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A brain computer interface BCI enables direct communication between a brain and a computer translating brain activity into computer commands using preprocessing, feature extraction, and classification operations. Feature extraction is crucial, as it has a substantial effect on the classification accuracy and speed. While fractal dimension has been successfully used in various domains to characterize data exhibiting fractal properties, its usage in motor imagery-based BCI has been more recent. In this study, commonly used fractal dimension estimation methods to characterize time series Katz's method, Higuchi's method, rescaled range method, and Renyi's entropy were evaluated for feature extraction in motor imagery-based BCI by conducting offline analyses of a two class motor imagery dataset. Different classifiers fuzzy k-nearest neighbours FKNN, support vector machine, and linear discriminant analysis were tested in combination with these methods to determine the methodology with the best performance. This methodology was then modified by implementing the time-dependent fractal dimension TDFD, differential fractal dimension, and differential signals methods to determine if the results could be further improved. Katz's method with FKNN resulted in the highest classification accuracy of 85%, and further improvements by 3% were achieved by implementing the TDFD method.

A brain computer interface (BCI) enables direct communication between a brain and a computer translating brain activity into computer commands, thus providing nonmuscular interaction with the environment. Sensorimotor rhythms (SMRs) are rhythmic brain waves found in the frequency range of 8 to 12 Hz over the left and right sensorimotor cortices. Movement, movement preparation, and motor imagery desynchronize SMRs, whereas during relaxation or postmovement, they are synchronized [

Feature extraction is the process of accurately simplifying the representation of data by reducing its dimensionality while extracting its relevant characteristics for the desired task. It has a substantial effect on the classification accuracy and speed, since classification carried out without a successful feature extraction process on a high dimensional and redundant data would be computationally complex and would overfit the training data. Fractal dimension is a statistical measure indicating the complexity of an object or a quantity that is self-similar over some region of space or time interval. It has been successfully used in various domains to characterize such objects and quantities [

In this study, Katz’s method [

The motor imagery dataset from the BCI Competition II (Data set III) provided by the Department of Medical Informatics, Institute for Biomedical Engineering, University of Technology Graz was analyzed. The data was acquired over seven runs from a healthy 25-year-old female subject during imagery left and right hand movements. The signals were recorded with a sampling rate of 128 Hz from three electrodes placed at the standard positions of the 10–20 international system (C3, Cz, and C4) and filtered between 0.5 and 30 Hz. Each run consisted of 40 trials and each trial was nine seconds long. During the first two seconds of each trial, neither a stimulus was presented nor did the subject perform any motor imagery task. After this period, an acoustic and a visual stimulus indicating the beginning of the motor imagery task were presented. Then, for six seconds, a cue (a left or right arrow) indicating the required motor imagery task was presented (in a random order for each trial), and the subject performed this task. During this period, a feedback bar was displayed. Both the training and testing sets consisted of 140 samples.

The samples from each electrode were zero phase filtered using a 6th-order bandpass digital Butterworth filter with cutoff frequencies of 0.5 and 30 Hz in both the forward and reverse directions. The last six seconds of each trial were extracted to discard the period without any motor imagery. Two different electrode configurations (C3 and C4, and C3, Cz, and C4) were tested.

In Katz’s method, Higuchi’s method, and the

Katz’s method [

Higuchi’s method [

The

Renyi’s entropy [

In TDFD method, a window (with size

The DFD method is a variation of the DS method. In the DFD method, first, the fractal dimensions of the samples from selected electrodes are estimated, and then, the pairwise differences of the fractal dimensions are calculated. However, in the DS method [

After constructing the feature vectors, the test samples were classified as imagery left or right hand movements using different classifiers. FKNN, SVM, and LDA were tested.

FKNN is a variation of KNN. The main difference between the two is that KNN assigns a class label to a sample that is most frequent among the k nearest neighbors of that sample, whereas FKNN assigns a membership value for each class in this neighborhood and classifies the sample as the class with the highest membership value. The membership value for a class was calculated by dividing the sum of the distances between the samples belonging to this class and the test sample by the sum of the distances between all the samples in the neighborhood and the testing sample. Number of nearest neighbors between one and the square root of the sample length were tested.

SVM separates the samples using a hyperplane that maximizes the margin between those belonging to different classes. SVM with a linear kernel was used.

LDA finds a linear combination of features that best separates the samples belonging to different classes and can be used as a classifier. To assign a class label to a sample, the probabilities of the sample belonging to each class were estimated using LDA. The label of the class with the highest probability was then assigned to the sample.

The classification accuracies (Table

Maximum classification accuracies (percentage of correctly classified testing samples) obtained by the combination of fractal dimension estimation methods and classifiers with the two and three electrode configurations (and the parameters (_{max}) used to obtain these values).

Classification accuracy (%) | ||||||||

Katz’s method | Higuchi’s method | Renyi’s entropy | ||||||

C3, C4 | C3, Cz, C4 | C3, C4 | C3, Cz, C4 | C3, C4 | C3, Cz, C4 | C3, C4 | C3, Cz, C4 | |

FKNN | 83 | 85 | 77 | 79 | 71 | 69 | 66 | 65 |

SVM | 77 | 79 | 78 | 81 | 71 | 70 | 59 | 55 |

LDA | 78 | 81 | 78 | 79 | 71 | 70 | 59 | 57 |

Computation times (time it took for feature extraction and classification) corresponding to the maximum classification accuracies obtained by the combination of fractal dimension estimation methods and classifiers with the two and three electrode configurations (and the parameters (_{max}) used to obtain these values).

Computation time (s) | ||||||||

Katz’s method | Higuchi’s method | Renyi’s entropy | ||||||

C3, C4 | C3, Cz, C4 | C3, C4 | C3, Cz, C4 | C3, C4 | C3, Cz, C4 | C3, C4 | C3, Cz, C4 | |

FKNN | 0.17 | 0.23 | 1.03 | 1.5 | 7.37 | 11.07 | 2.35 | 2.60 |

SVM | 0.34 | 0.34 | 1.07 | 1.4 | 7.36 | 10.99 | 1.90 | 4.87 |

LDA | 0.12 | 0.21 | 0.83 | 1.26 | 7.32 | 10.99 | 1.84 | 3.01 |

Table

Computation times and classification accuracies obtained by modifying the highest performing methodology (Katz’s Method with FKNN) (and the parameters (

TDFD method | DFD method | DS method | ||

C3, C4 | C3, Cz, C4 | C3, Cz, C4 | C3, Cz, C4 | |

Classification accuracy (%) | 88 ( | 85 ( | 84 ( | 71 ( |

Computation time (s) | 3.47 ( | 0.94 ( | 0.41 ( | 0.26 ( |

Mental activity may modulate FD of EEG signal which implies that it is timed-ependent in nature. By implementing TDFD method in Katz’s Method with FKNN, we may measure the fractality in short time intervals of time-sequential data from one end of the waveform to the other sequentially, and we may observe the dynamical changes in the FDs with respect the time series. These FDs, namely, are referred to the time-dependent fractal dimensions (TDFD) [

Katz’s algorithm is the most consistent method due to its exponential transformation of FD values and relative insensitivity to noise. Hiaguchi’s method, however, yields a more accurate estimation of signal FD, when tested on synthetic data, but it is more sensitive to noise. In the experiment, EEG datasets used are real data sets which contain noise, hence Katz’s method exhibits better result [

Since all fractal dimension estimation methods are not applicable to all types of data exhibiting fractal properties, commonly used fractal dimension estimation methods to characterize time series with different classifiers were evaluated to find the most suitable method for motor imagery data. Katz’s method with FKNN was determined to be the best methodology, and the results were further improved by implementing the TDFD method. The results warrant further research to use this methodology in online analysis of motor imagery data and analysis of other signals.

The authors would like to thank Alfonsius Geraldi, Umut Güçlüa, and Yağmur Güçlütürkaal and also gratefully acknowledge the helpful comments and suggestions of the reviewers.