [1] 0.00008 0.00040 0.00180 0.00880 0.02300 0.04580 0.05300 0.03872
##  [9] 0.02064 0.00588 0.00148 0.00040
## 
## $mids
##  [1] -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5
## [12]  27.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -25 -20 -15 -10  -5   0   5  10  15  20  25  30  35
## 
## $counts
##  [1]    1    7   72  235  691 1287 1312  892  366  113   21    3
## 
## $density
##  [1] 0.00004 0.00028 0.00288 0.00940 0.02764 0.05148 0.05248 0.03568
##  [9] 0.01464 0.00452 0.00084 0.00012
## 
## $mids
##  [1] -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5  27.5
## [12]  32.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -35 -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30
## 
## $counts
##  [1]    1    3   26  112  365  788 1263 1213  801  318   85   24    1
## 
## $density
##  [1] 0.00004 0.00012 0.00104 0.00448 0.01460 0.03152 0.05052 0.04852
##  [9] 0.03204 0.01272 0.00340 0.00096 0.00004
## 
## $mids
##  [1] -32.5 -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5
## [12]  22.5  27.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -25 -20 -15 -10  -5   0   5  10  15  20  25  30  35
## 
## $counts
##  [1]    3   34  174  505 1030 1332 1078  579  195   57   10    3
## 
## $density
##  [1] 0.00012 0.00136 0.00696 0.02020 0.04120 0.05328 0.04312 0.02316
##  [9] 0.00780 0.00228 0.00040 0.00012
## 
## $mids
##  [1] -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5  27.5
## [12]  32.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
## [1] -1  0
## 
## $counts
## [1] 5000
## 
## $density
## [1] 1
## 
## $mids
## [1] -0.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -40 -35 -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30
## 
## $counts
##  [1]    3   11   49  151  444  899 1234 1105  688  302   94   15    4    1
## 
## $density
##  [1] 0.00012 0.00044 0.00196 0.00604 0.01776 0.03596 0.04936 0.04420
##  [9] 0.02752 0.01208 0.00376 0.00060 0.00016 0.00004
## 
## $mids
##  [1] -37.5 -32.5 -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5
## [12]  17.5  22.5  27.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -20 -15 -10  -5   0   5  10  15  20  25  30  35  40
## 
## $counts
##  [1]    6   25  131  483 1110 1394 1088  562  158   37    5    1
## 
## $density
##  [1] 0.00024 0.00100 0.00524 0.01932 0.04440 0.05576 0.04352 0.02248
##  [9] 0.00632 0.00148 0.00020 0.00004
## 
## $mids
##  [1] -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5  27.5  32.5
## [12]  37.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -20 -15 -10  -5   0   5  10  15  20  25  30  35  40  45
## 
## $counts
##  [1]   11   41  172  522 1003 1272 1044  597  232   85   17    3    1
## 
## $density
##  [1] 0.00044 0.00164 0.00688 0.02088 0.04012 0.05088 0.04176 0.02388
##  [9] 0.00928 0.00340 0.00068 0.00012 0.00004
## 
## $mids
##  [1] -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5  27.5  32.5
## [12]  37.5  42.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -20 -15 -10  -5   0   5  10  15  20  25  30  35  40  45
## 
## $counts
##  [1]    1   11   37  222  578 1164 1316  958  474  186   38    9    6
## 
## $density
##  [1] 0.00004 0.00044 0.00148 0.00888 0.02312 0.04656 0.05264 0.03832
##  [9] 0.01896 0.00744 0.00152 0.00036 0.00024
## 
## $mids
##  [1] -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5  27.5  32.5
## [12]  37.5  42.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -25 -20 -15 -10  -5   0   5  10  15  20  25  30  35  40
## 
## $counts
##  [1]    3   20   77  313  750 1166 1153  857  446  167   39    7    2
## 
## $density
##  [1] 0.00012 0.00080 0.00308 0.01252 0.03000 0.04664 0.04612 0.03428
##  [9] 0.01784 0.00668 0.00156 0.00028 0.00008
## 
## $mids
##  [1] -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5  27.5
## [12]  32.5  37.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -25 -20 -15 -10  -5   0   5  10  15  20  25  30  35  40
## 
## $counts
##  [1]    1    5   31  136  420  934 1210 1125  749  279   89   16    5
## 
## $density
##  [1] 0.00004 0.00020 0.00124 0.00544 0.01680 0.03736 0.04840 0.04500
##  [9] 0.02996 0.01116 0.00356 0.00064 0.00020
## 
## $mids
##  [1] -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5  27.5
## [12]  32.5  37.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30  35  40
## 
## $counts
##  [1]    1    4   15   94  302  688 1105 1234  899  444  151   49   11    3
## 
## $density
##  [1] 0.00004 0.00016 0.00060 0.00376 0.01208 0.02752 0.04420 0.04936
##  [9] 0.03596 0.01776 0.00604 0.00196 0.00044 0.00012
## 
## $mids
##  [1] -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5
## [12]  27.5  32.5  37.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
## [1] -1  0
## 
## $counts
## [1] 5000
## 
## $density
## [1] 1
## 
## $mids
## [1] -0.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -15 -10  -5   0   5  10  15  20  25  30  35  40  45
## 
## $counts
##  [1]    2   18  133  436  960 1228 1162  672  293   75   16    5
## 
## $density
##  [1] 0.00008 0.00072 0.00532 0.01744 0.03840 0.04912 0.04648 0.02688
##  [9] 0.01172 0.00300 0.00064 0.00020
## 
## $mids
##  [1] -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5  27.5  32.5  37.5
## [12]  42.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -30 -25 -20 -15 -10  -5   0   5  10  15  20  25
## 
## $counts
##  [1]   28  111  319  885 1396 1273  686  243   50    8    1
## 
## $density
##  [1] 0.00112 0.00444 0.01276 0.03540 0.05584 0.05092 0.02744 0.00972
##  [9] 0.00200 0.00032 0.00004
## 
## $mids
##  [1] -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30
## 
## $counts
##  [1]    3   16  114  457 1011 1464 1213  540  154   23    4    1
## 
## $density
##  [1] 0.00012 0.00064 0.00456 0.01828 0.04044 0.05856 0.04852 0.02160
##  [9] 0.00616 0.00092 0.00016 0.00004
## 
## $mids
##  [1] -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5  22.5
## [12]  27.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -40 -35 -30 -25 -20 -15 -10  -5   0   5  10  15  20  25  30
## 
## $counts
##  [1]    1   10   44  208  549 1159 1326 1042  488  142   27    2    1    1
## 
## $density
##  [1] 0.00004 0.00040 0.00176 0.00832 0.02196 0.04636 0.05304 0.04168
##  [9] 0.01952 0.00568 0.00108 0.00008 0.00004 0.00004
## 
## $mids
##  [1] -37.5 -32.5 -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5
## [12]  17.5  22.5  27.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -35 -30 -25 -20 -15 -10  -5   0   5  10  15  20
## 
## $counts
##  [1]    2    5   62  268  774 1342 1364  818  299   60    6
## 
## $density
##  [1] 0.00008 0.00020 0.00248 0.01072 0.03096 0.05368 0.05456 0.03272
##  [9] 0.01196 0.00240 0.00024
## 
## $mids
##  [1] -32.5 -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5  17.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -40 -35 -30 -25 -20 -15 -10  -5   0   5  10  15  20
## 
## $counts
##  [1]    1    5   37  158  562 1088 1394 1110  483  131   25    6
## 
## $density
##  [1] 0.00004 0.00020 0.00148 0.00632 0.02248 0.04352 0.05576 0.04440
##  [9] 0.01932 0.00524 0.00100 0.00024
## 
## $mids
##  [1] -37.5 -32.5 -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5  12.5
## [12]  17.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## $breaks
##  [1] -45 -40 -35 -30 -25 -20 -15 -10  -5   0   5  10  15
## 
## $counts
##  [1]    5   16   75  293  672 1162 1228  960  436  133   18    2
## 
## $density
##  [1] 0.00020 0.00064 0.00300 0.01172 0.02688 0.04648 0.04912 0.03840
##  [9] 0.01744 0.00532 0.00072 0.00008
## 
## $mids
##  [1] -42.5 -37.5 -32.5 -27.5 -22.5 -17.5 -12.5  -7.5  -2.5   2.5   7.5
## [12]  12.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"

## $breaks
## [1] -1  0
## 
## $counts
## [1] 5000
## 
## $density
## [1] 1
## 
## $mids
## [1] -0.5
## 
## $xname
## [1] "fun(i, j)"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"