Instalación de Suministro Lavadora de Botellas

3.2.1 Denition and Notations

Formally, a domain is defined as D = {X , P(X)}, where X = {xi}

In

i=1 ⊂ X

denotes a sample of In feature vectors, X is the feature space and P(X) is the

marginal probability distribution of X. A classification task can be defined as a

pair T = {Y, f(·)} or T = {Y, P(Y|X)}, where Y = {y}In

i=1 ⊂ Y is the set of

labels that correspond to the sample X; Y denotes the label space; f : X 7→ Y the labeling function which is learned from the pair {X, Y}; and P(Y|X) is the conditional probability distribution of the labels.

We typically define two pairs of domain-task: (Ds,Ts)and (Dt,Tt), where

s and t denote the source and target respectively. To avoid confusion with the

terms “source” and “target” that respectively denote neural sources (or sources of variability) and target stimuli, we will use an emphasized font when we refer to the transfer learning-related terms source and target. Usually, the source labels are known while the target labels are unknown. Traditional Machine learning

approaches assume that Ds = Dt and Ts = Tt. Transfer learning addresses the

following issues:

Transfer Learning Methods 29

probability distributions of the source and target samples are different. The

latter is also known as covariate shift [Shimodaira,2000].

2. Ts6= Tt. This describes the case when there is a mismatch between the labels

due for example to unbalanced class labeling between the source and target, or when the conditional probability distributions of the labels have changed. Since transfer learning deals with a family of issues, it offers a family of solutions.

Weiss et al. [Weiss et al.,2016] define four general categories with respect to the

type of information transferred, each one related to one or more of the issues listed above:

1. Transfer learning through features. 2. Transfer learning through instances.

3. Transfer learning through shared parameters.

4. Transfer learning based on defined relationships between target and source. In the following section, we present some of the transfer learning methods that have been applied to BCI.

3.2.2 Transfer Learning in ERP-based BCI

In ERP-based BCI, all of the scenarios addressed by transfer learning can occur between source and target datasets. For example, if the number of electrodes is

different between two sessions, we have Xs 6= Xt. Mistakes from the side of the

user lead to differences in the labels between the target and the source domain. The non-stationarity of the signal, the effects of ERP variability and the changes in

the background brain activity or additive noise all result in covariate shift [Clerc

et al.,2016]. Moreover, the target dataset is not available immediately. On the contrary, if we consider that the online use of a BCI generates the target dataset, it becomes available trial by trial. This results in a large amount of imbalance between

sourceand target datasets. Out of the solutions proposed by transfer learning in the

taxonomy of Weiss et al. [Weiss et al.,2016], the following three have been applied

to ERP-based BCI: (i) transfer learning through features, (ii) transfer learning through instances and (iii) transfer learning through shared parameters.

Transfer learning trough features can be divided into two approaches. In the first one, one seeks to find a feature subspace where the source and target domains match. A promising approach that falls into that particular category is the Riemannian Geometry framework. Riemannian Geometry based algorithms were introduced in 2010 by Barachant et al. to classify features in Motor Imagery based

BCI [Barachant et al.,2010]. This approach proposes to use covariance matrices

as features, which are invariant to affine transformations when manipulated on the Riemannian manifold of symmetric positive definite matrices. This framework has been applied to ERP-based BCI by Congedo et al. and by Barachant et al. in [Congedo et al.,2013;Barachant and Congedo,2014], where a special form of the covariance matrix is used as a feature. The second family of transfer learning through features focuses on reweighting the features of the source domain so that it matches the target domain, or vice versa. An example would be the application of a noise reduction spatial filter, trained on the source dataset, over target data. This

approach was employed by Gayraud et al. in [Gayraud et al.,2017], where noise

reduction filters are learned over one P300-Speller session and applied on another.

Transfer learning through instances mostly apply to the covariance shift prob-

lem. Such solutions work by either reweighting the source dataset so that it matches the target dataset, or by reweighting the target dataset so that it matches the source. In comparison to transfer learning through features, the weights are particular to each feature vector, instead of each feature. Such solutions have been proposed in

the works of [Gayraud et al.,2017;Zanini et al.,2018]. In these works, the authors

compute transportation matrices to relocate a target dataset so that it matches a

sourcedataset.

Combined use of learned parameters involves parameters such as classifier

weights or distribution priors. Kindermans et al. propose a method in which they combine classification priors over multiple sources to train a classifier on the target

dataset in [Kindermans et al.,2012a] and [Kindermans et al.,2014]. This category

also includes ensemble learning methods which have been used in ERP-based BCI

to boost classification performance [Rakotomamonjy and Guigue,2008].

In the following section, we focus on three transfer learning frameworks that have been applied to ERP-based BCI: (i) Riemannian geometry (ii) Optimal trans-

Transfer Learning Methods 31

port and (iii) Ensemble Learning. Each comes from a different family of transfer

learning solutions and addresses a different domain adaptation issue. In chapter2,

we identified four main sources of ERP-based BCI variability: 1. peak amplitude variability, 2. peak latency variability, 3. scalp topography variability and 4. back- ground noise variability. We describe each transfer learning method and discuss their strengths and limitations in dealing with EEG variability.