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X ∈ X

. Give a speciﬁc domain, a task T consists of two components: a label space Y and a

prediction function f (X). From a probabilistic view point, f (X) can be written as the conditional

probability distribution PY|X. Given a source domain DS and a corresponding learning task TS ,

a target domain DT and a corresponding learning task TT , domain adaptation aims to improve

the learning of the target predictive function fT in DT using the knowledge in DS and TS , where

DT and TS

= TT , i.e., the tasks are the same but the domains are diﬀerent.

In real world applications of fault diagnosis, the working conditions (e.g. motor load and

speed) may change from time to time according to the requisite of the production. As a kind

of classiﬁcation problem, the goal of intelligent fault diagnosis is to train classiﬁer with sam-

ples collected and labeled in one working condition to be able to classify samples from another

working condition. Samples collected under di

ﬀerent working conditions can be regarded as dif-

ferent domains. Correspondingly, the fault diagnosis settings in domain adaptation situation are

as follows:

• The feature spaces between domains are the same, XS

= XT , e.g. the fast Fourier transform

(FFT) spectrum amplitudes of raw vibration temporal signals.

• The label spaces between domains are the same, YS

= YT = {1, ..., K}, where K is the

quantity of fault types.

• PS

XY and P

XY only diﬀer in the marginal probability distribution of the input data, i.e.,

X , while P

Y|X = P

Y|X .

which is similar to the assumptions in covariate shift [30, 31, 32] or sample selection bias [33].

2.2. Domain Divergence Measure

The main problem existing in domain adaptation is the divergence of distribution between

the target domain and source domain. Ben-David et al. [34, 35] deﬁnes a divergence measure

dH∆H(S, T ) between two domains S and T , which is widely used in the theory of nonconservative

domain adaptation. Using this notion, they established a probabilistic bound on the performance

T (h) of some label classiﬁer h from T evaluated on target domain given its performance

S (h)

on the source domain. Formally,

T (h) ≤

S (h)

dH∆H(S, T ) + λ

(1)

where λ is supposed to be a negligible term and dose not depend on classiﬁer h.

Eq. 1 tells us that to adapt well, one has to learn a label classiﬁer h which works well

on source domain while reducing the dH∆H(S, T ) divergence between S and T . Estimating

dH∆H(S, T ) for a ﬁnite sample is exactly the problem of minimizing the empirical risk of a do-

main classiﬁer hd that discriminates between instances drawn from S and instances drawn from

, respectively pseudo-labeled with 0 and 1. More speciﬁcally, it involves the following steps:

1. Pseudo-labeling the source and target instances with 0 and 1, respectively.

2. Randomly sampling two sets of instances as the training and testing set.

3. Learning a domain classiﬁer hd on the training set and verifying its performance on the

testing set.

4. Estimating the distance as ˆ

dH∆H(S, T ) = 1 − 2 (hd), where (hd) is the test error.

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