## 1 Introduction
In a different study [42], [PERSON]'s fractal dimension (HFD) was used, which showed superior sensitivity in differentiating minimally conscious state (MCS) and vegetative state (VS) patients compared to traditional linear EEG spectral power methods. [PERSON] et al. [43] conducted fractal dimension (FD)--particularly [PERSON]'s fractal dimension (HFD)--analysis of resting-state EEGs and revealed age-related changes in brain activity complexity. [PERSON] et al. [44] used [PERSON]'s fractal dimension (HFD) and the box-counting dimension (BCD) to provide insights into genetic heterogeneity across human populations and quantified genetic variations. Fractal dimension (FD) has recently emerged as a promising yet underexplored feature for signal analysis in BCI systems. [PERSON] et al. [14] used [PERSON]'s, [PERSON]'s, and rescaled range methods for FD estimation, revealing that [PERSON]'s method combined with fuzzy k-nearest neighbors (FKNNs) achieved the highest accuracy (85%), which further improved by 3% with the implementation of the time-dependent fractal dimension (TDFD). [PERSON] et al. [45] experimented with ALS patients suffering from impaired sensorimotor function. The Grassberger-Procaccia fractal dimension (GPFD) and [PERSON]'s fractal dimension (HFD) were applied to estimate the complexity of EEG signals, combined with [PERSON]'s criterion-based channel selection. [PERSON] et al. [46] showcased that FD methods provide a computationally efficient alternative to traditional methods, such as moment statistics and regression analysis, for EEG waveform analysis. [PERSON] et al. [37] introduced FD as a novel discriminative feature for the subject-independent classification of EEG signals, and their findings showed that fractal dimension features consistently outperformed event-related desynchronization (ERD) in classification accuracy observed in unilateral hand movement tasks. In this study, new fractal dimension (FD) algorithms are introduced. The usage of different combinations of FD algorithms is an extension of our previous work [36], with the aim of obtaining better results. New FDs, such as box-counting [47], DFA [48], MFDFA [49], and correlation dimension [50], are incorporated with the previously used Katz [51], Petrosian [52], and Higuchi [53], in groups of three, and the following four classifiers are used: GSVM [54], CART [55], LinearSVM, and SVM with polynomial kernels [56]. These are added to the latter stochastic gradient descent (SGD) classifier [57], with the aim of comparing its performance against other linear models--in our case, LinearSVM--with regards to the trade-off in accuracy. While fractal dimension (FD) methods have demonstrated their effectiveness in capturing nonlinear and complex characteristics of EEG signals, their applicability has been limited by several factors. Primarily, individual FD methods such as Katz, Petrosian, and Higuchi each capture different aspects of signal complexity at varying scales, but no single method effectively captures the comprehensive multi-scale complexity inherent to EEG signals during motor imagery (MI) tasks. For example, the Katz FD predominately captures waveform length changes, while the Higuchi FD effectively quantifies time series complexity over multiple temporal scales, and the box-counting FD reveals spatial self-similarity. Consequently, employing only a single fractal dimension approach restricts the analysis to specific types of complexity, thus limiting the overall classification performance. To address this limitation, our work innovatively proposes combining multiple fractal dimensions into feature sets, integrating both traditional and novel FD methods. This multi-dimensional fractal analysis significantly expands the representation of EEG signal complexity, bridging the existing gap by capturing distinct characteristics simultaneously, thus enhancing the classification accuracy. Section 2 presents the theoretical approaches utilized throughout this research.