Developer Guide

Contents

Details

Given
n
feature vectors
x
1
=(
x
11
,…,
x
1
p
),...,
x
n
=(
x
n
1
,…,
x
np
) of size
p
and a vector of class labels y=(
y
1
,…,
y
n
), where
y
i
K
= {0, ..., J-1} describes the class to which the feature vector
x
i
belongs and
J
is the number of classes, the problem is to build a multi-class LogitBoost classifier.

Training Stage

The LogitBoost model is trained using the Friedman method [Friedman00].
Let
y
i,j
=
I
{
x
i
j
} is the indicator that the
i
-th feature vector belongs to class
j
. The scheme below, which uses the stump weak learner, shows the major steps of the algorithm:
  1. Start with weights
    w
    ij
    = 1/
    n
    ,
    F
    j
    (
    x
    ) = 0,
    p
    j
    (
    x
    ) = 1/
    J
    ,
    i
    = 1,...,
    n
    ,
    j
    =0,...,
    J
    -1
  2. For
    m
    =1,...,
    M
    Do
    • For
      j
      = 1,...,
      J
    • Do
      • (i) Compute working responses and weights in the
        j
        -th class:
        w
        ij
        =
        p
        i
        (
        x
        i
        )(1-
        p
        i
        (
        x
        i
        )),
        w
        ij
        =
        max
        (
        z
        ij
        ,
        Thr
        1)
        z
        ij
        = (
        y
        ij
        -
        p
        i
        (
        x
        i
        )) /
        w
        ij
        ,
        z
        ij
        =
        min
        (
        max
        (
        z
        ij
        ,-
        Thr
        2),
        Thr
        2)
      • (ii) Fit the function
        f
        mj
        (
        x
        ) by a weighted least-squares regression of
        z
        ij
        to
        x
        i
        with weights
        w
        ij
        using the stump-based approach.
    • End do
    End do
The result of the model training is a set of
M
stumps.

Prediction Stage

Given the LogitBoost classifier and
r
feature vectors
x
1
,…,
x
r
, the problem is to calculate the labels
argmax
j
F
j
(
x
) of the classes to which the feature vectors belong.

Product and Performance Information

1

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Notice revision #20110804