cinbag: Modified Classification and Regression with Random Forest
In trimTrees: Trimmed opinion pools of trees in a random forest
Description
Usage
Arguments
Value
Note
Author(s)
References
See Also
Examples
Description
cinbag implements a modified random forest algorithm (based on the source code from the randomForest package by Andy Liaw and Matthew Wiener and on the original Fortran code by Leo Breiman and Adele Cutler) to return the number of times a row appears in a tree's bag. cinbag returns a randomForest object, e.g., rfobj, with an additional output, a matrix with inbag counts (rows) for each tree (columns). For instance, rfobj$inbagCount is similar to rfobj$inbag, but with inbag counts instead of inbag indicators.
Usage
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cinbag(x, y=NULL, xtest=NULL, ytest=NULL, ntree=500,
mtry=if (!is.null(y) && !is.factor(y))
max(floor(ncol(x)/3), 1) else floor(sqrt(ncol(x))),
replace=TRUE, classwt=NULL, cutoff, strata,
sampsize = if (replace) nrow(x) else ceiling(.632*nrow(x)),
nodesize = if (!is.null(y) && !is.factor(y)) 5 else 1,
maxnodes = NULL,
importance=FALSE, localImp=FALSE, nPerm=1,
proximity, oob.prox=proximity,
norm.votes=TRUE, do.trace=FALSE,
keep.forest=!is.null(y) && is.null(xtest), corr.bias=FALSE,
keep.inbag=FALSE, ...)
Arguments
x
a data frame or a matrix of predictors, or a formula
describing the model to be fitted (for the
print method, an randomForest object).
y
A response vector. If a factor, classification is assumed,
otherwise regression is assumed. If omitted, randomForest
will run in unsupervised mode.
xtest
a data frame or matrix (like x) containing
predictors for the test set.
ytest
response for the test set.
ntree
Number of trees to grow. This should not be set to too
small a number, to ensure that every input row gets predicted at
least a few times.
mtry
Number of variables randomly sampled as candidates at each
split. Note that the default values are different for
classification (sqrt(p) where p is number of variables in x)
and regression (p/3).
replace
Should sampling of cases be done with or without
replacement?
classwt
Priors of the classes. Need not add up to one.
Ignored for regression.
cutoff
(Classification only) A vector of length equal to
number of classes. The ‘winning’ class for an observation is the
one with the maximum ratio of proportion of votes to cutoff.
Default is 1/k where k is the number of classes (i.e., majority vote
wins).
strata
A (factor) variable that is used for stratified sampling.
sampsize
Size(s) of sample to draw. For classification, if
sampsize is a vector of the length the number of strata, then
sampling is stratified by strata, and the elements of sampsize
indicate the numbers to be drawn from the strata.
nodesize
Minimum size of terminal nodes. Setting this number
larger causes smaller trees to be grown (and thus take less time).
Note that the default values are different for classification (1)
and regression (5).
maxnodes
Maximum number of terminal nodes trees in the forest
can have. If not given, trees are grown to the maximum possible
(subject to limits by nodesize). If set larger than maximum
possible, a warning is issued.
importance
Should importance of predictors be assessed?
localImp
Should casewise importance measure be computed?
(Setting this to TRUE will override importance.)
nPerm
Number of times the OOB data are permuted per tree for
assessing variable importance. Number larger than 1 gives slightly
more stable estimate, but not very effective. Currently only
implemented for regression.
proximity
Should proximity measure among the rows be
calculated?
oob.prox
Should proximity be calculated only on “out-of-bag”
data?
norm.votes
If TRUE (default), the final result of votes
are expressed as fractions. If FALSE, raw vote counts are
returned (useful for combining results from different runs).
Ignored for regression.
do.trace
If set to TRUE, give a more verbose output as
randomForest is run. If set to some integer, then running
output is printed for every do.trace trees.
keep.forest
If set to FALSE, the forest will not be
retained in the output object. If xtest is given, defaults
to FALSE.
corr.bias
perform bias correction for regression? Note:
Experimental. Use at your own risk.
keep.inbag
Should an n by ntree matrix be
returned that keeps track of which samples are “in-bag” in which
trees (but not how many times, if sampling with replacement)
...
optional parameters to be passed to the low level function
cinbag.default.
Value
An object of class randomForest, which is a list with the
following components:
call
the original call to randomForest
type
one of regression, classification, or
unsupervised.
predicted
the predicted values of the input data based on
out-of-bag samples.
importance
a matrix with nclass + 2 (for classification)
or two (for regression) columns. For classification, the first
nclass columns are the class-specific measures computed as
mean descrease in accuracy. The nclass + 1st column is the
mean descrease in accuracy over all classes. The last column is the
mean decrease in Gini index. For Regression, the first column is
the mean decrease in accuracy and the second the mean decrease in MSE.
If importance=FALSE, the last measure is still returned as a
vector.
importanceSD
The “standard errors” of the permutation-based
importance measure. For classification, a p by nclass
+ 1 matrix corresponding to the first nclass + 1 columns
of the importance matrix. For regression, a length p vector.
localImp
a p by n matrix containing the casewise importance
measures, the [i,j] element of which is the importance of i-th
variable on the j-th case. NULL if localImp=FALSE.
ntree
number of trees grown.
mtry
number of predictors sampled for spliting at each node.
forest
(a list that contains the entire forest; NULL if
randomForest is run in unsupervised mode or if
keep.forest=FALSE.
err.rate
(classification only) vector error rates of the
prediction on the input data, the i-th element being the (OOB) error rate
for all trees up to the i-th.
confusion
(classification only) the confusion matrix of the
prediction (based on OOB data).
votes
(classification only) a matrix with one row for each
input data point and one column for each class, giving the fraction
or number of (OOB) ‘votes’ from the random forest.
oob.times
number of times cases are ‘out-of-bag’ (and thus used
in computing OOB error estimate)
proximity
if proximity=TRUE when
randomForest is called, a matrix of proximity measures among
the input (based on the frequency that pairs of data points are in
the same terminal nodes).
mse
(regression only) vector of mean square errors: sum of squared
residuals divided by n.
rsq
(regression only) “pseudo R-squared”: 1 - mse /
Var(y).
test
if test set is given (through the xtest or additionally
ytest arguments), this component is a list which contains the
corresponding predicted, err.rate, confusion,
votes (for classification) or predicted, mse and
rsq (for regression) for the test set. If
proximity=TRUE, there is also a component, proximity,
which contains the proximity among the test set as well as proximity
between test and training data.
inbag
An indicator (1 or 0) for each training set row and each tree. The indicator is 1 if the training set row is in the tree's bag and is 0 otherwise. Note that this value is not listed in the original randomForest function's output, although it is implemented.
inbagCount
A count for each training set row and each tree. The count is the number of times the training set row is in the tree's bag. This output is not available in the original randomForest package. The purpose of the cinbag function is to augment the randomForest function so that it returns inbag counts. These counts are necessary for computing and ensembling the trees' empirical cumulative distribution functions.
Note
cinbag's source files call the C functions classRFmod.c and regRFmod.c, which are slightly modified versions of the randomForest's source files classRF.c and regRF.c, respectively.
Author(s)
Yael Grushka-Cockayne, Victor Richmond R. Jose, Kenneth C. Lichtendahl Jr. and Huanghui Zeng, based on the source code from the randomForest package by Andy Liaw and Matthew Wiener and on the original Fortran code by Leo Breiman and Adele Cutler.
References
Breiman L (2001). Random forests. Machine Learning 45 5-32.
Breiman L (2002). Manual on setting up, using, and understanding random forests V3.1. http://oz.berkeley.edu/users/breiman/Using_random_forests_V3.1.pdf.
See Also
Examples
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17
18
# Load the data
set.seed(201) # Can be removed; useful for replication
data <- as.data.frame(mlbench.friedman1(500, sd=1))
summary(data)
# Prepare data for trimming
train <- data[1:400, ]
test <- data[401:500, ]
xtrain <- train[,-11]
ytrain <- train[,11]
xtest <- test[,-11]
ytest <- test[,11]
# Run cinbag
set.seed(201) # Can be removed; useful for replication
rf <- cinbag(xtrain, ytrain, ntree=500, nodesize=5, mtry=3, keep.inbag=TRUE)
rf$inbag[,1] # First tree's inbag indicators
rf$inbagCount[,1] # First tree's inbag counts
Example output
Loading required package: randomForest
randomForest 4.6-12
Type rfNews() to see new features/changes/bug fixes.
Loading required package: mlbench
x.1 x.2 x.3 x.4
Min. :0.001093 Min. :0.0009745 Min. :0.00216 Min. :0.00143
1st Qu.:0.263601 1st Qu.:0.2552209 1st Qu.:0.26539 1st Qu.:0.25687
Median :0.516836 Median :0.4793052 Median :0.51228 Median :0.53476
Mean :0.505285 Mean :0.4960016 Mean :0.50927 Mean :0.51471
3rd Qu.:0.748586 3rd Qu.:0.7622338 3rd Qu.:0.75783 3rd Qu.:0.77337
Max. :0.995597 Max. :0.9995513 Max. :0.99795 Max. :0.99771
x.5 x.6 x.7 x.8
Min. :0.002658 Min. :0.0007875 Min. :0.001599 Min. :0.002436
1st Qu.:0.263081 1st Qu.:0.2210349 1st Qu.:0.243378 1st Qu.:0.266648
Median :0.519615 Median :0.4380409 Median :0.471095 Median :0.493080
Mean :0.514470 Mean :0.4694873 Mean :0.485703 Mean :0.509182
3rd Qu.:0.766236 3rd Qu.:0.7247891 3rd Qu.:0.746310 3rd Qu.:0.786979
Max. :0.999698 Max. :0.9996735 Max. :0.999112 Max. :0.995875
x.9 x.10 y
Min. :0.000695 Min. :0.005041 Min. : 0.1456
1st Qu.:0.247458 1st Qu.:0.291609 1st Qu.:11.0023
Median :0.509853 Median :0.500964 Median :14.7441
Mean :0.495805 Mean :0.500816 Mean :14.5685
3rd Qu.:0.744925 3rd Qu.:0.728643 3rd Qu.:18.1207
Max. :0.998112 Max. :0.998811 Max. :27.1957
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
0 1 0 0 0 0 1 0 0 1 0 1 1 1 0 1 1 1 0 1
41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
0 0 1 0 0 1 0 0 1 1 1 1 1 0 1 1 1 1 0 1
61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
0 0 1 0 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0
81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
0 1 0 1 1 1 0 1 1 0 1 1 0 1 0 1 1 1 0 1
101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
1 1 0 1 1 1 1 1 1 1 0 1 0 0 1 1 0 1 1 1
121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140
1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0
141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160
1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1
161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180
0 1 1 0 1 0 1 0 1 1 1 0 0 1 0 1 1 0 0 0
181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200
0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220
0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0
221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240
0 1 0 0 0 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1
241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260
0 0 1 1 0 1 1 0 1 0 1 1 1 1 0 0 1 1 0 0
261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280
0 1 0 1 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0
281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300
1 0 1 1 0 1 0 1 0 1 0 0 0 0 1 1 1 1 1 1
301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320
1 0 1 1 1 1 1 1 0 0 0 1 0 1 1 0 0 0 1 1
321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340
1 0 0 0 1 0 0 1 1 1 1 0 0 1 1 0 0 1 1 1
341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360
0 1 1 1 1 0 1 1 1 1 0 1 1 0 1 0 0 1 1 0
361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380
1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 1 0 0 1 1
381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400
1 1 0 0 1 1 1 0 1 1 1 1 1 0 0 0 1 0 1 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 1 3 1 2 0 1 1 1 0 0 2 2 2 2 1 2 1 1
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
0 2 0 0 0 0 1 0 0 1 0 1 2 1 0 1 2 3 0 1
41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
0 0 2 0 0 2 0 0 1 3 1 1 5 0 1 2 1 1 0 2
61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
0 0 1 0 1 1 2 1 1 1 0 2 1 2 1 0 2 1 0 0
81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
0 2 0 2 1 1 0 2 1 0 1 2 0 3 0 1 2 1 0 1
101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
1 1 0 2 1 2 1 1 2 2 0 2 0 0 1 1 0 1 3 1
121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140
2 0 1 1 2 1 2 2 3 0 0 0 0 0 0 1 1 1 0 0
141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160
1 2 1 2 2 2 3 0 0 1 0 3 1 3 1 0 0 2 1 1
161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180
0 4 1 0 1 0 1 0 1 3 1 0 0 1 0 1 2 0 0 0
181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200
0 2 2 0 1 2 1 2 4 1 1 1 1 1 1 1 1 1 1 2
201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220
0 1 0 1 1 0 1 3 1 1 1 1 3 3 1 1 2 1 1 0
221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240
0 2 0 0 0 1 1 2 0 3 1 1 1 0 1 2 1 2 1 3
241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260
0 0 2 1 0 1 2 0 3 0 1 2 1 2 0 0 1 1 0 0
261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280
0 3 0 1 2 1 0 1 3 0 2 2 3 0 1 0 0 0 1 0
281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300
2 0 1 1 0 3 0 1 0 1 0 0 0 0 2 1 1 1 1 2
301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320
2 0 2 3 1 1 2 1 0 0 0 2 0 1 1 0 0 0 3 1
321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340
1 0 0 0 2 0 0 1 1 1 1 0 0 1 1 0 0 3 1 1
341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360
0 1 1 1 1 0 1 3 1 2 0 1 1 0 1 0 0 2 1 0
361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380
2 2 0 2 3 0 1 1 2 1 2 0 2 1 3 1 0 0 1 1
381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400
1 1 0 0 1 1 2 0 1 2 2 2 3 0 0 0 2 0 1 0
trimTrees documentation built on May 2, 2019, 6:09 a.m.
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Description Usage Arguments Value Note Author(s) References See Also Examples
Description
cinbag implements a modified random forest algorithm (based on the source code from the randomForest package by Andy Liaw and Matthew Wiener and on the original Fortran code by Leo Breiman and Adele Cutler) to return the number of times a row appears in a tree's bag. cinbag returns a randomForest object, e.g., rfobj, with an additional output, a matrix with inbag counts (rows) for each tree (columns). For instance, rfobj$inbagCount is similar to rfobj$inbag, but with inbag counts instead of inbag indicators.
Usage
1 2 3 4 5 6 7 8 9 10 11 12 | cinbag(x, y=NULL, xtest=NULL, ytest=NULL, ntree=500,
mtry=if (!is.null(y) && !is.factor(y))
max(floor(ncol(x)/3), 1) else floor(sqrt(ncol(x))),
replace=TRUE, classwt=NULL, cutoff, strata,
sampsize = if (replace) nrow(x) else ceiling(.632*nrow(x)),
nodesize = if (!is.null(y) && !is.factor(y)) 5 else 1,
maxnodes = NULL,
importance=FALSE, localImp=FALSE, nPerm=1,
proximity, oob.prox=proximity,
norm.votes=TRUE, do.trace=FALSE,
keep.forest=!is.null(y) && is.null(xtest), corr.bias=FALSE,
keep.inbag=FALSE, ...)
|
Arguments
x |
a data frame or a matrix of predictors, or a formula
describing the model to be fitted (for the
|
y |
A response vector. If a factor, classification is assumed,
otherwise regression is assumed. If omitted, |
xtest |
a data frame or matrix (like |
ytest |
response for the test set. |
ntree |
Number of trees to grow. This should not be set to too small a number, to ensure that every input row gets predicted at least a few times. |
mtry |
Number of variables randomly sampled as candidates at each
split. Note that the default values are different for
classification (sqrt(p) where p is number of variables in |
replace |
Should sampling of cases be done with or without replacement? |
classwt |
Priors of the classes. Need not add up to one. Ignored for regression. |
cutoff |
(Classification only) A vector of length equal to number of classes. The ‘winning’ class for an observation is the one with the maximum ratio of proportion of votes to cutoff. Default is 1/k where k is the number of classes (i.e., majority vote wins). |
strata |
A (factor) variable that is used for stratified sampling. |
sampsize |
Size(s) of sample to draw. For classification, if sampsize is a vector of the length the number of strata, then sampling is stratified by strata, and the elements of sampsize indicate the numbers to be drawn from the strata. |
nodesize |
Minimum size of terminal nodes. Setting this number larger causes smaller trees to be grown (and thus take less time). Note that the default values are different for classification (1) and regression (5). |
maxnodes |
Maximum number of terminal nodes trees in the forest
can have. If not given, trees are grown to the maximum possible
(subject to limits by |
importance |
Should importance of predictors be assessed? |
localImp |
Should casewise importance measure be computed?
(Setting this to |
nPerm |
Number of times the OOB data are permuted per tree for assessing variable importance. Number larger than 1 gives slightly more stable estimate, but not very effective. Currently only implemented for regression. |
proximity |
Should proximity measure among the rows be calculated? |
oob.prox |
Should proximity be calculated only on “out-of-bag” data? |
norm.votes |
If |
do.trace |
If set to |
keep.forest |
If set to |
corr.bias |
perform bias correction for regression? Note: Experimental. Use at your own risk. |
keep.inbag |
Should an |
... |
optional parameters to be passed to the low level function
|
Value
An object of class randomForest, which is a list with the
following components:
call |
the original call to |
type |
one of |
predicted |
the predicted values of the input data based on out-of-bag samples. |
importance |
a matrix with |
importanceSD |
The “standard errors” of the permutation-based
importance measure. For classification, a |
localImp |
a p by n matrix containing the casewise importance
measures, the [i,j] element of which is the importance of i-th
variable on the j-th case. |
ntree |
number of trees grown. |
mtry |
number of predictors sampled for spliting at each node. |
forest |
(a list that contains the entire forest; |
err.rate |
(classification only) vector error rates of the prediction on the input data, the i-th element being the (OOB) error rate for all trees up to the i-th. |
confusion |
(classification only) the confusion matrix of the prediction (based on OOB data). |
votes |
(classification only) a matrix with one row for each input data point and one column for each class, giving the fraction or number of (OOB) ‘votes’ from the random forest. |
oob.times |
number of times cases are ‘out-of-bag’ (and thus used in computing OOB error estimate) |
proximity |
if |
mse |
(regression only) vector of mean square errors: sum of squared
residuals divided by |
rsq |
(regression only) “pseudo R-squared”: 1 - |
test |
if test set is given (through the |
inbag |
An indicator (1 or 0) for each training set row and each tree. The indicator is 1 if the training set row is in the tree's bag and is 0 otherwise. Note that this value is not listed in the original |
inbagCount |
A count for each training set row and each tree. The count is the number of times the training set row is in the tree's bag. This output is not available in the original |
Note
cinbag's source files call the C functions classRFmod.c and regRFmod.c, which are slightly modified versions of the randomForest's source files classRF.c and regRF.c, respectively.
Author(s)
Yael Grushka-Cockayne, Victor Richmond R. Jose, Kenneth C. Lichtendahl Jr. and Huanghui Zeng, based on the source code from the randomForest package by Andy Liaw and Matthew Wiener and on the original Fortran code by Leo Breiman and Adele Cutler.
References
Breiman L (2001). Random forests. Machine Learning 45 5-32.
Breiman L (2002). Manual on setting up, using, and understanding random forests V3.1. http://oz.berkeley.edu/users/breiman/Using_random_forests_V3.1.pdf.
See Also
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | # Load the data
set.seed(201) # Can be removed; useful for replication
data <- as.data.frame(mlbench.friedman1(500, sd=1))
summary(data)
# Prepare data for trimming
train <- data[1:400, ]
test <- data[401:500, ]
xtrain <- train[,-11]
ytrain <- train[,11]
xtest <- test[,-11]
ytest <- test[,11]
# Run cinbag
set.seed(201) # Can be removed; useful for replication
rf <- cinbag(xtrain, ytrain, ntree=500, nodesize=5, mtry=3, keep.inbag=TRUE)
rf$inbag[,1] # First tree's inbag indicators
rf$inbagCount[,1] # First tree's inbag counts
|
Example output
Loading required package: randomForest
randomForest 4.6-12
Type rfNews() to see new features/changes/bug fixes.
Loading required package: mlbench
x.1 x.2 x.3 x.4
Min. :0.001093 Min. :0.0009745 Min. :0.00216 Min. :0.00143
1st Qu.:0.263601 1st Qu.:0.2552209 1st Qu.:0.26539 1st Qu.:0.25687
Median :0.516836 Median :0.4793052 Median :0.51228 Median :0.53476
Mean :0.505285 Mean :0.4960016 Mean :0.50927 Mean :0.51471
3rd Qu.:0.748586 3rd Qu.:0.7622338 3rd Qu.:0.75783 3rd Qu.:0.77337
Max. :0.995597 Max. :0.9995513 Max. :0.99795 Max. :0.99771
x.5 x.6 x.7 x.8
Min. :0.002658 Min. :0.0007875 Min. :0.001599 Min. :0.002436
1st Qu.:0.263081 1st Qu.:0.2210349 1st Qu.:0.243378 1st Qu.:0.266648
Median :0.519615 Median :0.4380409 Median :0.471095 Median :0.493080
Mean :0.514470 Mean :0.4694873 Mean :0.485703 Mean :0.509182
3rd Qu.:0.766236 3rd Qu.:0.7247891 3rd Qu.:0.746310 3rd Qu.:0.786979
Max. :0.999698 Max. :0.9996735 Max. :0.999112 Max. :0.995875
x.9 x.10 y
Min. :0.000695 Min. :0.005041 Min. : 0.1456
1st Qu.:0.247458 1st Qu.:0.291609 1st Qu.:11.0023
Median :0.509853 Median :0.500964 Median :14.7441
Mean :0.495805 Mean :0.500816 Mean :14.5685
3rd Qu.:0.744925 3rd Qu.:0.728643 3rd Qu.:18.1207
Max. :0.998112 Max. :0.998811 Max. :27.1957
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
0 1 0 0 0 0 1 0 0 1 0 1 1 1 0 1 1 1 0 1
41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
0 0 1 0 0 1 0 0 1 1 1 1 1 0 1 1 1 1 0 1
61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
0 0 1 0 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0
81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
0 1 0 1 1 1 0 1 1 0 1 1 0 1 0 1 1 1 0 1
101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
1 1 0 1 1 1 1 1 1 1 0 1 0 0 1 1 0 1 1 1
121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140
1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0
141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160
1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1
161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180
0 1 1 0 1 0 1 0 1 1 1 0 0 1 0 1 1 0 0 0
181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200
0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220
0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0
221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240
0 1 0 0 0 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1
241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260
0 0 1 1 0 1 1 0 1 0 1 1 1 1 0 0 1 1 0 0
261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280
0 1 0 1 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0
281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300
1 0 1 1 0 1 0 1 0 1 0 0 0 0 1 1 1 1 1 1
301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320
1 0 1 1 1 1 1 1 0 0 0 1 0 1 1 0 0 0 1 1
321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340
1 0 0 0 1 0 0 1 1 1 1 0 0 1 1 0 0 1 1 1
341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360
0 1 1 1 1 0 1 1 1 1 0 1 1 0 1 0 0 1 1 0
361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380
1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 1 0 0 1 1
381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400
1 1 0 0 1 1 1 0 1 1 1 1 1 0 0 0 1 0 1 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 1 3 1 2 0 1 1 1 0 0 2 2 2 2 1 2 1 1
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
0 2 0 0 0 0 1 0 0 1 0 1 2 1 0 1 2 3 0 1
41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
0 0 2 0 0 2 0 0 1 3 1 1 5 0 1 2 1 1 0 2
61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
0 0 1 0 1 1 2 1 1 1 0 2 1 2 1 0 2 1 0 0
81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
0 2 0 2 1 1 0 2 1 0 1 2 0 3 0 1 2 1 0 1
101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
1 1 0 2 1 2 1 1 2 2 0 2 0 0 1 1 0 1 3 1
121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140
2 0 1 1 2 1 2 2 3 0 0 0 0 0 0 1 1 1 0 0
141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160
1 2 1 2 2 2 3 0 0 1 0 3 1 3 1 0 0 2 1 1
161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180
0 4 1 0 1 0 1 0 1 3 1 0 0 1 0 1 2 0 0 0
181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200
0 2 2 0 1 2 1 2 4 1 1 1 1 1 1 1 1 1 1 2
201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220
0 1 0 1 1 0 1 3 1 1 1 1 3 3 1 1 2 1 1 0
221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240
0 2 0 0 0 1 1 2 0 3 1 1 1 0 1 2 1 2 1 3
241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260
0 0 2 1 0 1 2 0 3 0 1 2 1 2 0 0 1 1 0 0
261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280
0 3 0 1 2 1 0 1 3 0 2 2 3 0 1 0 0 0 1 0
281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300
2 0 1 1 0 3 0 1 0 1 0 0 0 0 2 1 1 1 1 2
301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320
2 0 2 3 1 1 2 1 0 0 0 2 0 1 1 0 0 0 3 1
321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340
1 0 0 0 2 0 0 1 1 1 1 0 0 1 1 0 0 3 1 1
341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360
0 1 1 1 1 0 1 3 1 2 0 1 1 0 1 0 0 2 1 0
361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380
2 2 0 2 3 0 1 1 2 1 2 0 2 1 3 1 0 0 1 1
381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400
1 1 0 0 1 1 2 0 1 2 2 2 3 0 0 0 2 0 1 0
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