A user of Brain Computer Interface (BCI) system must be able to control external computer devices with brain activity. Although the proof-of-concept was given decades ago, the reliable translation of user intent into device control commands is still a major challenge. There are problems associated with classification of different BCI tasks. In this paper we propose the use of chaotic indices of the BCI. We use largest Lyapunov exponent, mutual information, correlation dimension and minimum embedding dimension as the features for the classification of EEG signals which have been released by BCI Competition IV. A multi-layer Perceptron classifier and a KM- SVM(support vector machine classifier based on k-means clustering) is used for classification process, which lead us to an accuracy of 95.5%, for discrimination between two motor imagery tasks.
The automatic and online translation of the user intent to the computer language is the mission of brain computer interfacing. This intent is estimated from the Electroencephalogram (EEG) measurements.
Several linear and nonlinear techniques have been applied to the EEG signals for BCI applications in the past [2 , 3, …, 12]. For example [2,3] review the classification algorithms used in BCI, and some linear techniques are employed in [5,6]. [7] used the wavelet transform and [8] proposed the MVAR model. [9] used independent component analysis (ICA) for BCI-EEG recognition. Also some nonlinear and chaotic methods are presented in [10,11,12].
Linear methods of EEG analysis such as Fourier transform or power spectral density, in comparison to chaotic analysis, are more computationally efficient but less strong in the interpretation of results [13,14]. For example, [15] shows that through nonlinear time series analysis, one can discriminate even high-dimensional chaos from colored noise. Some of traditional linear methods have been found largely insensitive to task conditions associated with different brain dynamics [16,17].
Recent researches on the human EEG, revealed the chaotic nature of this signal. In this research we take the advantage of this chaotic behavior in order to classify the EEG signals of a BCI task.
In section II the important indices are introduced briefly. Section III contains a short explanation of the EEG signals which we have used for simulation. In section IV is dedicated to the proposed method. Results are presented in section V, and section VI is conclusion.
The chaotic indices used in this research are introduced in the following subsections.
The Lyapunov exponent is a quantitative measure of the dynamics of trajectory evolution in the phase space. It capsulizes the average rate of convergence or divergence of two neighboring trajectories in the phase space. It can be negative, zero or positive. Negative values mean that the two trajectories draw closer to one another. Positive exponents on the other hand, result from the trajectory divergence and appear only within the chaotic domain. In other words, a positive Lyapunov exponent is one of the chaos indicators. Here we use the popular Wolf et al. [18] method for the calculation of largest Lyapunov exponent. It is summarized in the following equation:
In this equation,
is the local slop of trajectory. Because it averages local divergences and/or convergences from many places over the entire attractor, a Lyapunov exponent is a global quantity, not a local quantity.
Because the neighboring trajectories represent changes in initial conditions of a system, 1 λ is an average or global measure of the sensitivity of the system to slight changes or perturbations. A system isn’t sensitive at all in the non-chaotic regime, since any two nearby trajectories converge. In contrast, a system is highly sensitive in the chaotic regime, in that two neighboring trajectories separates, sometimes rapidly.
If we denote Y X, as two random variables, then Y X H H , are their entropies and we have:
in which s N is the number of non-zero probabilities.
The mutual information for Y X, is defined as:
H ; is defined as:
After substituting (4) in (3) and by some mathematical simplifications, we will have:
For the calculation of x + . Bigger quantity of mutual information results in a less chaotic system. More details are reported in [19].
For computational costs, simplicity of interpretation and other reasons, we’d like to reconstruct an attractor in a small embedding dimension. There’s no theory or even a rule-of-thumb available in this regard. To solve the problem of false neighbor method, Cao proposed a method to choose the threshold value, which is often used to determine the embedding dimension. Different time series data may have different threshold values. These imply that it is difficult to give an independent reasonable threshold value which is independent of the dimension d and each trajectory point, as well as the considered time series data. In this method a new quantity is defined:
) (m E is dependent only on the dimension m and the lag τ and f is a function of m i, . To investigate its variation from m to m+1, E1(m) is defined as:
Cao found that E1(m) stops changing when m is greater than some value 0 m if the time series comes from an attractor. Then 1 0 + m is the minimum embedding dimension we look for. It is necessary to define another quantity which is useful to distinguish between deterministic and stochastic signals. Let
where:
More details are in [20].
The correlation dimension is the most popular noninteger dimension currently used. It probes the attractor to a much finer scale than does the information dimension and is also easier and faster to compute. Like the information dimension, it takes into account the frequency with which the system visits different phase space zones. Most other dimensions involve moving a mea
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