New PDF release: Analysis of Rare Categories

By Jingrui He

ISBN-10: 3642228127

ISBN-13: 9783642228124

In many real-world difficulties, infrequent different types (minority periods) play crucial roles regardless of their severe shortage. the invention, characterization and prediction of infrequent different types of infrequent examples may perhaps guard us from fraudulent or malicious habit, relief clinical discovery, or even shop lives.

This publication specializes in infrequent class research, the place the bulk periods have delicate distributions, and the minority sessions show the compactness estate. moreover, it specializes in the difficult circumstances the place the aid areas of the bulk and minority periods overlap. the writer has built potent algorithms with theoretical promises and reliable empirical effects for the similar concepts, and those are defined intimately. The publication is acceptable for researchers within the zone of man-made intelligence, specifically desktop studying and information mining.

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The maximum likelihood estimates βˆ1j and βˆ0i satisfy the following conditions: ∀j ∈ {1, . . , d} n (xjk )2 = k=1 n ˆj i=1 exp(β0i n xj (x (xjk −xji )2 )Eij ((xj )2 ) 2(σ j )2 n ˆj i=1 exp(β0i k=1 where Eij ((xj )2 ) = − j )2 √ 1 2πσ j exp(− − (xj −xji )2 j ) exp(βˆ0i + βˆ1j (xj )2 )dxj . 2(σ j )2 j Proof. 7, we have β0i = − log exp(− j ∂β0i ∂β1j (xj −xji )2 ) exp(β1j (xj )2 )dxj . 8) (xjk −xji )2 ) 2(σ j )2 √ 1 2πσ j exp(− xj −Eij ((xj )2 ) Then the log-likelihood of the data on the j th component is n l(β1j ) log(gβj (xjk )) = k=1 n = k=1 n 1 log( n n i=1 (xjk − xji )2 1 j √ exp(− ) · exp(β0i + β1j (xjk )2 )) j 2 j 2(σ ) 2πσ 1 log( √ exp(β1j (xjk )2 ) · = j n 2πσ k=1 n exp(− i=1 (xjk − xji )2 j ) exp(β0i )) 2(σ j )2 44 3 Rare Category Detection Taking the partial derivative of l(β1j ) with respect to β1j , we have: ∂l(β1j ) ∂β1j n = (xjk )2 + k=1 n = j n i=1 exp(β0i n j n i=1 exp(β0i k=1 (xjk )2 j n i=1 exp(β0i n − k=1 − − j (xjk −xji )2 ∂β0i ) j 2 2(σ ) ∂β j 1 − (xjk −xji )2 )Eij ((xj )2 ) 2(σj )2 j n i=1 exp(β0i k=1 (xjk −xji )2 ) 2(σ j )2 − (xjk −xji )2 ) 2(σ j )2 Setting the partial derivative to 0, we have that the maximum likelij j hood estimates βˆ1j and βˆ0i of β1j and β0i respectively satisfy nk=1 (xjk )2 = n k=1 n i=1 j j (x −x )2 i k )Eij ((xj )2 ) j 2(σ )2 j 2 j n ˆj (xk −xi ) i=1 exp(β0i − 2(σ j )2 ) j exp(βˆ0i − .

For example, in the Shuttle data set, the largest class has 580 times more examples than the smallest class. On the Page Blocks data set (Fig. 10), to discover all the classes, SEDER needs 36 label requests, MALICE needs 23 label requests, Interleave needs 77 label requests on average, RS needs 199 label requests on average, and Kernel needs more than 1000 label requests; on the Abalone data set (Fig. 11), to discover all the classes, SEDER needs 316 label requests, MALICE needs 179 label requests, Interleave needs 333 label requests on average, RS needs 483 label requests on average9 , and Kernel needs more than 1000 label requests; on the Shuttle data set (Fig.

2 24 3 Rare Category Detection the maximum score is selected for labeling by the oracle. If the example is from class c, stop the iteration; otherwise, enlarge the neighborhood where the scores of the examples are re-calculated and continue. Justification As before, next we prove that if the minority classes are concentrated in small regions and the pdf of the majority class is locally smooth, the proposed algorithm will repeatedly sample in the regions where the rare examples occur with a high probability.

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