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:
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