Contents

# Details

Given:
• n
feature vectors
X
= {
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
)
The problem is to build a decision forest regression model that minimizes the Mean-Square Error (MSE) between the predicted and true value.

## Training Stage

Decision forest regression follows the algorithmic framework of decision forest training algorithm based on the mean-squared error (MSE) [Breiman84]. If sample weights are provided as input, the library uses a weighted version of the algorithm.
MSE is an impurity metric (
D
is a set of observations that reach the node), calculated as follows:
Without sample weights
With sample weights
, which is equivalent to the number of elements in
S

## Prediction Stage

Given decision forest regression model and vectors
x
1
, ... ,
x
r
, the problem is to calculate the responses for those vectors. To solve the problem for each given query vector
x
i
, the algorithm finds the leaf node in a tree in the forest that gives the response by that tree as the mean of dependent variables. The forest predicts the response as the mean of responses from trees.

## Out-of-bag Error

Decision forest regression follows the algorithmic framework for calculating the decision forest out-of-bag (OOB) error, where aggregation of the out-of-bag predictions in all trees and calculation of the OOB error of the decision forest is done as follows:
• For each vector
x
i
in the dataset
X
, predict its response
as the mean of prediction from the trees that contain
x
i
in their OOB set.
• Calculate the OOB error of the decision forest
T
as the Mean-Square Error (MSE):
• If OOB error value per each observation is required, then calculate the prediction error for
x
i
.

#### Product and Performance Information

1

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Notice revision #20110804