ABB: Approximate Bayesian Bootstrap
In LaplacesDemon: Complete Environment for Bayesian Inference
Description
Usage
Arguments
Details
Value
Author(s)
References
See Also
Examples
Description
This function performs multiple imputation (MI) with the Approximate
Bayesian Bootstrap (ABB) of Rubin and Schenker (1986).
Usage
1
ABB(X, K=1)
Arguments
X
This is a vector or matrix of data that must include both
observed and missing values. When X is a matrix, missing
values must occur somewhere in the set, but are not required to
occur in each variable.
K
This is the number of imputations.
Details
The Approximate Bayesian Bootstrap (ABB) is a modified form of the
BayesianBootstrap (Rubin, 1981) that is used for
multiple imputation (MI). Imputation is a family of statistical
methods for replacing missing values with estimates. Introduced by
Rubin and Schenker (1986) and Rubin (1987), MI is a family of
imputation methods that includes multiple estimates, and therefore
includes variability of the estimates.
The data, X, are assumed to be independent and
identically distributed (IID), contain both observed and missing
values, and its missing values are assumed to be ignorable (meaning
enough information is available in the data that the missingness
mechanism can be ignored, if the information is used properly) and
Missing Completely At Random (MCAR). When ABB is used in
conjunction with a propensity score (described below), missing values
may be Missing At Random (MAR).
ABB does not add auxiliary information, but performs imputation
with two sampling (with replacement) steps. First,
X_star_obs is sampled from
X_obs. Then,
X_star_mis is sampled from
X_star_obs. The result is a sample of
the posterior predictive distribution of
(X_mis|X_obs). The first
sampling step is also known as hotdeck imputation, and the second
sampling step changes the variance. Since auxiliary information is not
included, ABB is appropriate for missing values that are
ignorable and MCAR.
Auxiliary information may be included in the process of imputation by
introducing a propensity score (Rosenbaum and Rubin, 1983; Rosenbaum
and Rubin, 1984), which is an estimate of the probability of
missingness. The propensity score is often the result of a binary
logit model, where missingness is predicted as a function of other
variables. The propensity scores are discretized into quantile-based
groups, usually quintiles. Each quintile must have both observed and
missing values. ABB is applied to each quintile. This is called
within-class imputation. It is assumed that the missing mechanism
depends only on the variables used to estimate the propensity score.
With K=1, ABB may be used in MCMC, such as in
LaplacesDemon, more commonly along with a propensity
score for missingness. MI is performed, despite K=1, because
imputation occurs at each MCMC iteration. The practical advantage of
this form of imputation is the ease with which it may be
implemented. For example, full-likelihood imputation should perform
better, but requires a chain to be updated for each missing value.
An example of a limitation of ABB with propensity scores is to
consider imputing missing values of income from age in a context where
age and income have a positive relationship, and where the highest
incomes are missing systematically. ABB with propensity scores
should impute these highest missing incomes given the highest observed
ages, but is unable to infer beyond the observed data.
ABB has been extended (Parzen et al., 2005) to reduce bias, by
introducing a correction factor that is applied to the MI variance
estimate. This correction may be applied to output from ABB.
Value
This function returns a list with K components, one for each set
of imputations. Each component contains a vector of imputations equal
in length to the number of missing values in the data.
ABB does not currently return the mean of the imputations, or
the between-imputation variance or within-imputation variance.
Author(s)
Statisticat, LLC software@bayesian-inference.com
References
Parzen, M., Lipsitz, S.R., and Fitzmaurice, G.M. (2005). "A Note on
Reducing the Bias of the Approximate Bayesian Bootstrap Imputation
Variance Estimator". Biometrika, 92, 4, p. 971–974.
Rosenbaum, P.R. and Rubin, D.B. (1983). "The Central Role of the
Propensity Score in Observational Studies for Causal
Effects". Biometrika, 70, p. 41–55.
Rosenbaum, P.R. and Rubin, D.B. (1984). "Reducing Bias in
Observational Studies Using Subclassification in the Propensity
Score". Journal of the American Statistical Association, 79,
p. 516–524.
Rubin, D.B. (1981). "The Bayesian Bootstrap". Annals of
Statistics, 9, p. 130–134.
Rubin, D.B. (1987). "Multiple Imputation for Nonresponse in
Surveys". John Wiley and Sons: New York, NY.
Rubin, D.B. and Schenker, N. (1986). "Multiple Imputation for Interval
Estimation from Simple Random Samples with Ignorable
Nonresponse". Journal of the American Statistical Association,
81, p. 366–374.
See Also
BayesianBootstrap,
LaplacesDemon, and
MISS.
Examples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
library(LaplacesDemon)
### Create Data
J <- 10 #Number of variables
m <- 20 #Number of missings
N <- 50 #Number of records
mu <- runif(J, 0, 100)
sigma <- runif(J, 0, 100)
X <- matrix(0, N, J)
for (j in 1:J) X[,j] <- rnorm(N, mu[j], sigma[j])
### Create Missing Values
M1 <- rep(0, N*J)
M2 <- sample(N*J, m)
M1[M2] <- 1
M <- matrix(M1, N, J)
X <- ifelse(M == 1, NA, X)
### Approximate Bayesian Bootstrap
imp <- ABB(X, K=1)
### Replace Missing Values in X (when K=1)
X.imp <- X
X.imp[which(is.na(X.imp))] <- unlist(imp)
X.imp
Example output
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 27.4479555 29.199600 124.164591 15.717739 46.4573213 97.267618
[2,] 29.7202987 186.538101 92.081068 4.063159 -17.8225758 -69.847897
[3,] 27.7102275 172.476328 62.959453 38.397134 2.5222753 -66.297545
[4,] 30.2898962 144.505717 55.979924 30.569144 38.1825057 110.850882
[5,] 32.6615055 72.577123 -2.437707 -24.874004 15.0867408 -18.862785
[6,] 15.1286486 199.062121 80.104398 33.168335 53.6240223 18.394285
[7,] 23.4929578 161.352597 5.806499 17.426940 21.1272725 79.060892
[8,] 28.3520628 77.671417 22.094706 -30.047490 6.0864001 -130.325092
[9,] 23.9676012 90.789308 32.805576 28.184561 -0.3074967 -42.832358
[10,] 28.9983927 111.162010 19.208856 17.549106 -39.1188892 -60.660768
[11,] 22.8781955 105.832515 120.265183 11.162739 -7.8482632 37.766315
[12,] 22.2760626 2.738746 53.893065 9.129157 45.5780280 -67.922736
[13,] 30.0323401 111.437885 105.820958 17.789519 -26.5269174 198.608419
[14,] 19.0657360 272.156214 66.562827 37.040025 10.7333052 35.367762
[15,] 13.9645675 157.736451 50.749252 56.834614 -1.7319273 209.727687
[16,] 20.5266618 137.345760 41.770118 33.542036 -12.1856838 58.582751
[17,] 28.1587886 -27.000097 48.546865 67.179022 59.5749657 -68.978821
[18,] 13.4312542 125.025906 29.218355 -18.652384 37.6765175 42.702075
[19,] 29.0078983 308.884069 51.240279 40.452247 3.3057440 80.333011
[20,] 14.0845672 2.738746 21.650372 21.740888 6.6220089 5.102486
[21,] 29.1918309 143.234386 -17.229724 7.512198 -6.0600364 -59.174359
[22,] 23.1451737 197.849629 -99.714565 13.264285 59.1097207 -20.898235
[23,] 25.5641067 29.961728 108.081188 31.416002 8.0758554 140.320916
[24,] 23.2496397 54.991400 79.889326 -2.262989 15.9912708 39.797582
[25,] 23.0961825 107.181817 -44.580881 -8.224126 11.0309395 -69.329590
[26,] 19.0657360 137.359676 -33.821053 66.111474 1.2584269 -43.726575
[27,] 19.2486644 34.844003 95.464418 -16.376266 21.1272725 233.693935
[28,] 24.3073132 90.755741 73.239456 -3.790246 24.8479993 21.733777
[29,] 14.7127080 47.659096 32.961227 41.991572 0.4734992 112.689749
[30,] 28.4034987 81.452669 99.272310 41.775893 59.5749657 -64.767935
[31,] 29.6663420 22.022469 99.870095 3.901195 21.9394485 153.446898
[32,] 0.2491529 198.407970 34.203042 46.194061 -16.7614461 -52.516128
[33,] 9.8535637 82.009603 85.364648 42.825741 13.2215368 79.138244
[34,] 17.9240326 92.637437 -39.837170 38.947679 21.1272725 79.060892
[35,] 24.3073132 29.929573 76.605078 8.962915 21.3454831 142.433243
[36,] 18.5800289 161.596980 78.734786 12.434474 75.6143081 103.123017
[37,] 28.1123864 137.849599 74.615540 -23.230711 51.0217125 92.302873
[38,] 31.2620869 81.793627 127.159579 25.320183 -11.0778830 -29.949833
[39,] 31.7291438 -52.586129 117.295781 -16.376266 69.8248144 127.997160
[40,] 16.1994694 42.049347 21.530445 3.807450 -15.8310900 127.997160
[41,] 13.9645675 143.158376 6.759298 36.591977 38.6337505 92.294658
[42,] 23.1451737 69.894617 98.895276 -9.297184 57.0860362 -74.718073
[43,] 19.5531642 172.476328 78.413789 14.718388 18.7160679 191.194212
[44,] 26.4009745 126.172298 -24.402567 -9.490415 -7.7369947 146.354162
[45,] 18.8902893 120.472277 141.984615 -17.432359 52.9504965 59.878062
[46,] 30.5881537 11.056665 53.031043 40.534146 87.1705786 62.311624
[47,] 26.9130475 104.717969 18.277497 37.304100 31.1407464 31.225120
[48,] 29.6335603 135.205036 -15.448668 55.735403 46.6558196 50.523129
[49,] 18.7859944 13.038355 47.690324 29.023077 -7.8482632 91.405343
[50,] 33.7413311 150.532821 -29.048349 54.658151 18.7485356 30.750731
[,7] [,8] [,9] [,10]
[1,] 22.4289007 107.68202 17.958639 -82.244383
[2,] -8.1868126 120.33030 -7.329413 77.122542
[3,] 15.1397611 106.56311 115.613275 -212.310149
[4,] 45.5430687 95.59442 100.181790 90.031150
[5,] 0.4156687 156.58683 59.717157 47.980456
[6,] 19.9475395 75.31008 61.286691 -57.222372
[7,] 26.3063055 100.90410 138.368175 -62.939258
[8,] 8.5353477 86.78485 58.153845 81.256030
[9,] 0.1204387 93.00113 71.013100 71.455358
[10,] 25.9217179 71.47643 -127.120920 63.931470
[11,] 9.7208983 88.70718 146.996307 -21.242962
[12,] -14.2384536 84.36056 60.866701 96.777360
[13,] 17.5591015 59.83607 13.627680 -18.901654
[14,] 12.0622153 84.27482 159.953018 55.638429
[15,] -3.1101647 84.94025 144.773018 38.977132
[16,] 11.8300385 118.60344 36.070018 27.017961
[17,] 20.8116965 81.41027 64.781799 49.160260
[18,] 14.6183014 94.25617 113.574047 -57.544182
[19,] 22.3901535 76.83697 191.380516 -72.825478
[20,] 21.6923615 104.21552 56.704791 -116.918462
[21,] 21.5956954 93.56764 135.972581 107.595679
[22,] -22.3168661 115.86241 122.647198 74.041987
[23,] 40.6628785 68.13787 80.026350 163.615126
[24,] 3.9154278 118.99069 217.008376 57.561552
[25,] 61.5639579 116.29137 19.108007 -7.238832
[26,] -11.9322184 86.39193 148.430820 81.256030
[27,] 32.1041391 91.71934 102.768745 123.250030
[28,] 8.5505864 104.66646 3.083909 223.392412
[29,] 9.7208983 75.95391 103.222110 -82.244383
[30,] 26.2366645 88.91749 211.821205 40.158435
[31,] 26.6985288 87.96929 52.851359 55.307285
[32,] 25.3672527 85.67475 146.468591 180.430553
[33,] -3.8168491 104.44860 116.889394 57.561552
[34,] 26.2366645 101.04614 16.295902 198.685646
[35,] -19.1006010 129.71035 165.154132 -29.247418
[36,] 13.3832374 76.46695 28.017739 113.875986
[37,] 18.1779059 97.50721 217.008376 66.871640
[38,] 0.2791158 84.41508 138.918558 -11.550056
[39,] 0.1174765 107.98680 -72.421144 21.990018
[40,] 2.0703579 80.30584 58.517290 99.788700
[41,] 13.2223361 101.99992 49.269747 68.579903
[42,] -0.3529732 111.41867 133.343950 30.762506
[43,] 39.5945161 126.48166 303.388024 93.031029
[44,] 19.5473831 82.35203 18.245547 129.643615
[45,] 0.2791158 90.67254 -59.296976 -95.330802
[46,] 32.0097487 88.74652 64.572335 68.746163
[47,] 7.8119702 110.81246 173.398315 133.812425
[48,] 42.9507905 117.01701 170.005925 -7.789793
[49,] 15.9368831 117.49441 138.847640 4.320419
[50,] -4.5792146 96.68661 -83.377163 43.277173
LaplacesDemon documentation built on July 9, 2021, 5:07 p.m.
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Description Usage Arguments Details Value Author(s) References See Also Examples
Description
This function performs multiple imputation (MI) with the Approximate Bayesian Bootstrap (ABB) of Rubin and Schenker (1986).
Usage
1 | ABB(X, K=1)
|
Arguments
X |
This is a vector or matrix of data that must include both
observed and missing values. When |
K |
This is the number of imputations. |
Details
The Approximate Bayesian Bootstrap (ABB) is a modified form of the
BayesianBootstrap (Rubin, 1981) that is used for
multiple imputation (MI). Imputation is a family of statistical
methods for replacing missing values with estimates. Introduced by
Rubin and Schenker (1986) and Rubin (1987), MI is a family of
imputation methods that includes multiple estimates, and therefore
includes variability of the estimates.
The data, X, are assumed to be independent and
identically distributed (IID), contain both observed and missing
values, and its missing values are assumed to be ignorable (meaning
enough information is available in the data that the missingness
mechanism can be ignored, if the information is used properly) and
Missing Completely At Random (MCAR). When ABB is used in
conjunction with a propensity score (described below), missing values
may be Missing At Random (MAR).
ABB does not add auxiliary information, but performs imputation
with two sampling (with replacement) steps. First,
X_star_obs is sampled from
X_obs. Then,
X_star_mis is sampled from
X_star_obs. The result is a sample of
the posterior predictive distribution of
(X_mis|X_obs). The first
sampling step is also known as hotdeck imputation, and the second
sampling step changes the variance. Since auxiliary information is not
included, ABB is appropriate for missing values that are
ignorable and MCAR.
Auxiliary information may be included in the process of imputation by
introducing a propensity score (Rosenbaum and Rubin, 1983; Rosenbaum
and Rubin, 1984), which is an estimate of the probability of
missingness. The propensity score is often the result of a binary
logit model, where missingness is predicted as a function of other
variables. The propensity scores are discretized into quantile-based
groups, usually quintiles. Each quintile must have both observed and
missing values. ABB is applied to each quintile. This is called
within-class imputation. It is assumed that the missing mechanism
depends only on the variables used to estimate the propensity score.
With K=1, ABB may be used in MCMC, such as in
LaplacesDemon, more commonly along with a propensity
score for missingness. MI is performed, despite K=1, because
imputation occurs at each MCMC iteration. The practical advantage of
this form of imputation is the ease with which it may be
implemented. For example, full-likelihood imputation should perform
better, but requires a chain to be updated for each missing value.
An example of a limitation of ABB with propensity scores is to
consider imputing missing values of income from age in a context where
age and income have a positive relationship, and where the highest
incomes are missing systematically. ABB with propensity scores
should impute these highest missing incomes given the highest observed
ages, but is unable to infer beyond the observed data.
ABB has been extended (Parzen et al., 2005) to reduce bias, by
introducing a correction factor that is applied to the MI variance
estimate. This correction may be applied to output from ABB.
Value
This function returns a list with K components, one for each set of imputations. Each component contains a vector of imputations equal in length to the number of missing values in the data.
ABB does not currently return the mean of the imputations, or
the between-imputation variance or within-imputation variance.
Author(s)
Statisticat, LLC software@bayesian-inference.com
References
Parzen, M., Lipsitz, S.R., and Fitzmaurice, G.M. (2005). "A Note on Reducing the Bias of the Approximate Bayesian Bootstrap Imputation Variance Estimator". Biometrika, 92, 4, p. 971–974.
Rosenbaum, P.R. and Rubin, D.B. (1983). "The Central Role of the Propensity Score in Observational Studies for Causal Effects". Biometrika, 70, p. 41–55.
Rosenbaum, P.R. and Rubin, D.B. (1984). "Reducing Bias in Observational Studies Using Subclassification in the Propensity Score". Journal of the American Statistical Association, 79, p. 516–524.
Rubin, D.B. (1981). "The Bayesian Bootstrap". Annals of Statistics, 9, p. 130–134.
Rubin, D.B. (1987). "Multiple Imputation for Nonresponse in Surveys". John Wiley and Sons: New York, NY.
Rubin, D.B. and Schenker, N. (1986). "Multiple Imputation for Interval Estimation from Simple Random Samples with Ignorable Nonresponse". Journal of the American Statistical Association, 81, p. 366–374.
See Also
BayesianBootstrap,
LaplacesDemon, and
MISS.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 | library(LaplacesDemon)
### Create Data
J <- 10 #Number of variables
m <- 20 #Number of missings
N <- 50 #Number of records
mu <- runif(J, 0, 100)
sigma <- runif(J, 0, 100)
X <- matrix(0, N, J)
for (j in 1:J) X[,j] <- rnorm(N, mu[j], sigma[j])
### Create Missing Values
M1 <- rep(0, N*J)
M2 <- sample(N*J, m)
M1[M2] <- 1
M <- matrix(M1, N, J)
X <- ifelse(M == 1, NA, X)
### Approximate Bayesian Bootstrap
imp <- ABB(X, K=1)
### Replace Missing Values in X (when K=1)
X.imp <- X
X.imp[which(is.na(X.imp))] <- unlist(imp)
X.imp
|
Example output
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 27.4479555 29.199600 124.164591 15.717739 46.4573213 97.267618
[2,] 29.7202987 186.538101 92.081068 4.063159 -17.8225758 -69.847897
[3,] 27.7102275 172.476328 62.959453 38.397134 2.5222753 -66.297545
[4,] 30.2898962 144.505717 55.979924 30.569144 38.1825057 110.850882
[5,] 32.6615055 72.577123 -2.437707 -24.874004 15.0867408 -18.862785
[6,] 15.1286486 199.062121 80.104398 33.168335 53.6240223 18.394285
[7,] 23.4929578 161.352597 5.806499 17.426940 21.1272725 79.060892
[8,] 28.3520628 77.671417 22.094706 -30.047490 6.0864001 -130.325092
[9,] 23.9676012 90.789308 32.805576 28.184561 -0.3074967 -42.832358
[10,] 28.9983927 111.162010 19.208856 17.549106 -39.1188892 -60.660768
[11,] 22.8781955 105.832515 120.265183 11.162739 -7.8482632 37.766315
[12,] 22.2760626 2.738746 53.893065 9.129157 45.5780280 -67.922736
[13,] 30.0323401 111.437885 105.820958 17.789519 -26.5269174 198.608419
[14,] 19.0657360 272.156214 66.562827 37.040025 10.7333052 35.367762
[15,] 13.9645675 157.736451 50.749252 56.834614 -1.7319273 209.727687
[16,] 20.5266618 137.345760 41.770118 33.542036 -12.1856838 58.582751
[17,] 28.1587886 -27.000097 48.546865 67.179022 59.5749657 -68.978821
[18,] 13.4312542 125.025906 29.218355 -18.652384 37.6765175 42.702075
[19,] 29.0078983 308.884069 51.240279 40.452247 3.3057440 80.333011
[20,] 14.0845672 2.738746 21.650372 21.740888 6.6220089 5.102486
[21,] 29.1918309 143.234386 -17.229724 7.512198 -6.0600364 -59.174359
[22,] 23.1451737 197.849629 -99.714565 13.264285 59.1097207 -20.898235
[23,] 25.5641067 29.961728 108.081188 31.416002 8.0758554 140.320916
[24,] 23.2496397 54.991400 79.889326 -2.262989 15.9912708 39.797582
[25,] 23.0961825 107.181817 -44.580881 -8.224126 11.0309395 -69.329590
[26,] 19.0657360 137.359676 -33.821053 66.111474 1.2584269 -43.726575
[27,] 19.2486644 34.844003 95.464418 -16.376266 21.1272725 233.693935
[28,] 24.3073132 90.755741 73.239456 -3.790246 24.8479993 21.733777
[29,] 14.7127080 47.659096 32.961227 41.991572 0.4734992 112.689749
[30,] 28.4034987 81.452669 99.272310 41.775893 59.5749657 -64.767935
[31,] 29.6663420 22.022469 99.870095 3.901195 21.9394485 153.446898
[32,] 0.2491529 198.407970 34.203042 46.194061 -16.7614461 -52.516128
[33,] 9.8535637 82.009603 85.364648 42.825741 13.2215368 79.138244
[34,] 17.9240326 92.637437 -39.837170 38.947679 21.1272725 79.060892
[35,] 24.3073132 29.929573 76.605078 8.962915 21.3454831 142.433243
[36,] 18.5800289 161.596980 78.734786 12.434474 75.6143081 103.123017
[37,] 28.1123864 137.849599 74.615540 -23.230711 51.0217125 92.302873
[38,] 31.2620869 81.793627 127.159579 25.320183 -11.0778830 -29.949833
[39,] 31.7291438 -52.586129 117.295781 -16.376266 69.8248144 127.997160
[40,] 16.1994694 42.049347 21.530445 3.807450 -15.8310900 127.997160
[41,] 13.9645675 143.158376 6.759298 36.591977 38.6337505 92.294658
[42,] 23.1451737 69.894617 98.895276 -9.297184 57.0860362 -74.718073
[43,] 19.5531642 172.476328 78.413789 14.718388 18.7160679 191.194212
[44,] 26.4009745 126.172298 -24.402567 -9.490415 -7.7369947 146.354162
[45,] 18.8902893 120.472277 141.984615 -17.432359 52.9504965 59.878062
[46,] 30.5881537 11.056665 53.031043 40.534146 87.1705786 62.311624
[47,] 26.9130475 104.717969 18.277497 37.304100 31.1407464 31.225120
[48,] 29.6335603 135.205036 -15.448668 55.735403 46.6558196 50.523129
[49,] 18.7859944 13.038355 47.690324 29.023077 -7.8482632 91.405343
[50,] 33.7413311 150.532821 -29.048349 54.658151 18.7485356 30.750731
[,7] [,8] [,9] [,10]
[1,] 22.4289007 107.68202 17.958639 -82.244383
[2,] -8.1868126 120.33030 -7.329413 77.122542
[3,] 15.1397611 106.56311 115.613275 -212.310149
[4,] 45.5430687 95.59442 100.181790 90.031150
[5,] 0.4156687 156.58683 59.717157 47.980456
[6,] 19.9475395 75.31008 61.286691 -57.222372
[7,] 26.3063055 100.90410 138.368175 -62.939258
[8,] 8.5353477 86.78485 58.153845 81.256030
[9,] 0.1204387 93.00113 71.013100 71.455358
[10,] 25.9217179 71.47643 -127.120920 63.931470
[11,] 9.7208983 88.70718 146.996307 -21.242962
[12,] -14.2384536 84.36056 60.866701 96.777360
[13,] 17.5591015 59.83607 13.627680 -18.901654
[14,] 12.0622153 84.27482 159.953018 55.638429
[15,] -3.1101647 84.94025 144.773018 38.977132
[16,] 11.8300385 118.60344 36.070018 27.017961
[17,] 20.8116965 81.41027 64.781799 49.160260
[18,] 14.6183014 94.25617 113.574047 -57.544182
[19,] 22.3901535 76.83697 191.380516 -72.825478
[20,] 21.6923615 104.21552 56.704791 -116.918462
[21,] 21.5956954 93.56764 135.972581 107.595679
[22,] -22.3168661 115.86241 122.647198 74.041987
[23,] 40.6628785 68.13787 80.026350 163.615126
[24,] 3.9154278 118.99069 217.008376 57.561552
[25,] 61.5639579 116.29137 19.108007 -7.238832
[26,] -11.9322184 86.39193 148.430820 81.256030
[27,] 32.1041391 91.71934 102.768745 123.250030
[28,] 8.5505864 104.66646 3.083909 223.392412
[29,] 9.7208983 75.95391 103.222110 -82.244383
[30,] 26.2366645 88.91749 211.821205 40.158435
[31,] 26.6985288 87.96929 52.851359 55.307285
[32,] 25.3672527 85.67475 146.468591 180.430553
[33,] -3.8168491 104.44860 116.889394 57.561552
[34,] 26.2366645 101.04614 16.295902 198.685646
[35,] -19.1006010 129.71035 165.154132 -29.247418
[36,] 13.3832374 76.46695 28.017739 113.875986
[37,] 18.1779059 97.50721 217.008376 66.871640
[38,] 0.2791158 84.41508 138.918558 -11.550056
[39,] 0.1174765 107.98680 -72.421144 21.990018
[40,] 2.0703579 80.30584 58.517290 99.788700
[41,] 13.2223361 101.99992 49.269747 68.579903
[42,] -0.3529732 111.41867 133.343950 30.762506
[43,] 39.5945161 126.48166 303.388024 93.031029
[44,] 19.5473831 82.35203 18.245547 129.643615
[45,] 0.2791158 90.67254 -59.296976 -95.330802
[46,] 32.0097487 88.74652 64.572335 68.746163
[47,] 7.8119702 110.81246 173.398315 133.812425
[48,] 42.9507905 117.01701 170.005925 -7.789793
[49,] 15.9368831 117.49441 138.847640 4.320419
[50,] -4.5792146 96.68661 -83.377163 43.277173
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