Contents

# Details

Given:
• n
feature vectors
x
1
=(
x
11
, ...,
x
1
p
), ...,
x
n
=(
x
n
1
, ...,
x
np
) of size
p
• Their non-negative sample weights
w
=(
w
1
, ...,
w
n
• The vector of responses
y
=(
y
1
, ...,
y
n
), where
y
i
R describes the dependent variable for independent variables
x
i
.
The problem is to build a regression decision tree.

## Split Criterion

The library provides the decision tree regression algorithm based on the mean-squared error (MSE) [Breiman84]
If sample weights are provided as input, the library will use weighted MSE split criterion.
MSE split criterion (
D
is a set of observations that reach the node):
Without sample weights
With sample weights
To find the best test using MSE, each possible test
is examined using
Where
• O
(
τ
) is the set of all possible outcomes of test
τ
• D
v
is the subset of
D
, for which outcome of
τ
is
v
, for example,
.
• operator
for arbitrary set of observations
S
is defined in the table below:
Without sample weights
With sample weights
equivalently the number of elements in
S
The test used in the node is selected as
. For binary decision tree with "true" and "false" branches,

## Training Stage

The regression decision tree follows the algorithmic framework of decision tree training described in Classification and Regression > Decision tree >Training stage.

## Prediction Stage

The regression decision tree follows the algorithmic framework of decision tree prediction described in Classification and Regression > Decision tree > Prediction stage.
Given the regression decision tree and vectors
x
1
, …,
x
r
, the problem is to calculate the responses for those vectors.

## Examples

#### Product and Performance Information

1

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Notice revision #20110804