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matching algorithms that exploit the geometrical structure of the data.
Problems related to DA are the sample selection bias
and covariate shift [ 23]–[25 ]. The sample selection bias
originates when the available training samples are not in -
dependently and randomly selected from the underlying distribution (i.e., the training set is not a random sample of the population). This situation is very likely to occur in
remote sensing problems where training points are usually
selected and labeled through photointerpretation or field surveys. The covariate shift is a particular case of sample se -
lection bias where the bias depends only on the input vari -
able
X (and not on .)Y Different operational conditions
that result in biased training samples are discussed in [ 26].
Clearly, a training set xx{( ,),, (, )} Ty ynn 11f = obtained un -
der a sample selection bias leads to skewed estimation of
the true underlying distribution of the classes, resulting in an estimated
(, )( ,) . PXYP XY! t For this reason, the effect
of a sample selection bias is similar to the DA problem de -
scribed previously. The covariate shift problem is generally not as severe as the general sample selection bias and the
DA problem. Let us consider that both training sets on the
source and target are samples with bias (a bias depending on X only, i.e., covariate shift) from the same joint distribution.
The joint probabilities on the two domains can be factorized
as
(, )( )( |) PX YP XP YXss s= and (, )( )( |) . PX YP XP YXtt t=
In this particular case of covariate shift, the estimated con -
ditional probabilities will be approximately equal, while
the marginal distributions will generally be different, i.e.,
(| )( |) PY XP YXst. tt and () () . PX PXst!tt
Figure 3 illustrates the two different issues of DA
and the sample selection bias with two explanatory ex -
amples. The plots report the distributions of the labeled
samples from two domains in a bidimensional feature space. A four-class classification problem is considered.
In the case of DA, the class-conditional densities may
change from the source to the target domain, possibly resulting in a significantly different classification prob -
lem. Note that the distribution of the classes in the target domain may overlap with different classes on the source domain (see the green circle in the target domain that
represents the location of the green class in the source).
In this situation, training samples of the source domain
can be misleading for the classification of the target
domain [ 27], [28 ]. In the  case of sample selection bias, Source Domain
DA
Sample
Selection BiasTarget Domain
FiguRe 3. Explanatory examples of DA and sample selection bias
problems. The plots represent labeled samples in a bidimensional feature space on the source and target domain. A four-class
classification problem is considered. In the case of DA, the class-
conditional densities may vary from the source domain to the target
domain, resulting in a significantly different classification problem
(the green circles correspond to the location of the green classes in the source domain). In the case of sample selection bias, the
aforementioned issue will not occur, resulting in a milder type of shift
from the source domain.
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june 2016 ieee Geoscience and remote sensin G ma Gazine 45
source- and target-domain samples are drawn (with bias)
from the same underlying distribution, generally resulting
in a milder type of shift with respect to DA. (The problem
of cross-domain class overlap is not likely to occur in sam -
ple selection bias problems.)
In real remote sensing problems, it is expected that the
two aforementioned issues may occur at the same time and can be profoundly entangled, meaning that 1) the two
theoretical underlying distributions
(, ) PX Ys and (, ) PX Yt
associated with the source and target domains, respec -
tively, may differ because of changes in the image acquisi -
tion conditions (e.g., radiometric conditions), and 2) the
available training samples could be not fully representative
of the statistical populations in their respective domains
and will, therefore, lead to biased estimations of the class probabilities and to inaccurate classification models. In
the remote sensing literature, several techniques have been
presented to solve the transfer-learning problem irrespec -
tive of the cause of the data set shift between source and
target domains. The next section will present the different
families of techniques that have been proposed in remote sensing image processing.
a taXonom Y oF aDaptation met HoDS
Adapting a model trained on one image to another (or a
series of new images) can be performed in different ways.
In this section, we detail recent approaches proposed in the remote sensing literature by grouping them into the
following four categories:
◗The selection of invariant features: A set of the input
features (i.e., the original bands or additional features
extracted from the remote sensing image) that are not
affected by shifting factors are identified and selected before training the classification algorithm. According -
ly, the features affected by the most severe data set shift are removed, and the classifier considers a feature space showing higher stability across domains. An alternative
approach is to encode invariance by including additional
synthetic labeled samples in the training set to extract features that better model the intraclass variability across
the domains.
◗The adaptation of the data distributions: The data
distributions of the target and source domains are made as similar as possible to keep the classifier un -
changed. With respect to the previous family, these
methods work on the original input spaces and try
to extract a common space where all domains can be
treated equally. This is generally achieved by means of joint feature extraction.
◗The adaptation of the classifier: Here, the classifica -
tion model defined by training on the source domain is
adapted to the target domain by considering unlabeled