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.
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Each question is worth 5 points.
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.
library(ggplot2)
library(tidyverse)
library(rethinking)
# All models must be fit to exactly the same observations because the number of observations directly effects the value of the information criterion and deviance determines information criteria. The 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 could calculate WAIC for models fit to increasingly small sub samples of the same data. The information criteria should decrease alongside the sample size. Howell1 data set is used as it is a large data set.
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 1819.807 28.41008 0.0000 NA 3.274282 1.000000e+00
## m_400 2449.790 31.32177 629.9824 46.51640 3.134012 1.588796e-137
## m_500 3070.801 35.33610 1250.9935 53.30373 3.270270 2.239865e-2727-2. What happens to the effective number of parameters, as measured by PSIS or WAIC, as a prior becomes more concentrated? Why? Perform some simulations.
# The effective number of parameters is the penalty term of our regularisation approaches. When priors become more concentrated on certain prior assumptions, the effective number of parameters generally decreases because there is variance in the log-likelihoods for each observation in the training data WAIC. More concentrated priors limit this likelihood and subsequent measure of variance, therefore reducing it.
# In terms of PSIS, effective number of parameters displays the flexibility of the model. Increasing concentrated priors decreases the flexibility of the model.
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)
m1 <- 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
)
m2 <- 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(m1, refresh = 0)## WAIC lppd penalty std_err
## 1 -103.0982 55.60892 4.059833 36.4611WAIC(m2, refresh = 0)## WAIC lppd penalty std_err
## 1 -102.8002 55.61789 4.217797 36.56129PSIS(m1)## PSIS lppd penalty std_err
## 1 -102.2317 51.11586 4.586719 36.81524PSIS(m2)## PSIS lppd penalty std_err
## 1 -102.7351 51.36756 4.299613 36.653577-3. 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?
##First Island
p1 <- c(0.2, 0.2, 0.2, 0.2, 0.2)
-sum(p1 * log(p1))## [1] 1.609438##second island
p2 <- c(0.8, 0.1, 0.05, 0.025, 0.025)
-sum(p2 * log(p2))## [1] 0.7430039##third island
p3 <- c(0.05, 0.15, 0.7, 0.05, 0.05)
-sum(p3 * log(p3))## [1] 0.9836003# There is an uncertainty in data of Entropy and a positive relationship between the value and the uncertainties: the higher the value of entropy, the more uncertain is the probability density distribution.
# predict the other two bird distribution
# mean: Island 2 and 3
(m <- apply(cbind(p2, p3), 1, mean))## [1] 0.4250 0.1250 0.3750 0.0375 0.0375# divergence of Island 1 from combined Islands 2 & 3
(D_pq <- sum(p1 * log(p1 / m)))## [1] 0.4871152# divergence of combined Islands 2 & 3 from Island 1
(D_qp <- sum(m * log(m / p1)))## [1] 0.3717826# When we combine knowledge of Islands 2 & 3 to approximate Island 1, we bring in an additional 0.49 of uncertainty. On the other hand, by using our knowledge of Island 1 to approximate the combination of Islands 2 & 3, we add an additional 0.37 of uncertainty.
# repeating for the other two target islands
output <- data.frame(
Approximated = D_pq,
Approximator = D_qp
)
# when target is Island 2
m <- apply(cbind(p1, p3), 1, mean)
D_pq <- sum(p2 * log(p2 / m))
D_qp <- sum(m * log(m / p2))
output <- rbind(output, c(D_pq, D_qp))
# when target is 3
m <- apply(cbind(p1, p2), 1, mean)
D_pq <- sum(p3 * log(p3 / m))
D_qp <- sum(m * log(m / p3))
output <- rbind(output, c(D_pq, D_qp))
# output
rownames(output) <- c("Island 1", "Island 2", "Island 3")
output## Approximated Approximator
## Island 1 0.4871152 0.3717826
## Island 2 1.2387437 1.2570061
## Island 3 1.0097143 1.11840607-4. 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?
# 6.21
d <- sim_happiness(seed = 1977, N_years = 1000)
# 6.22
d2 <- d[d$age > 17, ]
d2$A <- (d2$age - 18) / (65 - 18)
# 6.23
d2$mid <- d2$married + 1
m6.9 <- quap(
alist(
happiness ~ dnorm(mu, sigma),
mu <- a[mid] + bA * A,
a[mid] ~ dnorm(0, 1),
bA ~ dnorm(0, 2),
sigma ~ dexp(1)
),
data = d2
)
# 6.24
m6.10 <- quap(
alist(
happiness ~ dnorm(mu, sigma),
mu <- a + bA * A,
a ~ dnorm(0, 1),
bA ~ dnorm(0, 2),
sigma ~ dexp(1)
),
data = d2
)
# m6.9 vs m6.10
compare(m6.9, m6.10)## WAIC SE dWAIC dSE pWAIC weight
## m6.9 2713.971 37.54465 0.0000 NA 3.738532 1.000000e+00
## m6.10 3101.906 27.74379 387.9347 35.40032 2.340445 5.768312e-85# Model m6.9 make better predictions and Model m6.10 provides the correct causal inference about the influence of age on happiness.7-5. Revisit the urban fox data, data(foxes), from the previous chapter’s practice problems. Use WAIC or PSIS based model comparison on five different models, each using weight as the outcome, and containing these sets of predictor variables:
Can you explain the relative differences in WAIC scores, using the fox DAG from the previous chapter? Be sure to pay attention to the standard error of the score differences (dSE).
data(foxes)
x <- foxes
x$area <- scale(x$area)
x$avgfood <- scale(x$avgfood)
x$weight <- scale(x$weight)
x$groupsize <- scale(x$groupsize)
#models
# 1: avgfood + groupsize + area
bx_1 <- 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 = x
)
# 2: avgfood + groupsize
bx_2 <- 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 = x
)
# 3: groupsize + area
bx_3 <- 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 = x
)
# 4: avgfood
bx_4 <- quap(
alist(
weight ~ dnorm(mu, sigma),
mu <- a + bFood * avgfood,
a ~ dnorm(0, .2),
bFood ~ dnorm(0, 5),
sigma ~ dexp(1)
),
data = x
)
# 5: area
bx_5 <- quap(
alist(
weight ~ dnorm(mu, sigma),
mu <- a + bArea * area,
a ~ dnorm(0, .2),
bArea ~ dnorm(0, 5),
sigma ~ dexp(1)
),
data = x
)
# compare models
compare(bx_1, bx_2, bx_3, bx_4, bx_5)## WAIC SE dWAIC dSE pWAIC weight
## bx_1 323.3416 16.88472 0.0000000 NA 5.187854 0.410865167
## bx_2 323.9809 16.81807 0.6393574 3.894043 4.130907 0.298445216
## bx_3 324.0666 16.18771 0.7250103 4.207249 3.945774 0.285933698
## bx_4 333.5084 13.79238 10.1668387 8.659625 2.454047 0.002546821
## bx_5 333.7929 13.79707 10.4513608 8.704578 2.684646 0.002209099# There is not much differences between model 1, 2, 3 but these 3 models are significantly different from model 4 and 5. For the standard errors, model 1, 2, and 3 are distinct from model 4 and 5. Both Model 4 and Model 5 exclude groupsize. However, because area and avgfood largely contain the same data, their WAIC estimations are comparable.