Assignment #6

Chapter 7

Chapter 7 - Ulysses’ Compass

The chapter began with the problem of overfitting, a universal phenomenon by which models with more parameters fit a sample better, even when the additional parameters are meaningless. Two common tools were introduced to address overfitting: regularizing priors and estimates of out-of-sample accuracy (WAIC and PSIS). Regularizing priors reduce overfitting during estimation, and WAIC and PSIS help estimate the degree of overfitting. Practical functions compare in the rethinking package were introduced to help analyze collections of models fit to the same data. If you are after causal estimates, then these tools will mislead you. So models must be designed through some other method, not selected on the basis of out-of-sample predictive accuracy. But any causal estimate will still overfit the sample. So you always have to worry about overfitting, measuring it with WAIC/PSIS and reducing it with regularization.

Place each answer inside the code chunk (grey box). The code chunks should contain a text response or a code that completes/answers the question or activity requested. Make sure to include plots if the question requests them.

Finally, upon completion, name your final output .html file as: YourName_ANLY505-Year-Semester.html and publish the assignment to your R Pubs account and submit the link to Canvas. Each question is worth 5 points.

Questions

7-1. When comparing models with an information criterion, why must all models be fit to exactly the same observations? What would happen to the information criterion values, if the models were fit to different numbers of observations? Perform some experiments.

#Information criteria are based on deviance, which is accrued over observations without being divided by the #number of observations (page 182). Thus, it is a sum and not an average. So, all else being equal, a model #with more observations will have a higher deviance and thus worse accuracy according to information criteria. #It would be an unfair comparison to contrast models fit to different numbers of observations.

#As an experiment, we can calculate WAIC for models fit to increasingly small subsamples of the same data. The #information criteria should decrease alongside the sample size. In order to get a large sample to begin with, #I will return to the Howell1 database from last chapter.
library(maps)
library(rethinking)
data(Howell1)
str(Howell1)

## 'data.frame':    544 obs. of  4 variables:
##  $ height: num  152 140 137 157 145 ...
##  $ weight: num  47.8 36.5 31.9 53 41.3 ...
##  $ age   : num  63 63 65 41 51 35 32 27 19 54 ...
##  $ male  : int  1 0 0 1 0 1 0 1 0 1 ...

d <- Howell1[complete.cases(Howell1), ]
d

##       height    weight   age male
## 1   151.7650 47.825606 63.00    1
## 2   139.7000 36.485807 63.00    0
## 3   136.5250 31.864838 65.00    0
## 4   156.8450 53.041914 41.00    1
## 5   145.4150 41.276872 51.00    0
## 6   163.8300 62.992589 35.00    1
## 7   149.2250 38.243476 32.00    0
## 8   168.9100 55.479971 27.00    1
## 9   147.9550 34.869885 19.00    0
## 10  165.1000 54.487739 54.00    1
## 11  154.3050 49.895120 47.00    0
## 12  151.1300 41.220173 66.00    1
## 13  144.7800 36.032215 73.00    0
## 14  149.9000 47.700000 20.00    0
## 15  150.4950 33.849303 65.30    0
## 16  163.1950 48.562694 36.00    1
## 17  157.4800 42.325803 44.00    1
## 18  143.9418 38.356873 31.00    0
## 19  121.9200 19.617854 12.00    1
## 20  105.4100 13.947954  8.00    0
## 21   86.3600 10.489315  6.50    0
## 22  161.2900 48.987936 39.00    1
## 23  156.2100 42.722696 29.00    0
## 24  129.5400 23.586784 13.00    1
## 25  109.2200 15.989118  7.00    0
## 26  146.4000 35.493574 56.00    1
## 27  148.5900 37.903281 45.00    0
## 28  147.3200 35.465224 19.00    0
## 29  137.1600 27.328918 17.00    1
## 30  125.7300 22.679600 16.00    0
## 31  114.3000 17.860185 11.00    1
## 32  147.9550 40.312989 29.00    1
## 33  161.9250 55.111428 30.00    1
## 34  146.0500 37.506388 24.00    0
## 35  146.0500 38.498621 35.00    0
## 36  152.7048 46.606578 33.00    0
## 37  142.8750 38.838815 27.00    0
## 38  142.8750 35.578623 32.00    0
## 39  147.9550 47.400364 36.00    0
## 40  160.6550 47.882306 24.00    1
## 41  151.7650 49.413179 30.00    1
## 42  162.8648 49.384829 24.00    1
## 43  171.4500 56.557252 52.00    1
## 44  147.3200 39.122310 42.00    0
## 45  147.9550 49.895120 19.00    0
## 46  144.7800 28.803092 17.00    0
## 47  121.9200 20.411640  8.00    1
## 48  128.9050 23.359988 12.00    0
## 49   97.7900 13.267566  5.00    0
## 50  154.3050 41.248522 55.00    1
## 51  143.5100 38.555320 43.00    0
## 52  146.7000 42.400000 20.00    1
## 53  157.4800 44.650463 18.00    1
## 54  127.0000 22.010552 13.00    1
## 55  110.4900 15.422128  9.00    0
## 56   97.7900 12.757275  5.00    0
## 57  165.7350 58.598416 42.00    1
## 58  152.4000 46.719976 44.00    0
## 59  141.6050 44.225220 60.00    0
## 60  158.8000 50.900000 20.00    0
## 61  155.5750 54.317642 37.00    0
## 62  164.4650 45.897841 50.00    1
## 63  151.7650 48.024053 50.00    0
## 64  161.2900 52.219779 31.00    1
## 65  154.3050 47.627160 25.00    0
## 66  145.4150 45.642695 23.00    0
## 67  145.4150 42.410852 52.00    0
## 68  152.4000 36.485807 79.30    1
## 69  163.8300 55.933563 35.00    1
## 70  144.1450 37.194544 27.00    0
## 71  129.5400 24.550667 13.00    1
## 72  129.5400 25.627948 14.00    0
## 73  153.6700 48.307548 38.00    1
## 74  142.8750 37.336292 39.00    0
## 75  146.0500 29.596878 12.00    0
## 76  167.0050 47.173568 30.00    1
## 77  158.4198 47.286966 24.00    0
## 78   91.4400 12.927372  0.60    1
## 79  165.7350 57.549485 51.00    1
## 80  149.8600 37.931631 46.00    0
## 81  147.9550 41.900561 17.00    0
## 82  137.7950 27.584063 12.00    0
## 83  154.9400 47.201918 22.00    0
## 84  160.9598 43.204638 29.00    1
## 85  161.9250 50.263663 38.00    1
## 86  147.9550 39.377456 30.00    0
## 87  113.6650 17.463292  6.00    1
## 88  159.3850 50.689000 45.00    1
## 89  148.5900 39.434154 47.00    0
## 90  136.5250 36.287360 79.00    0
## 91  158.1150 46.266384 45.00    1
## 92  144.7800 42.269104 54.00    0
## 93  156.8450 47.627160 31.00    1
## 94  179.0700 55.706767 23.00    1
## 95  118.7450 18.824068  9.00    0
## 96  170.1800 48.562694 41.00    1
## 97  146.0500 42.807745 23.00    0
## 98  147.3200 35.068331 36.00    0
## 99  113.0300 17.888534  5.00    1
## 100 162.5600 56.755699 30.00    0
## 101 133.9850 27.442316 12.00    1
## 102 152.4000 51.255896 34.00    0
## 103 160.0200 47.230267 44.00    1
## 104 149.8600 40.936678 43.00    0
## 105 142.8750 32.715323 73.30    0
## 106 167.0050 57.067543 38.00    1
## 107 159.3850 42.977842 43.00    1
## 108 154.9400 39.944446 33.00    0
## 109 148.5900 32.460178 16.00    0
## 110 111.1250 17.123098 11.00    1
## 111 111.7600 16.499409  6.00    1
## 112 162.5600 45.954540 35.00    1
## 113 152.4000 41.106775 29.00    0
## 114 124.4600 18.257078 12.00    0
## 115 111.7600 15.081934  9.00    1
## 116  86.3600 11.481547  7.60    1
## 117 170.1800 47.598810 58.00    1
## 118 146.0500 37.506388 53.00    0
## 119 159.3850 45.019006 51.00    1
## 120 151.1300 42.269104 48.00    0
## 121 160.6550 54.856282 29.00    1
## 122 169.5450 53.523856 41.00    1
## 123 158.7500 52.191429 81.75    1
## 124  74.2950  9.752228  1.00    1
## 125 149.8600 42.410852 35.00    0
## 126 153.0350 49.583275 46.00    0
## 127  96.5200 13.097469  5.00    1
## 128 161.9250 41.730464 29.00    1
## 129 162.5600 56.018612 42.00    1
## 130 149.2250 42.155707 27.00    0
## 131 116.8400 19.391058  8.00    0
## 132 100.0760 15.081934  6.00    1
## 133 163.1950 53.098613 22.00    1
## 134 161.9250 50.235314 43.00    1
## 135 145.4150 42.524250 53.00    0
## 136 163.1950 49.101334 43.00    1
## 137 151.1300 38.498621 41.00    0
## 138 150.4950 49.810071 50.00    0
## 139 141.6050 29.313383 15.00    1
## 140 170.8150 59.760746 33.00    1
## 141  91.4400 11.708343  3.00    0
## 142 157.4800 47.939005 62.00    1
## 143 152.4000 39.292407 49.00    0
## 144 149.2250 38.130077 17.00    1
## 145 129.5400 21.999212 12.00    0
## 146 147.3200 36.882700 22.00    0
## 147 145.4150 42.127357 29.00    0
## 148 121.9200 19.787951  8.00    0
## 149 113.6650 16.782904  5.00    1
## 150 157.4800 44.565414 33.00    1
## 151 154.3050 47.853956 34.00    0
## 152 120.6500 21.177076 12.00    0
## 153 115.6000 18.900000  7.00    1
## 154 167.0050 55.196477 42.00    1
## 155 142.8750 32.998818 40.00    0
## 156 152.4000 40.879979 27.00    0
## 157  96.5200 13.267566  3.00    0
## 158 160.0000 51.200000 25.00    1
## 159 159.3850 49.044635 29.00    1
## 160 149.8600 53.438808 45.00    0
## 161 160.6550 54.090846 26.00    1
## 162 160.6550 55.366574 45.00    1
## 163 149.2250 42.240755 45.00    0
## 164 125.0950 22.367756 11.00    0
## 165 140.9700 40.936678 85.60    0
## 166 154.9400 49.696674 26.00    1
## 167 141.6050 44.338618 24.00    0
## 168 160.0200 45.954540 57.00    1
## 169 150.1648 41.957260 22.00    0
## 170 155.5750 51.482692 24.00    0
## 171 103.5050 12.757275  6.00    0
## 172  94.6150 13.012420  4.00    0
## 173 156.2100 44.111822 21.00    0
## 174 153.0350 32.205032 79.00    0
## 175 167.0050 56.755699 50.00    1
## 176 149.8600 52.673371 40.00    0
## 177 147.9550 36.485807 64.00    0
## 178 159.3850 48.846188 32.00    1
## 179 161.9250 56.954146 38.70    1
## 180 155.5750 42.099007 26.00    0
## 181 159.3850 50.178615 63.00    1
## 182 146.6850 46.549879 62.00    0
## 183 172.7200 61.801910 22.00    1
## 184 166.3700 48.987936 41.00    1
## 185 141.6050 31.524644 19.00    1
## 186 142.8750 32.205032 17.00    0
## 187 133.3500 23.756881 14.00    0
## 188 127.6350 24.408919  9.00    1
## 189 119.3800 21.517270  7.00    1
## 190 151.7650 35.295127 74.00    0
## 191 156.8450 45.642695 41.00    1
## 192 148.5900 43.885026 33.00    0
## 193 157.4800 45.557646 53.00    0
## 194 149.8600 39.008912 18.00    0
## 195 147.9550 41.163474 37.00    0
## 196 102.2350 13.125818  6.00    0
## 197 153.0350 45.245802 61.00    0
## 198 160.6550 53.637254 44.00    1
## 199 149.2250 52.304828 35.00    0
## 200 114.3000 18.342126  7.00    1
## 201 100.9650 13.749507  4.00    1
## 202 138.4300 39.093961 23.00    0
## 203  91.4400 12.530479  4.00    1
## 204 162.5600 45.699394 55.00    1
## 205 149.2250 40.398038 53.00    0
## 206 158.7500 51.482692 59.00    1
## 207 149.8600 38.668718 57.00    0
## 208 158.1150 39.235708 35.00    1
## 209 156.2100 44.338618 29.00    0
## 210 148.5900 39.519203 62.00    1
## 211 143.5100 31.071052 18.00    0
## 212 154.3050 46.776675 51.00    0
## 213 131.4450 22.509503 14.00    0
## 214 157.4800 40.624834 19.00    1
## 215 157.4800 50.178615 42.00    1
## 216 154.3050 41.276872 25.00    0
## 217 107.9500 17.576690  6.00    1
## 218 168.2750 54.600000 41.00    1
## 219 145.4150 44.990657 37.00    0
## 220 147.9550 44.735511 16.00    0
## 221 100.9650 14.401546  5.00    1
## 222 113.0300 19.050864  9.00    1
## 223 149.2250 35.805419 82.00    1
## 224 154.9400 45.217453 28.00    1
## 225 162.5600 48.109102 50.00    1
## 226 156.8450 45.671045 43.00    0
## 227 123.1900 20.808533  8.00    1
## 228 161.0106 48.420946 31.00    1
## 229 144.7800 41.191823 67.00    0
## 230 143.5100 38.413573 39.00    0
## 231 149.2250 42.127357 18.00    0
## 232 110.4900 17.661738 11.00    0
## 233 149.8600 38.243476 48.00    0
## 234 165.7350 48.335898 30.00    1
## 235 144.1450 38.923864 64.00    0
## 236 157.4800 40.029494 72.00    1
## 237 154.3050 50.206964 68.00    0
## 238 163.8300 54.289293 44.00    1
## 239 156.2100 45.600000 43.00    0
## 240 153.6700 40.766581 16.00    0
## 241 134.6200 27.130471 13.00    0
## 242 144.1450 39.434154 34.00    0
## 243 114.3000 20.496689 10.00    0
## 244 162.5600 43.204638 62.00    1
## 245 146.0500 31.864838 44.00    0
## 246 120.6500 20.893581 11.00    1
## 247 154.9400 45.444249 31.00    1
## 248 144.7800 38.045029 29.00    0
## 249 106.6800 15.989118  8.00    0
## 250 146.6850 36.088913 62.00    0
## 251 152.4000 40.879979 67.00    0
## 252 163.8300 47.910655 57.00    1
## 253 165.7350 47.712209 32.00    1
## 254 156.2100 46.379782 24.00    0
## 255 152.4000 41.163474 77.00    1
## 256 140.3350 36.599204 62.00    0
## 257 158.1150 43.091240 17.00    1
## 258 163.1950 48.137451 67.00    1
## 259 151.1300 36.712603 70.00    0
## 260 171.1198 56.557252 37.00    1
## 261 149.8600 38.697068 58.00    0
## 262 163.8300 47.485413 35.00    1
## 263 141.6050 36.202312 30.00    0
## 264  93.9800 14.288148  5.00    0
## 265 149.2250 41.276872 26.00    0
## 266 105.4100 15.223681  5.00    0
## 267 146.0500 44.763860 21.00    0
## 268 161.2900 50.433760 41.00    1
## 269 162.5600 55.281525 46.00    1
## 270 145.4150 37.931631 49.00    0
## 271 145.4150 35.493574 15.00    1
## 272 170.8150 58.456669 28.00    1
## 273 127.0000 21.488921 12.00    0
## 274 159.3850 44.423667 83.00    0
## 275 159.4000 44.400000 54.00    1
## 276 153.6700 44.565414 54.00    0
## 277 160.0200 44.622113 68.00    1
## 278 150.4950 40.483086 68.00    0
## 279 149.2250 44.083472 56.00    0
## 280 127.0000 24.408919 15.00    0
## 281 142.8750 34.416293 57.00    0
## 282 142.1130 32.772022 22.00    0
## 283 147.3200 35.947166 40.00    0
## 284 162.5600 49.554900 19.00    1
## 285 164.4650 53.183662 41.00    1
## 286 160.0200 37.081146 75.90    1
## 287 153.6700 40.511435 73.90    0
## 288 167.0050 50.603857 49.00    1
## 289 151.1300 43.970075 26.00    1
## 290 147.9550 33.792604 17.00    0
## 291 125.3998 21.375523 13.00    0
## 292 111.1250 16.669506  8.00    0
## 293 153.0350 49.890000 88.00    1
## 294 139.0650 33.594158 68.00    0
## 295 152.4000 43.856676 33.00    1
## 296 154.9400 48.137451 26.00    0
## 297 147.9550 42.751046 56.00    0
## 298 143.5100 34.841535 16.00    1
## 299 117.9830 24.097075 13.00    0
## 300 144.1450 33.906002 34.00    0
## 301  92.7100 12.076887  5.00    0
## 302 147.9550 41.276872 17.00    0
## 303 155.5750 39.717650 74.00    1
## 304 150.4950 35.947166 69.00    0
## 305 155.5750 50.915702 50.00    1
## 306 154.3050 45.756093 44.00    0
## 307 130.6068 25.259404 15.00    0
## 308 101.6000 15.337079  5.00    0
## 309 157.4800 49.214732 18.00    0
## 310 168.9100 58.825212 41.00    1
## 311 150.4950 43.459784 27.00    0
## 312 111.7600 17.831836  8.90    1
## 313 160.0200 51.964633 38.00    1
## 314 167.6400 50.688906 57.00    1
## 315 144.1450 34.246196 64.50    0
## 316 145.4150 39.377456 42.00    0
## 317 160.0200 59.562300 24.00    1
## 318 147.3200 40.312989 16.00    1
## 319 164.4650 52.163080 71.00    1
## 320 153.0350 39.972795 49.50    0
## 321 149.2250 43.941725 33.00    1
## 322 160.0200 54.601137 28.00    0
## 323 149.2250 45.075705 47.00    0
## 324  85.0900 11.453198  3.00    1
## 325  84.4550 11.765042  1.00    1
## 326  59.6138  5.896696  1.00    0
## 327  92.7100 12.105237  3.00    1
## 328 111.1250 18.313777  6.00    0
## 329  90.8050 11.368149  5.00    0
## 330 153.6700 41.333571 27.00    0
## 331  99.6950 16.244263  5.00    0
## 332  62.4840  6.803880  1.00    0
## 333  81.9150 11.878440  2.00    1
## 334  96.5200 14.968536  2.00    0
## 335  80.0100  9.865626  1.00    1
## 336 150.4950 41.900561 55.00    0
## 337 151.7650 42.524000 83.40    1
## 338 140.6398 28.859791 12.00    1
## 339  88.2650 12.785625  2.00    0
## 340 158.1150 43.147939 63.00    1
## 341 149.2250 40.823280 52.00    0
## 342 151.7650 42.864444 49.00    1
## 343 154.9400 46.209685 31.00    0
## 344 123.8250 20.581737  9.00    0
## 345 104.1400 15.875720  6.00    0
## 346 161.2900 47.853956 35.00    1
## 347 148.5900 42.524250 35.00    0
## 348  97.1550 17.066399  7.00    0
## 349  93.3450 13.182517  5.00    1
## 350 160.6550 48.505994 24.00    1
## 351 157.4800 45.869491 41.00    1
## 352 167.0050 52.900167 32.00    1
## 353 157.4800 47.570461 43.00    1
## 354  91.4400 12.927372  6.00    0
## 355  60.4520  5.669900  1.00    1
## 356 137.1600 28.916490 15.00    1
## 357 152.4000 43.544832 63.00    0
## 358 152.4000 43.431434 21.00    0
## 359  81.2800 11.509897  1.00    1
## 360 109.2200 11.708343  2.00    0
## 361  71.1200  7.540967  1.00    1
## 362  89.2048 12.700576  3.00    0
## 363  67.3100  7.200773  1.00    0
## 364  85.0900 12.360382  1.00    1
## 365  69.8500  7.796112  1.00    0
## 366 161.9250 53.212012 55.00    0
## 367 152.4000 44.678812 38.00    0
## 368  88.9000 12.558829  3.00    1
## 369  90.1700 12.700576  3.00    1
## 370  71.7550  7.370870  1.00    0
## 371  83.8200  9.213587  1.00    0
## 372 159.3850 47.201918 28.00    1
## 373 142.2400 28.632995 16.00    0
## 374 142.2400 31.666391 36.00    0
## 375 168.9100 56.443855 38.00    1
## 376 123.1900 20.014747 12.00    1
## 377  74.9300  8.504850  1.00    1
## 378  74.2950  8.306404  1.00    0
## 379  90.8050 11.623295  3.00    0
## 380 160.0200 55.791816 48.00    1
## 381  67.9450  7.966209  1.00    0
## 382 135.8900 27.215520 15.00    0
## 383 158.1150 47.485413 45.00    1
## 384  85.0900 10.801160  3.00    1
## 385  93.3450 14.004653  3.00    0
## 386 152.4000 45.160753 38.00    0
## 387 155.5750 45.529297 21.00    0
## 388 154.3050 48.874538 50.00    0
## 389 156.8450 46.578229 41.00    1
## 390 120.0150 20.128145 13.00    0
## 391 114.3000 18.143680  8.00    1
## 392  83.8200 10.914558  3.00    1
## 393 156.2100 43.885026 30.00    0
## 394 137.1600 27.158821 12.00    1
## 395 114.3000 19.050864  7.00    1
## 396  93.9800 13.834556  4.00    0
## 397 168.2750 56.046962 21.00    1
## 398 147.9550 40.086193 38.00    0
## 399 139.7000 26.563482 15.00    1
## 400 157.4800 50.802304 19.00    0
## 401  76.2000  9.213587  1.00    1
## 402  66.0400  7.569317  1.00    1
## 403 160.7000 46.300000 31.00    1
## 404 114.3000 19.419407  8.00    0
## 405 146.0500 37.903281 16.00    1
## 406 161.2900 49.356479 21.00    1
## 407  69.8500  7.314171  0.00    0
## 408 133.9850 28.151053 13.00    1
## 409  67.9450  7.824462  0.00    1
## 410 150.4950 44.111822 50.00    0
## 411 163.1950 51.029100 39.00    1
## 412 148.5900 40.766581 44.00    1
## 413 148.5900 37.563088 36.00    0
## 414 161.9250 51.596090 36.00    1
## 415 153.6700 44.820560 18.00    0
## 416  68.5800  8.022908  0.00    0
## 417 151.1300 43.403084 58.00    0
## 418 163.8300 46.719976 58.00    1
## 419 153.0350 39.547553 33.00    0
## 420 151.7650 34.784836 21.50    0
## 421 132.0800 22.792998 11.00    1
## 422 156.2100 39.292407 26.00    1
## 423 140.3350 37.449689 22.00    0
## 424 158.7500 48.676091 28.00    1
## 425 142.8750 35.606972 42.00    0
## 426  84.4550  9.383684  2.00    1
## 427 151.9428 43.714929 21.00    1
## 428 161.2900 48.194150 19.00    1
## 429 127.9906 29.852024 13.00    1
## 430 160.9852 50.972401 48.00    1
## 431 144.7800 43.998424 46.00    0
## 432 132.0800 28.292801 11.00    1
## 433 117.9830 20.354941  8.00    1
## 434 160.0200 48.194150 25.00    1
## 435 154.9400 39.179009 16.00    1
## 436 160.9852 46.691626 51.00    1
## 437 165.9890 56.415505 25.00    1
## 438 157.9880 48.591043 28.00    1
## 439 154.9400 48.222499 26.00    0
## 440  97.9932 13.295915  5.00    1
## 441  64.1350  6.662133  1.00    0
## 442 160.6550 47.485413 54.00    1
## 443 147.3200 35.550273 66.00    0
## 444 146.7000 36.600000 20.00    0
## 445 147.3200 48.959587 25.00    0
## 446 172.9994 51.255896 38.00    1
## 447 158.1150 46.521529 51.00    1
## 448 147.3200 36.967748 48.00    0
## 449 124.9934 25.117657 13.00    1
## 450 106.0450 16.272613  6.00    1
## 451 165.9890 48.647742 27.00    1
## 452 149.8600 38.045029 22.00    0
## 453  76.2000  8.504850  1.00    0
## 454 161.9250 47.286966 60.00    1
## 455 140.0048 28.349500 15.00    0
## 456  66.6750  8.136306  0.00    0
## 457  62.8650  7.200773  0.00    1
## 458 163.8300 55.394923 43.00    1
## 459 147.9550 32.488527 12.00    1
## 460 160.0200 54.204244 27.00    1
## 461 154.9400 48.477645 30.00    1
## 462 152.4000 43.062891 29.00    0
## 463  62.2300  7.257472  0.00    0
## 464 146.0500 34.189497 23.00    0
## 465 151.9936 49.951819 30.00    0
## 466 157.4800 41.305222 17.00    1
## 467  55.8800  4.847765  0.00    0
## 468  60.9600  6.236890  0.00    1
## 469 151.7650 44.338618 41.00    0
## 470 144.7800 33.452410 42.00    0
## 471 118.1100 16.896302  7.00    0
## 472  78.1050  8.221355  3.00    0
## 473 160.6550 47.286966 43.00    1
## 474 151.1300 46.124637 35.00    0
## 475 121.9200 20.184844 10.00    0
## 476  92.7100 12.757275  3.00    1
## 477 153.6700 47.400364 75.50    1
## 478 147.3200 40.851630 64.00    0
## 479 139.7000 50.348712 38.00    1
## 480 157.4800 45.132404 24.20    0
## 481  91.4400 11.623295  4.00    0
## 482 154.9400 42.240755 26.00    1
## 483 143.5100 41.645415 19.00    0
## 484  83.1850  9.156889  2.00    1
## 485 158.1150 45.217453 43.00    1
## 486 147.3200 51.255896 38.00    0
## 487 123.8250 21.205426 10.00    1
## 488  88.9000 11.594945  3.00    1
## 489 160.0200 49.271431 23.00    1
## 490 137.1600 27.952607 16.00    0
## 491 165.1000 51.199197 49.00    1
## 492 154.9400 43.856676 41.00    0
## 493 111.1250 17.690088  6.00    1
## 494 153.6700 35.521923 23.00    0
## 495 145.4150 34.246196 14.00    0
## 496 141.6050 42.885420 43.00    0
## 497 144.7800 32.545226 15.00    0
## 498 163.8300 46.776675 21.00    1
## 499 161.2900 41.872211 24.00    1
## 500 154.9000 38.200000 20.00    1
## 501 161.3000 43.300000 20.00    1
## 502 170.1800 53.637254 34.00    1
## 503 149.8600 42.977842 29.00    0
## 504 123.8250 21.545620 11.00    1
## 505  85.0900 11.424848  3.00    0
## 506 160.6550 39.774349 65.00    1
## 507 154.9400 43.346385 46.00    0
## 508 106.0450 15.478827  8.00    0
## 509 126.3650 21.914164 15.00    1
## 510 166.3700 52.673371 43.00    1
## 511 148.2852 38.441922 39.00    0
## 512 124.4600 19.277660 12.00    0
## 513  89.5350 11.113004  3.00    1
## 514 101.6000 13.494362  4.00    0
## 515 151.7650 42.807745 43.00    0
## 516 148.5900 35.890467 70.00    0
## 517 153.6700 44.225220 26.00    0
## 518  53.9750  4.252425  0.00    0
## 519 146.6850 38.073378 48.00    0
## 520  56.5150  5.159609  0.00    0
## 521 100.9650 14.316498  5.00    1
## 522 121.9200 23.218241  8.00    1
## 523  81.5848 10.659412  3.00    0
## 524 154.9400 44.111822 44.00    1
## 525 156.2100 44.026773 33.00    0
## 526 132.7150 24.975910 15.00    1
## 527 125.0950 22.594552 12.00    0
## 528 101.6000 14.344847  5.00    0
## 529 160.6550 47.882306 41.00    1
## 530 146.0500 39.405805 37.40    0
## 531 132.7150 24.777463 13.00    0
## 532  87.6300 10.659412  6.00    0
## 533 156.2100 41.050076 53.00    1
## 534 152.4000 40.823280 49.00    0
## 535 162.5600 47.031821 27.00    0
## 536 114.9350 17.519991  7.00    1
## 537  67.9450  7.229122  1.00    0
## 538 142.8750 34.246196 31.00    0
## 539  76.8350  8.022908  1.00    1
## 540 145.4150 31.127751 17.00    1
## 541 162.5600 52.163080 31.00    1
## 542 156.2100 54.062497 21.00    0
## 543  71.1200  8.051258  0.00    1
## 544 158.7500 52.531624 68.00    1

d_550 <- d[sample(1:nrow(d), size = 200, replace = FALSE), ]
d_450 <- d[sample(1:nrow(d), size = 300, replace = FALSE), ]
d_350 <- d[sample(1:nrow(d), size = 400, replace = FALSE), ]
m_550 <- map(
  alist(
    height ~ dnorm(mu, sigma),
    mu <- a + b * log(weight)
  ),
  data = d_550,
  start = list(a = mean(d_550$height), b = 0, sigma = sd(d_550$height))
)
m_450 <- map(
  alist(
    height ~ dnorm(mu, sigma),
    mu <- a + b * log(weight)
  ),
  data = d_450,
  start = list(a = mean(d_450$height), b = 0, sigma = sd(d_450$height))
)
m_350 <- map(
  alist(
    height ~ dnorm(mu, sigma),
   mu <- a + b * log(weight)
  ),
  data = d_350,
  start = list(a = mean(d_350$height), b = 0, sigma = sd(d_350$height))
)
(model.compare <- compare(m_550, m_450, m_350))

##           WAIC       SE     dWAIC      dSE    pWAIC        weight
## m_550 1232.439 20.30732    0.0000       NA 2.929953  1.000000e+00
## m_450 1847.189 30.02313  614.7493 31.22239 3.489963 3.227676e-134
## m_350 2452.368 32.89998 1219.9290 30.69325 3.463851 1.246770e-265

#Different numbers of observations found for at least two models.
#Information criteria only valid for comparing models fit to exactly same observations.
#Number of observations for each model:
#m_550 250 
#m_450 350 
#m_350 450 
#longer object length is not a multiple of shorter object lengthlonger object length is not a multiple of #shorter object lengthlonger object length is not a multiple of shorter object length

7-2. What happens to the effective number of parameters, as measured by PSIS or WAIC, as a prior becomes more concentrated? Why? Perform some experiments.

d <- Howell1[complete.cases(Howell1), ]
d$height.log <- log(d$height)
d$height.log.z <- (d$height.log - mean(d$height.log)) / sd(d$height.log)
d$weight.log <- log(d$weight)
d$weight.log.z <- (d$weight.log - mean(d$weight.log)) / sd(d$weight.log)
m_wide <- map(
  alist(
    height.log.z ~ dnorm(mu, sigma),
    mu <- a + b * weight.log.z,
    a ~ dnorm(0, 10),
    b ~ dnorm(1, 10),
    sigma ~ dunif(0, 10)
  ),
  data = d
)
m_narrow <- map(
  alist(
    height.log.z ~ dnorm(mu, sigma),
    mu <- a + b * weight.log.z,
    a ~ dnorm(0, 0.10),
    b ~ dnorm(1, 0.10),
    sigma ~ dunif(0, 1)
  ),
  data = d
)
WAIC(m_wide, refresh = 0)

##        WAIC     lppd  penalty  std_err
## 1 -102.5901 55.66528 4.370222 36.59395

7-3. Consider three fictional Polynesian islands. On each there is a Royal Ornithologist charged by the king with surveying the bird population. They have each found the following proportions of 5 important bird species:

height	weight	age	male	height.log	height.log.z	weight.log	weight.log.z
151.7650	47.825606	63.00	1	5.022333	0.4910353	3.867561	0.7383006
139.7000	36.485807	63.00	0	4.939497	0.1484141	3.596923	0.2684103
136.5250	31.864838	65.00	0	4.916508	0.0533263	3.461503	0.0332893
156.8450	53.041914	41.00	1	5.055258	0.6272167	3.971082	0.9180376
145.4150	41.276872	51.00	0	4.979592	0.3142503	3.720302	0.4826249
163.8300	62.992589	35.00	1	5.098829	0.8074334	4.143017	1.2165561
149.2250	38.243476	32.00	0	5.005455	0.4212254	3.643973	0.3500994
168.9100	55.479971	27.00	1	5.129366	0.9337375	4.016022	0.9960632
147.9550	34.869885	19.00	0	4.996908	0.3858736	3.551624	0.1897593
165.1000	54.487739	54.00	1	5.106551	0.8393729	3.997976	0.9647305
154.3050	49.895120	47.00	0	5.038931	0.5596864	3.909923	0.8118509
151.1300	41.220173	66.00	1	5.018140	0.4736930	3.718928	0.4802384
144.7800	36.032215	73.00	0	4.975215	0.2961490	3.584413	0.2466901
149.9000	47.700000	20.00	0	5.009968	0.4398925	3.864931	0.7337346
150.4950	33.849303	65.30	0	5.013930	0.4562776	3.521918	0.1381843
163.1950	48.562694	36.00	1	5.094946	0.7913707	3.882856	0.7648553
157.4800	42.325803	44.00	1	5.059298	0.6439284	3.745397	0.5261950
143.9418	38.356873	31.00	0	4.969409	0.2721334	3.646934	0.3552400
121.9200	19.617854	12.00	1	4.803365	-0.4146473	2.976440	-0.8088931
105.4100	13.947954	8.00	0	4.657858	-1.0164868	2.635333	-1.4011346
86.3600	10.489315	6.50	0	4.458525	-1.8409552	2.350357	-1.8959188
161.2900	48.987936	39.00	1	5.083204	0.7428050	3.891574	0.7799925
156.2100	42.722696	29.00	0	5.051201	0.6104372	3.754730	0.5423999
129.5400	23.586784	13.00	1	4.863990	-0.1638955	3.160687	-0.4889983
109.2200	15.989118	7.00	0	4.693364	-0.8696262	2.771908	-1.1640077
146.4000	35.493574	56.00	1	4.986343	0.3421729	3.569352	0.2205394
148.5900	37.903281	45.00	0	5.001191	0.4035872	3.635038	0.3345857
147.3200	35.465224	19.00	0	4.992607	0.3680837	3.568553	0.2191521
137.1600	27.328918	17.00	1	4.921148	0.0725195	3.307945	-0.2333227
125.7300	22.679600	16.00	0	4.834137	-0.2873715	3.121466	-0.5570946
114.3000	17.860185	11.00	1	4.738827	-0.6815876	2.882574	-0.9718665
147.9550	40.312989	29.00	1	4.996908	0.3858736	3.696674	0.4416002
161.9250	55.111428	30.00	1	5.087133	0.7590570	4.009357	0.9844913
146.0500	37.506388	24.00	0	4.983949	0.3322727	3.624511	0.3163094
146.0500	38.498621	35.00	0	4.983949	0.3322727	3.650622	0.3616444
152.7048	46.606578	33.00	0	5.028507	0.5165692	3.841742	0.6934719
142.8750	38.838815	27.00	0	4.961970	0.2413650	3.659420	0.3769193
142.8750	35.578623	32.00	0	4.961970	0.2413650	3.571745	0.2246948
147.9550	47.400364	36.00	0	4.996908	0.3858736	3.858630	0.7227938
160.6550	47.882306	24.00	1	5.079259	0.7264888	3.868746	0.7403577
151.7650	49.413179	30.00	1	5.022333	0.4910353	3.900217	0.7949989
162.8648	49.384829	24.00	1	5.092920	0.7829934	3.899643	0.7940025
171.4500	56.557252	52.00	1	5.144292	0.9954721	4.035253	1.0294534
147.3200	39.122310	42.00	0	4.992607	0.3680837	3.666693	0.3895465
147.9550	49.895120	19.00	0	4.996908	0.3858736	3.909923	0.8118509
144.7800	28.803092	17.00	0	4.975215	0.2961490	3.360483	-0.1421056
121.9200	20.411640	8.00	1	4.803365	-0.4146473	3.016105	-0.7400250
128.9050	23.359988	12.00	0	4.859076	-0.1842206	3.151025	-0.5057736
97.7900	13.267566	5.00	0	4.582822	-1.3268427	2.585322	-1.4879644
154.3050	41.248522	55.00	1	5.038931	0.5596864	3.719615	0.4814321
143.5100	38.555320	43.00	0	4.966405	0.2597071	3.652094	0.3641996
146.7000	42.400000	20.00	1	4.988390	0.3506399	3.747148	0.5292359
157.4800	44.650463	18.00	1	5.059298	0.6439284	3.798865	0.6190274
127.0000	22.010552	13.00	1	4.844187	-0.2458019	3.091522	-0.6090841
110.4900	15.422128	9.00	0	4.704925	-0.8218091	2.735803	-1.2266944
97.7900	12.757275	5.00	0	4.582822	-1.3268427	2.546102	-1.5560607
165.7350	58.598416	42.00	1	5.110390	0.8552506	4.070708	1.0910102
152.4000	46.719976	44.00	0	5.026509	0.5083052	3.844172	0.6976912
141.6050	44.225220	60.00	0	4.953042	0.2044349	3.789295	0.6024126
158.8000	50.900000	20.00	0	5.067646	0.6784531	3.929863	0.8464709
155.5750	54.317642	37.00	0	5.047128	0.5935894	3.994849	0.9593020
164.4650	45.897841	50.00	1	5.102698	0.8234340	3.826418	0.6668665
151.7650	48.024053	50.00	0	5.022333	0.4910353	3.871702	0.7454900
161.2900	52.219779	31.00	1	5.083204	0.7428050	3.955461	0.8909157
154.3050	47.627160	25.00	0	5.038931	0.5596864	3.863403	0.7310813
145.4150	45.642695	23.00	0	4.979592	0.3142503	3.820844	0.6571879
145.4150	42.410852	52.00	0	4.979592	0.3142503	3.747404	0.5296802
152.4000	36.485807	79.30	1	5.026509	0.5083052	3.596923	0.2684103
163.8300	55.933563	35.00	1	5.098829	0.8074334	4.024165	1.0102006
144.1450	37.194544	27.00	0	4.970820	0.2779682	3.616162	0.3018132
129.5400	24.550667	13.00	1	4.863990	-0.1638955	3.200739	-0.4194579
129.5400	25.627948	14.00	0	4.863990	-0.1638955	3.243683	-0.3448963
153.6700	48.307548	38.00	1	5.034807	0.5426302	3.877588	0.7557091
142.8750	37.336292	39.00	0	4.961970	0.2413650	3.619966	0.3084174
146.0500	29.596878	12.00	0	4.983949	0.3322727	3.387669	-0.0949042
167.0050	47.173568	30.00	1	5.118024	0.8868243	3.853834	0.7144665
158.4198	47.286966	24.00	0	5.065249	0.6685385	3.856235	0.7186352
91.4400	12.927372	0.60	1	4.515683	-1.6045401	2.559347	-1.5330639
165.7350	57.549485	51.00	1	5.110390	0.8552506	4.052645	1.0596495
149.8600	37.931631	46.00	0	5.009702	0.4387886	3.635785	0.3358838
147.9550	41.900561	17.00	0	4.996908	0.3858736	3.735299	0.5086630
137.7950	27.584063	12.00	0	4.925767	0.0916241	3.317238	-0.2171882
154.9400	47.201918	22.00	0	5.043038	0.5766727	3.854434	0.7155096
160.9598	43.204638	29.00	1	5.081155	0.7343286	3.765948	0.5618762
161.9250	50.263663	38.00	1	5.087133	0.7590570	3.917282	0.8246282
147.9550	39.377456	30.00	0	4.996908	0.3858736	3.673194	0.4008330
113.6650	17.463292	6.00	1	4.733256	-0.7046302	2.860101	-1.0108847
159.3850	50.689000	45.00	1	5.071323	0.6936621	3.925709	0.8392586
148.5900	39.434154	47.00	0	5.001191	0.4035872	3.674632	0.4033311
136.5250	36.287360	79.00	0	4.916508	0.0533263	3.591470	0.2589411
158.1150	46.266384	45.00	1	5.063323	0.6605729	3.834416	0.6807522
144.7800	42.269104	54.00	0	4.975215	0.2961490	3.744056	0.5238676
156.8450	47.627160	31.00	1	5.055258	0.6272167	3.863403	0.7310813
179.0700	55.706767	23.00	1	5.187777	1.1753325	4.020102	1.0031463
118.7450	18.824068	9.00	0	4.776978	-0.5237866	2.935136	-0.8806061
170.1800	48.562694	41.00	1	5.136857	0.9647200	3.882856	0.7648553
146.0500	42.807745	23.00	0	4.983949	0.3322727	3.756719	0.5458528
147.3200	35.068331	36.00	0	4.992607	0.3680837	3.557298	0.1996123
113.0300	17.888534	5.00	1	4.727653	-0.7278019	2.884160	-0.9691128
162.5600	56.755699	30.00	0	5.091047	0.7752454	4.038756	1.0355347
133.9850	27.442316	12.00	1	4.897728	-0.0243499	3.312086	-0.2261333
152.4000	51.255896	34.00	0	5.026509	0.5083052	3.936831	0.8585685
160.0200	47.230267	44.00	1	5.075299	0.7101080	3.855035	0.7165521
149.8600	40.936678	43.00	0	5.009702	0.4387886	3.712026	0.4682560
142.8750	32.715323	73.30	0	4.961970	0.2413650	3.487844	0.0790224
167.0050	57.067543	38.00	1	5.118024	0.8868243	4.044236	1.0450484
159.3850	42.977842	43.00	1	5.071323	0.6936621	3.760685	0.5527381
154.9400	39.944446	33.00	0	5.043038	0.5766727	3.687490	0.4256544
148.5900	32.460178	16.00	0	5.001191	0.4035872	3.480014	0.0654285
111.1250	17.123098	11.00	1	4.710656	-0.7981062	2.840428	-1.0450412
111.7600	16.499409	6.00	1	4.716354	-0.7745384	2.803325	-1.1094619
162.5600	45.954540	35.00	1	5.091047	0.7752454	3.827653	0.6690100
152.4000	41.106775	29.00	0	5.026509	0.5083052	3.716173	0.4754554
124.4600	18.257078	12.00	0	4.823984	-0.3293631	2.904553	-0.9337061
111.7600	15.081934	9.00	1	4.716354	-0.7745384	2.713498	-1.2654224
86.3600	11.481547	7.60	1	4.458525	-1.8409552	2.440741	-1.7389910
170.1800	47.598810	58.00	1	5.136857	0.9647200	3.862808	0.7300475
146.0500	37.506388	53.00	0	4.983949	0.3322727	3.624511	0.3163094
159.3850	45.019006	51.00	1	5.071323	0.6936621	3.807085	0.6332994
151.1300	42.269104	48.00	0	5.018140	0.4736930	3.744056	0.5238676
160.6550	54.856282	29.00	1	5.079259	0.7264888	4.004717	0.9764345
169.5450	53.523856	41.00	1	5.133118	0.9492578	3.980128	0.9337418
158.7500	52.191429	81.75	1	5.067331	0.6771506	3.954918	0.8899728
74.2950	9.752228	1.00	1	4.308044	-2.4633652	2.277496	-2.0224231
149.8600	42.410852	35.00	0	5.009702	0.4387886	3.747404	0.5296802
153.0350	49.583275	46.00	0	5.030667	0.5255033	3.903654	0.8009654
96.5200	13.097469	5.00	1	4.569750	-1.3809106	2.572419	-1.5103677
161.9250	41.730464	29.00	1	5.087133	0.7590570	3.731231	0.5016004
162.5600	56.018612	42.00	1	5.091047	0.7752454	4.025684	1.0128386
149.2250	42.155707	27.00	0	5.005455	0.4212254	3.741370	0.5192034
116.8400	19.391058	8.00	0	4.760806	-0.5906798	2.964812	-0.8290821
100.0760	15.081934	6.00	1	4.605930	-1.2312666	2.713498	-1.2654224
163.1950	53.098613	22.00	1	5.094946	0.7913707	3.972151	0.9198925
161.9250	50.235314	43.00	1	5.087133	0.7590570	3.916718	0.8236487
145.4150	42.524250	53.00	0	4.979592	0.3142503	3.750075	0.5343163
163.1950	49.101334	43.00	1	5.094946	0.7913707	3.893886	0.7840069
151.1300	38.498621	41.00	0	5.018140	0.4736930	3.650622	0.3616444
150.4950	49.810071	50.00	0	5.013930	0.4562776	3.908217	0.8088889
141.6050	29.313383	15.00	1	4.953042	0.2044349	3.378044	-0.1116149
170.8150	59.760746	33.00	1	5.140581	0.9801246	4.090349	1.1251121
91.4400	11.708343	3.00	0	4.515683	-1.6045401	2.460302	-1.7050294
157.4800	47.939005	62.00	1	5.059298	0.6439284	3.869930	0.7424125
152.4000	39.292407	49.00	0	5.026509	0.5083052	3.671031	0.3970789
149.2250	38.130077	17.00	1	5.005455	0.4212254	3.641003	0.3449435
129.5400	21.999212	12.00	0	4.863990	-0.1638955	3.091007	-0.6099789
147.3200	36.882700	22.00	0	4.992607	0.3680837	3.607743	0.2871950
145.4150	42.127357	29.00	0	4.979592	0.3142503	3.740697	0.5180354
121.9200	19.787951	8.00	0	4.803365	-0.4146473	2.985073	-0.7939039
113.6650	16.782904	5.00	1	4.733256	-0.7046302	2.820361	-1.0798831
157.4800	44.565414	33.00	1	5.059298	0.6439284	3.796958	0.6157172
154.3050	47.853956	34.00	0	5.038931	0.5596864	3.868154	0.7393295
120.6500	21.177076	12.00	0	4.792894	-0.4579581	3.052919	-0.6761073
115.6000	18.900000	7.00	1	4.750136	-0.6348104	2.939162	-0.8736166
167.0050	55.196477	42.00	1	5.118024	0.8868243	4.010899	0.9871686
142.8750	32.998818	40.00	0	4.961970	0.2413650	3.496472	0.0940029
152.4000	40.879979	27.00	0	5.026509	0.5083052	3.710640	0.4658496
96.5200	13.267566	3.00	0	4.569750	-1.3809106	2.585322	-1.4879644
160.0000	51.200000	25.00	1	5.075174	0.7095911	3.935739	0.8566741
159.3850	49.044635	29.00	1	5.071323	0.6936621	3.892731	0.7820009
149.8600	53.438808	45.00	0	5.009702	0.4387886	3.978537	0.9309808
160.6550	54.090846	26.00	1	5.079259	0.7264888	3.990665	0.9520374
160.6550	55.366574	45.00	1	5.079259	0.7264888	4.013976	0.9925108
149.2250	42.240755	45.00	0	5.005455	0.4212254	3.743386	0.5227027
125.0950	22.367756	11.00	0	4.829073	-0.3083140	3.107620	-0.5811334
140.9700	40.936678	85.60	0	4.948547	0.1858455	3.712026	0.4682560
154.9400	49.696674	26.00	1	5.043038	0.5766727	3.905938	0.8049317
141.6050	44.338618	24.00	0	4.953042	0.2044349	3.791856	0.6068588
160.0200	45.954540	57.00	1	5.075299	0.7101080	3.827653	0.6690100
150.1648	41.957260	22.00	0	5.011733	0.4471926	3.736651	0.5110109
155.5750	51.482692	24.00	0	5.047128	0.5935894	3.941246	0.8662340
103.5050	12.757275	6.00	0	4.639620	-1.0919200	2.546102	-1.5560607
94.6150	13.012420	4.00	0	4.549816	-1.4633613	2.565904	-1.5216787
156.2100	44.111822	21.00	0	5.051201	0.6104372	3.786728	0.5979550
153.0350	32.205032	79.00	0	5.030667	0.5255033	3.472123	0.0517273
167.0050	56.755699	50.00	1	5.118024	0.8868243	4.038756	1.0355347
149.8600	52.673371	40.00	0	5.009702	0.4387886	3.964110	0.9059318
147.9550	36.485807	64.00	0	4.996908	0.3858736	3.596923	0.2684103
159.3850	48.846188	32.00	1	5.071323	0.6936621	3.888676	0.7749614
161.9250	56.954146	38.70	1	5.087133	0.7590570	4.042247	1.0415949
155.5750	42.099007	26.00	0	5.047128	0.5935894	3.740024	0.5168666
159.3850	50.178615	63.00	1	5.071323	0.6936621	3.915589	0.8216879
146.6850	46.549879	62.00	0	4.988287	0.3502170	3.840524	0.6913584
172.7200	61.801910	22.00	1	5.151672	1.0259972	4.123934	1.1834239
166.3700	48.987936	41.00	1	5.114214	0.8710676	3.891574	0.7799925
141.6050	31.524644	19.00	1	4.953042	0.2044349	3.450770	0.0146534
142.8750	32.205032	17.00	0	4.961970	0.2413650	3.472123	0.0517273
133.3500	23.756881	14.00	0	4.892977	-0.0439991	3.167872	-0.4765223
127.6350	24.408919	9.00	1	4.849175	-0.2251728	3.194949	-0.4295114
119.3800	21.517270	7.00	1	4.782312	-0.5017272	3.068856	-0.6484377
151.7650	35.295127	74.00	0	5.022333	0.4910353	3.563745	0.2108048
156.8450	45.642695	41.00	1	5.055258	0.6272167	3.820844	0.6571879
148.5900	43.885026	33.00	0	5.001191	0.4035872	3.781573	0.5890054
157.4800	45.557646	53.00	0	5.059298	0.6439284	3.818979	0.6539497
149.8600	39.008912	18.00	0	5.009702	0.4387886	3.663790	0.3845066
147.9550	41.163474	37.00	0	4.996908	0.3858736	3.717551	0.4778485
102.2350	13.125818	6.00	0	4.627274	-1.1429841	2.574581	-1.5066137
153.0350	45.245802	61.00	0	5.030667	0.5255033	3.812110	0.6420242
160.6550	53.637254	44.00	1	5.079259	0.7264888	3.982244	0.9374164
149.2250	52.304828	35.00	0	5.005455	0.4212254	3.957089	0.8937411
114.3000	18.342126	7.00	1	4.738827	-0.6815876	2.909200	-0.9256368
100.9650	13.749507	4.00	1	4.614774	-1.1946865	2.621003	-1.4260146
138.4300	39.093961	23.00	0	4.930365	0.1106409	3.665968	0.3882879
91.4400	12.530479	4.00	1	4.515683	-1.6045401	2.528164	-1.5872047
162.5600	45.699394	55.00	1	5.091047	0.7752454	3.822085	0.6593434
149.2250	40.398038	53.00	0	5.005455	0.4212254	3.698781	0.4452592
158.7500	51.482692	59.00	1	5.067331	0.6771506	3.941246	0.8662340
149.8600	38.668718	57.00	0	5.009702	0.4387886	3.655031	0.3692986
158.1150	39.235708	35.00	1	5.063323	0.6605729	3.669587	0.3945717
156.2100	44.338618	29.00	0	5.051201	0.6104372	3.791856	0.6068588
148.5900	39.519203	62.00	1	5.001191	0.4035872	3.676787	0.4070717
143.5100	31.071052	18.00	0	4.966405	0.2597071	3.436277	-0.0105099
154.3050	46.776675	51.00	0	5.038931	0.5596864	3.845385	0.6997970
131.4450	22.509503	14.00	0	4.878589	-0.1035129	3.113938	-0.5701654
157.4800	40.624834	19.00	1	5.059298	0.6439284	3.704379	0.4549793
157.4800	50.178615	42.00	1	5.059298	0.6439284	3.915589	0.8216879
154.3050	41.276872	25.00	0	5.038931	0.5596864	3.720302	0.4826249
107.9500	17.576690	6.00	1	4.681668	-0.9180027	2.866574	-0.9996469
168.2750	54.600000	41.00	1	5.125599	0.9181588	4.000034	0.9683040
145.4150	44.990657	37.00	0	4.979592	0.3142503	3.806455	0.6322057
147.9550	44.735511	16.00	0	4.996908	0.3858736	3.800768	0.6223314
100.9650	14.401546	5.00	1	4.614774	-1.1946865	2.667336	-1.3455705
113.0300	19.050864	9.00	1	4.727653	-0.7278019	2.947112	-0.8598127
149.2250	35.805419	82.00	1	5.005455	0.4212254	3.578099	0.2357273
154.9400	45.217453	28.00	1	5.043038	0.5766727	3.811483	0.6409360
162.5600	48.109102	50.00	1	5.091047	0.7752454	3.873471	0.7485620
156.8450	45.671045	43.00	0	5.055258	0.6272167	3.821464	0.6582660
123.1900	20.808533	8.00	1	4.813728	-0.3717854	3.035363	-0.7065889
161.0106	48.420946	31.00	1	5.081470	0.7356338	3.879932	0.7597800
144.7800	41.191823	67.00	0	4.975215	0.2961490	3.718240	0.4790439
143.5100	38.413573	39.00	0	4.966405	0.2597071	3.648411	0.3578046
149.2250	42.127357	18.00	0	5.005455	0.4212254	3.740697	0.5180354
110.4900	17.661738	11.00	0	4.704925	-0.8218091	2.871401	-0.9912660
149.8600	38.243476	48.00	0	5.009702	0.4387886	3.643973	0.3500994
165.7350	48.335898	30.00	1	5.110390	0.8552506	3.878175	0.7567278
144.1450	38.923864	64.00	0	4.970820	0.2779682	3.661608	0.3807171
157.4800	40.029494	72.00	1	5.059298	0.6439284	3.689617	0.4293472
154.3050	50.206964	68.00	0	5.038931	0.5596864	3.916154	0.8226686
163.8300	54.289293	44.00	1	5.098829	0.8074334	3.994327	0.9583956
156.2100	45.600000	43.00	0	5.051201	0.6104372	3.819908	0.6555631
153.6700	40.766581	16.00	0	5.034807	0.5426302	3.707863	0.4610268
134.6200	27.130471	13.00	0	4.902456	-0.0047937	3.300657	-0.2459762
144.1450	39.434154	34.00	0	4.970820	0.2779682	3.674632	0.4033311
114.3000	20.496689	10.00	0	4.738827	-0.6815876	3.020263	-0.7328057
162.5600	43.204638	62.00	1	5.091047	0.7752454	3.765948	0.5618762
146.0500	31.864838	44.00	0	4.983949	0.3322727	3.461503	0.0332893
120.6500	20.893581	11.00	1	4.792894	-0.4579581	3.039442	-0.6995071
154.9400	45.444249	31.00	1	5.043038	0.5766727	3.816486	0.6496226
144.7800	38.045029	29.00	0	4.975215	0.2961490	3.638770	0.3410666
106.6800	15.989118	8.00	0	4.669834	-0.9669516	2.771908	-1.1640077
146.6850	36.088913	62.00	0	4.988287	0.3502170	3.585986	0.2494200
152.4000	40.879979	67.00	0	5.026509	0.5083052	3.710640	0.4658496
163.8300	47.910655	57.00	1	5.098829	0.8074334	3.869338	0.7413854
165.7350	47.712209	32.00	1	5.110390	0.8552506	3.865187	0.7341790
156.2100	46.379782	24.00	0	5.051201	0.6104372	3.836864	0.6850025
152.4000	41.163474	77.00	1	5.026509	0.5083052	3.717551	0.4778485
140.3350	36.599204	62.00	0	4.944032	0.1671721	3.600026	0.2737981
158.1150	43.091240	17.00	1	5.063323	0.6605729	3.763320	0.5573131
163.1950	48.137451	67.00	1	5.094946	0.7913707	3.874061	0.7495849
151.1300	36.712603	70.00	0	5.018140	0.4736930	3.603120	0.2791693
171.1198	56.557252	37.00	1	5.142364	0.9874985	4.035253	1.0294534
149.8600	38.697068	58.00	0	5.009702	0.4387886	3.655764	0.3705711
163.8300	47.485413	35.00	1	5.098829	0.8074334	3.860423	0.7259063
141.6050	36.202312	30.00	0	4.953042	0.2044349	3.589123	0.2548670
93.9800	14.288148	5.00	0	4.543082	-1.4912142	2.659430	-1.3592957
149.2250	41.276872	26.00	0	5.005455	0.4212254	3.720302	0.4826249
105.4100	15.223681	5.00	0	4.657858	-1.0164868	2.722852	-1.2491806
146.0500	44.763860	21.00	0	4.983949	0.3322727	3.801401	0.6234313
161.2900	50.433760	41.00	1	5.083204	0.7428050	3.920661	0.8304939
162.5600	55.281525	46.00	1	5.091047	0.7752454	4.012439	0.9898418
145.4150	37.931631	49.00	0	4.979592	0.3142503	3.635785	0.3358838
145.4150	35.493574	15.00	1	4.979592	0.3142503	3.569352	0.2205394
170.8150	58.456669	28.00	1	5.140581	0.9801246	4.068286	1.0868052
127.0000	21.488921	12.00	0	4.844187	-0.2458019	3.067537	-0.6507267
159.3850	44.423667	83.00	0	5.071323	0.6936621	3.793772	0.6101860
159.4000	44.400000	54.00	1	5.071417	0.6940514	3.793239	0.6092608
153.6700	44.565414	54.00	0	5.034807	0.5426302	3.796958	0.6157172
160.0200	44.622113	68.00	1	5.075299	0.7101080	3.798230	0.6179247
150.4950	40.483086	68.00	0	5.013930	0.4562776	3.700884	0.4489106
149.2250	44.083472	56.00	0	5.005455	0.4212254	3.786085	0.5968388
127.0000	24.408919	15.00	0	4.844187	-0.2458019	3.194949	-0.4295114
142.8750	34.416293	57.00	0	4.961970	0.2413650	3.538530	0.1670260
142.1130	32.772022	22.00	0	4.956622	0.2192465	3.489575	0.0820289
147.3200	35.947166	40.00	0	4.992607	0.3680837	3.582050	0.2425871
162.5600	49.554900	19.00	1	5.091047	0.7752454	3.903081	0.7999715
164.4650	53.183662	41.00	1	5.102698	0.8234340	3.973751	0.9226712
160.0200	37.081146	75.90	1	5.075299	0.7101080	3.613109	0.2965117
153.6700	40.511435	73.90	0	5.034807	0.5426302	3.701584	0.4501261
167.0050	50.603857	49.00	1	5.118024	0.8868243	3.924028	0.8363398
151.1300	43.970075	26.00	1	5.018140	0.4736930	3.783509	0.5923669
147.9550	33.792604	17.00	0	4.996908	0.3858736	3.520242	0.1352736
125.3998	21.375523	13.00	0	4.831507	-0.2982484	3.062247	-0.6599132
111.1250	16.669506	8.00	0	4.710656	-0.7981062	2.813581	-1.0916542
153.0350	49.890000	88.00	1	5.030667	0.5255033	3.909821	0.8116727
139.0650	33.594158	68.00	0	4.934941	0.1295706	3.514352	0.1250475
152.4000	43.856676	33.00	1	5.026509	0.5083052	3.780927	0.5878834
154.9400	48.137451	26.00	0	5.043038	0.5766727	3.874061	0.7495849
147.9550	42.751046	56.00	0	4.996908	0.3858736	3.755394	0.5435516
143.5100	34.841535	16.00	1	4.966405	0.2597071	3.550810	0.1883472
117.9830	24.097075	13.00	0	4.770541	-0.5504142	3.182091	-0.4518361
144.1450	33.906002	34.00	0	4.970820	0.2779682	3.523592	0.1410901
92.7100	12.076887	5.00	0	4.529476	-1.5474890	2.491293	-1.6512205
147.9550	41.276872	17.00	0	4.996908	0.3858736	3.720302	0.4826249
155.5750	39.717650	74.00	1	5.047128	0.5935894	3.681796	0.4157684
150.4950	35.947166	69.00	0	5.013930	0.4562776	3.582050	0.2425871
155.5750	50.915702	50.00	1	5.047128	0.5935894	3.930171	0.8470064
154.3050	45.756093	44.00	0	5.038931	0.5596864	3.823325	0.6614962
130.6068	25.259404	15.00	0	4.872191	-0.1299727	3.229198	-0.3700456
101.6000	15.337079	5.00	0	4.621043	-1.1687545	2.730273	-1.2362957
157.4800	49.214732	18.00	0	5.059298	0.6439284	3.896193	0.7880121
168.9100	58.825212	41.00	1	5.129366	0.9337375	4.074571	1.0977170
150.4950	43.459784	27.00	0	5.013930	0.4562776	3.771836	0.5720994
111.7600	17.831836	8.90	1	4.716354	-0.7745384	2.880985	-0.9746247
160.0200	51.964633	38.00	1	5.075299	0.7101080	3.950563	0.8824117
167.6400	50.688906	57.00	1	5.121819	0.9025213	3.925707	0.8392554
144.1450	34.246196	64.50	0	4.970820	0.2779682	3.533575	0.1584237
145.4150	39.377456	42.00	0	4.979592	0.3142503	3.673194	0.4008330
160.0200	59.562300	24.00	1	5.075299	0.7101080	4.087023	1.1193370
147.3200	40.312989	16.00	1	4.992607	0.3680837	3.696674	0.4416002
164.4650	52.163080	71.00	1	5.102698	0.8234340	3.954375	0.8890295
153.0350	39.972795	49.50	0	5.030667	0.5255033	3.688199	0.4268862
149.2250	43.941725	33.00	1	5.005455	0.4212254	3.782864	0.5912471
160.0200	54.601137	28.00	0	5.075299	0.7101080	4.000055	0.9683402
149.2250	45.075705	47.00	0	5.005455	0.4212254	3.808343	0.6354847
85.0900	11.453198	3.00	1	4.443709	-1.9022325	2.438269	-1.7432833
84.4550	11.765042	1.00	1	4.436219	-1.9332149	2.465133	-1.6966418
59.6138	5.896696	1.00	0	4.087887	-3.3739632	1.774392	-2.8959280
92.7100	12.105237	3.00	1	4.529476	-1.5474890	2.493638	-1.6471496
111.1250	18.313777	6.00	0	4.710656	-0.7981062	2.907654	-0.9283224
90.8050	11.368149	5.00	0	4.508714	-1.6333635	2.430816	-1.7562243
153.6700	41.333571	27.00	0	5.034807	0.5426302	3.721675	0.4850082
99.6950	16.244263	5.00	0	4.602116	-1.2470433	2.787740	-1.1365206
62.4840	6.803880	1.00	0	4.134911	-3.1794677	1.917493	-2.6474716
81.9150	11.878440	2.00	1	4.405682	-2.0595190	2.474725	-1.6799872
96.5200	14.968536	2.00	0	4.569750	-1.3809106	2.705950	-1.2785261
80.0100	9.865626	1.00	1	4.382152	-2.1568444	2.289057	-2.0023508
150.4950	41.900561	55.00	0	5.013930	0.4562776	3.735299	0.5086630
151.7650	42.524000	83.40	1	5.022333	0.4910353	3.750069	0.5343061
140.6398	28.859791	12.00	1	4.946202	0.1761459	3.362449	-0.1386912
88.2650	12.785625	2.00	0	4.480344	-1.7507086	2.548322	-1.5522066
158.1150	43.147939	63.00	1	5.063323	0.6605729	3.764635	0.5595962
149.2250	40.823280	52.00	0	5.005455	0.4212254	3.709252	0.4634399
151.7650	42.864444	49.00	1	5.022333	0.4910353	3.758043	0.5481509
154.9400	46.209685	31.00	0	5.043038	0.5766727	3.833189	0.6786232
123.8250	20.581737	9.00	0	4.818869	-0.3505199	3.024404	-0.7256163
104.1400	15.875720	6.00	0	4.645736	-1.0666224	2.764791	-1.1763653
161.2900	47.853956	35.00	1	5.083204	0.7428050	3.868154	0.7393295
148.5900	42.524250	35.00	0	5.001191	0.4035872	3.750075	0.5343163
97.1550	17.066399	7.00	0	4.576308	-1.3537883	2.837112	-1.0507998
93.3450	13.182517	5.00	1	4.536302	-1.5192559	2.578892	-1.4991299
160.6550	48.505994	24.00	1	5.079259	0.7264888	3.881687	0.7628269
157.4800	45.869491	41.00	1	5.059298	0.6439284	3.825800	0.6657938
167.0050	52.900167	32.00	1	5.118024	0.8868243	3.968406	0.9133915
157.4800	47.570461	43.00	1	5.059298	0.6439284	3.862212	0.7290131
91.4400	12.927372	6.00	0	4.515683	-1.6045401	2.559347	-1.5330639
60.4520	5.669900	1.00	1	4.101850	-3.3162120	1.735172	-2.9640243
137.1600	28.916490	15.00	1	4.921148	0.0725195	3.364412	-0.1352835
152.4000	43.544832	63.00	0	5.026509	0.5083052	3.773791	0.5754938
152.4000	43.431434	21.00	0	5.026509	0.5083052	3.771183	0.5709664
81.2800	11.509897	1.00	1	4.397900	-2.0917070	2.443207	-1.7347093
109.2200	11.708343	2.00	0	4.693364	-0.8696262	2.460302	-1.7050294
71.1200	7.540967	1.00	1	4.264369	-2.6440113	2.020350	-2.4688872
89.2048	12.700576	3.00	0	4.490935	-1.7069020	2.541647	-1.5637945
67.3100	7.200773	1.00	0	4.209309	-2.8717461	1.974188	-2.5490353
85.0900	12.360382	1.00	1	4.443709	-1.9022325	2.514496	-1.6109349
69.8500	7.796112	1.00	0	4.246350	-2.7185383	2.053625	-2.4111145
161.9250	53.212012	55.00	0	5.087133	0.7590570	3.974284	0.9235965
152.4000	44.678812	38.00	0	5.026509	0.5083052	3.799499	0.6201294
88.9000	12.558829	3.00	1	4.487512	-1.7210588	2.530424	-1.5832810
90.1700	12.700576	3.00	1	4.501697	-1.6623892	2.541647	-1.5637945
71.7550	7.370870	1.00	0	4.273257	-2.6072454	1.997536	-2.5084988
83.8200	9.213587	1.00	0	4.428672	-1.9644312	2.220679	-2.1210697
159.3850	47.201918	28.00	1	5.071323	0.6936621	3.854434	0.7155096
142.2400	28.632995	16.00	0	4.957516	0.2229411	3.354560	-0.1523894
142.2400	31.666391	36.00	0	4.957516	0.2229411	3.455256	0.0224427
168.9100	56.443855	38.00	1	5.129366	0.9337375	4.033246	1.0259687
123.1900	20.014747	12.00	1	4.813728	-0.3717854	2.996469	-0.7741176
74.9300	8.504850	1.00	1	4.316554	-2.4281638	2.140637	-2.2600425
74.2950	8.306404	1.00	0	4.308044	-2.4633652	2.117027	-2.3010347
90.8050	11.623295	3.00	0	4.508714	-1.6333635	2.453011	-1.7176873
160.0200	55.791816	48.00	1	5.075299	0.7101080	4.021627	1.0057950
67.9450	7.966209	1.00	0	4.218699	-2.8329089	2.075209	-2.3736404
135.8900	27.215520	15.00	0	4.911846	0.0340436	3.303787	-0.2405419
158.1150	47.485413	45.00	1	5.063323	0.6605729	3.860423	0.7259063
85.0900	10.801160	3.00	1	4.443709	-1.9022325	2.379653	-1.8450535
93.3450	14.004653	3.00	0	4.536302	-1.5192559	2.639390	-1.3940911
152.4000	45.160753	38.00	0	5.026509	0.5083052	3.810228	0.6387576
155.5750	45.529297	21.00	0	5.047128	0.5935894	3.818356	0.6528689
154.3050	48.874538	50.00	0	5.038931	0.5596864	3.889257	0.7759688
156.8450	46.578229	41.00	1	5.055258	0.6272167	3.841133	0.6924155
120.0150	20.128145	13.00	0	4.787617	-0.4797847	3.002119	-0.7643083
114.3000	18.143680	8.00	1	4.738827	-0.6815876	2.898322	-0.9445237
83.8200	10.914558	3.00	1	4.428672	-1.9644312	2.390097	-1.8269203
156.2100	43.885026	30.00	0	5.051201	0.6104372	3.781573	0.5890054
137.1600	27.158821	12.00	1	4.921148	0.0725195	3.301702	-0.2441629
114.3000	19.050864	7.00	1	4.738827	-0.6815876	2.947112	-0.8598127
93.9800	13.834556	4.00	0	4.543082	-1.4912142	2.627169	-1.4153081
168.2750	56.046962	21.00	1	5.125599	0.9181588	4.026190	1.0137170
147.9550	40.086193	38.00	0	4.996908	0.3858736	3.691032	0.4318048
139.7000	26.563482	15.00	1	4.939497	0.1484141	3.279537	-0.2826456
157.4800	50.802304	19.00	0	5.059298	0.6439284	3.927942	0.8431352
76.2000	9.213587	1.00	1	4.333361	-2.3586473	2.220679	-2.1210697
66.0400	7.569317	1.00	1	4.190261	-2.9505321	2.024103	-2.4623723
160.7000	46.300000	31.00	1	5.079539	0.7276472	3.835142	0.6820133
114.3000	19.419407	8.00	0	4.738827	-0.6815876	2.966273	-0.8265456
146.0500	37.903281	16.00	1	4.983949	0.3322727	3.635038	0.3345857
161.2900	49.356479	21.00	1	5.083204	0.7428050	3.899069	0.7930056
69.8500	7.314171	0.00	0	4.246350	-2.7185383	1.989814	-2.5219061
133.9850	28.151053	13.00	1	4.897728	-0.0243499	3.337585	-0.1818618
67.9450	7.824462	0.00	1	4.218699	-2.8329089	2.057255	-2.4048123
150.4950	44.111822	50.00	0	5.013930	0.4562776	3.786728	0.5979550
163.1950	51.029100	39.00	1	5.094946	0.7913707	3.932396	0.8508690
148.5900	40.766581	44.00	1	5.001191	0.4035872	3.707863	0.4610268
148.5900	37.563088	36.00	0	5.001191	0.4035872	3.626022	0.3189321
161.9250	51.596090	36.00	1	5.087133	0.7590570	3.943446	0.8700541
153.6700	44.820560	18.00	0	5.034807	0.5426302	3.802667	0.6256291
68.5800	8.022908	0.00	0	4.228001	-2.7944329	2.082301	-2.3613266
151.1300	43.403084	58.00	0	5.018140	0.4736930	3.770531	0.5698327
163.8300	46.719976	58.00	1	5.098829	0.8074334	3.844172	0.6976912
153.0350	39.547553	33.00	0	5.030667	0.5255033	3.677504	0.4083167
151.7650	34.784836	21.50	0	5.022333	0.4910353	3.549182	0.1855194
132.0800	22.792998	11.00	1	4.883408	-0.0835797	3.126453	-0.5484351
156.2100	39.292407	26.00	1	5.051201	0.6104372	3.671031	0.3970789
140.3350	37.449689	22.00	0	4.944032	0.1671721	3.622998	0.3136827
158.7500	48.676091	28.00	1	5.067331	0.6771506	3.885188	0.7689048
142.8750	35.606972	42.00	0	4.961970	0.2413650	3.572541	0.2260777
84.4550	9.383684	2.00	1	4.436219	-1.9332149	2.238973	-2.0893085
151.9428	43.714929	21.00	1	5.023504	0.4958781	3.777690	0.5822627
161.2900	48.194150	19.00	1	5.083204	0.7428050	3.875238	0.7516287
127.9906	29.852024	13.00	1	4.851957	-0.2136652	3.396253	-0.0800008
160.9852	50.972401	48.00	1	5.081312	0.7349812	3.931284	0.8489388
144.7800	43.998424	46.00	0	4.975215	0.2961490	3.784154	0.5934860
132.0800	28.292801	11.00	1	4.883408	-0.0835797	3.342607	-0.1731414
117.9830	20.354941	8.00	1	4.770541	-0.5504142	3.013324	-0.7448546
160.0200	48.194150	25.00	1	5.075299	0.7101080	3.875238	0.7516287
154.9400	39.179009	16.00	1	5.043038	0.5766727	3.668141	0.3920609
160.9852	46.691626	51.00	1	5.081312	0.7349812	3.843565	0.6966373
165.9890	56.415505	25.00	1	5.111922	0.8615846	4.032744	1.0250964
157.9880	48.591043	28.00	1	5.062519	0.6572493	3.883439	0.7658685
154.9400	48.222499	26.00	0	5.043038	0.5766727	3.875826	0.7526497
97.9932	13.295915	5.00	1	4.584898	-1.3182570	2.587457	-1.4842584
64.1350	6.662133	1.00	0	4.160990	-3.0715984	1.896440	-2.6840252
160.6550	47.485413	54.00	1	5.079259	0.7264888	3.860423	0.7259063
147.3200	35.550273	66.00	0	4.992607	0.3680837	3.570948	0.2233108
146.7000	36.600000	20.00	0	4.988390	0.3506399	3.600048	0.2738358
147.3200	48.959587	25.00	0	4.992607	0.3680837	3.890995	0.7789875
172.9994	51.255896	38.00	1	5.153288	1.0326826	3.936831	0.8585685
158.1150	46.521529	51.00	1	5.063323	0.6605729	3.839915	0.6903007
147.3200	36.967748	48.00	0	4.992607	0.3680837	3.610046	0.2911940
124.9934	25.117657	13.00	1	4.828261	-0.3116747	3.223571	-0.3798162
106.0450	16.272613	6.00	1	4.663863	-0.9916451	2.789483	-1.1334932
165.9890	48.647742	27.00	1	5.111922	0.8615846	3.884605	0.7678933
149.8600	38.045029	22.00	0	5.009702	0.4387886	3.638770	0.3410666
76.2000	8.504850	1.00	0	4.333361	-2.3586473	2.140637	-2.2600425
161.9250	47.286966	60.00	1	5.087133	0.7590570	3.856235	0.7186352
140.0048	28.349500	15.00	0	4.941677	0.1574286	3.344609	-0.1696655
66.6750	8.136306	0.00	0	4.199830	-2.9109515	2.096336	-2.3369580
62.8650	7.200773	0.00	1	4.140990	-3.1543240	1.974188	-2.5490353
163.8300	55.394923	43.00	1	5.098829	0.8074334	4.014488	0.9933996
147.9550	32.488527	12.00	1	4.996908	0.3858736	3.480887	0.0669442
160.0200	54.204244	27.00	1	5.075299	0.7101080	3.992759	0.9556735
154.9400	48.477645	30.00	1	5.043038	0.5766727	3.881103	0.7618119
152.4000	43.062891	29.00	0	5.026509	0.5083052	3.762662	0.5561705
62.2300	7.257472	0.00	0	4.130837	-3.1963156	1.982032	-2.5354177
146.0500	34.189497	23.00	0	4.983949	0.3322727	3.531919	0.1555467
151.9936	49.951819	30.00	0	5.023838	0.4972608	3.911059	0.8138228
157.4800	41.305222	17.00	1	5.059298	0.6439284	3.720989	0.4838170
55.8800	4.847765	0.00	0	4.023206	-3.6414909	1.578518	-3.2360117
60.9600	6.236890	0.00	1	4.110218	-3.2815998	1.830482	-2.7985436
151.7650	44.338618	41.00	0	5.022333	0.4910353	3.791856	0.6068588
144.7800	33.452410	42.00	0	4.975215	0.2961490	3.510124	0.1177061
118.1100	16.896302	7.00	0	4.771616	-0.5459643	2.827095	-1.0681913
78.1050	8.221355	3.00	0	4.358054	-2.2565152	2.106735	-2.3189034
160.6550	47.286966	43.00	1	5.079259	0.7264888	3.856235	0.7186352
151.1300	46.124637	35.00	0	5.018140	0.4736930	3.831347	0.6754247
121.9200	20.184844	10.00	0	4.803365	-0.4146473	3.004932	-0.7594244
92.7100	12.757275	3.00	1	4.529476	-1.5474890	2.546102	-1.5560607
153.6700	47.400364	75.50	1	5.034807	0.5426302	3.858630	0.7227938
147.3200	40.851630	64.00	0	4.992607	0.3680837	3.709947	0.4646452
139.7000	50.348712	38.00	1	4.939497	0.1484141	3.918973	0.8275635
157.4800	45.132404	24.20	0	5.059298	0.6439284	3.809601	0.6376673
91.4400	11.623295	4.00	0	4.515683	-1.6045401	2.453011	-1.7176873
154.9400	42.240755	26.00	1	5.043038	0.5766727	3.743386	0.5227027
143.5100	41.645415	19.00	0	4.966405	0.2597071	3.729191	0.4980582
83.1850	9.156889	2.00	1	4.421067	-1.9958849	2.214506	-2.1317872
158.1150	45.217453	43.00	1	5.063323	0.6605729	3.811483	0.6409360
147.3200	51.255896	38.00	0	4.992607	0.3680837	3.936831	0.8585685
123.8250	21.205426	10.00	1	4.818869	-0.3505199	3.054257	-0.6737846
88.9000	11.594945	3.00	1	4.487512	-1.7210588	2.450569	-1.7219272
160.0200	49.271431	23.00	1	5.075299	0.7101080	3.897344	0.7900112
137.1600	27.952607	16.00	0	4.921148	0.0725195	3.330510	-0.1941445
165.1000	51.199197	49.00	1	5.106551	0.8393729	3.935724	0.8566468
154.9400	43.856676	41.00	0	5.043038	0.5766727	3.780927	0.5878834
111.1250	17.690088	6.00	1	4.710656	-0.7981062	2.873004	-0.9884813
153.6700	35.521923	23.00	0	5.034807	0.5426302	3.570150	0.2219257
145.4150	34.246196	14.00	0	4.979592	0.3142503	3.533575	0.1584237
141.6050	42.885420	43.00	0	4.953042	0.2044349	3.758532	0.5490004
144.7800	32.545226	15.00	0	4.975215	0.2961490	3.482631	0.0699716
163.8300	46.776675	21.00	1	5.098829	0.8074334	3.845385	0.6997970
161.2900	41.872211	24.00	1	5.083204	0.7428050	3.734622	0.5074879
154.9000	38.200000	20.00	1	5.042780	0.5756047	3.642835	0.3481245
161.3000	43.300000	20.00	1	5.083266	0.7430614	3.768153	0.5657042
170.1800	53.637254	34.00	1	5.136857	0.9647200	3.982244	0.9374164
149.8600	42.977842	29.00	0	5.009702	0.4387886	3.760685	0.5527381
123.8250	21.545620	11.00	1	4.818869	-0.3505199	3.070172	-0.6461517
85.0900	11.424848	3.00	0	4.443709	-1.9022325	2.435791	-1.7475863
160.6550	39.774349	65.00	1	5.079259	0.7264888	3.683222	0.4182452
154.9400	43.346385	46.00	0	5.043038	0.5766727	3.769223	0.5675632
106.0450	15.478827	8.00	0	4.663863	-0.9916451	2.739473	-1.2203229
126.3650	21.914164	15.00	1	4.839175	-0.2665345	3.087133	-0.6167041
166.3700	52.673371	43.00	1	5.114214	0.8710676	3.964110	0.9059318
148.2852	38.441922	39.00	0	4.999137	0.3950941	3.649149	0.3590855
124.4600	19.277660	12.00	0	4.823984	-0.3293631	2.958947	-0.8392653
89.5350	11.113004	3.00	1	4.494630	-1.6916199	2.408116	-1.7956360
101.6000	13.494362	4.00	0	4.621043	-1.1687545	2.602272	-1.4585360
151.7650	42.807745	43.00	0	5.022333	0.4910353	3.756719	0.5458528
148.5900	35.890467	70.00	0	5.001191	0.4035872	3.580472	0.2398464
153.6700	44.225220	26.00	0	5.034807	0.5426302	3.789295	0.6024126
53.9750	4.252425	0.00	0	3.988521	-3.7849551	1.447489	-3.4635073
146.6850	38.073378	48.00	0	4.988287	0.3502170	3.639515	0.3423599
56.5150	5.159609	0.00	0	4.034506	-3.5947543	1.640861	-3.1277696
100.9650	14.316498	5.00	1	4.614774	-1.1946865	2.661412	-1.3558542
121.9200	23.218241	8.00	1	4.803365	-0.4146473	3.144938	-0.5163411
81.5848	10.659412	3.00	0	4.401643	-2.0762255	2.366443	-1.8679895
154.9400	44.111822	44.00	1	5.043038	0.5766727	3.786728	0.5979550
156.2100	44.026773	33.00	0	5.051201	0.6104372	3.784798	0.5946043
132.7150	24.975910	15.00	1	4.888204	-0.0637420	3.217912	-0.3896421
125.0950	22.594552	12.00	0	4.829073	-0.3083140	3.117709	-0.5636177
101.6000	14.344847	5.00	0	4.621043	-1.1687545	2.663391	-1.3524195
160.6550	47.882306	41.00	1	5.079259	0.7264888	3.868746	0.7403577
146.0500	39.405805	37.40	0	4.983949	0.3322727	3.673913	0.4020825
132.7150	24.777463	13.00	0	4.888204	-0.0637420	3.209935	-0.4034924
87.6300	10.659412	6.00	0	4.473123	-1.7805726	2.366443	-1.8679895
156.2100	41.050076	53.00	1	5.051201	0.6104372	3.714793	0.4730589
152.4000	40.823280	49.00	0	5.026509	0.5083052	3.709252	0.4634399
162.5600	47.031821	27.00	0	5.091047	0.7752454	3.850824	0.7092416
114.9350	17.519991	7.00	1	4.744367	-0.6586726	2.863343	-1.0052567
67.9450	7.229122	1.00	0	4.218699	-2.8329089	1.978118	-2.5422132
142.8750	34.246196	31.00	0	4.961970	0.2413650	3.533575	0.1584237
76.8350	8.022908	1.00	1	4.341660	-2.3243223	2.082301	-2.3613266
145.4150	31.127751	17.00	1	4.979592	0.3142503	3.438100	-0.0073445
162.5600	52.163080	31.00	1	5.091047	0.7752454	3.954375	0.8890295
156.2100	54.062497	21.00	0	5.051201	0.6104372	3.990141	0.9511272
71.1200	8.051258	0.00	1	4.264369	-2.6440113	2.085828	-2.3552023
158.7500	52.531624	68.00	1	5.067331	0.6771506	3.961415	0.9012532

## [1] 1.609438

## [1] 0.7430039

## [1] 0.9836003

##          Approximated Approximator
## Island 1    1.0024795   -0.5377298
## Island 2   -0.4136578    0.7218202
## Island 3   -0.1759094    0.7218202

Notice that each row sums to 1, all the birds. This problem has two parts. It is not computationally complicated. But it is conceptually tricky. First, compute the entropy of each island’s bird distribution. Interpret these entropy values. Second, use each island’s bird distribution to predict the other two. This means to compute the KL divergence of each island from the others, treating each island as if it were a statistical model of the other islands. You should end up with 6 different KL divergence values. Which island predicts the others best? Why?

#Island 1 is the best predictor for orger islands.

7-4. Recall the marriage, age, and happiness collider bias example from Chapter 6. Run models m6.9 and m6.10 again (page 178). Compare these two models using WAIC (or PSIS, they will produce identical results). Which model is expected to make better predictions? Which model provides the correct causal inference about the influence of age on happiness? Can you explain why the answers to these two questions disagree?

d <- sim_happiness(seed = 1977, N_years = 1000)
## R code 6.22
d2 <- d[d$age > 17, ] # only adults
d2$A <- (d2$age - 18) / (65 - 18)
## R code 6.23
d2$mid <- d2$married + 1
m6.9 <- quap(
  alist(
    happiness ~ dnorm(mu, sigma),
    mu <- a[mid] + bA * A,
    a[mid] ~ dnorm(0, 1),
    bA ~ dnorm(0, 2),
    sigma ~ dexp(1)
  ),
  data = d2
)
## R code 6.24
m6.10 <- quap(
  alist(
    happiness ~ dnorm(mu, sigma),
    mu <- a + bA * A,
    a ~ dnorm(0, 1),
    bA ~ dnorm(0, 2),
    sigma ~ dexp(1)
  ),
  data = d2
)
## Comparison
compare(m6.9, m6.10)

##           WAIC       SE    dWAIC      dSE    pWAIC       weight
## m6.9  2713.971 37.54465   0.0000       NA 3.738532 1.000000e+00
## m6.10 3101.906 27.74379 387.9347 35.40032 2.340445 5.768312e-85

#Based on this, it appears that mc.9 does a better job at predicting as it contains both age and marriage status.
#However, m6.10 is better as mc.9 fails to find casuality.

7-5. Revisit the urban fox data, data(foxes), from the previous chapter’s practice problems. Use WAIC or PSIS based model comparison on five different models, each using weight as the outcome, and containing these sets of predictor variables:

• avgfood + groupsize + area
• avgfood + groupsize
• groupsize + area
• avgfood
• area

Can you explain the relative differences in WAIC scores, using the fox DAG from the previous chapter? Be sure to pay attention to the standard error of the score differences (dSE).

# data loading and prepping
data(foxes)
data = foxes
data$area = scale( data$area )
data$food = scale( data$avgfood )
data$groupsize = scale( data$groupsize )
data$weight = scale( data$weight)

## Model for`avgfood + groupsize + area`
m1 = quap(
  alist(
    weight ~ dnorm(mu, sigma),
    mu <- a + bFood * avgfood + bGroup * groupsize + bArea * area,
    a ~ dnorm(0, .2),
    c(bFood, bGroup, bArea) ~ dnorm(0, 5),
    sigma ~ dexp(1)
  ),
  data = data
)
# Model for`avgfood + groupsize`
m2 = quap(
  alist(
    weight ~ dnorm(mu, sigma),
    mu <- a + bFood * avgfood + bGroup * groupsize,
    a ~ dnorm(0, .2),
    c(bFood, bGroup) ~ dnorm(0, 5),
    sigma ~ dexp(1)
  ),
  data = data
)
# Model for`groupsize + area`
m3 = quap(
  alist(
    weight ~ dnorm(mu, sigma),
    mu <- a + bGroup * groupsize + bArea * area,
    a ~ dnorm(0, .2),
    c(bGroup, bArea) ~ dnorm(0, 5),
    sigma ~ dexp(1)
  ),
  data = data
)
# `Model for avgfood`
m4 = quap(
  alist(
    weight ~ dnorm(mu, sigma),
    mu <- a + bFood * avgfood,
    a ~ dnorm(0, .2),
    bFood ~ dnorm(0, 5),
    sigma ~ dexp(1)
  ),
  data = data
)
# Model for `area`
m5 = quap(
  alist(
    weight ~ dnorm(mu, sigma),
    mu <- a + bArea * area,
    a ~ dnorm(0, .2),
    bArea ~ dnorm(0, 5),
    sigma ~ dexp(1)
  ),
  data = data
)
## Comparison
(model.compare=compare(m1, m2, m3, m4, m5))

##        WAIC       SE     dWAIC       dSE    pWAIC      weight
## m1 323.7983 16.03770 0.0000000        NA 3.961414 0.518084659
## m3 324.0666 16.18771 0.2682736 0.4015293 3.945774 0.453049710
## m2 330.4839 14.82790 6.6855673 6.5270897 3.105662 0.018308309
## m4 332.3905 13.78571 8.5922089 7.4951036 1.946376 0.007057100
## m5 333.7929 13.79707 9.9946241 7.4644674 2.684646 0.003500223

#DAG
library(dagitty)

library(tidygraph)
quest <- dagitty( "dag {area -> avgfood
    avgfood -> groupsize
    avgfood -> weight
    groupsize -> weight
}")

coordinates (quest) <- list(x=c(area=1,avgfood=0,groupsize=2, weight=1),
                          y=c(area=0, avgfood=1, groupsize=1, weight=2))

drawdag(quest, lwd = 2)

#Thus from the above, we can see that WAIC scores all fall well within the 99% intervals of the differences. # Models m1, m2, and m3 are mostly identical in their out-of-sample deviance.All three models use groupsize and two of them use area and two use avgfood. From DAG, we can see that the effect of area while adjusting for groupsize is the same as the effect of avgfood while adjusting for groupsize, because the effect of area is routed entirely through avgfood.

#Model m4 and m5 are nearly identical because these two only contain area or avgfood in isolation and all information of area onto weight must pass through avgfood.