NONSTATIONARY STREAM DATA LEARNING WITH IMBALANCED CLASS DISTRIBUTION - Imbalanced Learning: Foundations, Algorithms, and Applications - page 162

a i

n

+ a j
a i a j

1
2

= n +

i = 1 ,j = 1

i = j

n
2

× 2 = n + n 2 − n = n 2 ≥ n +

Therefore, according to Equation 7.23 and Inequation 7.24,

n

1
n 2

σ η C σ η i C

≤ (7.26)

i = 1

The single classifier developed by MuSeRA/REA is based on the big picture
of the training data chunk augmented by the previous minority class examples,
while the single classifier developed by UB is based on a partial view. Thus, it
is valid that the error rate of the base hypothesis h i created by MuSeRA/REA
should be equal to or less than that created by UB, that is,

σ η i C ≤ σ b i

(7.27)

which naturally leads to the same result regarding the comparison between their
averaged error rates, that is,

n k

1
n

1
k

σ η i C σ b i

≤ (7.28)

i = 1 j = 1

Given the arrival of sufficient training data chunks, it is inevitable that n>k ;
thus,

k 2 n

1
n 2

1
k 2

σ η i C σ b i ≤ (7.29)

i = 1 j = 1

On the basis of Equations 7.18, 7.26, and 7.29, there is

σ η C ≤ σ b E (7.30)

which proves that MuSeRA / REA can provide less erroneous prediction results
than UB.