Course-Grained Multi-scale EMD Based Fuzzy Entropy for Multi-target Classification During Simultaneous SSVEP-RSVP Hybrid BCI Paradigm

Abstract

Hybridization in brain-computer interface (BCI) is an active research assessed by integrating more than two modalities to control multiple user commands and enhance the BCI performance. However, evaluation of these hybrid modalities is substantially challenging which necessitates robust techniques that can efficiently analyze the underlying dynamics of the brain signal. Fuzzy entropy based approaches are the objective measurements of evaluating the dynamic electroencephalography (EEG) complexity reflecting its ability to adapt to the variabilities of the brain. In this study, we proposed an empirical mode decomposition (EMD) featured fuzzy entropy method by applying coarse-graining to the time series signal at a multi-scale level (EMFuzzyEn) to efficiently discriminate the target and non-target events, and improve classification performance during simultaneous steady-state visual evoked potential (SSVEP) and rapid serial visual presentation (RSVP) BCI system framework. The proposed hybrid BCI framework with applied canonical correlation analysis (CCA) and EMFuzzyEn algorithm to the SSVEP and RSVP classifications achieved an offline accuracy of 87.41% for 12-channel combination and an online accuracy of 84.69% for 2-channel combination. The results demonstrated that the proposed algorithm enhanced the BCI performance by 5.85% (offline) and 6.78% (online) compared to our previous ERP-based classification results. Further, the EMFuzzyEn algorithm achieved highest classification accuracy across all channel combinations compared to the multi-scale fuzzy (MFuzzyEn) and non-fuzzy (ShEn, AppEn, CondEn) entropy algorithms. Overall, our empirical results confirmed that the proposed EMFuzzyEn algorithm significantly increased the EEG complexity during target tasks thereby enhancing the discrimination between target and non-target tasks, which might be a potential indicator to measure the brain dynamics and enhance the BCI performance.

Appendices

Appendix A

Theorem

For a given time series ${\{X}_{i }:1\le i\le N\},$ given $d, n$ and$r$, the entropy of the sequence when fuzziness is incorporated is$\mathrm{FuzzyEn}=\mathrm{ln}({\Phi }_{\left\{\mathrm{d}+1\right\}})-\mathrm{ln}({\Phi }_{\left\{\mathrm{d}+1\right\}})$.

Proof

Let ${A}_{ji}$ denote the number of vectors of $d$ dimension ${Z}_{j\{d\}}$ within $r$ of ${Z}_{i\{d\}}$ and ${B}_{ji}$ denote the number of vectors of $d+1$ dimension ${Z}_{j\{d+1\}}$ within $r$ of ${Z}_{i\{d+1\}}$.

Given $n$ and $r$, the probability of difference between any two vectors ${Z}_{j\{d\}}$ and ${Z}_{i\{d\}}$ that lies within $r$ denoted by ${D}_{ji\{d\}}$ is defined as ${D}_{ji\left\{d\right\}}$ = ${A}_{ji}/(N-d)$.

Similarly, the probability of difference between any two vectors ${Z}_{j\{d+1\}}$ and ${Z}_{i\{d+1\}}$ that lies within $r$ denoted by ${D}_{ji\left\{d+1\right\}}$ is defined as ${D}_{ji\left\{d+1\right\}}$ = ${B}_{ji}/(N-d+1)$.

Then, from the work of Grassberger et al. [50], the function ${\Phi }_{\left\{\mathrm{d}\right\}}$ and ${\Phi }_{\left\{\mathrm{d}+1\right\}}$ are called the average of the natural logarithms of functions ${D}_{ji\left\{d\right\}}$ and ${D}_{ji\left\{d+1\right\}}$ which are denoted by

$$\Phi _{{\left\{ d \right\}}} = \frac{1}{{N - d}}\sum\limits_{{j = 1}}^{{N - d}} {{\text{ln}}} [D_{{ji\left\{ d \right\}}} ]$$

$$\Phi _{{\left\{ {d + 1} \right\}}} = \frac{1}{{N - d + 1}}\sum\limits_{{j = 1}}^{{N - d + 1}} {{\text{ln}}} [D_{{ji\left\{ {d + 1} \right\}}} ]$$

With sufficiently large $N$ value the above equations can be approximated to

$$\Phi _{{\left\{ d \right\}}} = \frac{1}{{N - d}}\sum\limits_{{j = 1}}^{{N - d}} {{\text{ln}}} [D_{{ji\left\{ d \right\}}} ]$$

$$\Phi _{{\left\{ {d + 1} \right\}}} = \frac{1}{{N - d}}\sum\limits_{{j = 1}}^{{N - d}} {{\text{ln}}} [D_{{ji\left\{ {d + 1} \right\}}} ]$$

From Pincus work [51], the calculation of difference between ${\Phi }_{\left\{\mathrm{d}\right\}}$ and ${\Phi }_{\left\{\mathrm{d}+1\right\}}$ for fixed parameters $d, r$ and $N$ provide inherent measure of regularity and complexity, which is often named as entropy.

$$\begin{array}{*{20}l} {{\text{Entropy}}} \hfill & { = \Phi _{{\left\{ d \right\}}} - \Phi _{{\left\{ {d + 1} \right\}}} } \hfill \\ {} \hfill & { = \frac{1}{{N - d}}\sum\limits_{{{\text{j}} = 1}}^{{N - d}} {\left( {\ln \left[ {D_{{ji\left\{ d \right\}}} } \right] - \ln \left[ {D_{{ji\left\{ {d + 1} \right\}}} } \right]} \right)} } \hfill \\ {} \hfill & { = {\text{Avg}}\left( { - \ln \left( {\frac{{B_{{ji}} }}{{A_{{ji}} }}} \right)} \right)} \hfill \\ \end{array}$$

Similarly, the equation $\mathrm{FuzzyEn}=\mathrm{ln}({\Phi }_{\left\{\mathrm{d}+1\right\}})-\mathrm{ln}({\Phi }_{\left\{\mathrm{d}+1\right\}})$ gives

$$\mathrm{FuzzyEn}= \frac{1}{\mathrm{N}-\mathrm{d}}\left\{\sum_{\mathrm{j}=1}^{\mathrm{N}-\mathrm{d}}\mathrm{ln}[{D}_{ji\left\{d\right\}}]-\mathrm{ln}{[D}_{ji\left\{d+1\right\}}]\right\}$$

that is equal to the average of the negative natural logarithm over i $-\mathrm{ln}({{B}_{ji}/A}_{ji})$.

Hence, the ratio ${{[B}_{ji}/A}_{ji}]$ represents an entropy.

Appendix B

Pseudocode of multi-scale EMD fuzzy entropy algorithm.

2.1 Required

Preprocessed EEG signal X ϵ R^N×C, N = data length, C = number of channels, τ = scale-factor, r = width and n = gradient of the fuzzy function.

2.2 Pseudocode