Assignment #6
Chapter 7
Ran Ding
2022-10-28
Chapter 7 - Ulysses’ Compass
This week 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.
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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 simulations.
data(Howell1)
d = Howell1[complete.cases(Howell1), ]
d_500 = d[sample(1:nrow(d), size = 500, replace = FALSE), ]
d_400 = d[sample(1:nrow(d), size = 400, replace = FALSE), ]
d_300 = d[sample(1:nrow(d), size = 300, replace = FALSE), ]
m_500 = quap(
alist(
height ~ dnorm(mu, sigma),
mu <- a + b * log(weight)
),
data = d_500,
start = list(a = mean(d_500$height), b = 0, sigma = sd(d_500$height))
)
m_400 = quap(
alist(
height ~ dnorm(mu, sigma),
mu <- a + b * log(weight)
),
data = d_400,
start = list(a = mean(d_400$height), b = 0, sigma = sd(d_400$height))
)
m_300 = quap(
alist(
height ~ dnorm(mu, sigma),
mu <- a + b * log(weight)
),
data = d_300,
start = list(a = mean(d_300$height), b = 0, sigma = sd(d_300$height))
)
(model.compare = compare(m_500, m_400, m_300))## WAIC SE dWAIC dSE pWAIC weight
## m_300 1846.918 30.08244 0.0000 NA 3.464280 1.000000e+00
## m_400 2442.033 33.49658 595.1149 49.20764 3.325174 5.921665e-130
## m_500 3068.314 35.61066 1221.3967 54.13279 3.281983 5.985294e-266#A model with more observations tend to have higher deviance and worse
#accuracy according to the information criterion. According to the 3 models
#with 200, 300 and 400 observations, it shows that the WAIC increases
#along with the number of observations.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 simulations (at least 4).
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 = quap(
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 = quap(
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 -102.6587 55.64129 4.311925 36.48315WAIC(m_narrow, refresh = 0)## WAIC lppd penalty std_err
## 1 -102.7126 55.63829 4.281981 36.67874#For PSIS: the model becomes less flexible as the prior become more
#concentrated on certain prior assumptions. And for WAIC: there is a
#measure of the variance in the log-likelihood for each observation in
#the training date. 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 |
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?
p1 <- c(0.2, 0.2, 0.2, 0.2, 0.2)
-sum(p1 * log(p1))## [1] 1.609438p2 <- c(0.8, 0.1, 0.05, 0.025, 0.025)
-sum(p2 * log(p2))## [1] 0.7430039p3 <- c(0.05, 0.15, 0.7, 0.05, 0.05)
-sum(p3 * log(p3))## [1] 0.9836003D_kl <- function(p, q) sum(p * (log(p) - log(q)))
D_kl(p1, p2)## [1] 0.9704061D_kl(p1, p3)## [1] 0.6387604D_kl(p2, p1)## [1] 0.866434D_kl(p2, p3)## [1] 2.010914D_kl(p3, p1)## [1] 0.6258376D_kl(p3, p2)## [1] 1.838845#Island 1 the most tough to predict due to they has five bird species evenly #distributed. And Island 2 is much easier to predict because the vast majority
#is species A, leading to the lowest number of entropy. and also, Island 1has
#the highest entropy based on the KL distance is shorter. 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)
d2 <- d[d$age>17,] # only adults
d2$A <- (d2$age - 18) / (65 - 18)
d2$mid <- d2$married + 1
# m6.9
m6.9 <- quap(
alist(
happiness ~ dnorm(mu, sigma),
mu <- a[mid] + bA*A,
a[mid] ~ dnorm(0, 1),
bA ~ dnorm(0, 2),
sigma ~ dexp(1)
) , data=d2)
precis(m6.9, depth = 2)## mean sd 5.5% 94.5%
## a[1] -0.2350877 0.06348986 -0.3365568 -0.1336186
## a[2] 1.2585517 0.08495989 1.1227694 1.3943340
## bA -0.7490274 0.11320112 -0.9299447 -0.5681102
## sigma 0.9897080 0.02255800 0.9536559 1.0257600m6.10 <- quap(
alist(
happiness ~ dnorm(mu, sigma),
mu <- a + bA*A,
a ~ dnorm(0, 1),
bA ~ dnorm(0, 2),
sigma ~ dexp(1)
) , data=d2 )
precis(m6.10)## mean sd 5.5% 94.5%
## a 1.649248e-07 0.07675015 -0.1226614 0.1226617
## bA -2.728620e-07 0.13225976 -0.2113769 0.2113764
## sigma 1.213188e+00 0.02766080 1.1689803 1.2573949compare(m6.9, m6.10)## WAIC SE dWAIC dSE pWAIC weight
## m6.9 2713.971 37.54465 0.0000 NA 3.738532 1.000000e+00
## m6.10 3101.906 27.74379 387.9347 35.40032 2.340445 5.768312e-85#m6.9 performs better for prediction. and m6.10 provides reflects the
#true causal relationship between age and happiness.7-5. Revisit the urban fox data, data(foxes), from the previous chapter’s practice problems. Use WAIC or PSIS based model comparison on five different models, each using weight as the outcome, and containing these sets of predictor variables:
- avgfood + groupsize + area
- avgfood + groupsize
- groupsize + area
- avgfood
- area
Can you explain the relative differences in WAIC scores, using the fox DAG from the previous chapter? Be sure to pay attention to the standard error of the score differences (dSE).
data(foxes)
fox_dat <- foxes[,-1] %>%
as_tibble() %>%
mutate(across(everything(), standardize))
set.seed(1234)
N<-100
a <- rnorm(N, 0, 0.3)
b <- rnorm(N, 0, 0.5)
plot(NULL, xlim=range(fox_dat$area),ylim=c(-4,4))
xbar<-mean(fox_dat$area)
for(i in 1:N) curve(a[i] + b[i]*(x- xbar),
from=min(fox_dat$area), to = max(fox_dat$area), add = TRUE,
col = col.alpha("blue",0.2)
)
m1 = quap(
alist(
weight ~ dnorm(mu, sigma),
mu <- a + b_avgfood*avgfood + b_groupsize*groupsize + b_area*area,
a ~ dnorm(0,3),
b_avgfood ~ dnorm(0,5),
b_groupsize ~ dnorm(0,5),
b_area ~ dnorm(0,5),
sigma ~ dexp(1)
), data = fox_dat
)
m2 = quap(
alist(
weight ~ dnorm(mu, sigma),
mu <- a + b_avgfood*avgfood + b_groupsize*groupsize,
a ~ dnorm(0,3),
b_avgfood ~ dnorm(0,5),
b_groupsize ~ dnorm(0,5),
sigma ~ dexp(1)
), data = fox_dat
)
m3 = quap(
alist(
weight ~ dnorm(mu, sigma),
mu <- a + b_area*area + b_groupsize*groupsize,
a ~ dnorm(0,3),
b_area ~ dnorm(0,5),
b_groupsize ~ dnorm(0,5),
sigma ~ dexp(1)
), data = fox_dat
)
m4 = quap(
alist(
weight ~ dnorm(mu, sigma),
mu <- a + b_avgfood*avgfood,
a ~ dnorm(0,3),
b_avgfood ~ dnorm(0,5),
sigma ~ dexp(1)
), data = fox_dat
)
m5 = quap(
alist(
weight ~ dnorm(mu, sigma),
mu <- a + b_area*area,
a ~ dnorm(0,3),
b_area ~ dnorm(0,5),
sigma ~ dexp(1)
), data = fox_dat
)
compare(m1, m2, m3, m4, m5)## WAIC SE dWAIC dSE pWAIC weight
## m1 323.7790 16.97973 0.0000000 NA 5.394304 0.395584053
## m2 324.0854 16.82870 0.3063355 3.890751 4.194372 0.339405493
## m3 324.6152 16.13735 0.8361358 4.277576 4.220335 0.260419919
## m4 333.9423 13.86435 10.1632843 8.712388 2.663152 0.002456460
## m5 334.2237 13.90559 10.4446596 8.760104 2.886085 0.002134075plot(compare(m1, m2, m3, m4, m5))
#M1, m2, and m3 have dWAIC that are close to each other, and then,
#M4 and m5 have dWAIC that are close to each other.