result of rpart is a root, but data shows Information Gain - r

Question

I have a dataset with an event rate of less than 3% (i.e. there are about 700 records with class 1 and 27000 records with class 0).
ID V1 V2 V3 V5 V6 Target
SDataID3 161 ONE 1 FOUR 0 0
SDataID4 11 TWO 2 THREE 2 1
SDataID5 32 TWO 2 FOUR 2 0
SDataID7 13 ONE 1 THREE 2 0
SDataID8 194 TWO 2 FOUR 0 0
SDataID10 63 THREE 3 FOUR 0 1
SDataID11 89 ONE 1 FOUR 0 0
SDataID13 78 TWO 2 FOUR 0 0
SDataID14 87 TWO 2 THREE 1 0
SDataID15 81 ONE 1 THREE 0 0
SDataID16 63 ONE 3 FOUR 0 0
SDataID17 198 ONE 3 THREE 0 0
SDataID18 9 TWO 3 THREE 0 0
SDataID19 196 ONE 2 THREE 2 0
SDataID20 189 TWO 2 ONE 1 0
SDataID21 116 THREE 3 TWO 0 0
SDataID24 104 ONE 1 FOUR 0 0
SDataID25 5 ONE 2 ONE 3 0
SDataID28 173 TWO 3 FOUR 0 0
SDataID29 5 ONE 3 ONE 3 0
SDataID31 87 ONE 3 FOUR 3 0
SDataID32 5 ONE 2 THREE 1 0
SDataID34 45 ONE 1 FOUR 0 0
SDataID35 19 TWO 2 THREE 0 0
SDataID37 133 TWO 2 FOUR 0 0
SDataID38 8 ONE 1 THREE 0 0
SDataID39 42 ONE 1 THREE 0 0
SDataID43 45 ONE 1 THREE 1 0
SDataID44 45 ONE 1 FOUR 0 0
SDataID45 176 ONE 1 FOUR 0 0
SDataID46 63 ONE 1 THREE 3 0
I am trying to find out split using the decision tree. But result of tree is only 1 root.
> library(rpart)
> tree <- rpart(Target ~ ., data=subset(train, select=c( -Record.ID) ),method="class")
> printcp(tree)
Classification tree:
rpart(formula = Target ~ ., data = subset(train, select = c(-Record.ID)), method = "class")
Variables actually used in tree construction:
character(0)
Root node error: 749/18239 = 0.041066
n= 18239
CP nsplit rel error xerror xstd
1 0 0 1 0 0
After reading most of the resources on StackOverflow, I loosened/tweaked the control parameters which gave me the desired decision tree.
> tree <- rpart(Target ~ ., data=subset(train, select=c( -Record.ID) ),method="class" ,control =rpart.control(minsplit = 1,minbucket=2, cp=0.00002))
> printcp(tree)
Classification tree:
rpart(formula = Target ~ ., data = subset(train, select = c(-Record.ID)),
method = "class", control = rpart.control(minsplit = 1, minbucket = 2,
cp = 2e-05))
Variables actually used in tree construction:
[1] V5 V2 V1
[4] V3 V6
Root node error: 749/18239 = 0.041066
n= 18239
CP nsplit rel error xerror xstd
1 0.00024275 0 1.00000 1.0000 0.035781
2 0.00019073 20 0.99466 1.0267 0.036235
3 0.00016689 34 0.99199 1.0307 0.036302
4 0.00014835 54 0.98798 1.0334 0.036347
5 0.00002000 63 0.98665 1.0427 0.036504
When I pruned the tree it resulted in a tree with a single node.
> pruned.tree <- prune(tree, cp = tree$cptable[which.min(tree$cptable[,"xerror"]),"CP"])
> printcp(pruned.tree)
Classification tree:
rpart(formula = Target ~ ., data = subset(train, select = c(-Record.ID)),
method = "class", control = rpart.control(minsplit = 1, minbucket = 2,
cp = 2e-05))
Variables actually used in tree construction:
character(0)
Root node error: 749/18239 = 0.041066
n= 18239
CP nsplit rel error xerror xstd
1 0.00024275 0 1 1 0.035781
The tree should not be giving out only root node because mathematically, on a given node (example provided) we are getting Information Gain. I don't know if I am making a mistake by pruning or there is an issue with rpart in handling low event rate dataset?
NODE p 1-p Entropy Weights Ent*Weight # Obs
Node 1 0.032 0.968 0.204324671 0.351398601 0.071799404 10653
Node 2 0.05 0.95 0.286396957 0.648601399 0.185757467 19663
Sum(Ent*wght) 0.257556871
Information gain 0.742443129

Answer 1

The data you provided does not reflect the ratio of the two target classes, so I've tweaked the data to better reflect that (see Data section):
> prop.table(table(train$Target))
0 1
0.96707581 0.03292419
> 700/27700
[1] 0.02527076
The ratios are now relatively close...
library(rpart)
tree <- rpart(Target ~ ., data=train, method="class")
printcp(tree)
Results in:
Classification tree:
rpart(formula = Target ~ ., data = train, method = "class")
Variables actually used in tree construction:
character(0)
Root node error: 912/27700 = 0.032924
n= 27700
CP nsplit rel error xerror xstd
1 0 0 1 0 0
Now, the reason that you are seeing only the root node for your first model, is probably due to the fact that you have extremely imbalanced target classes, and so, your independent variables could not provide enough information to grow the tree. My sample data has 3.3% event rate, but yours has only about 2.5%!
As you have mentioned, there is a way to force rpart to grow the tree. That is to override the default complexity parameter (cp). The complexity measure is a combination of the size of the tree and how well the tree separates the target classes. From ?rpart.control, "Any split that does not decrease the overall lack of fit by a factor of cp is not attempted". This means that your model at this point does not have a split beyond the root node that decreases the complexity level enough for rpart to take into consideration. We can relax this threshold of what is considered "enough" by either setting a low or a negative cp (negative cp basically forces the tree to grow to its full size).
tree <- rpart(Target ~ ., data=train, method="class" ,parms = list(split = 'information'),
control =rpart.control(minsplit = 1,minbucket=2, cp=0.00002))
printcp(tree)
Results in:
Classification tree:
rpart(formula = Target ~ ., data = train, method = "class", parms = list(split = "information"),
control = rpart.control(minsplit = 1, minbucket = 2, cp = 2e-05))
Variables actually used in tree construction:
[1] ID V1 V2 V3 V5 V6
Root node error: 912/27700 = 0.032924
n= 27700
CP nsplit rel error xerror xstd
1 4.1118e-04 0 1.00000 1.0000 0.032564
2 3.6550e-04 30 0.98355 1.0285 0.033009
3 3.2489e-04 45 0.97807 1.0702 0.033647
4 3.1328e-04 106 0.95504 1.0877 0.033911
5 2.7412e-04 116 0.95175 1.1031 0.034141
6 2.5304e-04 132 0.94737 1.1217 0.034417
7 2.1930e-04 149 0.94298 1.1458 0.034771
8 1.9936e-04 159 0.94079 1.1502 0.034835
9 1.8275e-04 181 0.93640 1.1645 0.035041
10 1.6447e-04 193 0.93421 1.1864 0.035356
11 1.5664e-04 233 0.92654 1.1853 0.035341
12 1.3706e-04 320 0.91228 1.2083 0.035668
13 1.2183e-04 344 0.90899 1.2127 0.035730
14 9.9681e-05 353 0.90789 1.2237 0.035885
15 2.0000e-05 364 0.90680 1.2259 0.035915
As you can see, the tree has grown to a size that reduces the complexity level by a minimum of cp. Two things to note:
At zero nsplit, CP is already as low as 0.0004, where as the default cp in rpart is set to 0.01.
Starting from nsplit == 0, the cross validation error (xerror) increases as you increase the number of splits.
Both of these indicate that your model is overfitting the data at nsplit == 0 and beyond, since adding more independent variables into your model does not add enough information (insufficient reduction in CP) to reduce the cross validation error. With this being said, your root node model is the best model in this case, which explains why your initial model has only the root node.
pruned.tree <- prune(tree, cp = tree$cptable[which.min(tree$cptable[,"xerror"]),"CP"])
printcp(pruned.tree)
Results in:
Classification tree:
rpart(formula = Target ~ ., data = train, method = "class", parms = list(split = "information"),
control = rpart.control(minsplit = 1, minbucket = 2, cp = 2e-05))
Variables actually used in tree construction:
character(0)
Root node error: 912/27700 = 0.032924
n= 27700
CP nsplit rel error xerror xstd
1 0.00041118 0 1 1 0.032564
As for the pruning part, it is now clearer why your pruned tree is the root node tree, since a tree that goes beyond 0 splits has increasing cross validation error. Taking the tree with the minimum xerror would leave you with root node tree as expected.
Information gain basically tells you how much "information" is added for each split. So technically, every split has some degree of information gain since you are adding more variables into your model (information gain is always non-negative). What you should think about is whether that additional gain (or no gain) reduces the errors enough for you to warrant a more complex model. Hence, the tradeoff between bias and variance.
In this case, it doesn't really make sense for you to reduce cp and later prune the resulting tree. since by setting a low cp, you are telling rpart to make splits even if it overfits, while pruning "cuts" all the nodes that overfits.
Data:
Note that I am shuffling the rows for each column and sample instead of sampling the row indices. This is because the data you provided is probably not a random sample of your original dataset (likely biased), so I am basically randomly creating new observations with combinations of your existing rows which would hopefully reduce that bias.
init_train = structure(list(ID = structure(c(16L, 24L, 29L, 30L, 31L, 1L,
2L, 3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L, 15L,
17L, 18L, 19L, 20L, 21L, 22L, 23L, 25L, 26L, 27L, 28L), .Label = c("SDataID10",
"SDataID11", "SDataID13", "SDataID14", "SDataID15", "SDataID16",
"SDataID17", "SDataID18", "SDataID19", "SDataID20", "SDataID21",
"SDataID24", "SDataID25", "SDataID28", "SDataID29", "SDataID3",
"SDataID31", "SDataID32", "SDataID34", "SDataID35", "SDataID37",
"SDataID38", "SDataID39", "SDataID4", "SDataID43", "SDataID44",
"SDataID45", "SDataID46", "SDataID5", "SDataID7", "SDataID8"), class = "factor"),
V1 = c(161L, 11L, 32L, 13L, 194L, 63L, 89L, 78L, 87L, 81L,
63L, 198L, 9L, 196L, 189L, 116L, 104L, 5L, 173L, 5L, 87L,
5L, 45L, 19L, 133L, 8L, 42L, 45L, 45L, 176L, 63L), V2 = structure(c(1L,
3L, 3L, 1L, 3L, 2L, 1L, 3L, 3L, 1L, 1L, 1L, 3L, 1L, 3L, 2L,
1L, 1L, 3L, 1L, 1L, 1L, 1L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L
), .Label = c("ONE", "THREE", "TWO"), class = "factor"),
V3 = c(1L, 2L, 2L, 1L, 2L, 3L, 1L, 2L, 2L, 1L, 3L, 3L, 3L,
2L, 2L, 3L, 1L, 2L, 3L, 3L, 3L, 2L, 1L, 2L, 2L, 1L, 1L, 1L,
1L, 1L, 1L), V5 = structure(c(1L, 3L, 1L, 3L, 1L, 1L, 1L,
1L, 3L, 3L, 1L, 3L, 3L, 3L, 2L, 4L, 1L, 2L, 1L, 2L, 1L, 3L,
1L, 3L, 1L, 3L, 3L, 3L, 1L, 1L, 3L), .Label = c("FOUR", "ONE",
"THREE", "TWO"), class = "factor"), V6 = c(0L, 2L, 2L, 2L,
0L, 0L, 0L, 0L, 1L, 0L, 0L, 0L, 0L, 2L, 1L, 0L, 0L, 3L, 0L,
3L, 3L, 1L, 0L, 0L, 0L, 0L, 0L, 1L, 0L, 0L, 3L), Target = c(0L,
1L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L
)), .Names = c("ID", "V1", "V2", "V3", "V5", "V6", "Target"
), class = "data.frame", row.names = c(NA, -31L))
set.seed(1000)
train = as.data.frame(lapply(init_train, function(x) sample(x, 27700, replace = TRUE)))