Feature Extraction

Let <sup>Ψ</sup> <sup>=</sup> {**X***<sup>n</sup>* <sup>∈</sup> <sup>R</sup>*C*×*M*}*<sup>N</sup> <sup>n</sup>*=<sup>1</sup> be an EEG set holding *N* trials from either an MI or a WM dataset, recorded from a single subject, where *C* stands for the number of channels and *<sup>M</sup>* corresponds to the number of samples. In addition, let {*ln*}*<sup>N</sup> <sup>n</sup>*=<sup>1</sup> be a set whose *n*-th element is the label associated with trial **X***n*. For the MI database *ln* can take the values of 1 and 2, corresponding to right hand and left hand motor imagination, respectively. Similarly, for the WM database, *ln* can take the values of 1, 2, and 3 corresponding to low, medium, and high memory loads. In both cases, our goal is to estimate the class label from relevant effective connectivity features extracted from **X***n*. Because of the results obtained for the simulated data (see Section 4.1 for details), here we consider features from only three approaches for phase-based effective connectivity estimation, namely, TE*<sup>θ</sup> κα*, GC*θ*, and PSI. Additionally, we also characterize the data through the real-valued versions of TE*κα* and GC.

For the real-valued effective connectivity measures considered, we do the following: let *<sup>λ</sup>*(**x***<sup>c</sup>* <sup>→</sup> **<sup>x</sup>***<sup>c</sup>* ) be a measure of effective connectivity between channels **<sup>x</sup>***c*, **<sup>x</sup>***<sup>c</sup>* <sup>∈</sup> <sup>R</sup>*M*. By computing *λ*(**x***<sup>c</sup>* → **x***<sup>c</sup>* ) for each pairwise combination of channels in **X***<sup>n</sup>* we obtain a connectivity matrix **<sup>Λ</sup>** <sup>∈</sup> <sup>R</sup>*C*×*C*. In the case when *<sup>c</sup>* <sup>=</sup> *<sup>c</sup>* , we set *λ*(**x***<sup>c</sup>* → **x***<sup>c</sup>* ) = 0. Then, we normalize **Λ** to the range [0, 1]. After performing the above procedure for the *N* trials, we get set of connectivity matrices {**Λ***<sup>n</sup>* <sup>∈</sup> <sup>R</sup>*C*×*C*}*<sup>N</sup> <sup>n</sup>*=1. Then, we apply vector concatenation to **<sup>Λ</sup>***<sup>n</sup>* to yield a vector *<sup>φ</sup><sup>n</sup>* <sup>∈</sup> <sup>R</sup>1×(*C*×*C*). Next, we stack the *<sup>N</sup>* vectors *<sup>φ</sup>n*, corresponding to each trial, to obtain a matrix **<sup>Φ</sup>** <sup>∈</sup> <sup>R</sup>*N*×(*C*×*C*) holding all directed interactions, estimated through *λ*, for the EEG set Ψ. A graphical representation of the above-described steps, as well as of our overall classification setup, is depicted in Figure 4.

**Figure 4.** Schematic representation of our overall classification setup.

For the phase-based effective connectivity measures of interest, we proceed in a similar fashion: let *<sup>λ</sup><sup>θ</sup>* (**x***<sup>c</sup>* <sup>→</sup> **<sup>x</sup>***<sup>c</sup>* , *<sup>f</sup>*) be a measure of phase-based effective connectivity between channels **<sup>x</sup>***c*, **<sup>x</sup>***<sup>c</sup>* at frequency *<sup>f</sup>* . By computing *<sup>λ</sup><sup>θ</sup>* (**x***<sup>c</sup>* <sup>→</sup> **<sup>x</sup>***<sup>c</sup>* , *<sup>f</sup>*) for each pairwise combination of channels in **<sup>X</sup>***<sup>n</sup>* we obtain a connectivity matrix **<sup>Λ</sup>**(*f*) <sup>∈</sup> <sup>R</sup>*C*×*<sup>C</sup>* (when *<sup>c</sup>* <sup>=</sup> *<sup>c</sup>* we set *<sup>λ</sup><sup>θ</sup>* (**x***<sup>c</sup>* <sup>→</sup> **<sup>x</sup>***<sup>c</sup>* , *<sup>f</sup>*) = 0). For the MI database, we vary the values of *<sup>f</sup>* in the range from 8 Hz to 18 Hz, in 2 Hz steps, since activity in that frequency range has been associated with MI tasks [43]. Then we define two bandwidths of interest Δ*f* ∈ {*α* ∈ [8 − 12], *β<sup>l</sup>* ∈ [14 − 18]} Hz. Afterward, we average the matrices **Λ**(*f*) within each bandwidth, normalize the resulting matrices to the range [0, 1], and stack them together, so that for each trial we have a connectivity matrix **<sup>Λ</sup>** <sup>∈</sup> <sup>R</sup>*C*×*C*×2. Therefore, for the *<sup>N</sup>* trials, we get set of connectivity matrices {**Λ** *<sup>n</sup>* <sup>∈</sup> <sup>R</sup>*C*×*C*×2}*<sup>N</sup> <sup>n</sup>*=1. Then, we apply vector concatenation to **Λ** *<sup>n</sup>* to yield a vector *<sup>φ</sup><sup>n</sup>* <sup>∈</sup> <sup>R</sup>1×(*C*×*C*×2). After that, we stack the *<sup>N</sup>* vectors *<sup>φ</sup><sup>n</sup>* in order to obtain a single matrix **<sup>Φ</sup>** <sup>∈</sup> <sup>R</sup>*N*×(*C*×*C*×2) characterizing <sup>Ψ</sup> for the MI data. For the WM we follow the same steps, only that in this case we vary the values of *f* in the range from 4 Hz to 18 Hz, in 2 Hz steps, since oscillatory activity at those frequencies has been shown to play a role in the interactions between different brain regions during WM [50,51]. Next, we define three bandwidths of interest Δ*f* ∈ {*θ* ∈ [4 − 6], *α* ∈ [8 − 12], *β<sup>l</sup>* ∈ [14 − 18]} Hz, which leads to a connectivity matrix **<sup>Λ</sup>** <sup>∈</sup> <sup>R</sup>*C*×*C*×<sup>3</sup> for each trial and ultimately to a matrix **<sup>Φ</sup>** <sup>∈</sup> <sup>R</sup>*N*×(*C*×*C*×3) characterizing Ψ for the WM data. Note that since the PSI is an antisymmetric connectivity measure, we only use the upper triangular part of the connectivity matrix associated with each trial to build **Φ**.

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